MORSE THEORY AND HIGHER TORSION INVARIANTS I

Sebastian Goette

Abstract. We compare the higher analytic torsion T of Bismut and Lott of a fibre bundle p: M → B equipped with a flat vector bundle F → M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with Bismut. Under additional conditions on F, the torsion T is related to Igusa’s higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles p_{i}: M → B with diffeomorphic fibres that are homeomorphic but not diffeomorphic as bundles.

This is the first of two papers devoted to a comparison of Igusa’s and Klein’s higher Franz-Reidemeister torsion τ with Bismut’s and Lott’s higher analytic torsion T . In this paper, we evaluate the Bismut-Lott torsion form T for families that carry a fibre-wise Morse function, thus extending earlier work with Bismut ([BG1]). In the second part of the series, we relate τ and T for families with fibre-wise Morse functions in those situations where both invariants are defined. Let us recall the development of higher torsion invariants.

Franz and Reidemeister constructed a numerical invariant τ_{FR} of chain complexes in [R], [F], and used it to detect homeomorphism types of homotopy-equivalent Lense-spaces. The higher Franz-Reidemeister torsion τ is an extension of τ_{FR} to families of manifolds p: M → B, which takes values in the cohomology of B. It was first constructed by John Klein in [K] using a variation of Waldhausen’s A-theory. Other describtions of τ were later given by Igusa and Klein, see [I2] for more detailled references. In [IK], they computed τ in the special case of circle bundles.

We will constantly refer to the construction of τ in [I2]. Given a family p: M → B of smooth manifolds, one first finds a function h: M → ℝ that has only Morse-type and cubical singularities along each fibre of p, such that the unstable manifolds of the fibre-wise singularities are trivialised in a compatible way. By [I1], such a “framed” function always exists if dim M > 2 dim B. If dim M ≤ 2 dim B, one may replace M by M ×ℝP^{2N} for some sufficiently large N. Let F → M be a unitarily flat complex vector bundle that is fibre-wise acyclic, then F and h give rise to a functor from the category of generic small simplices on B to the simplicial Whitehead category Wh^{h}(M(ℂ), U) associated to the infinite matrix ring M(ℂ) = lim_{→} M_{n}(ℂ) and the infinite unitary group U = lim_{→} U(n). Now, the k-th higher torsion τ_{k}(M/B; F) is defined as the pull-back of a certain cohomology class D_{2k} ∈ H^{2k}(Wh^{h}(M(ℂ), U), ℝ).

On the other hand, Ray and Singer defined an analytic torsion T_{RS} of unitarily flat complex vector bundles on compact manifolds in [RS] and conjectured that T_{RS} = τ_{FR}. This conjecture was established independendly by Cheeger ([C]) and Müller ([M1]), and further generalised by Müller ([M2]) to unimodular flat bundles, and by Bismut and Zhang ([BZ1]) to arbitrary flat complex vector bundles.

2000 Mathematics Subject Classification. Primary 58J52; Secondary 57R22, 57Q10.

Key words and phrases. analytic torsion forms, higher Franz-Reidemeister torsion, Cheeger-Müller theorem, Whitehead space, Witten deformation.

In [BL], Bismut and Lott constructed characteristic differential forms ch^{o}(∇^{F} , g^{F}) of flat vector bundles (F, ∇^{F}) with arbitrary metrics g^{F}, which provide obstructions towards finding a parallel metric. In fact, the forms defined in [BL] have different coefficients, but we prefer to state the results of [BL] in the so-called “Chern normalisation” that was introduced in [BG1], for reasons explained below. Let p: M → B be a family of compact manifolds with a horizontal distribution T^{H}M ⊂ T M and a metric g^{T X} on the vertical tangent bundle T X, and let (F, ∇^{F}) → M be a flat vector bundle with metric g^{F}. Then Bismut and Lott defined a higher analytic torsion form T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) on B such that

(*)  d^{B}T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = ∫_{M/B} e(∇^{T X}) ch^{o}(∇^{F} , g^{F} ) − ch^{o}(∇^{H}, g^{H}) ,

where e(∇^{T X}) is the Euler form of the vertical tangent bundle with its natural connection ∇^{T X}, and H = H^{∗}(M/B; F) → B is the bundle of the fibre-wise cohomology of F, equipped with its natural connection ∇^{H} and the L_{2}-metric g^{H} on harmonic forms. They also showed that the component of T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) in degree 0 equals the Ray-Singer analytic torsion of the fibres.

Formula (∗) above in particular implies a “Grothendieck-Riemann-Roch” theorem for flat vector bundles at a cohomological level. In [DWW], Dwyer, Weiss and Williams proved such a theorem at the level of algebraic K-theory. They also suggested another higher torsion invariant. We do not know wether their higher torsion is related to one of the above.

In [L], Lott defined a secondary family index using the analytic torsion form, which is a real analogue of the push-forward in Arakelov geometry. Ma has studied the behaviour of T under iterated fibrations in [Ma]. As a consequence, Bunke showed in [Bu3] that Lott’s secondary index is indeed functorial. One can use Theorem 0.1 below to evaluate Lott’s index in certain cases.

If the family p: M → B has compact structure group G and if F is induced from a G-equivariant flat vector bundle on the typical fibre X, then the higher analytic torsion of Bismut and Lott can be treated with equivariant methods, see [BL], Chapter 4. In [Bu1] and [Bu2], Bunke computed both the equivariant torsion and the higher analytic torsion for compact structure groups in degree > 0 for fibre-wise acyclic, unitarily flat vector bundles from the equivariant Euler characteristic of X. This was used in [Bu4] to construct cohomology classes of Diff (S^{2n−1}) both with the smooth and with the discrete topology. Bunke’s results also imply a relation between higher analytic torsion for families with compact structure group and equivariant torsion in the special case above, which was proved in general in [BG2]. Since this result depends on the Chern-normalisation of the analytic torsion forms, while all other results discussed here hold for any possible choice of coefficients, we have chosen to work with the particular normalisation T .

In [BG1], we calculated T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) for families p: M → B with a function h: M → ℝ that is fibre-wise Morse, such that there exists a fibre-wise gradient field for h that satisfies Smale’s transversality condition on every fibre of p. In this setting, the fibre-wise Thom-Smale complexes form a flat ℤ-graded vector bundle (V, ∇^{V}) over B, the metric g^{F} defines a metric g^{V} on V, and the coboundary map v becomes a parallel section of End V. In particular, there is a flat superconnection A′ = ∇^{V} + v of total degree 1, and we showed that T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) is related to the torsion form T (A′, g^{V}) defined in [BL], section 2. If H still denotes the fibre-wise cohomology and g_{V}^{H} denotes the metric induced by identifying H with a subbundle of V using finite-dimensional Hodge theory, this torsion form satisfies

d^{B}T (A′, g^{V} ) = ch^{o}(∇^{V} , g^{V} ) − ch^{o}(∇^{H}, g_{V}^{H}) .

When [BG1] was written, Igusa’s framing principle of [I2], chapter 6, was not yet available. Thus in order to compare our result with the theory of Igusa and Klein, we could only regard unitarily flat bundles F → M such that the fibre-wise cohomology forms a trivial flat bundle over B, and only framed Morse functions h, i.e., Morse functions whose fibre-wise unstable manifolds are trivialised. Moreover, we still needed the existence of a fibre-wise Smale gradient field, in which case there always is a deformation of h to a self-indexing Morse function on M → B. In this very special case, both the higher Franz-Reidemeister torsion of Igusa and Klein and the higher analytic torsion form vanish.

In the present paper, we generalise the results of [BG1] to families of compact manifolds M → B that carry a fibre-wise Morse function h: M → ℝ, but we do not require the existence of a fibre-wise Smale gradient field. Let p̂: C → B denote the covering of B by the fibre-wise critical points of h, and let o(T^{u}X) → C denote the orientation bundle of the unstable manifolds along C. We still define a vector bundle

V = π̂_{∗}(F |_{C} ⊗ o(T^{u}X))

of fibre-wise Thom-Smale complexes. A generic fibre-wise gradient field will satisfy the Smale condition everywhere outside a stratified subset of B of codimension 1. Whenever one passes this subset, the coboundary map v on the bundle V will change by an elementary isomorphism. This elementary isomorphism will change whenever we pass the stratum of codimension 2, and the composition of the elementary isomorphisms that we encounter when we go around the stratum of codimension 2 will be homotopic to the identity, but not necessarily equal to it. If we fix homotopies on the stratum of codimension 2, then these will change when we pass the stratum of codimension 3, and the process continues. The algebraic structure arising from these coboundary maps, elementary isomophisms, homotopies etc. is called a “system of higher homotopies” in [I2]. We encounter such structures in Section 1, where we interpret them as “simplicial superconnections”, since they behave similar as classical superconnections, except that the exterior differential d gets replaced by the simplicial coboundary operator δ.

In order to define an analytic torsion form for a family V of Thom-Smale complexes where the fibre-wise coboundary map v jumps as described above, we replace the superconnection ∇^{V} + v of [BG1] by a flat superconnection A′ of total degree 1 on V that may now contain terms of arbitrarily high degree with respect to Ω^{∗}(B). This unfortunately implies that the analytic torsion form T (A′, g^{V}) of Bismut-Lott is no longer defined. However, the Morse function h induces an endomorphism h^{V} of V by multiplication with h|_{C}, and we can construct an analytic torsion form T (A′, g^{V} , h^{V}) that depends explicitly on h^{V}, such that still

d^{B}T (A′, g^{V} , h^{V} ) = ch^{o}(∇^{V} , g^{V} ) − ch^{o}(∇^{H}, g_{V}^{H}) .

In contrast to the situation in [BG1], the form T (A′, g^{V} , h^{V}) can be non-trivial even if the bundle F is fibre-wise acyclic and g^{F} is parallel. For example, it will follow from [I2] and our results in [G] that T (A′, g^{V} , h^{V}) restricts to the Borel regulator classes on the Volodin K-theory space.

In order to state the main theorem of this paper, let c̃h^{o}(H, g_{V}^{H}, g_{L_{2}}^{H}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) be the natural Chern-Simons class satisfying

d^{B}c̃h^{o}(H, g_{V}^{H}, g_{L_{2}}^{H}) = ch^{o}(H, g_{L_{2}}^{H}) − ch^{o}(H, g_{V}^{H}) .

Let ψ(∇^{T X}, g^{T X}) denote the Mathai-Quillen current on the total space of T X → M, such that

d^{M}((∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X})) = e(T X, ∇^{T X}) − (−1)^{ind^{h}} δ_{C} ,

where δ_{C} denotes the current of integration over the critical set C. Finally, let ^{0}J denote the additive genus defined in [BG1], cf. (3.6) below.

0.1. Theorem. Modulo exact forms on B,

T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) − T (A′, g^{V} , h^{V} ) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H})

= − ∫_{M/B} ch^{o}(∇^{F} , g^{F} ) (∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) + p̂_{∗}((−1)^{ind^{h}} ^{0}J(T^{s}X − T^{u}X)) rk F .

The formulation of Theorem 0.1 precisely equals Theorem 0.1 in [BG1] in the non-equivariant case, only T (A′, g^{V}) has been replaced by T (A′, g^{V} , h^{V}). In fact, a large portion of the proof of Theorem 0.1 is identical to the corresponding parts of [BG1] and has therefore been omitted. However, the computation of the limit of the analytic torsion under the Witten deformation in [BG1] relies on Smale’s transversality condition; we give a replacement for this part of the proof in Section 4.

As an application of Theorem 0.1, we use the higher analytic torsion to detect homeomorphic, but not diffeomorphic fibre bundles. More precisely, we say that two maps p, p′: M → B are homeomorphic (diffeomorphic) as maps iff there exist homeomorphisms (diffeomorphisms) Φ of M and Ψ of B such that the diagramme

M ⟶_{∼}^{Φ} M
p↓  ↓^{p′}
B ⟶_{∼}^{Ψ} B

commutes. If p and p′ are continuous (smooth) fibre bundles, we say that p and p′ are homeomorphic (diffeomorphic) as fibre bundles iff they are equivalent as continuous (smooth) maps and we can take Ψ = id_{B} in the diagramme above. The following result generalises Igusa’s Corollary 6.5.8 in [I2] for smooth bundles that are homeomorphic to trivial bundles.

0.2. Theorem. For each integer k > 0 there exists l_{0} > 0 such that the following holds. Let p: M → B be a smooth fibre bundle with compact, connected, oriented base B of dimension 4k and with compact, connected fibres of dimension 2l − 1, where l ≥ l_{0}. Then there exist smooth fibre bundles p_{j}: M → B for all j ∈ Z with p_{0} = p, which are all homeomorphic to p as fibre bundles and have diffeomorphic fibres, but which are pairwise not diffeomorphic as fibre bundles.

Moreover, if the fibre-wise cohomology bundle H^{∗}(M/B; ℂ) admits a parallel metric with respect to the Gauß-Manin connection, then the p_{j} are pairwise not diffeomorphic as maps.

We give a more general “quantitative” version in Theorem 5.31 below. The bundles p_{i} are constructed by a gluing construction involving homeomorphically trivial bundles with base S^{4k} and fibre S^{2l−1}. The construction of these bundles goes back to Hatcher and is discussed in Section 5. Note that the second statement of Theorem 0.2 can be proved using an extension of Igusa’s higher Franz-Reidemeister torsion that we will introduce in [G]. Our proof of the first statement however relies on the fact that the higher analytic torsion forms can be defined even if the cohomology bundle does not admit a parallel metric.

In the second paper [G] of this series, we will show that the construction of the bundle (V, A′, h^{V}) over B gives rise to a map ξ_{p,h,F} from B to Igusa’s Whitehead space Wh(M_{r}(ℂ), GL(r, ℂ)), which is homotopic to a similar map constructed in Chapter 4 of [I2]. Conversely, each such functor ξ_{p,h,F} up to homotopy determines a ℤ-graded flat vector bundle V → B with a flat superconnection A′ = ∇^{V} + a′ and an endomorphism h^{V}. Thus, ξ_{p,h,F} is the analogue of a classifying map for flat vector bundles with the extra structure described above. The bundle V admits a parallel metric g^{V} that is adapted to h^{V} and A′ and the fibre-wise cohomology H = H^{∗}(V, a′_{0}) → B vanishes iff the image of ξ_{p,h,F} lies in the subcategory Wh^{h}(M_{r}(ℂ), U(r)) ⊂ Wh(M_{r}(ℂ), GL(r, ℂ)). In this case, the torsion form T (A′, g^{V} , h^{V}) becomes a cohomology class on B, which is the pull-back of a cohomology class T ∈ H^{∗}(Wh^{h}(M_{r}(ℂ), U(r))). This class T will be shown to equal Igusa’s torsion class τ.

We also construct a Whitehead space Wh^{u}(M_{r}(ℂ), U(r))) that classifies complexes (V, A′) as above where both F and H carry parallel metrics. Then T = τ extends to a cohomology class of Wh^{u}(M_{r}(ℂ), U(r))). Igusa’s general construction gives a map ξ: B → Wh^{u}(M_{r}(ℂ), U(r)), and the pull-back ξ^{∗}T yields a new class τ(M/B; F) ∈ H^{∗}(B) for all families M → B of smooth manifolds with a unitarily flat vector bundles (F, ∇^{F} , g^{F}) → M with unitarily flat fibre-wise cohomology (H, ∇^{H}, g^{H})—the existence of a fibre-wise Morse function is no longer required, and compactness of the fibres can be relaxed as in [I2].

In the case that the fibres of p: M → B are compact and the bundles F → M and H → F admit parallel metrics, the form

T (M/B; F) = (T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) + c̃h^{o}(∇^{H}, g^{H}, g_{L_{2}}^{H}))^{[≥2]}

of Definition 2.88 is closed, and its cohomology classes in H^{∗}(B) is independent of T^{H}M, g^{T X}, g^{F}, and g^{H}. Using Igusa’s framing principle and Theorem 0.1, we will prove the following result.

0.3. Theorem ([G]). Let M → B be a family of compact manifolds, and let (F, ∇^{F}) → M be a flat complex vector bundle with a parallel Hermitian metric, such that (H, ∇^{H}) also admits a parallel metric. Assume that there exists a function h: M → ℝ that is fibre-wise Morse, then

T (M/B; F) = τ(M/B; F)^{[≥2]} + ∫_{M/B} e(T X) ^{0}J(T X) rk F ∈ H^{∗}(B) .

Note that the fibre-wise Morse function does not appear in the conclusion. We conjecture that Theorem 0.3 holds without the assumption that such a function exists. In fact, by [I1], after taking the fibre-wise product with ℝP^{2N} for N sufficiently large, there exists a framed function h: M → ℝ whose fibre-wise singularities are either of Morse type or cubical. In order to treat the general case, it thus remains to incorporate cubical singularities into Theorems 0.1 and 0.3.

The present paper is organised as follows. In Section 1, we construct a CW-structure on M over a fixed simplicial structure on B, such that the cells of M over each simplex of B give a decomposition of the fibres that is adapted to h. We then describe how to turn the dicrete coboundary operator of the CW-complex into a smooth flat superconnection A′ on the bundle V. We also construct a de-Rham or “integration” map I that gives a quasi-isomorphism from the total de Rham complex (Ω^{∗}(M; F), ∇^{F}) to (Ω^{∗}(B; V); A′) and has certain geometric properties that are needed in Section 4.

In Section 2, we recall the definition of the characteristic form ch(∇^{F} , g^{F}) and of the analytic torsion classes T (A′, g^{V}) and T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) from [BL] and [BG1]. We describe the behaviour of the superconnection A′ and the endomorphism h^{V} on V, and use it to define the torsion class T (A′, g^{V} , h^{V}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B). This definition already involves a finitedimensional Witten-deformation on V, which is similar to the Witten deformation used in the proof of Theorem 0.1. We also define the higher torsion classes T (M/B; F, h) and T (M/B; F) if F and H are unitarily flat.

In Section 3, we show that Theorem 0.1 is compatible with several known results on higher analytic torsion. We apply our result to study Lott’s \overline{K}_{R}^{0}-pushdown. We then recall the outline of the proof of Theorem 0.1 in [BG1], stating all the intermediate results needed to prove it, and noting that all but one of these intermediate steps hold unchanged in our situation.

In Section 4, we study the behaviour of the analytic torsion form T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) under Witten deformation, without using the Smale property. Here, we use all the properties of the superconnection A′ on V, the map I: Ω^{∗}(M; F) → Ω^{∗}(B; V), and the higher torsion class T (A′, g^{V} , h^{V}) that were established in Sections 1 and 2.

Finally, in Section 5, we use Hatcher’s example of a non-linear disc bundle to construct a certain sphere bundle p: M → B. Using the higher torsion invariants, we see that p: M → B is homeomorphic but not diffeomorphic to a trivial sphere bundle. Thus we recover a result by Bökstedt, cf. [I2], Chapter 6. We then use Hatcher’s example prove Theorem 0.2 above. We also discuss Lott’s conjecture that the \overline{K}_{R}^{0}-pushdown is related to a Becker Gottlieb transfer.

We have not attempted to keep this paper self-contained, because we rely heavily on the methods and results of [BG1]. However, at the beginnings of Sections 1 and 2, we recall all necessary constructions from [BL] and [BG1], so that at least the part of this paper that constructs T (A′, g^{V} , h^{V}) is reasonably self-contained.

We are grateful to J.-M. Bismut, U. Bunke, and P. Teichner for fruitful discussions and their interest in this project. We are in great debt to K. Igusa for giving us some topological background of his work [I2] on higher FR-torsion, and for explaining Hatcher’s example, cf. Section 5. We also thank the Mathematical Sciences and Research Institute at Berkeley and Igusa’s family for their hospitality.

1. Generalised Thom-Smale complexes

Let M → B be a smooth bundle with compact fibres, and let h: M → ℝ be a Morse function on each fibre. We show that one can patch together locally defined Thom-Smale complexes for h to obtain a finite-dimensional vector bundle V → B with a flat superconnection A′, and a cochain map I: (Ω^{∗}(M; F), d^{M}) → (Ω^{∗}(B; V), A′) Together with the action of the Morse function on V, we obtain an example of a family Thom-Smale complex as defined in Definition 2.35.

We start by reviewing in Section 1.a the construction of a Thom-Smale complex (Ω^{∗}(B; V), A′) associated to a family of fibre-wise Morse-Smale gradient fields, cf. [BG1], Chapter 5. In Section 1.b, we state some simplifying assumptions on the local geometry of M and F near the set of fibrewise critical points. Dropping the Smale transversality condition, we construct in Section 1.c a particular CW-complex that is adapted to a simplicial complex S on the base space B. The coboundary operator of the associated cochain complex naturally leads to the notion of a flat “simplicial” superconnection δ + a, see Proposition 1.33. This is nothing but a “system of higher homotopies” as defined in chapter 3 of [I2]. We also construct a quasi-isomorphism relating ∇^{F} to δ + a.

In the next step, we lift the simplicial superconnection to a neighbourhood of the diagonal in B × B in Section 1.d. In Section 1.e, we extend it to a flat “mixed” simplicial and differential superconnection. The construction in Proposition 1.51 is reminiscent of the spectral sequence proof of de Rham’s theorem that simiplicial cohomology and de Rham cohomology are isomorphic. Nevertheless, the situation here is more complicated since flat superconnections obey a quadratic equation, while the equation for cohomology is linear. Note also that we have to construct an explicit quasi-isomorphism from Ω^{∗}(M; F) to Ω^{∗}(B; F) with certain geometric properties that will be crucial in Section 4; this is done in Proposition 1.59. Finally in Section 1.f, we pull the purely differential part of the mixed superconnection back to B by a partition of unity, to obtain a flat smooth superconnection on the Thom-Smale bundle V → B in Definition 1.65. The main properties of our construction are summarized in Theorem 1.66.

1.a. Local Thom-Smale complexes. Let p: M → B be a fibre-bundle with compact fibre X, and let F → M be a flat vector bundle, giving rise to a de Rham complex Ω^{∗}(M; F). Let h: M → ℝ be a family of Morse functions on the fibres of p. and let C ⊂ M be the fibre-wise critical set of h. Then C forms a finite covering p̂ = p|_{C}: C → B. If C^{j} denotes the critical set of index j, then p^{j} = p|_{C^{j}}: C^{j} → B is also a covering, and p̂ is the disjoint union of the coverings p̂_{0}, p̂_{1}, …

Let T X → M denote the vertical tangent bundle, and let T^{s}X ⊕ T^{u}X → C be any decomposition of T X|_{C} such that the vertical second derivative d^{2}h: (T X)^{2} → ℝ of h becomes positive (negative) definite on T^{s}X (on T^{u}X), and such that d^{2}h vanishes on T^{s}X × T^{u}X. The orientation bundle o(T^{u}X) = Λ^{max}T^{u}X is independent of the choice of such splitting.

1.1. Definition. Define vector bundles

V̂^{j} = F |_{C^{j}} ⊗ o(T^{u}X) ⟶ C^{j} ,  V̂ = ⋃_{j≥0} V̂^{j} = F |_{C} ⊗ o(T^{u}X) ⟶ C ,
V^{j} = p_{∗}^{j}V̂^{j} ⟶ B ,  V = ⨁_{j≥0} V^{j} = p̂_{∗}V̂ ⟶ B ,
and  End^{k} V = ⨁_{j≥0} Hom(V^{j}, V^{j+k}) ⊂ End V .

Let ∇V̂, ∇^{V} and ∇^{End V} be the flat connections on V̂ and V respectively that are induced by ∇^{F}.

1.2. Definition. Let N V̂ ∈ Γ(End V̂) denote multiplication by j on V̂^{j}, and let N^{V} ∈ Γ(End^{0} V) be its pushdown to B. Multiplication by h|_{C} on V̂ will be denoted by hV̂, and its pushdown to B by h^{V} ∈ Γ(End^{0} V). Both N^{V} and h^{V} extend to Ω^{∗}(B)-linear endomorphisms of Ω^{∗}(B, V).

An endomorphism e ∈ Ω^{∗}(B, End V) will be called strictly h^{V} -increasing if e maps the λeigenspace of h^{V} to the sum of eigenspaces of eigenvalues strictly greater than λ. In other words, such an operator maps subspaces V̂_{c} ⊂ V_{b} for c ∈ p^{−1}b to the sum of subspaces V̂_{c′} with c′ ∈ p^{−1}b and h(c′) > h(c).

1.3. Definition. For b ∈ B, let End_{+} V_{b} = ⨁_{k∈ℤ} End_{+}^{k} V_{b} be the space of strictly h^{V} -increasing endomorphisms of V_{b}. Let

Γ(End_{+} V) = { e ∈ Γ(End V) | e_{b} ∈ End_{+} V_{b} for all b ∈ B } ,
and  Ω^{∗}(B; End_{+} V) = Ω^{∗}(B) ⊗_{C^{∞}(B)} Γ(End_{+} V) .

Note however that the spaces End_{+} V_{b} may not form vector bundles over B because h is not locally constant on C.

1.4. Definition. Let F^{0}(M; F) ⊃ F^{1}(M; F) ⊃ … and F^{0}(B; V) ⊃ F^{1}(B; V) ⊃ … be the filtrations of Ω^{∗}(M; F) and Ω^{∗}(B, V) respectively by horizontal degree, given by

F^{p}(M; F) = ⨁_{k} F^{p,k}(M; F) ,
with F^{p,k}(M; F) = { α ∈ Ω^{k}(M; F) | ι_{X_{0}} … ι_{X_{k−p}}α = 0 for all X_{0}, …, X_{k−p} ∈ Γ(T X) }
and  F^{p}(B; V) = ⨁_{q≥p} Ω^{q}(B; V) .

We can choose an open cover U = { U_{i} | i ∈ I }, such that we have vertical metrics g_{i}^{T X} on T X|_{p^{−1}(U_{i})} over U_{i}, for which h becomes a fibre-wise Morse-Smale function. Let M_{i}^{u} → p^{−1}(U_{i}) ⊂ C with M_{i}^{u}|_{c} ⊂ M_{p̂(c)} denote the bundle of unstable cells of the fibre-wise negative gradient flow with respect to the metric g_{i}^{T X}.

1.5. Definition. Set

Î_{i}: Ω^{∗}(M; F) ⟶ Ω^{∗}(p^{−1}(U_{i}); V̂|_{p^{−1}(U_{i})}) ,  α ⟼ ∫_{M_{i}^{u}/C} α ,
and  I_{i}: Ω^{∗}(M; F) ⟶ Ω^{∗}(U_{i}; V |_{U_{i}}) ,  α ⟼ ∫_{M_{i}^{u}/B} α .

Note that Î_{i} and I_{i} are Ω^{∗}(B)-linear in the sense that

Î_{i}(p^{∗}α ∧ β) = p^{∗}α ∧ Î_{i}(β)  and  I_{i}(p^{∗}α ∧ β) = α ∧ I_{i}(β) .

By [BG1], Theorem 5.5, these maps are C^{∞} under the simplifying assumptions of Section 1.b.

Summarizing the results of [BG1], Sections 5.4 and 5.5, in particular Theorem 5.8, we obtain the following characterisation of I_{i}.

1.6. Theorem. There exists a unique coboundary operator ∂_{i} ∈ Γ(End_{+}^{1} V |_{U_{i}}) such that the superconnection A′_{i} = ∂_{i} + ∇^{V} on V is flat and

I_{i}(Ω^{∗}(p^{−1}(U_{i}), F |_{p^{−1}(U_{i})}), ∇^{F} ) ⟶ (Ω^{∗}(U_{i}, V |_{U_{i}}), A′_{i})

is a quasi-isomorphism of complexes respecting the filtrations by horizontal degree. If we denote the spectral sequences associated to the filtered complexes above by (E_{k}^{∗}(M; F), d_{k}) and (E_{k}^{∗}(U_{i}; V), d_{k}) respectively, then I_{i} induces isomorphisms

I_{i∗}: (E_{k}^{∗}(M; F), d_{k}) ⟶^{∼} (E_{k}^{∗}(U_{i}; V), d_{k}) for k ≥ 1.

We will be mainly interested in the E_{0}- and E_{1}-terms of these spectral sequences. Let T^{H}M ⊂ T M be a horizontal subbundle complementary to T X, and let Ω ∈ Ω^{2}(B, p_{∗}T X) be the associated fibre bundle curvature. Then the operator ∇^{F} on Ω^{∗}(M; F) splits naturally as

∇^{F} = ∇^{F |_{X}} + ∇^{Ω^{∗}(X;F)} + ι_{Ω}

with respect to the splitting Ω^{∗}(M; F) = Ω^{∗}(B; Ω^{∗}X; F) defined by T^{H}M. The operator ∇^{F |_{X}} induces an operator d_{0}^{p} on the quotient E_{0}^{p} = F^{p}(M; F)/F^{p+1}(M; F), which is independent of T^{H}M. We have

(1.7) (E_{0}^{p}(M; F), d_{0}) ≅ (Ω^{p}(B; Ω^{∗}(X; F)), ∇^{F |_{X}} ) , and (E_{0}^{p}(U_{i}; V), d_{0}) ≅ (Ω^{p}(U_{i}; V), ∂_{i}) .

1.8. Definition. Let H = H_{dR}^{∗}(M/B; F) denote the bundle of the fibre-wise cohomology with its natural flat Gauß-Manin connection ∇^{H}. Similarly, let H_{i} = H^{∗}(V; ∂_{i}), and let ∇^{H_{i}} be the connection induced by ∇^{V}.

By flatness of A′_{i}, the operator ∂_{i} is parallel. If follows that ∇^{H_{i}} is well-defined and flat. As in [BG1], we have

(E_{1}^{∗}(M; F), d_{1}) ≅ (Ω^{∗}(B; H), ∇^{H}) ,  and  (E_{1}^{∗}(U_{i}; V), d_{1}) ≅ (Ω^{∗}(U_{i}; H_{i}), ∇^{H_{i}}) .

By Theorem 1.6, the integration map gives an isomorphism of these E_{1}-terms. In particular, there is an isomorphism of flat vector bundles

I_{i∗}: (H|_{U_{i}}, ∇^{H}) ⟶ (H_{i}, ∇^{H_{i}}) .

1.b. Some simplifying assumptions. In this section, we construct vertical metrics g^{T X}, horizontal subbundles T^{H}M, and metrics g^{F} on F, satisfying certain properties that we will use throughout this section and Section 4.

Let p: M → B be a smooth bundle with compact fibres, and let h: M → ℝ be a smooth function that is Morse on each fibre of p. Let p̂ = p|_{C}: C → B be the finite covering of B by the fibre-wise critical points of h. Let (F, ∇^{F}) → M be a flat vector bundle.

For a Euclidean vector bundle π: (E, g^{E}) → M, let D_{r}E denote the disc bundle of radius r,

D_{r}E = { e ∈ E | |e| < r } .

A Euclidean connection ∇^{E} on E with curvature F^{E} naturally defines a horizontal subbundle T^{H}E of the tangent bundle T E to its total space, such that the fibre bundle curvature Ω^{E}: π^{∗}T M → E takes the form

(1.9)  Ω^{E}(v, w)_{e} = −[v̄, w̄]_{e}^{⊥} = F_{v,w}^{E}e ,

where v̄, w̄ ∈ Γ(T^{H}E) are horizontal lifts of vector fields v, w on M, and e ∈ E is a point on the total space.

If g^{T X} is a vertical metric on T X → M, let T^{s}X ⊂ T X|_{C} (T^{u}X ⊂ T X|_{C}) denote the sum of all positive (negative) eigenspaces of the second derivative ∇^{2}h ∈ Γ(End T X|_{C}), so the bundle π: T X|_{C} → C splits as T X|_{C} = T^{s}X ⊕ T^{u}X. Let U_{r}(C) denote the fibre-wise neighbourhood of C of radius r, and let exp^{X}: T X → M be the fibre-wise Riemannian exponential map.

1.10. Proposition. There exists a horizontal subbbundle T^{H}M ⊂ T M, a vertical metric g^{T X} on T X, a metric g^{F} on F, and a Euclidean connection ∇^{T X|_{C}} respecting the splitting T X|_{C} = T^{s}X ⊕ T^{u}X, with the following property. For each compact subset K ⊂ B there exists an ε > 0, such that

(1) over K, the fibre-wise exponential map exp^{X} |_{D_{2ε}T X|_{C}} induces an isometry onto its image U_{2ε}(C), and its differential maps T^{H}(T X) to T^{H}M|_{U_{2ε}(C)};

(2) the Morse function h|_{U_{2ε}(C)} is for c ∈ C|_{K} and (z^{s}, z^{u}) ∈ D_{2ε}T X_{c} given by

h(exp_{c}^{X}(z^{s}, z^{u})) = h(c) + \frac{|z^{s}|^{2} − |z^{u}|^{2}}{2} ;

(3) over K, the flat bundle (F, ∇^{F} , g^{F})|_{U_{2ε}(C)} is isometric to π^{∗}(F |_{C}, ∇^{F} |_{C}, g^{F} |_{C}).

Proof. This follows easily because the classical Morse lemma holds for families. □

If conditions (1) – (3) above hold for T^{H}M, g^{T X} and g^{F}, we will say that T^{H}M, g^{T X} and g^{F} satify the simplifying assumptions of Section 1.b. We assume throughout this section that this is the case. For simplicity, we may assume (at least for the purpose of estimates) that B is compact, so that we may fix an ε > 0 as in Proposition 1.10 for all of B.

In particular, the closures of the 2ε-tubular neighbourhoods of different leaves of the critical covering p̂: C → B do not intersect, and h differs by at most 2ε^{2} from h|_{C} along the fibres of these tubular neighbourhoods. For each flow line γ of the negative gradient flow, this implies that h(c_{−}) − h(c_{+}) > 4ε^{2}, where c_{±} = lim_{t→±∞} γ(t) ∈ C.

1.c. A CW complex for a fibre-wise Morse function. In Section 1.a, we recalled from [BG1] how to associate a family of CW complexes to a fibre-wise Morse function h in the special case that there exists a fibre-wise Morse-Smale gradient field for h. In this subsection, we similarly construct a CW complexes on M over a simplicial complex S on B for general fibre-wise Morse functions.

With respect to these complexes, the map p: M → B will be cellular, and the preimage of any simplex |σ| in M will consist of one cell M_{α}(σ) of dimension k + ind h for each leaf C_{α}(σ) of the critical covering p̂: C → B.

Let the standard k-simplex be a fixed closed affine k-simplex Δ_{k} ⊂ ℝ^{k}, whose corners will be enumerated Δ_{k}(0), …, Δ_{k}(k). Then for each finite sequence of integers 0 ≤ i_{0} ≤ ⋯ ≤ i_{l} ≤ k let f_{i_{0}...i_{l}}: Δ_{l} → Δ_{k} be the affine map that maps Δ_{l}(j) to Δ_{k}(i_{j}).

Let x_{0}, …, x_{k} be barycentric coordinates on Δ_{k}, i.e., x_{j} is an affine function on Δ_{k} ⊂ ℝ^{k} that takes the value δ_{ij} at Δ_{k}(i), and X_{0} + ⋯ + x_{k} = 1. Let w_{ij} denote the constant vector field on ℝ^{k} with w_{ij} = Δ_{k}(j) − Δ_{k}(i) ∈ ℝ^{k}. We recall that Δ_{k} is oriented by fixing the oriented base w_{0,1}, …, w_{0,k}, and that ∂Δ_{k} is oriented such that prefixing an oriented base of ∂Δ_{k} with an outward normal vector gives an oriented base of Δ_{k}. Then we have

∂Δ_{k} = ∑_{j=0}^{k}(−1)^{j}f_{0...ĵ...k}Δ_{k−1} .

1.11. Definition. A smoothly embedded k-simplex in B is a map σ: Δ_{k} → B that extends to a smooth embedding of some neighbourhood of Δ_{k} in ℝ^{k} into B. Let |σ| be the image of σ, and write σ_{j} = σ(Δ_{k}(j)) ∈ B for 0 ≤ j ≤ k.

By a smooth simplicial complex S on B we mean a simplicial set S together with a homeomorphic identification of B with the geometric realisation |S| of B, such that the characteristic maps σ: Δ_{k} → B are smooth embeddings in the sense above.

We will now fix a smooth simplicial complex S on B. Let us assume that S is so fine that each simplex in S is uniquely determined by the set of its vertices. We will write shortly σ_{i_{0},...,i_{l}} for the generalised face σ ∘ f_{i_{0},...,i_{l}}: Δ^{l} → B, thereby identifying vertices σ_{j} with their images in B. Note that we will not need degenerated simplices or degeneracy maps for most of the following constructions.

1.12. Remark. By the definition of a smoothly embedded k-simplex σ in B, we can pull the bundle M → B back to a neighbourhood U(Δ_{k}) of Δ_{k} in ℝ^{k}. In particular, we can say a vector field on p^{−1}(|σ|) ⊂ M is smooth if its pullback extends smoothly to the pullback of M to U(Δ_{k}). If two smooth submanifolds of p^{−1}(|σ|) intersect, we can say that they are transversal if their pullbacks extend to smooth submanifolds over U(Δ_{k}) that intersect transversally. This makes sense even if the original intersection took place inside the boundary of p^{−1}(|σ|).

For each k-simplex σ ∈ S_{k}, let P (σ) be the set of leaves of the covering p̂: C → B restricted to |σ|. For α ∈ P (σ), let C_{α}(σ) be the corresponding leaf of C|_{|σ|}. Let h_{α} be restriction of h to C_{α}(σ), regarded as a function on |σ|, and let ind_{α} denote the fibre-wise Morse index of h at C_{α}(σ). We choose the simplical structure on B fine enough such that

(1.13)  |h_{α}(q′) − h_{α}(q)| < \frac{ε^{2}}{2}  for all α ∈ P (σ) and all q, q′ ∈ |σ|.

Let us define a relation ≺_{σ} on P (σ) by setting

(1.14)  α ≺_{σ} β  iff  (h_{β} − h_{α})(q) > 2ε^{2}  for some q ∈ |σ|.

1.15. Remark.

(1) Assumption (1.13) guarantees that ≺_{σ} is a partial ordering on P (σ).

(2) If a σ′ is a face of σ, then ≺_{σ} is finer than ≺_{σ′}.

(3) Because of the simplifying assumptions of Section 1.b, the unstable cell M^{u}(C_{β}(σ)) can only intersect the closed 2ε-tubular neighbourhood of C_{α}(σ) if α ≺_{σ} β. In particular, there is no negative gradient flow line with respect to g_{0}^{T X} from C_{β}(σ) to C_{α}(σ) over some q ∈ |σ| unless α ≺_{σ} β.

Using barycentric coordinates and the vector fields w_{ij} = Δ_{k}(j)−Δ_{k}(i), we construct compatible vector fields on all simplices of S.

1.16. Proposition. The vector fields

W_{k} = ∑_{0≤i<j≤k} x_{i} x_{j} w_{ij}

on the standard simplices Δ_{k} have the following properties:

(1) whenever 0 ≤ i_{0} ≤ ⋯ ≤ i_{l} ≤ k, we have

W_{k} ∘ f_{i_{0},...,i_{l}} = df_{i_{0},...,i_{l}} ∘ W_{l} ,

in particular, W_{k} is tangential to all faces of Δ_{k};

(2) W_{k} vanishes only at the vertices of Δ_{k}, where it has non-degenerate zeros; and

(3) the stable and unstable cells of W_{k} at Δ_{k}(j) are given respectively by

|f_{0,...,j}| \ |f_{0,...,j−1}|  and  |f_{j,...,k}| \ |f_{j+1,...,k}| .

By [S], the vector field W_{k} can be obtained as a gradient field of a Morse function on Δ_{k} with respect to some Riemannian metric.

0  0  1  0  2  1  0  1 2 3

Figure 1.17. The vector fields W_{k} on Δ_{k} for k = 0, 1, 2, 3.

Proof. By construction w_{ij} is tangent to all faces, in particular, it vanishes at the vertices. To prove that W_{k} in nondegenerate at the vertices and has no other zeros, regard the affine function h_{k} = ∑_{j=0}^{k} j x_{j}. We have

(1.18)  W_{k}(h_{k}) = ∑_{0≤i<j≤k}(j − i) x_{i} x_{j} ≥ 0 ,

with “=” only at the vertices of Δ_{k}. With (1.18) and (1), it is easy to check (2) and (3). □

Property (1) guarantees the existence of a piece-wise smooth vector field W_{B} on B such that

W_{B} ∘ σ = Dσ ∘ W_{k}

for all k and all σ ∈ S_{k}. Note that W_{B} looks like the negative gradient vector field of a Morse function when restricted to the image of a single simplex.

Let \overline{W}_{B} be the horizontal lift of W_{B} to M, so \overline{W}_{B} is now a piece-wise smooth horizontal vector field on M. We fix a smooth cut-off function φ_{M}: M → [0, 1] with support in U_{ε}(C) and φ_{M}|_{U_{ε/2}(C)} = 1, and we define a vector field

(1.19)  W = −∇^{T X}h + φ_{M} \overline{W}_{B} .

Then W is smooth on p^{−1}(|σ|) in the sense of Remark 1.12 for each simplex σ ∈ S. We denote by M_{W}^{s} (x, σ), M_{W}^{u} (x, σ) the stable and unstable cells of the flow of W restricted to p^{−1}(|σ|) at a critical point x of W, while M^{s}, M^{u} still denotes the fibre-wise stable and unstable cells of −∇^{T X}h. For c ∈ ℝ, we define the sets

h^{−1}(h_{α} + c) = { x ∈ p^{−1}(|σ|) | h(x) = h_{α}(p(x)) + c } ,
h^{−1}(h_{α} + c, ∞) = { x ∈ p^{−1}(|σ|) | h(x) > h_{α}(p(x)) + c } ,
and  h^{−1}(−∞, h_{α} + c) = { x ∈ p^{−1}(|σ|) | h(x) < h_{α}(p(x)) + c } ⊂ p^{−1}(|σ|) .

1.20. Proposition. The zeros of W are precisely the points p^{−1}(|σ|) ∈ C for σ ∈ S_{0}. For each k and each k-simplex σ ∈ S_{k}, the flow of W preserves both p^{−1}(|σ|) ⊂ M and C(|σ|). If W vanishes at x ∈ p^{−1}(|σ|), then there is a coordinate chart ψ: U → V of M around x that maps p^{−1}(|σ|) ∩ U diffeomorphically onto ([0, ∞)^{k} × ℝ^{n}) ∩ V, such that W is in this coordinates given as

(1)  (Dψ ∘ W)|_{p^{−1}(|σ|)∩U} = (±ψ^{1} \frac{∂}{∂ψ^{1}} ± ⋯ ± ψ^{k} \frac{∂}{∂ψ^{k}} ) ∘ ψ|_{p^{−1}(|σ|)∩U} .

For α ∈ P (σ), let x ∈ C_{α}(|σ|) project to the j-th vertex σ_{j} of |σ|. Then we have

\begin{aligned}M_{W}^{s} (x, σ) ∩ h^{−1}(−∞, h_{α} + \frac{3}{2} ε^{2}) & = D_{\sqrt{3} ε}T^{s}X|_{C_{α}(|σ_{0,...,j}|\|σ_{0,...,j−1}|)} , \\ and M_{W}^{u} (x, σ) ∩ h^{−1}(h_{α} − \frac{3}{2} ε^{2}, ∞) & = D_{\sqrt{3} ε}T^{u}X|_{C_{α}(|σ_{j,...,k}|\|σ_{j+1,...,k}|)} . \end{aligned} (2)

Moreover, if y is another critical point of W and M_{W}^{u} (x, σ) ∩ M_{W}^{s} (y, σ) ≠ ∅, then either β = α or β ≺_{σ} α.

Proof. The first two claims of the proposition follow directly from our construction. Let x be a zero of W with p(x) = σ_{j}. We can choose a coodinate chart of B that restricts to the barycentric coordinates of |σ|, except the jth. Composing with p̂, we obtain a chart ψ_{0} of C around x. Finally, we obtain the desired chart in (1) by trivialising T^{s}X and T^{u}X → C by parallel translation with respect to the connection ∇^{T X|_{C}} of Proposition 1.10, along paths starting from x that are radial in the chart ψ_{0}.

Let σ ∈ S_{k} be a k-simplex, and let h_{k} be the function constructed in the proof of Proposition 1.16, which we now regard as a function on |σ| ⊂ B. Then for some sufficiently large C > 0, we clearly have

W (h − C h_{k} ∘ p) ≤ 0 ,

with “=” only at the zeros of W. This shows that each non-constant flow line γ(t) converges to a zero of W for t → ±∞.

We will restrict our attention to the unstable cells of W and −∇^{T X}h, the stable case being completely analogous. By our simplifying assumptions and because φ_{M} \overline{W}_{B} is horizontal, W is tangential to the subset D_{2ε}T^{u}X of the 2ε-tubular neighbourhood U_{2ε}(C_{α}(|σ|)), and restricted to U_{2ε}(C_{α}(|σ|)), the unstable cells of x is

D_{2ε}T^{u}X|_{C_{α}(|σ_{0,...,j}|\|σ_{j+1,...,k}|)} .

T^{u}X T^{u}XC

Figure 1.21. The vector field W near the fibre-wise critical set of h.

Now regard a typical flow-line γ(t) of W with lim_{t→−∞} γ(t) = x. Unless γ(t) ∈ C_{α}(|σ|) for one t ∈ ℝ (and thus for all t), the flow-line γ will leave U_{2ε}(C_{α}(|σ|)) at a certain times t_{0} with

h(γ(t_{0})) < h_{α}(p(γ(t_{0}))) − 2ε^{2} .

Because supp(φ_{M}) ⊂ U_{ε}(C), the trajectory γ will now follow the fibre-wise flow of −∇^{T X}h until it enters U_{ε}(C_{β}(|σ|)) for some β Let t_{1} > t_{0} be the time where γ(t) enters the larger neighbourhood U_{2ε}(C_{β}(|σ|)), and let γ(t_{1}) = (z^{s}, z^{u}) ∈ ∂D_{2ε}T X|_{C_{β}(|σ|)}. Since γ(t) hits U_{ε}(C_{β}(|σ|)), clearly

|z^{u}| < ε  and  |z^{s}| = (4ε^{2} − |z^{u}|^{2})^{\frac{1}{2}} > \sqrt{3} ε ,

so by Proposition 1.10 (2),

h_{α}(p(γ(t_{0}))) − ε^{2} > h(γ(t_{0})) > h(γ(t_{1})) > h_{β}(p(γ(t_{1}))) + ε^{2} ,

where of course p(γ(t_{0})) = p(γ(t_{1})) ∈ |σ|. In particular, we have β ≺_{σ} α by (1.14).

Repeating this argument starting from the time where γ(t) leaves U_{2ε}(C_{β}(|σ|)), we check that γ(t) passes the ε-tubular neighbourhoods of a finite number of critical levels that form a decreasing sequence with respect to the partial ordering ≺_{σ}. The vertical movement inside the larger 2εtubular neighbourhood of any of these fixpoints will decrement h(γ(t)) by at least 2ε^{2}, while the horizontal movement introduced by φ_{M} W_{B} can increment h(γ(t)) by at most ε^{2}/2 by (1.13). This shows in particular that h(γ(t)) < h_{α}(p(γ(t))) for t > t_{0}, and as claimed,

M_{W}^{u} (x, σ) ∩ h^{−1}(h_{α} − \frac{3}{2} ε^{2}, ∞) = D_{\sqrt{3} ε}T^{u}X|_{C_{α}(|σ_{0,...,j}|\|σ_{j+1,...,k}|)} . □

In order to describe the closures of the stable and unstable cells geometrically, we need the notion of submanifolds with conical singularities. We recall the recursive definition in [La].

1.22. Definition. A stratified subset Σ = (Σ_{0}, …, Σ_{k}) is a submanifold with conical singularities of dimension k, if

(1) for any 0 ≤ i ≤ k, the set Σ_{i} \ Σ_{i+1} is a smooth submanifold of M,

(2) around each point x ∈ Σ_{i} \ Σ_{i+1}, there is a C^{∞}-coordinate chart U_{x} → ℝ^{m} of M that

maps U_{x} ∩ (Σ_{0}, …, Σ_{i}) to

ℝ^{k−i} × (T_{0}, …, T_{i}) ,

where T = (T_{0}, …, T_{i},) ⊂ ℝ^{m−k+i} is a submanifold with conical singularities, and

(3) around each x ∈ Σ_{k}, there is a C^{1}-coordinate chart U′_{x} → ℝ^{m} of M that maps Σ ∩ U′_{x} to

the cone of a (k − 1)-dimensional submanifold of S^{m−1} with conical singularities. Two submanifolds Σ and T ⊂ M with conical singularities are said to intersect transversally iff Σ_{i} \ Σ_{i+1} intersects T_{j} \ T_{j+1} transversally for all i, j.

1.23. Remark. Let Σ = (Σ_{0}, …, Σ_{k}) be a submanifold with conical singularities of M, and let T ⊂ M be a submanifold. Suppose that T intersects a stratum Σ_{j} \ Σ_{j+1} transversally. Then there is a neighbourhood U of T ∩ (Σ_{j} \ Σ_{j+1}) in M such that T ∩ U intersects Σ_{i} ∩ U transversally in U for all i ≤ j.

Note that |σ| ⊂ B and p^{−1}(|σ|) ⊂ M are submanifolds with conical singularities. We now want to take the unstable cells of W to form a CW-complex on M whose cells are submanifolds with conical singularities. By the following generalisation of a result of Laudenbach, it suffices to deform W in such a way that a suitable version of the Smale transversality condition along the lines of Remark 1.12 holds. For a fixed simplex σ ∈ S, let h′: |σ| → ℝ be a smooth function such that h′(q) is a regular value of h|_{p^{−1}(q)} for all q ∈ |σ|.

1.24. Proposition. Let σ ∈ S be a fixed simplex. If the stable cells M_{W}^{u} (x, σ) intersect the unstable cells M_{W}^{s} (y, σ) of W transversally in p^{−1}(|σ|) ⊂ M in the sense of Remark 1.12, then \overline{M_{W}^{u} (x, σ)} and \overline{M_{W}^{s} (y, σ)} are submanifolds of M with conical singularities.

If for a fixed zero x of W over |σ|, all M_{W}^{u} (x′, σ) with x′ ∈ \overline{M_{W}^{u} (x, σ)} ∩ h^{−1}(h′, ∞) meet all M_{W}^{s} (y, σ) with y ∈ h^{−1}(h′, ∞) transversally in the sense of Remark 1.12, then \overline{M}_{W}^{u} (x, σ) ∩ h^{−1}(h′) is a submanifold of h^{−1}(h′) with conical singularities. The strata of \overline{M}_{W}^{u} (x, σ)∩h^{−1}(h′) are of the form M_{W}^{u} (x′, σ′)∩h^{−1}(h′), where σ′ is a face of σ and x′ ∈ \overline{M_{W}^{u} (x, σ)}∩p^{−1}(|σ′|)∩h^{−1}(h′, ∞).

Proof. First, assume that for one of the unstable cells M_{W}^{u} (x, σ), all M_{W}^{u} (x′, σ) with x′ ∈ \overline{M_{W}^{u} (x, σ)} meet all M_{W}^{s} (y, σ) transversally in the sense of Remark 1.12, for a fixed σ ∈ S_{k} and a zero x of W over p^{−1}(|σ|). In particular, we can pull M and W back to Δ_{k} by σ, and we can extend the pullbacks p̄: \overline{M} → Δ_{k} and \overline{W} smoothly over a neighbourhood U of Δ_{k} in ℝ^{k}, preserving all transversal intersections. Because \overline{W} takes the form of Proposition 1.20 (1) near zeros of \overline{W}, Laudenbach’s arguments in [La] imply that the closure \overline{\overline{M} u ; \overline{W} (x̄, U)} of the unstable cell at the pullback x̄ of x is a submanifold with conical singularities whose strata are unions of other unstable cells.

Let H be an affine hyperplane that contains a codimension 1 face of Δ_{k}. Me may assume that p̄^{−1}(H) is preserved by the flow. Assume first that p̄(x̄) ∉ H, then \overline{\overline{M} u ; \overline{W} (x̄, U)} may hit p̄^{−1}(H), but it cannot extend across p̄^{−1}(H). On the other hand, assume that p̄(x̄) ∈ H. Then it is easy to see that either \overline{M} u ; \overline{W} (x̄, U) is contained in p̄^{−1}(H), or \overline{M} u ; \overline{W} (x̄, U) intersects p̄^{−1}(H) transversally. Similar statement hold for all zeros x̄′ ∈ \overline{\overline{M} u ; \overline{W} (x̄, U)}. This implies that \overline{\overline{M} u ; \overline{W} (x̄, U)} ∩ p̄^{−1}(Δ_{k}) is again a submanifold with conical singularities, which has some extra strata given as intersections of unstable cells with p̄-preimages of faces of Δ_{k}.

Going back to M, we know now that the strata of \overline{M_{W}^{u} (x, σ)} are unions of sets of the form

(1.25)  M_{W}^{u} (x′, σ) ∩ p^{−1}(|σ′|) = M_{W}^{u} (x′, σ′) ,

where σ′ is a face of σ, and x′ ∈ \overline{M_{W}^{u} (x, σ)} ∩ p^{−1}(|σ′|) is a zero of W. A similar statement holds if we reverse the roles of the stable and unstable cells.

Finally assume that h′: |σ| → ℝ is a smooth function such that h′(q) is a regular value of h|_{p^{−1}(q)} for all q ∈ |σ|. We also assume that for one of the unstable cells M_{W}^{u} (x, σ), all M_{W}^{u} (x′, σ) with x′ ∈ \overline{M_{W}^{u} (x, σ)} ∩ h^{−1}(h′, ∞) meet all M_{W}^{s} (y, σ) with y ∈ h^{−1}(h′, ∞) transversally in the sense of Remark 1.12. Then the same argument as above implies that \overline{M}_{W}^{u} (x, σ) ∩ h^{−1}(h′) is a submanifold of h^{−1}(h′) with conical singularities, whose strata are of the form M_{W}^{u} (x′, σ′)∩h^{−1}(h′), for σ′ a face of σ and x′ ∈ \overline{M_{W}^{u} (x, σ)} ∩ p^{−1}(|σ′|) ∩ h^{−1}(h′, ∞). □

We use Proposition 1.24 inductively to deform W vertically such that Smale’s transversality condition holds in p^{−1}(|σ|) in the sense of Remark 1.12.

1.26. Proposition. After a small vertical perturbation of W outside U_{2ε}(C) that is smooth on each p^{−1}(|σ|), the closures of the stable and unstable cells \overline{M_{W}^{u} (x, σ)}, \overline{M_{W}^{s} (y, σ)} are submanifolds of M with conical singularities. The strata of \overline{M_{W}^{u} (x, σ)} are unions of sets M_{W}^{u} (x′, σ′), where σ′ is a face of σ, and x′ is a zero of W. If W already satisfies the transversality condition of Remark 1.12 over each closed simplex of a simplicial subset B_{0} ⊂ B, then W can be deformed in such a way that W |_{π^{−1}(B_{0})} is not affected.

Proof. By Proposition 1.24, it suffices to establish transversality in the sense above. We proceed by induction of the degree k of the simplices of S, modifying W only over those k-simplices where the the stable and unstable cells do not yet meet transversally. Then the relativity statement above is automatically satisfied. For k = 0, we can achieve transversality by the classical argument of Smale ([S]).

Thus assume that σ ∈ S_{k} with k ≥ 1, and that transversality in the sense of Remark 1.12 already holds over each face of ∂ |σ|. Let x, y be two zeros of W on p^{−1}(|σ|) with {x} = C_{α}(σ_{i}) and {y} = C_{β}(σ_{j}). If M_{W}^{u} (x, σ) ∩ M_{W}^{s} (y, σ) ≠ ∅, then 0 ≤ i ≤ j ≤ k, and

p(M_{W}^{u} (x, σ) ∩ M_{W}^{s} (y, σ)) ⊂ |σ_{i,...,j}| \ (|σ_{i+1,...,j}| ∪ |σ_{i,...,j−1}|) .

Unless i = 0 and j = k, the simplex σ_{i,...,j} is a proper face of σ, and we know by induction that the intersection of M_{W}^{u} (x, σ) and M_{W}^{s} (y, σ) is transversal over |σ_{i,...,j}|. Let γ(t) ∈ M_{W}^{u} (x, σ)∩M_{W}^{s} (y, σ) for some flow-line γ(t) of W, then lim_{t→−∞} = x and lim_{t→∞} = y. By definition of W_{k} and W and using Proposition 1.16, it is easy to check that

T_{γ(t)}M_{W}^{u} (x, σ) ≡ T_{γ(t)}(p^{−1} |σ_{i,...,k}|)  mod T_{γ(t)}(p^{−1} |σ_{i,...,j}|)
and  T_{γ(t)}M_{W}^{s} (y, σ) ≡ T_{γ(t)}(p^{−1} |σ_{0,...,j}|)  mod T_{γ(t)}(p^{−1} |σ_{i,...,j}|)

(1.27)

for all t ∈ ℝ. Together with transversality over |σ_{i,...,j}|, i.e.,

T_{γ(t)}(p^{−1} |σ_{i,...,j}|) ⊂ T_{γ(t)}M_{W}^{u} (x, σ) + T_{γ(t)}M_{W}^{s} (y, σ) ,

we obtain transversality in p^{−1}(|σ|).

Now assume that x ∈ C_{α}(|σ|) and y ∈ C_{β}(|σ|) with p(x) = σ_{0} and p(y) = σ_{k}. By Proposition 1.20, we must have either β = α or β ≺_{σ} α. In the first case, the intersection takes place along

(1.28)  C_{α}(|σ| \ (|σ_{1,...,k}| ∪ |σ_{0,...,k−1}|))

and is clearly transversal by Proposition 1.20.

We are left with the case {x} = C_{α}(σ_{0}), {y} = C_{β}(σ_{k}) with β ≺_{σ} α. We start another induction over α, where we complete ≺_{σ} to a total ordering <_{σ}. Fix α and assume that M_{W}^{u} (x′, σ) and M_{W}^{s} (y, σ) intersect transversally for all zeros x, y of W with x ∈ C_{α′}(|σ|) for some α′ <_{σ} α, and with y ∈ C(|σ|). This is in particular true for the smallest α.

Let S_{2ε}T^{u}X = ∂D_{2ε}T^{u}X be the sphere bundle in T^{u}X of radius 2ε. We may fix a small open neighbourhood U_{α} of S_{2ε}T^{u}X|_{C_{α}(|σ|)} in the hypersurface

h^{−1}(h_{α} − 2ε^{2}) ∩ p^{−1}(|σ|) = { z ∈ p^{−1}(|σ|) | h(z) > h_{α}(p(z)) − 2ε^{2} } ,

such that U_{α} ∩ U_{2ε}((C′_{α}(|σ|)) = ∅ for all α′ <_{σ} α. Then we have in particular U_{α} ∩ M_{W}^{u} (x′, σ) = ∅ for all x′ ∈ C_{α′}(|σ|) with α′ <_{σ} α, so a small modification of W along U_{α} over the interior σ̊ of |σ| will not destroy any transversal intersection established in a previous step. Note that for U_{α} small enough, we get

M_{W}^{u} (x, σ) ∩ U_{α} = S_{2ε}T^{u}X|_{C_{α}(|σ|)} .

Our goal is to find a diffeomorphism F_{α} of U_{α} that is an arbitrarily small perturbation of id_{U_{α}}, that equals id_{U_{α}} outside some compact subset of U_{α} ∩ p^{−1}(σ̊), and that preserves fibres of p, such that the intersections of

F_{α}(S_{2ε}T^{u}X|_{C_{α}(|σ|)})

with all M_{W}^{s} (y, σ) ∩ U_{α} become transversal in U_{α}. We will then cut M along U_{α} ⊂ h^{−1}(h_{α} − 2ε^{2}) and glue the two sides back using F_{α}. Clearly, we can get the same effect by a suitable perturbation of W “just below” h^{−1}(h_{α} − 2ε^{2}), so that W |_{U_{2ε}(C)} is unaffected. Moreover, Proposition 1.20 still holds for the modified vector field W.

By induction over α and by Proposition 1.24, we know that M_{W}^{s} (y, σ) ∩ U_{α} is a submanifold of h^{−1}(h_{α} −2ε^{2}) with conical singularities for all zeros y of W, whose strata are unions of sets of the form M_{W}^{s} (y′, σ′)∩U_{α}. Let σ′ be a proper face of σ, then we know by induction that all M_{W}^{s} (y, σ′)∩ U_{α} intersect S_{2ε}T^{u}X|_{C(|σ|′)} transversally in p^{−1}(|σ′|). As above, this implies that M_{W}^{s} (y, σ′) ∩ U_{α} intersects S_{2ε}T^{u}X|_{C(|σ|)} transversally in p^{−1}(|σ|).

By Remark 1.23, the intersection of M_{W}^{s} (y, σ) ∩ U_{α} with S_{2ε}T^{u}X|_{C(|σ|)} is transversal in a neighbourhood of p^{−1}(∂ |σ|). There clearly is a diffeomorphism F_{α} as above such that F_{α}(S_{2ε}T^{u}X|_{C_{α}(|σ|)}) intersects all M_{W}^{s} (y′, σ) transversally, if we allow F_{α} to mix the fibres of p. However, because p restricts to a submersion S_{2ε}T^{u}X|_{C_{α}(|σ|)} → |σ|, we can always take F_{α} such that it preserves the fibres of p. After the gluing construction described above, M_{W}^{u} (x, σ) now intersects all M^{s}(y, σ) transversally in a neighbourhood of U_{α}. Moreover, the glueing construction is equivalent to a small vertical modification of W over σ̊, and “just below” h^{−1}(h_{α} −2ε^{2}). Since transversality is preserved by the flow of W, these intersections are everywhere transversal. This finishes the last step of both inductions. □

For each k, each k-simplex σ ∈ S_{k} and each α ∈ P (σ), let

(1.29)  M_{α}(σ) = \overline{M_{W}^{u} (x, σ)}

denote the closure of the W -unstable manifold of the critical point x ∈ C_{α}(σ_{0}) over |σ|. Then M_{α}(σ) is a submanifold of M with conical singularities whose interior is homeomorphic to a k + ind_{α}- dimensional disc. The following proposition shows that the M_{α}(σ) form a CW-structure on M that behaves nicely with respect to the projection p: M → B and the simplicial complex S on B. We write M_{α}(σ) for the generators of the corresponding chain complex, and use ∂ to denote its boundary operator. Let us for the moment assume that the bundle T^{u}X → C is oriented, so that all cells M_{α}(σ) carry a natural orientation that agrees with the orientation of |σ| × T^{u}X_{σ_{0}} at σ_{0}.

Abusing notation, we use M_{∗} as an abbreviation for the collection of cells M_{α}(σ). To describe the boundary operator ∂ of the CW complex, let us write

(1.30)  ∂_{S}M_{α}(σ) = ∑_{j}(−1)^{j}M_{α}(σ_{0...ĵ...k}) .

If a(σ′) = (a(σ′))_{αβ} is an integer matrix for each face σ′ of σ, we define a product

(1.31)  (a ∗ M_{∗})_{α}(σ) = ∑_{j} ∑_{β}(−1)^{(ind_{α} − ind_{β})(k−j)}a_{αβ}(σ_{0...j}) M_{β}(σ_{j...k}) .

We similarly define (∂_{S}a)_{αβ}(σ) and (a ∗ b)_{αβ}(σ) for systems of matrices a and b as above. Then it is easy to check that ∂_{S}^{2} = 0, that“∗” is associative and that

(1.32)  ∂_{S}(a ∗ b) = (∂_{S}a) ∗ b + (−1)^{deg a}a ∗ (∂_{S}b) ,

where the total degree of a component a_{αβ}(σ′) is deg(a_{αβ}(σ′)) = ind_{α} − ind_{β} +j if σ′ ∈ S_{j}.

1.33. Proposition. For all α, β ∈ P (σ), there exist an integer a_{αβ}(σ), such that

(1)  a_{αβ}(σ) = 0  unless β ≺_{σ} α and ind_{α} +k = ind_{β} +1,

(2)  ∂M_{α}(σ) = ∂_{S}M_{α}(σ) + (a ∗ M)_{α}(σ) ,
(3)  and  0 = ∂_{S}a_{αβ}(σ) + (a ∗ a)_{αβ}(σ) .

Note that equation (1) makes sure that only cells of dimension k +ind_{α} −1 enter the second sum in (2). The right hand side of (2) reads like the dual of a “simplicial” superconnection, consisting of the simplical boundary operator on B and a sum of products with simplicial cochains valued in the endomorphisms of a bundle p̂_{∗}(C × ℤ) of free abelian groups. Then (3) simply says that this superconnection is flat.

1.34. Remark. In [I2], Definition 3.1.1, Igusa defines system of higher homotopies to be systems of endomorphism e(v_{0}, …, v_{k}) ∈ End(V) for n ≥ v_{0} ≥ ⋯ ≥ v_{k} ≥ 0, satisfying a non-degeneracy axiom and axioms similar to Proposition 1.33 (1) and (3), where (3) above is replaced by

0 = ∑_{j=0}^{k}(−1)^{j}(e(v_{0}, …, v_{j}) e(v_{j}, …, v_{k}) − e(v_{0}, …, v̂_{j}, …, v_{k})) .

In particular, if we fix a simplex σ ∈ S_{n} and set

e_{σ}(v_{0}, …, v_{k}) = \{ \begin{matrix}(−1)^{\frac{k(k−1)}{2}} a(σ_{v_{0},...,v_{k}}) & if v_{0} < ⋯ < v_{k}, and \\ 0 & otherwise, \end{matrix}

we obtain a system of higher homotopies in the sense of Igusa.

If T^{u}X → C is orientable, let G = {1} be the trivial group, otherwise take G = {±1}. Then the structure group of o(T^{u}X) is G, and the structure group of p̂_{∗}o(T^{u}X) → B is a subgroup of the group of G-monomial matrices, i.e., the group of invertible matrices having precisely one non-zero element in each row and each column, which is an element of G. Putting the e_{σ} together for all simplices σ ∈ S, we get a functor ξ from the category of simplices of S to Igusa’s Whitehead space Wh(ℤ, G) (which is constructed as a simplicial category), and ξ is G-monomial in the sense of Chapter 3.3 in [I2].

Proof of Proposition 1.33. Let σ ∈ S_{k}, and let x be the critical point in C_{α} over the vertex σ_{0}. Recall that by Proposition 1.26, the strata of \overline{M_{W}^{u} (x, σ)} are unions of sets M_{W}^{u} (x′, σ′), with σ′ a face of σ, and x′ ∈ \overline{M_{W}^{u} (x, σ)} ∩ p^{−1}(|σ′|). We now have to identify the codimension-1-stratum of \overline{M_{W}^{u} (x, σ)}. It is a union of two types of unstable cells M_{W}^{u} (x′, σ′). The first type has σ′ = σ, so assume that x′ ∈ C_{β} lies over the vertex σ_{j}. Then the dimension of M_{W}^{u} (x′, σ) = M_{W}^{u} (x′, σ_{j,...,k}) is ind_{β} +k − j. In particular, M^{u}(x′, σ) is of codimension 1 iff ind_{α} +j = ind_{β} +1 as in (1). This way we obtain all terms in (a ∗ M)_{α}(σ) and the very first term M_{α}(σ_{1,...,k}) of ∂_{S}M_{α}(σ) in (2), cf. (1.28).

The remaining contributions to the codimension-1-stratum arise as intersections of unstable cells with p^{−1}(∂ |σ|). The only such intersections of codimension 1 in \overline{M_{W}^{u} (x, σ)} are of the form \overline{M_{W}^{u} (x, σ′)} with σ′ = σ_{0,...,ĵ,...,k} for 1 ≤ j ≤ k, accounting for the rest of ∂_{S}M_{α}(σ) in (2). Note that j = 0 is impossible here, because for a transversal intersection, we had to assume that p(x) = σ_{0} ∈ |σ′| in the proof of Proposition 1.26.

The preceding arguments show that the boundary of M_{α}(σ) as a chain is given by (2), with constants a_{αβ}(σ_{0...j}) as in (1), which might, however, still depend on σ. To see that this is not the case, and to explain the sign factor (−1)^{(ind_{α} − ind_{β})(k−j)}, we define a_{αβ}(σ_{0...j}) to be the coefficient of M_{β}(σ_{j}) in the (chain) boundary of M_{α}(σ_{0...j}). Suppose that we orient both M_{α}(σ) and M_{β}(σ_{j...k}) by placing an oriented base of σ_{j...k} to the right of oriented bases of M_{α}(σ_{0...j}) and M_{β}(σ_{j}), respectively. By the arguments after (1.27) in the proof of Proposition 1.26, the coefficient of M_{β}(σ_{j...k}) in ∂M_{α}(σ) is still a_{αβ}(σ_{0...j}) with respect to these orientations. However, since we actually have to fit the oriented base of σ_{j...k} to the left of the bases of T^{u}X|_{C_{α}} and T^{u}X|_{C_{β}} according to the conventions fixed at the beginning of this subsection and before Proposition 1.33, these new orientations differ from the actual ones by factors of (−1)^{(k−j) ind_{α}} and (−1)^{(k−j) ind_{β}}, respectively.

Finally, because ∂^{2} = 0,

0 = ∂^{2}M_{α}(σ) = ((∂_{S} + a∗)^{2}M_{∗})_{α}(σ) = ((∂_{S}a + a ∗ a) ∗ M_{∗})_{α}(σ)

for each choice of α and σ, which implies (3) by comparison of coefficients. □

1.d. Simplicial complexes and good coverings. In the previous subsections, we constructed a CW complex on M over a simplicial complex S on B and gave an adapted basis M_{α}(σ) for the associated chain complex. Each chain M_{α}(σ) lived over a simplex σ ∈ S of B. For the following constructions however, we need families of cells and chains over subsets of B, such that we can integrate over the fibres of these cells and chains locally on B. We therefore modify the constructions of the previous subsections. We will need a sufficiently fine simplicial complex, in the following sense.

1.35. Remark. Let π_{1}, π_{2}: B × B → B denote the two product projections, and let ε > 0 be defined as in Proposition 1.10. Then there exists a neighbourhood U_{ΔB} of the diagonal ΔB in B × B and a diffeomorphism F: (π_{2}^{∗}M)|_{U_{ΔB}} → (π_{1}^{∗}M)|_{U_{ΔB}} of fibre-bundles extending id_{M}: (π_{2}^{∗}M)|_{ΔB} → (π_{1}^{∗}M)|_{ΔB}, such that for all (q_{1}, q_{2}) ∈ U_{ΔB},

(1) |h ∘ F_{q_{1},q_{2}} − h| < \frac{ε^{2}}{2} ,

(2) F_{q_{1},q_{2}} maps p^{−1}(q_{2}) to p^{−1}(q_{1}), and

(3) F_{q_{1},q_{2}}(exp_{c}^{X}(z^{s}, z^{u})) = exp_{F_{q_{1},q_{2}} c}^{X}(F_{q_{1},q_{2},∗c}z^{s}, F_{q_{1},q_{2},∗c}z^{u}) for all c ∈ p^{−1}(q_{2}), and (z^{s}, z^{u}) ∈

D_{2ε}T_{c}X. We may choose the smooth simplicial complex S so fine that

(1.36)  |σ| × |σ| ⊂ U_{ΔB}  for all σ ∈ S.

For σ ∈ S, let σ̊ denote the interior of |σ| as a submanifold of M, and let

(1.37)  U(σ) = ⋃_{σ′∈S σ is a face of σ′} σ̊′

denote the star of σ. We assume that S is so fine that the closure \overline{U(σ)} is contractible for all σ ∈ S. In particular,

(1.38)  U(S) = { U(σ) | σ ∈ S_{0} }

is a good open cover of B. Let us define

(1.39)  \overline{B}(σ) = \overline{U(σ)} × |σ|  and  \overline{B} = ⋃_{σ∈S} \overline{B}(σ) ⊂ B × B ,

then \overline{B}(σ) is a manifold with corners for all σ. Note that by (1.36),

\overline{B} = ⋃_{σ∈S} |σ| × |σ| ⊂ U_{ΔB} .

S on B U(S)=U(S_{0}) U(S_{1})  ^{π_{2}  π_{1}  \overline{B}}

Figure 1.40. A one-dimensional simplical complex S, the cover U(S), and \overline{B}.

Consider the bundle \overline{M} = (π_{1}^{∗}M)|\overline{B} → \overline{B} with its fibre-wise Morse function \overline{h} = π_{1}^{∗}h. Then

(1.41)  \overline{M}(σ) = (π_{1}^{∗}M)|_{\overline{B}(σ)} ⟶ \overline{U(σ)}

is a bundle with fibre |σ| × X. For each σ, each α ∈ P (σ), and each q_{1} ∈ \overline{U(σ)}, we define a submanifold

\begin{aligned}\overline{M}_{α}(σ)|_{q_{1}} & = F_{q_{1},·}(π_{2}^{∗}M_{α}(σ)) \\ & =\{ F_{q_{1},q_{2}}x | x ∈ M_{α}(σ) and q_{2} = p(x) \} ⊂ \overline{M}(σ)|_{q_{1}} \end{aligned} (1.42)

with conical singularities.

We still assume that the bundle T^{u}X|_{C} is oriented. We write \overline{M}_{∗} for the collection of the \overline{M}_{α}(σ), and we define ∂_{S} and ∗ as in (1.30) and (1.31).

1.43. Proposition. The submanifolds \overline{M}_{α}(σ)|_{q_{1}} ⊂ \overline{M}(σ)|_{q_{1}} with conical singularities form a bundle \overline{M}_{α}(σ) → \overline{U(σ)}, such that

(1) for each σ ∈ S_{k}, the restrictions of the \overline{M}_{α}(σ′) to \overline{U(σ)} for all α and all faces σ′ of σ form a family of fibre-wise CW-complexes on \overline{M}(σ) → \overline{U(σ)};

(2) let a_{αβ}(σ) be the coefficients of Proposition 1.33, then the boundary of \overline{M}_{α}(σ) as a fibre-wise chain is given by

∂\overline{M}_{α}(σ) = (∂_{S}\overline{M}_{α}(σ) + (a ∗ \overline{M}_{∗})_{α}(σ))|\overline{U(σ)} ;

(3) we have

\overline{M}_{α}(σ) ∩ \{ (q_{1}, q_{2}, x) ∈ \overline{M} | h(x) > h_{α}(q_{1}) − ε^{2} \} = (π_{1}^{∗}D_{\sqrt{2} ε}T^{u}X)|_{\overline{B}(σ)} .

Proof. From the construction (1.42), it follows that \overline{M}_{α}(σ) → \overline{U(σ)} is a bundle with fibre diffeomorphic to the M_{α}(σ) of Proposition 1.33. Also, the \overline{M}_{α}(σ_{i_{0}...i_{j}})|\overline{U(σ)} form a fibre-wise CW-

complex on \overline{M}(σ) → \overline{U(σ)} with fibre diffeomorphic to the CW-complex on p^{−1}(|σ|) formed by the M_{α}(σ_{i_{0}...i_{j}}) for all α and all 0 ≤ i_{0} < ⋯ < i_{j} ≤ k, so we get (1). Moreover, from Proposition 1.33 (2), we get (2). Finally, (3) follows from Proposition 1.20 by the specifications in Remark 1.35 and (1.42). □

We want to state a dual version of Proposition 1.33 and Proposition 1.43. Let F → M be a flat vector bundle. We no longer assume that the bundle T^{u}X → C is oriented, and set V_{α}(σ) = o(T^{u}X) ⊗ F |_{C_{α}(σ)} → B over any simplex |σ|. We define V → B as in Definition 1.1, so V |_{|σ|} = ⨁_{α} V_{α}(σ).

Let \overline{F} = (π_{1}^{∗}F)|\overline{B} → \overline{M} be the flat bundle over \overline{M} induced by F, and let \overline{F} (σ) = \overline{F} |_{\overline{M}(σ)}. We have natural pull-back maps

π_{1}^{∗}(σ): Ω^{∗}(p^{−1}(\overline{U(σ)}); F ) ⟶ Ω^{∗}(\overline{M}(σ); \overline{F} ) ,

and we define integration maps by integrating over the fibres of \overline{M}_{α}(σ)/\overline{U(σ)},

(1.44)  \begin{aligned}I_{α}(σ) & = ∫_{\overline{M}_{α}(σ)/\overline{U(σ)}} ∘ π_{1}^{∗}(σ) : Ω^{∗}(p^{−1}(\overline{U(σ)}); F ) ⟶ Ω^{∗}(\overline{U(σ)}; V_{α}) \\ and  I(σ) & = ⨁_{α} I_{α}(σ) : Ω^{∗}(p^{−1}(\overline{U(σ)}); F ) ⟶ Ω^{∗}(\overline{U(σ)}; V ) . \end{aligned}

We extend ∂_{S} and ∗ of (1.30) and (1.31) to the present situation.

1.45. Corollary. For all k and all σ ∈ S_{k}, there exist ∇^{V} -parallel endomorphisms a(σ) ∈ Γ(End^{1−k} V |\overline{U(σ)}) such that we have

(1)  ∇^{V} ∘ I(σ) = (−1)^{k} I(σ) ∘ ∇^{F} + (∂_{S}I(σ) + (a ∗ I)(σ))|\overline{U(σ)}
(2)  and  0 = (∂_{S}a(σ) + (a ∗ a)(σ))|\overline{U(σ)} .

Suppose that v ∈ V_{β}|\overline{U(σ)}, then

(3)  a(σ) v ∈ ⨁_{β≺_{σ}α} V_{α} .

Suppose that ω ∈ Ω^{∗}(\overline{U(σ)}; F) has support in h^{−1}(h_{α} − ε^{2}, ∞) ⊂ M|_{U_{σ}}, then

(4)  I_{α}(σ) ω = \{ \begin{matrix}∫_{D_{2ε}T^{u}X|_{Cα(σ)}/\overline{U(σ)}} ω & if σ ∈ S_{0}, and \\ 0 & otherwise. \end{matrix}

Note that (4) implies in particular that if supp ω ⊂ h^{−1}(h_{α}−ε^{2}, ∞) and β ≺_{σ} α, then I_{β}(σ) ω = 0.

1.46. Remark. For each simplex σ ∈ S, the endomorphisms a(σ′) for all faces σ′ of σ give rise to a system e_{σ} of higher homotopies in the sense of Igusa, just as in Remark 1.34. Suppose that R is an associative ring with unit, with a fixed complex representation ρ: End(ℂ^{r}) for some r ≥ 1, and that G is a subgroup of GL_{s}(R) for some s ≥ 1, such that the flat bundle F has structure group ρ(G) ⊂ GL_{rs}(ℂ) (if T^{u}X → C is not orientable, we assume that −1 ∈ G). Then the coefficients a_{αβ}(σ): V_{β} → V_{α} of a(σ) naturally take values in the matrices M_{s}(R) over R, and the structure group of V is a subgroup of the group of G-monomial matrices (matrices having exactly one non-zero entree in each row or column, which is an element of G). Then the e_{σ} above form a G-monomial functor from the category of simplices of S to the Whitehead space Wh(R, G).

Proof of Corollary 1.45. Consider the families of cochain-complexes associated to the bundles of fibre-wise CW-complex on \overline{M}(σ) → \overline{U(σ)} constructed in Proposition 1.43, with values in \overline{F}. Since we now have to take the possible orientations of T^{u}X into account, each cell \overline{M}_{α}(σ′) contributes a copy V_{α}(σ′) of V_{α}|\overline{U(σ)} → \overline{U(σ)} to the family of complexes over \overline{U(σ)}, where σ′ is a face of σ. For any fibre-wise cochain s, let s_{α}(σ′) denote the component of s in V_{α}(σ′). By Proposition 1.33 and Proposition 1.43, the coboundary map of this complex takes the form

(1.47)  (δs)_{α}(σ′) = ∂_{S}s_{α}(σ′) + (a ∗ s)(σ′)

if σ′ ∈ S_{k} is a face of σ ∈ S. Here a(σ′′) is a matrix of parallel homomorphisms a_{αβ}(σ′′): V_{β} → V_{α} for each face σ′′ of σ′, and a satisfies Proposition 1.33 (1) and (3). Now (1) follows from (1.47) and Stokes’ theorem, while (2) and (3) follow from Proposition 1.33 (1) and (3).

If supp ω ⊂ h^{−1}(h_{α} − ε^{2}, ∞), then by Proposition 1.43 (3) and (1.44),

I_{α}(σ), ω = ∫_{(π_{1}^{∗}D_{2ε}T^{u}X)|_{\overline{B}(σ)}/\overline{U(σ)}} π_{1}^{∗}(σ) ω ,

and π_{1}^{∗}(σ)ω vanishes in the directions of |σ|, so this integral vanishes unless σ ∈ S_{0}. □

1.e. A “mixed” superconnection. Starting from the endomorphisms a(σ) of the previous subsection, we construct a family of compatible flat superconnections ∇\overline{V} + a′(σ, ∅) on V |_{|σ|}, where a′(σ, ∅) ∈ Ω^{∗}(|σ| , End_{+} V) is an End V -valued differential form of total degree 1. We also construct a family of compatible “integration maps” I′(σ, ∅): Ω^{∗}(M; F) → Ω^{∗}(|σ|; V) of total degree zero that satisfy I′(σ, ∅) ∘ ∇^{F} = (∇\overline{V} + a′(σ, ∅)) ∘ I′(σ, ∅). Using a partition of unity, we pull ∇\overline{V} + a′(σ, ∅) and I′(σ, ∅) back to a flat superconnection A′ on V → B of total degree one and an integration map I: Ω^{∗}(M; F) → Ω^{∗}(B; V) satisfying I ∘ ∇^{F} = A′ ∘ I.

In the following, we will work with “mixed” simplicial and differential cochains living on an approximation of the diagonal in B × B. If σ′ = σ_{i_{0}...i_{j}} is a face of σ ∈ S_{k}, set

(1.48)  |σ \ σ′| = |σ_{i_{j},i_{j}+1,...,k}| ,

so |σ \ σ′| is the simplex spanned by all vertices of σ that are “higher than or equal to” all vertices of σ′. For convenience, we introduce the empty simplex ∅ of formal dimension −1, which we regard as a face ∅ = σ_{∅} of all σ ∈ S, so that we may write |σ| = |σ \ ∅|.

1.49. Definition. A mixed system of End V -valued differential forms a′ is a system of differential forms a′(σ, σ′) = (a′_{αβ}(σ, σ′))_{α,β∈P (σ)} for all pairs of simplices σ, σ′ ∈ S where σ′ is a face of σ, with

a′_{αβ}(σ, σ′) ∈ \{ \begin{matrix}Ω^{∗}(|σ \ σ′| , V_{β}^{∗} ⊗ V_{α}) & if σ′ = ∅, and \\ Ω^{∗}(|σ \ σ′| , V_{β}^{∗}) ⊗ V_{α}|_{σ′_{0}} & otherwise. \end{matrix}

We say that a′ is of total degree l if a′_{αβ}(σ, σ′) ∈ Ω^{l−k−(ind_{h} α−ind_{h} β)}(|σ \ σ′|) for all σ′ ∈ S_{k−1}. Similarly, we define mixed systems of V -valued differential forms η′.

In this definition, the bundles V_{α} → |σ| are regarded as trivialised by the connection ∇^{V}.

For such systems a′, b′ and σ′ ∈ S_{k−1} a face of σ with k ≥ 1, we define

\begin{aligned}(∂′_{S}a′)(σ, σ′) & = ∑_{j=0}^{k−1}(−1)^{j}a′(σ, σ′_{0,...,ĵ,...,k−1})|_{|σ\σ′|} + (−1)^{k} a′(σ, σ′_{0,...,k−2})|_{σ′_{k−1}} , \\ (d′a′)(σ, σ′) & = (−1)^{k}[∇^{V} , a′(σ, σ′)] , \\ and  (a′ ∗′ b′)(σ, σ′) & = ∑_{j=0}^{k−1}(−1)^{(k−j) deg^{V} a′(σ,σ_{0...j−1})} a′(σ, σ_{0...j−1})|_{σ′_{j}} b′(σ, σ_{j,...,k−1}) \\ & + a′(σ, σ′) b′(σ, ∅) , \end{aligned} (1.50)

assuming that a′ is of total degree 1. The term a′(σ, σ′_{0,...,k−2})|_{σ_{k−1}} in the definition of ∂′_{S}a′ denotes a ∇^{V} -parallel section on |σ \ σ′| equal to the value of a′(σ, σ′_{0,...,k−2})^{[0]} at the vertex σ_{k−1}. The connection ∇^{V} in the definition of d′ has to be replaced by d when applied to a V_{α}|_{σ′_{0}}-valued differential form. It is easy to check that ∂_{S}^{′2} = d^{′2} = 0, that ∗′ is associative, and that both ∂′_{S} and d′ satisfy a Leibniz rule for ∗′.

Let a(σ) ∈ Γ(End^{1−k} V |\overline{U(σ)}) be the parallel endomorphisms constructed in Corollary 1.45.

1.51. Proposition. There exists a system of mixed End V -valued differential forms a′ of total degree 1 such that

(1) if σ is a face of τ, then

a′(τ, σ′)|_{|σ\σ′|} = a′(σ, σ′) ;

(2) the form a′(σ, σ) on |σ_{l}| vanishes;

(3) if σ ∈ S_{l}, then

a′(σ, σ_{0,...,l−1})|_{|σ_{l}|} = a(σ)|_{|σ_{l}|} ;

(4) we have 0 = (∇^{V} + a′(σ, ∅))^{2}, and for σ′ ∈ S_{k−1} with k ≥ 1,

0 = (∂′_{S}a′ + d′a′ + a′ ∗′ a′)(σ, σ′) ;

(5) the component a′_{αβ}(σ, σ′) in Ω^{∗}(|σ \ σ′|; Hom(V_{β}, V_{α})) vanishes unless β ≺_{σ} α.

1.52. Remark. If a′(σ, σ′) was a function, Proposition 1.51 (1) would mean that a′(σ, σ′) is the restriction to |σ \ σ′| of a piece-wise smooth function a′(σ′) on the union of all |σ \ σ′| for all σ having σ′ as a face. Note however that this union is not a differentiable manifold (not even a manifold with corners). Moreover, we are dealing with differential forms. Suppose that σ is a codimension-1-face of two simplices τ and of τ′. Then in local coordinates, those coefficients of a′(τ, σ′) and a′(τ′, σ′) that contain the direction normal to |σ| may jump at |σ|, so that a′(σ′) would not even be well-defined along |σ|. We will nevertheless later on write a′(σ′) instead of a′(σ, σ′) as long as no confusion about σ arises.

If we introduce the notation

a(σ_{0,...,k−1, ·}) = a′(σ, σ_{0,...,k−1})  and  a(σ_{·}) = a′(σ, ∅)

then (4) above without the term involving ∇\overline{V} looks precisely as Corollary 1.45 (2), where the index “k” has been replaced by “ · ”. The placement of “ · ” to the right of k − 1 in the index above is compatible with the fact that we may only evaluate a′(σ_{0,...,k−1}) on the set of points “behind” σ′_{k−1} in σ. In fact, these two analogies with the coefficients a(σ) of Corollary 1.45 were the motivation for our construction of a′(σ′).

While we could regard the a(σ) of Corollary 1.45 as coefficients of a purely combinatorial superconnection, (or as a “system of higher homotopies” as in. [I2], chapter 3), we now have a mixed superconnection which is partly combinatorial and partly differential. Although the forms a′(σ′) do not pull back from endomorphisms defined on B, we can pull them back to B using a section of π_{1}: \overline{B} → B constructed from a partition of unity, to obtain a differential superconnection on V → B.

Proof of Proposition 1.51. We construct the forms a′(σ, σ′) on |σ \ σ′| for σ ∈ S_{l} by induction over l. For all σ ∈ S_{0}, we find that a′(σ, σ) = 0 and a′(σ, ∅) = a(σ) are determined by (2) and (3), and (4), (5) hold at the point σ by Corollary 1.45 (2).

Fix σ ∈ S_{l} for l > 0 and suppose that all a(τ, σ′) have been constructed on all proper faces τ of σ, and that (1)–(5) are satisfied on each proper face. If σ′ = σ_{i_{0},...,i_{k−1}} ∈ S_{k−1} is a face of σ with i_{k−1} > k − 1, then the simplex τ = σ_{i_{0},...,i_{k−1},i_{k−1}+1,...,l} is a proper face of σ, and |σ \ σ′| = |τ \ σ′|. In particular, the form a′(σ, σ′) = a′(τ, σ′) is already determined by (1) and induction, and (4) holds for the pair (σ, σ′). By Remark 1.15 (2), claim (5) also holds.

It remains to construct a′(σ, σ_{0,...,k−1}) for k = 0, …, l + 1. This will be done by induction on l − k. For k = l + 1, we have a′(σ, σ) = 0 by (2), and by (3), (4) reduces to the trivial equation

a′(σ, σ_{0,...,l−1})|_{|σ\σ|} − a′(σ, σ_{0,...,l−1})|_{σ_{l}} = a(σ) − a(σ) = 0

on |σ \ σ| = σ_{l}. Moreover, (5) follows from Corollary 1.45 (3).

Now fix σ′ = σ_{0,...,k−1} for 1 ≤ k ≤ l, and assume that a′(σ, σ_{0,...,k}) has already been constructed. The restriction of a′(σ, σ′) to each of the sets |σ_{k−1,...,ĵ,...,l}| with j ≥ k is already determined by the induction hypothesis applied to the proper face τ = σ_{0,...,ĵ,...,l} of σ. Thus, the restriction of a′(σ, σ′) to the set

(1.53)  |∂σ_{k−1,...,l}| \ σ̊_{k,...,l}

is already defined, cf. Figure 1.56 (a). Similarly if k = 0, the restriction of a′(σ, ∅) to ∂ |σ| is already known.

Claim (4) for the pair (σ, σ_{0,...,k}) involves the coefficient a′(σ, σ′) and can be used to determine a′(σ, σ′)|_{|σ_{k,...,l}|}. Since we do not know yet if the form obtained this way is compatible with the form a′(σ, σ′) determined above on |∂σ_{k−1,...,l}|\σ̊_{k,...,l} by induction, we will denote it by a′′(σ, σ′). Formally, we write

(1.54)  −(−1)^{k}a′′(σ, σ′) = ((∂′_{S}a′)(σ, σ_{0,...,k}) − (−1)^{k}a′(σ, σ′)) + (d′a′ + a′ ∗′ a′)(σ, σ_{0,...,k})

on |σ_{k,...,l}|, see Figure 1.56 (b), and we note that all remaining terms on the right hand side have already been constructed.

Then a′′(σ′) clearly satisfies (5). For j > k, the induction hypothesis applied to the face σ_{0...ĵ...l} shows that a′(σ′) satisfies equation (1.54) on |σ_{k...ĵ...l}|, thus a′′(σ′) coincides with a′(σ′) on |∂σ_{k...l}|\ σ̊_{k+1,...,l}, see Figure 1.56 (c). It remains to prove that

(1.55)  a′(σ, σ′)|_{|σ_{k+1,...,l}|} = a′′(σ, σ′)|_{|σ_{k+1,...,l}|} .

σ^{k−1}  σ_{k} (a) |∂σ^{k−1,...,l}|\σ̊_{k,...,l}  σ^{k−1}  σ_{k} (b) |σ^{k,...,l}|  σ_{k−1}  σ_{k} (c) |∂σ_{k,...,l}|\σ̊_{k+1,...,l}

Figure 1.56. The domains of definition of a′(σ, σ′) and a′′(σ, σ′).

Most terms on the right hand side of (1.54) have already been defined on some proper face of σ, with the exception of

(−1)^{k+1}(d + a(σ_{0})) a′(σ, σ_{0,...,k})|_{|σ_{k+1,...,l}|} + a′(σ, σ_{0,...,k})|_{|σ_{k+1,...,l}|} (∇^{V} + a′(σ, ∅)) .

By induction, the analogue of equation (1.54) holds for a′(σ, σ_{0,...,k})|_{|σ_{k+1,...,l}|}, and the term above above becomes

(−1)^{k+1}[∇^{V} + a′(σ, ∅), a′(σ, σ_{0,...,k})]|_{|σ_{k+1,...,l}|}

= [∇^{V} + a′(σ, ∅), ((∂′_{S}a′)(σ, σ_{0,...,k+1}) + (−1)^{k}a′(σ, σ_{0,...,k})) + (d′a′ + a′ ∗ a′)(σ, σ_{0,...,k+1})]

= [∇^{V} + a′(σ, ∅), ((∂′_{S}a′)(σ, σ_{0,...,k+1}) + (−1)^{k}a′(σ, σ_{0,...,k}))
+ (a′ ∗ a′)(σ, σ_{0,...,k+1}) − (−1)^{k} a(σ_{0}) a′(σ, σ_{0,...,k+1}) − a′(σ, σ_{0,...,k+1}) a′(σ, ∅)] ,

where we have used the flatness of the superconnections d + a(σ_{0}) and ∇^{V} + a′(σ, ∅) on each proper face of σ. The expression above can be expanded using (1.50). Using (4) inductively, a long and tedious calculation shows that finally

(−1)^{k+1}[∇^{V} + a′(σ, ∅), a′(σ, σ_{0,...,k})]|_{|σ_{k+1,...,l}|}

= −(∂′_{S}a′ + a′ ∗ a′)(σ, σ_{0,...,k}) + (−1)^{k+1}[a′(σ, ∅), a′(σ, σ_{0,...,k})]|_{|σ_{k+1,...,l}|} .

Inserting this into (1.54) proves (1.55).

In particular, we can now extend a′(σ, σ′) to |∂σ_{0,...,k−1}| by a′′(σ, σ′). We choose a smooth continuation a′(σ, σ′) of a′(σ′)|_{|∂σ_{0,...,k−1}|} to |σ_{0,...,k−1}| that is of total degree 1−k, cf. [I2], Lemma 2.2.3. By Remark 1.15 (2), the relation ≺_{σ} refines on the corresponding relations for the faces of σ, so we can choose a′(σ, σ′) such that (5) still holds.

In the last step, we define a′′(σ, ∅) on |σ| by

(1.57)  ∇^{V} + a′′(σ, ∅) = (id +a′(σ, σ_{0}))^{−1} (d + a(σ_{0})) (id +a′(σ, σ_{0})) .

As above, we automatically get a′′(σ, ∅) = a′(σ, ∅) on ∂ |σ| \ σ̊_{1...l}, and a similar calculation as above proves a′′(σ, ∅) = a′(σ, ∅) on the whole boundary of |σ|, so a′(σ, ∅) naturally extends to |σ|. Moreover, since a(σ_{0}) is a parallel endomorphism with a(σ_{0})^{2} = 0, the superconnection d + a(σ_{0}) on |σ| is flat, and thus the superconnection ∇\overline{V} + a′(σ, ∅) on |σ| is also flat, which means that (4) is satisfied also for a′(σ, ∅). □

Given the coefficients a′(σ, σ′) of a “mixed” superconnection, we now have to construct the coefficients I′(σ, σ′) of a “mixed” chain homomorphism in such a way that the analogue of Corollary 1.45 (1) holds. More precisely, we look for a system of maps (I′(σ, σ′))_{σ′⊂σ} that maps ω ∈ Ω^{∗}(M; F) to a system (I′(σ, σ′)ω)_{σ′⊂σ} to a mixed system of V -valued differential forms in the sense of Definition 1.49. Let I(σ) be the integration maps of Corollary 1.45. We define operators ∂′_{S}, d′ and a′ ∗′ as in (1.50) that act on such systems (I′(σ, σ′))_{σ′⊂σ}, where now

(1.58)  (d′I′)(σ, σ′) = (−1)^{k} ∇^{V} ∘ I′(σ, σ′) − I′(σ, σ′) ∘ ∇^{V}  if σ′ ∈ S_{k},

assuming that I′ is of total degree 0.

1.59. Proposition. There exists a system (I′(σ, σ′))_{σ′⊂σ} of maps

I′(σ, σ′): Ω^{∗}(M; F) ⟶ Ω^{∗}(|σ \ σ′|; V )

preserving the total degree, such that (1) if σ is a face of τ, then

I′(σ, σ′) ω = (I′(τ, σ′) ω)|_{|σ\σ′|} for all ω ∈ Ω^{∗}(M; F);

(2) the map I′(σ, σ) vanishes;

(3) if σ ∈ S_{l}, then

(I′(σ, σ_{0,...,l−1}) ω)|_{|σ_{l}|} = (I(σ) ω) ∈ V_{σ_{l}} for all ω ∈ Ω^{∗}(M; F);

(4) we have I′(σ, ∅) ∘ ∇^{F} = (∇^{V} + a′(σ, ∅)) ∘ I′(σ, ∅), and

0 = (∂′_{S}I′ + d′I′ + a′ ∗′ I′)(σ, σ′)

for all faces σ′ ∈ S_{k} of σ with k ≥ 1;

(5) for each face σ′′ ∈ S_{j} of σ and α, β ∈ P (σ), there exist differential forms

b_{αβ}(σ, σ′, σ′′) ∈ Ω^{ind_{β} − ind_{α} +j−k}(|σ \ σ′|) ,

with b_{αβ}(σ, σ′, σ′′) = 0 unless β ≺_{σ} α or β = α, such that

I′_{α}(σ, σ′) = ∑_{β} ∑_{σ′′} b_{αβ}(σ, σ′, σ′′) I_{β}(σ′′) ,

where I′_{α}(σ, σ′) is the component of I(σ, σ′) with values in in Ω^{∗}(|σ \ σ′|; V_{α});

(6) if ω ∈ Ω^{∗}(M, F) has supp(ω) ⊂ h^{−1}(h_{α} − ε^{2}, ∞), then on any |σ \ σ′|, we have

I′_{α}(σ, σ′) ω = \{ \begin{matrix}∫_{D_{2ε}T^{u}X|_{Cα(|σ\σ′|)}/|σ\σ′|} ω & if σ′ = ∅, and \\ 0 & otherwise. \end{matrix}

Note that the analogue of Remark 1.52 applies to the maps I′(σ, σ′) as well: We may regard I′(σ, σ′) as a restriction of a map I′(σ′) from Ω^{∗}(M|_{U(σ′)}; F) to the \overline{V} -valued forms on the union of all \overline{B}(σ, σ′). Moreover, if we introduce the notation I(σ′_{0,...,k−1, ·}) = I′(σ′) for σ′ ∈ S_{k−1}, then (4) above is related to Corollary 1.45 (1) just as Proposition 1.51 (4) is related to Corollary 1.45 (2). Finally, properties (5) and (6) will be crucial when we prove Theorem 0.1 in Section 4.

Proof. We basically repeat the proof of Proposition 1.51. For all σ ∈ S_{0}, we set I′(σ, σ) = 0 and I′(σ, ∅) = I(σ). By Corollary 1.45, this agrees with (2)–(6) above.

Fix σ ∈ S_{l} and assume that all I′(τ, σ′) have already been defined for all proper faces τ of σ and σ′ of τ. Then by the same argument as in the previous proof, for any face σ′ = σ_{i_{0},...,i_{k−1}} of σ with i_{k−1} > k − 1, the map I′(σ, σ′) is already inductively defined by (1), and (4)–(6) hold for I′(σ, σ′) on |σ \ σ′|. Note also that I′(σ, σ) = 0 by (2), which is compatible with (3)–(6).

Now fix σ′ = σ_{0,...,k−1} for 1 ≤ k ≤ l and assume that I′(σ, σ_{0,...,k}) has already been constructed. Then I′(σ, σ′) is already known on |∂σ_{k−1...l}|\σ̊_{k...l}, cf. (1.53) (a). We define I′′(σ, σ′): Ω^{∗}(M; F) → Ω^{∗}(|σ_{k...l}|; V) formally by

(1.60)  −(−1)^{k} I′′(σ, σ′) = (∂′_{S}I′)(σ, σ_{0,...,k}) − (−1)^{k} I′(σ, σ′) + (d′I′ + a′ ∗′ I′)(σ, σ_{0,...,k}) ,

cf. (1.54).

Note that since (5) holds for I′(σ_{0...k}) by the construction in the previous step, we have

(1.61) (d′I′)_{α}(σ, σ_{0...k}) = ∑_{β} ∑_{σ′′⊂σ}((−1)^{k+1} db_{αβ}(σ, σ_{0...k}, σ′′) I_{β}(σ, σ′′)

+ b_{αβ}(σ, σ_{0...k}, σ′′) ((−1)^{j+ind_{β} − ind_{α}} ∇^{V} I_{β}(σ′′) + I_{β}(σ, σ′′) ∇^{F} )) .

By Corollary 1.45, (1.60) and (1.61), we see that I′′(σ′) satisfies (5), where the degrees of the coefficient forms can be calculated using the fact that I′′(σ′) is of total degree −k.

As in the proof of Proposition 1.51, I′(σ′) and I′′(σ′) coincide on |∂σ_{k...l}|\σ̊_{k+1...l}, cf. Figure 1.56. In order to check that I′(σ′) = I′′(σ′) on |σ_{k+1...l}|, we proceed by computations similar to those in the previous proof. In fact, comparing (1.54) with its analogue (1.60), we find that we can re-use the computations in the previous proof after making the following changes.

(1) First, replace each occurrence of ∇^{V} + a′(σ, ∅) on the right hand side of a summand by ∇^{F}.

(2) Then replace the rightmost occurrence of a′( ·) or a( ·) by I′( ·) or I( ·), respectively. Then all calculations remain valid and show that we can use I′′(σ′) to extend I′(σ′) to |∂σ_{k−1...l}|. We now choose smooth continuations of the coefficients b_{αβ}(σ, σ′, σ′′) in (5) to |∂σ_{k−1...l}|, thus extending I′(σ′) to |σ \ σ′|.

In order to determine I′(∅), we may rewrite (4) as

(1.62)  I′′(σ, ∅) = (id +a′(σ, σ_{0}))^{−1} ((d + a(σ, σ_{0})) ∘ I′(σ, σ_{0}) + I′(σ, σ_{0}) ∘ ∇^{F} + I(σ, σ_{0})) .

As before, we can show that I′′(∅) = I′(∅) on ∂ |σ|, so we can set I′(∅) = I′′(∅) on |σ|. By the argument following (1.61), property (5) then also holds for I′(∅). Moreover, using Corollary 1.45, (1.57) and (4), we finally find that

I′(σ, ∅) ∘ ∇^{F} = (id +a′(σ, σ_{0}))^{−1} ((∇^{V} + a(σ, σ_{0})) ∘ I′(σ, σ_{0}) + I(σ, σ_{0})) ∘ ∇^{F}
= (id +a′(σ, σ_{0}))^{−1} (∇^{V} + a(σ, σ_{0}))
((id +a′(σ, σ_{0})) I′(σ, ∅) − I(σ, σ_{0}) − (∇^{V} + a(σ, σ_{0})) I′(σ, σ_{0}) + I(σ, σ_{0}))
= (∇^{V} + a′(σ, ∅)) I′(σ, ∅) .

We have now proved Proposition 1.59 (1)–(5). Claim (6) follows from (5) by Corollary 1.45 (4), where we use that for σ′ ∈ S_{k−1} and σ′′ ∈ S_{j}, b_{αα}(σ, σ′, σ′′) = 0 unless j ≥ k. □

1.f. A smoothing construction. We note that by Proposition 1.51 (4) and Proposition 1.59 (4), we obtain a superconnection ∇^{V} + a′(σ, ∅) on Ω^{∗}(|σ|; V) and a cochain-map from Ω^{∗}(M; F) to Ω^{∗}(|σ|; V). By Remark 1.52, these objects do not patch together to yield a smooth flat superconnection and a smooth cochain-map over all of B. There are several possible ways out.

For example, one can define an algebra of “S-piece-wise smooth” differential forms, given as families α = (α_{σ})_{σ∈S} with α_{σ} ∈ Ω^{∗}(|σ|), such that α_{σ}|_{τ} = α_{τ} whenever τ is a face of σ. This means that near a simplex σ ∈ S, the components of α tangential to |σ| are continuous, while the remaining components are only bounded with all their derivatives. One can show that it is possible to do analysis in such a setup. E.g., the equation (∇^{V} + a′( · , ∅))^{2} = 0 is still meaningful. It is possible to define an S-piece-wise smooth torsion form on B.

We will instead give a smoothing construction that leads to a smooth superconnection A′ on V. Choose a partition of unity { φ_{σ}: M → [0, 1] | σ ∈ S_{0} } subordinate to the open covering U(S) of (1.38), i.e., with supp(φ_{σ}) ⊂ U_{σ} for all σ ∈ S_{0}. If σ ∈ S_{k}, then all these functions vanish on |σ| except φ_{σ_{0}}, …, φ_{σ_{k}}, and of course φ_{σ_{0}} + ⋯ + φ_{σ_{k}} = 1 on |σ|. If we interpret φ_{σ_{0}}, …, φ_{σ_{k}} as barycentric coordinates on Δ_{k}, we obtain a smooth map φ: B → B by setting

(1.63)  φ̄(b) = σ(φ_{σ_{0}}(b), φ_{σ_{1}}(b), …) ∈ |σ|  for b ∈ σ̊.

Note that φ̄ maps a small neighbourhood

U′(σ) = { b ∈ B | b ∉ supp(φ_{σ′}) unless σ′ is a vertex of σ }

\begin{matrix}φ_{·} \\ S on B \end{matrix}  \overline{B} ; graph φ̄

Figure 1.64. A partition of unity and the corresponding section of π_{1}^{∗}.

of each simplex σ ∈ S to |σ|.

1.65. Definition. Define a superconnection A′ = ∇^{V} + a′ on V → B with a′ ∈ Ω^{∗}(B, End_{+}^{∗} V), and a map I: Ω^{∗}(M; F) → Ω^{∗}(B; V), such that restricted to |σ|,

A′ = ∇^{V} + φ̄^{∗}a′(σ, ∅) ,
and  I = ∑_{σ′′ is a face of σ} φ̄^{∗}b(σ, ∅, σ′′) · I(σ′′) ,

where the coefficients b(σ, ∅, σ′′) are given in Proposition 1.59 (5).

Let h^{V} denote the endomorphism of V → B induced by h|_{C}, then we will call (V, A′, h^{V}) a family Thom-Smale complex for p: M → B and F → M. The complexes Ω^{∗}(M; F) and Ω^{∗}(B; V) are filtered by F^{∗}Ω^{∗}(M; F) and F^{∗}Ω^{∗}(B; V) according to the horizontal degree, see Definition 1.4. Recall that (E_{k}^{∗,∗}(M; F), d_{k}) and (E_{k}^{∗,∗}(B; V), d_{k}) denote the accociated spectral sequences, and that E_{∗}^{∗,∗}(M; F) is the Leray spectral sequence in de Rham theory with coefficients in F → M.

1.66. Theorem. The superconnection A′ = ∇^{V} + a′ is flat and of total degree one, and I is a cochain map of total degree zero. The endomorphism a′ decomposes locally over |σ| into components a′_{αβ} ∈ Ω^{∗}(B, Hom(V_{β}, V_{α})) with

(1)  a′_{αβ} = 0  unless β ≺_{σ} α.

Locally on |σ|, the map I is an Ω^{∗}(B; End V)-linear combination of the maps I_{α}(σ′) of (1.44), with σ′ a face of σ. If ω ∈ Ω^{∗}(M; F) has support in h^{−1}(h_{α} − ε^{2}, ∞) locally on \overline{U(σ)} ⊂ B, then the component I_{α}ω of I ω in Ω^{∗}(\overline{U(σ)}; V_{α}) is given by

(2)  I_{α}ω = ∫_{D_{2ε}T^{u}X|_{Cα(σ)}/\overline{U(σ)}} ω ∈ Ω^{∗}(\overline{U(σ)}; V_{α}) .

The map I commutes with the action of Ω^{∗}(B), maps F^{p}Ω^{∗}(M; F) to F^{p}Ω^{∗}(B; V) for all integers p ≥ 0, and induces cochain homomorphisms

(3)  I_{k}^{p,q}: (E_{k}^{∗,∗}(M; F), d_{k}) ⟶ (E_{k}^{∗,∗}(B; V), d_{k})

for k, p, q ≥ 0 that are isomorphisms for k ≥ 1. In particular, I: (Ω^{∗}(M; F), ∇^{F}) → (Ω^{∗}(B; V), A′) is a quasi-isomorphism.

Proof. We know that near a simplex σ ∈ S, the components of a′( · , ∅) and I′( · , ∅) that are tangential to |σ| are continuous, together with all their higher derivatives in tangential directions. On the other hand, since φ̄ is smooth and maps an open neighbourhood of |σ| to |σ|, pulling back by φ̄ kills all components and all higher derivatives of a′( · , ∅) and I′( · , ∅) that are not purely tangential to |σ|. This implies that A′ and I are smooth.

It is immediate from Proposition 1.51 that A′ is a flat superconnection of total degree 1, and that (1) holds.

Furthermore, (1.62) and Definition 1.65 imply that for all σ, there exists a map

I′_{σ} = ∑_{σ′′ is a face of σ} φ̄^{∗}b(σ, σ_{0}, σ′′) I(σ′′) ,

such that

(1.67)  I = (id +φ̄^{∗}a′(σ_{0}))^{−1} ((∇^{V} + a(σ_{0})) ∘ I′_{σ} + I′_{σ} ∘ ∇^{F} + I(σ_{0})) ,
and  I ∘ ∇^{F} = (∇^{V} + φ̄^{∗}a′(∅)) ∘ I = A′ ∘ I .

From the construction and Proposition 1.59 (5) and (6), we conclude that I|_{|σ|} is indeed an Ω^{∗}(B; End V)-linear combination of the maps I_{α}(σ′) of (1.44), and that (2) holds. Since the maps I_{α}(σ′) are defined by integration over the fibres of certain smooth families of submanifolds with conical singularities, these maps clearly commute with Ω^{∗}(B), and so does I.

As an Ω^{∗}(B)-homomorphism, I maps F^{p}Ω^{∗}(M; F) to F^{p}Ω^{∗}(B; V) for all p, so it induces cochain homomorphisms of spectral sequences as claimed. It suffices to show that I_{1}^{∗,∗} is an isomorphism, then all I_{k}^{∗,∗} with k ≥ 1 are also isomorphisms, and I is thus a quasi-isomorphism. By (1.67), for each σ ∈ S_{k}, we have a diagramme of cochain maps

\begin{aligned}(Ω^{∗}(M; F), ∇^{F} ) & (Ω^{∗}(|σ|; V), A′) \\ & (Ω^{∗}(|σ|; V), ∇^{V} + a(σ, σ_{0})) \end{aligned}_{I}_{I(σ_{0})}_{id +φ̄^{∗}a′(σ,σ_{0})} ≅

that commutes up to the cochain homotopy I′_{σ}. Note that by classical Morse-theory, the diagonal arrow induces an isomorphism from H^{∗}(X; F) to H^{∗}(V, a(σ_{0})) for each fibre X over |σ|.

Passing to E_{1}^{p,q}, the map

(∇^{V} + φ̄^{∗}a(σ_{j})) I′_{σ} + I′_{σ} ∇^{F}: Ω^{∗}(M; F) ⟶ Ω^{∗}(|σ|; V )

induces 0, because in the grading induced by the filtrations in Definition 1.4, it splits into a fibre-wise homotopy and a part that increases the horizontal degree. In particular, the diagramme

Ω^{p}(B; H^{q}(M/B; F)) Ω^{p}(|σ|; H^{q}(V, a′_{0})) Ω^{p}(|σ|; H^{q}(V, a(σ, σ_{0})^{[0]})) I_{1}^{p,q} ∼ ∼

commutes, where a′_{0} and a(σ, σ_{0})^{[0]} denote the components of horizontal degree 0. It follows that I_{1}^{p,q} is an isomorphism for all p, q, which finishes the proof of Theorem 1.66. □

1.68. Remark. As we will see in Section 4, the Witten deformation will give another proof of the fact that I_{1}^{∗,∗} is a quasi-isomorphism.

1.69. Remark. Let σ′ ∈ S_{k} be a face of σ ∈ S_{l}. It is clear from the proof of Proposition 1.51 that we can always choose

a′(σ, σ′) ∈ ⨁_{j=0}^{l−k−1} Ω^{j}(|σ \ σ′|; End_{+}^{1−j−k} V ) .

However, this means that A′ may have components of horizontal degree up to dim X + 1. If we assume that the dimension of the fibres is higher than the dimension of the base—which we are forced to assume when we apply the main theorem of [I1] in order to construct h—then A′ can have components in any possible horizontal degree.

The construction so far has involved many choices. We had to pick a diffeomorphism F of the bundles π_{1}^{∗}M and π_{2}^{∗}M over a neighbourhood U of ΔB in B × B, a sufficiently fine smooth simplical complex S on B, a continuous vector field W that over each simpex of S is smooth and satisfies the Smale transversality condition, and finally, we had to choose smooth continuations of the coefficients of a′(σ, σ′) and I′(σ, σ′) from the boundary of certain simplices to their interiors. Still, the superconnections A′ and integration maps I of Definition 1.65 coming from different choices are not completely unrelated.

By Theorem 1.66, any two pairs of complexes and integration maps

I_{i}: (Ω^{∗}(M; F), ∇^{F} ) ⟶ (Ω^{∗}(B; V_{i}), A′_{i})

for i = 0, 1 are related by a quasi-isomorphism. One possible right inverse for I_{i} is the inverse of the restriction of I_{i} to the space of very small eigenvalues under the Witten deformation, cf. Section 4.b. However, we can make the relation of I_{0}, A′_{0} and I_{1}, A′_{1} much more concrete.

1.70. Proposition. Assume that A′_{0} and A′_{1} are two superconnections and I_{0}, I_{1} are integration maps for these superconnections, all constructed using the methods of this chapter. Then the same methods allow us to construct a superconnection \overline{A}′ on the pull-back V × [0, 1] of V to B × [0, 1], together with an integration map \overline{I}: Ω^{∗}(M × [0, 1]; F × [0, 1]) → Ω^{∗}(B × [0, 1]; V × [0, 1]), such that Theorem 1.66 holds for \overline{A}′ and \overline{I}, and \overline{A}′|_{B×\{s\}} = A′_{s} and \overline{I}|_{B×\{s\}} = I_{s} for s = 0, 1.

Proof. One has to check that all choices made so far on B × {0, 1} can be extended to the interior of \overline{B} = B × [0, 1], still satisfying all required properties. For the diffeomorphism F of Remark 1.35, this is clear because it is just a perturbation of id_{π_{1}^{∗}M} |_{ΔB}. Suppose that \overline{F} extends F on a neighbourhood U_{Δ\overline{B}} of the diagonal in \overline{B} ×\overline{B}, then we can construct a smooth simplicial complex \overline{S} as fine as (1.36) that restricts to the given complexes on B × {0} and B × {1}.

In the next step, we extend the vertical gradient field −∇^{T X}h and the cut-off function φ_{M} over B × {0, 1} that were used in (1.19) smoothly to B × [0, 1], thus defining a preliminary vector field W on M × [0, 1]. Now the constructions of Proposition 1.26, Proposition 1.51, and Proposition 1.59 are all by induction over the degree of simplices, so we can keep every choice we made over B × \{0, 1\}. For Definition 1.65, it suffices to find a partition of unity on \overline{B} that restricts to the given partitions on B × {0} and B × {1}. This finishes the proof of our proposition. □

We will need this proposition in [G], to show that a certain “classifying” map from B to the Whitehead space is well-defined up to homotopy. Proposition 1.70 can also be used to compare the torsion forms T (A′_{s}, g^{V} , h^{V}) of Section 2 for s = 0, 1 coming from superconnections A′_{s} related as above. Of course, a formula relating T (A′_{0}, g^{V} , h^{V}) and T (A′_{1}, g^{V} , h^{V}) will ultimately follow from Theorem 0.1.

2. Torsion forms for family Thom-Smale complexes

Inspired by the previous section, we define the notion of a family Thom-Smale complex and construct torsion forms for it. These torsion forms generalise the torsion forms for families of finite-dimensional complexes of [BL], Section 2.f.

Before we recall the definitions of odd Chern classes and transgression forms for flat superconnections from [BL] in Sections 2.b, and 2.c, we introduce some sign conventions in Section 2.a.

In Section 2.d, we define family Thom-Smale complexes (V, A′, h^{V}) in Definition 2.35. Their torsion forms T (A′, g^{V} , h^{V}) are constructed in Definition 2.46 using a finite-dimensional version of the Witten deformation. Theorem 2.48 states the most important properties of T (A′, g^{V} , h^{V}), which are similar to those of T (A′, g^{V}) in [BL]. We show that these forms are rigid under deformation of the underlying flat superconnection in Section 2.e, cf. [BG1]. In Section 2.f, we use T (A′, g^{V} , h^{V}) to define the higher torsion classes T (M/B; F, h) when F and the fibrewise cohomology H carry parallel metrics, see Definition 2.64.

In Section 2.g, we collect some facts about generalised metrics (cf. [BG1]) that will be needed in Section 4. We recall the construction of analytic torsion forms from [BL] and the Chern normalisation of the odd classes and torsion forms from [BG1] in Sections 2.h and 2.i. We also define a higher analytic torsion class T (M/B; F) if F and H are unitarily flat in Definition 2.88.

2.a. Notes on Sign Conventions. We recall the sign conventions of [BL] and [BG1] regarding superconnections and their adjoints. In particular, we hope to disarm a few well-concealed traps here and in Section 2.g.

For most of this paper, we strictly follow the Koszul sign convention, by which swapping to terms of degrees p and q modulo 2 introduces a sign factor of (−1)^{pq}. However, there are a few exceptions in connection with metrics and their generalisations. This is best motivated by the fact that a Hermitian sesquilinear form in the ℤ_{2}-graded sense would have to be Hermitian on the even part and skew-Hermitian on the odd degree part of a vector space, whereas for analytic reasons, we need positive definite symmetric forms both in even and in odd degree.

Let V = V^{+} ⊕ V^{−} → M be ℤ_{2}-graded complex vector bundle, and let \overline{V}^{∗} be its antidual. We define Ω^{∗}(M; V) = Γ(Λ^{∗}M ˆ⊗V ) to be the space of sections of a graded tensor product. The natural ℂ-sesquilinear pairing \overline{V}^{∗} ⊗ V → ℂ extends to a pairing

Ω^{∗}(M; \overline{V}^{∗}) ⊗ Ω^{∗}(M; V) ⟶ Ω^{∗}(M)
with  (α ˆ⊗ f) ⊗ (β ˆ⊗ v) ⟼ (α ˆ⊗ f, β ˆ⊗ v) = (−1)^{[\frac{deg β}{2} ]} α\overline{β} f(v)

(2.1)

for all α, β ∈ Ω^{∗}(M), f ∈ Γ(\overline{V}^{∗}) and v ∈ Γ(V). The factor (−1)^{[ \frac{deg β}{2} ]} comes from reversing the one-form components of \overline{β}. Note the absence of a sign factor (−1)^{deg V ·deg β}; in particular, the pairing ( · , ·) violates the sign convention.

A metric g^{V} = g^{V^{+}} ⊕g^{V^{−}} on V which respect the ℤ_{2}-grading can be regarded as an even complex linear isomorphism g^{V}: V → \overline{V}^{∗}. Together with (2.1), this induces a pairing

g^{V}: Ω^{∗}(M, V) ⊗ Ω^{∗}(M; V) ⟶ Ω^{∗}(M)
with  (α ˆ⊗ v) ⊗ (β ˆ⊗ w) ⟼ ⟨α ˆ⊗ v, β ˆ⊗ w⟩ = (−1)^{[\frac{deg β}{2} ]} α\overline{β} ⟨v, w⟩ .

(2.2)

The fact that g^{V} is Hermitian is equivalent to the relation

(2.3)  ⟨β ˆ⊗ w, α ˆ⊗ v⟩ = (−1)^{[\frac{deg(αβ)}{2}]} \overline{⟨α ˆ⊗ v, β ˆ⊗ w⟩} ,

where the sign factor is once more equivalent to a reversal of the order of the one-form factors in each decomposable summand.

As in [BL] and [BG1], we extend complex conjugation to Ω^{∗}(M; ℂ) and Ω^{∗}(M; End V).

2.4. Definition. Define a ℂ-anti-linear operator on Ω^{∗}(M; ℂ) by

(2.5)  \overline{α}^{∗} = (−1)^{[\frac{N^{B}+1}{2}]}\overline{α} .

2.6. Remark. This operator has the following properties:

\overline{f}^{∗} = \overline{f}  for all f ∈ Ω^{0}(B; ℂ),
\overline{α}^{∗} = −\overline{α}  for all α ∈ Ω^{1}(B; ℂ),
and  \overline{(α ∧ β)}^{∗} = \overline{β}^{∗} ∧ \overline{α}^{∗} .  for all α, β ∈ Ω^{∗}(B; ℂ).

Note that this operator differs from the operator (−1)^{[^{\frac{deg( ·)}{2}}]} \overline{·} encountered in (2.1)–(2.3) by a factor of (−1)^{deg( ·)}.

We let E^{∗} denote the classical (ungraded) adjoint of an endomorphism E ∈ End V with respect to g^{V}, and we let ∇^{V,∗} denote the adjoint of a connection ∇^{V}.

2.7. Definition. The adjoint of a differential form α ˆ⊗ E ∈ Ω^{∗}(B; End V) is defined as

(α ˆ⊗ E)^{∗} = (−1)^{deg α·deg E} \overline{α}^{∗} E^{∗} .

The adjoint of a superconnection A = ∇^{V} + a with a ∈ Ω^{∗}(M; End V) is given by

A^{∗} = ∇^{V,∗} + a^{∗} .

We call E or A self-adjoint (skew-adjoint) iff E^{∗} = E or A^{∗} = A (E^{∗} = −E or A^{∗} = −A) respectively.

2.8. Remark. The definition of A^{∗} is independent of the splitting A = ∇^{V} +a. For odd elements E ∈ Ω^{∗}(M; End V)^{odd} and for superconnections A on V, Definition 2.7 is compatible with the sign convention of (2.2) in the sense that

0 = ⟨E^{∗}v, w⟩ + (−1)^{deg E (deg v+deg w)} ⟨v, Ew⟩
and  d⟨v, w⟩ = ⟨A^{∗}v, w⟩ + (−1)^{deg E (deg v+deg w)} ⟨v, Aw⟩

for all v, w ∈ Ω^{∗}(M; V).

Symmetry of g^{V} together with Definition 2.7 implies that (E^{∗})^{∗} = E and (A^{∗})^{∗} = A for all superconnections A and all E ∈ Ω^{∗}(M; End V).

2.9. Remark. Sometimes it is useful to regard g^{V} as an element of Γ(Hom(V, \overline{V}^{∗})). In this setting, the adjoint of a superconnection A is given as

(2.10)  A^{∗} = (g^{V} )^{−1} \overline{A}^{∗} g^{V} ,

where \overline{A}^{∗} is the superconnection that is naturally induced on \overline{V}^{∗} by A, cf. [BL], Sections 1(c), 1(d). Note that once more \overline{A}^{∗} is compatible with (2.1) in the sense that

d(f, v) = (\overline{A}^{∗}f, v) + (−1)^{deg f+deg v} (f, Av)

for all f ∈ Ω^{∗}(M; \overline{V}^{∗}) and all v ∈ Ω^{∗}(M, V).

2.b. Odd characteristic classes for flat superconnections. We recall the construction of Kamber-Tondeur classes in de Rham theory, following [BL] and [BG1] (cf. [KT], [Du]). Let B be a smooth manifold, and let N^{B} denote the number operator on Ω^{∗}(B).

We will fix for the rest of the paper a generating function

(2.11)  f(z) = z e^{z^{2}} ,

although in a few explicitly marked steps, we need a different choice of f. Then f is an odd holomorphic function, which is real, i.e., f(ℝ) ⊂ ℝ, and there exists c > 0 such that

(2.12)  |\frac{∂^{k}f(z)}{∂z^{k}} | = O((1 + |Re z|)^{−l})  uniformly on \{ z ∈ ℂ | |Re(z)| < c \}

for all pairs of integers k, l ≥ 0.

Now let A′ be a flat superconnection on a ℤ_{2}-graded vector bundle V → B. Fix a Hermitian metric g^{V} on V, and let A′′ = (A′)^{∗} be the adjoint superconnection in the sense of Definition 2.7. Define a self-adjoint superconnection A and a skew-adjoint operator X by

(2.13)  A = \frac{1}{2} (A′′ + A′)  and  X = \frac{1}{2} (A′′ − A′) ∈ Ω^{∗}(B; End V ) ,

then clearly

[A, X] = 0 ∈ Ω^{∗}(B; End V )

because A′ and A′′ are flat.

2.14. Definition. Let f(A′, g^{V}) ∈ Ω^{∗}(B) be defined by

f(A′, g^{V} ) = (2πi)^{\frac{1−N^{B}}{2}} str_{V} (f(X)) .

Because supertraces vanish on supercommutators, we have str_{V} (X^{2k}) = \frac{1}{2} str_{V} ([X, X^{2k−1}]) = 0 for k ≥ 1, so we have lost no information by specifying f to be an odd function. Since f and X are odd and str_{V} vanishes on End^{odd} V, only the part of f(X) in Ω^{odd}(B; End^{even} V) contributes, so f(A′, g^{V}) is an odd form.

Because [A, X] = 0, we have

d^{B} str_{V} (f(X)) = str_{V} ([A, f(X)]) = 0 ,

which means that f(A′, g^{V}) is closed.

Finally, since f is odd and real, it follows that f(X) is skew-adjoint in the sense of Definition 2.7. Using also that str_{V} (f(X)) is odd, we find

str_{V} (f(X)) = − str_{V} (\overline{f(X)}^{∗}) = (−1)^{\frac{N^{B}−1}{2}} \overline{str_{V} (f(X))}

by (2.5), so the form f(A′, g^{V}) is real.

2.15. Theorem ([BL], Theorem 1.8). The form f(A′, g^{V}) is odd, closed and real. Its cohomology class does not depend on g^{V}.

Let g_{0}^{V} and g_{1}^{V} be two metrics on V. Then there exists a path (g_{s}^{V})_{s∈[0,1]} of metrics connecting g_{0}^{V} and g_{1}^{V}. Let A′′_{s} denote the adjoint of A′ with respect to g_{s}^{V}, and define A_{s} and X_{s} as in (2.13). Let ḡ^{V} be the metric on the pullback of V to B ×[0, 1] that restricts to g_{s}^{V} on B ×\{s\}. We write \overline{A}′ for the pull-back of A′ to B × [0, 1]. Then we have

f(\overline{A}′, ḡ^{V} )_{p,s} = f(A′, g_{s}^{V} )_{p} + (2πi)^{− \frac{N^{B}}{2}} str_{V} ((g_{s}^{V} )^{−1} \frac{∂g_{s}^{V}}{∂s} f′(X_{s}))_{p} ds .

2.16. Definition. Let f̃(A′, g_{0}^{V} , g_{1}^{V}) be defined by

f̃(A′, g_{0}^{V} , g_{1}^{V} ) = (2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{1} str_{V} (\frac{1}{2} (g_{s}^{V} )^{−1} \frac{∂g_{s}^{V}}{∂s} f′(X_{s})) ds ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

Then by a standard argument, Theorem 2.15 immediately implies [BL], Theorem 1.11.

2.17. Corollary. The class f̃(A′, g_{0}^{V} , g_{1}^{V}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) is independent of the path g_{s}^{V} chosen in its definition. It is even and real, and satisfies

d^{B}f̃(A′, g_{0}^{V} , g_{1}^{V} ) = f(A′, g_{1}^{V} ) − f(A′, g_{0}^{V} ) .

We recall the behaviour of f(A′, g^{V}) under smooth changes of A′, following [BG1], Chapter 2. Let (A′_{s})_{s∈[0,1]} be a family of flat superconnections on V → B, let (g_{s}^{V})_{s∈[0,1]} be a family of metrics that respect the ℤ_{2}-grading, and define A_{s}, X_{s} as in (2.13). Define an odd holomorphic function k by

k(z) = \frac{f′(z) − f′(0)}{2z} .

On \overline{B} = B × [0, 1], we calculate

d\overline{B}(^{− \frac{N^{B}}{2}} str_{V} (k(X_{s}) \frac{∂A_{s}}{∂s} ) ds) = (2πi)^{\frac{1−N^{B}}{2}} str_{V} ([A_{s}, k(X_{s}) \frac{∂A_{s}}{∂s} ]) ds

= −(2πi)^{\frac{1−N^{B}}{2}} str_{V} (k(X_{s}) \frac{∂A_{s}^{2}}{∂s} ) ds

= \frac{∂}{∂s} (f(A′_{s}, g_{s}^{V} ) − str_{V} (X_{s}) f′(0)) ds .

(2.18)

This motivates the following definition.

2.19. Definition. Set

L_{k}(A′_{s}, g_{s}^{V} ) = (2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} str_{V} (k(X_{s}) \frac{∂A_{s}}{∂s} ) ds ∈ Ω^{∗}(B; ℂ) .

These classes resemble the Bott-Chern classes in complex geometry. We now state Theorem 2.5 of [BG1] without proof.

2.20. Theorem. The form L_{k}(A′_{s}, g_{0}^{V} , g_{1}^{V}) is even and real. If deg k ≥ 3, then L_{k}(A′_{s}, g_{s}^{V}) depends modulo d^{B}Ω^{∗}(B) only on (A′_{s}, g_{s}^{V}) at s = 0, 1 and on the homotopy class of the path (A′_{s})_{s∈[0,1]} in the space of flat superconnections on V. For k(z) = \frac{f′(z)−f′(0)}{2z},

d^{B}L_{k}(A′_{s}, g_{0}^{V} , g_{1}^{V} ) = f(A′_{1}, g_{1}^{V} )^{[≥3]} − f(A′_{0}, g_{0}^{V} )^{[≥3]} .

2.c. Superconnections of total degree 1 and the form S_{f}(A′, g^{V}). We recall the definition of a superconnection of total degree 1. A flat superconnection of degree 1 on a Hermitian vector bundle V → B gives rise to a fibre-wise cochain complex with a metric g^{H} and a connection ∇^{H} on its cohomology bundle H → B. Then we define a natural form S_{f}(A′, g^{V}) ∈ Ω^{∗}(B) relating the form f(∇^{H}, g^{H}) to f(A′, g^{V}).

2.21. Definition. A superconnection A on a ℤ-graded vector bundle V = ⨁_{j} V^{j} is of total degree 1 iff

A(Ω^{j}(V^{k})) ⊂ ⨁_{l≥0} Ω^{j+l}(V^{k+1−l}) ,

and A is of total degree −1 iff

A(Ω^{j}(V^{k})) ⊂ ⨁_{l≥0} Ω^{j+l}(V^{k−1+l}) .

2.22. Remark. Let V be a ℤ-graded vector bundle with a Hermition metric g^{V}. We say that V is a ℤ-graded Hermitian vector bundle iff g^{V} respects the grading, i.e., iff V^{j} ⊥ V^{k} for all j ≠ k. In this cases, a superconnection is of total degree 1 iff its adjoint is of total degree −1.

Now suppose that V → B is a ℤ-graded Hermition vector bundle, and let ∇^{V} be any connection on V with respect to which all subbundles V^{j} are parallel. Let A′ be a flat superconnection of total degree 1, then

(2.23)  A′ = ∇^{V} + ∑_{j} a′_{j}  with  a′_{j} ∈ ⨁_{k} Ω^{j}(B; Hom(V^{k}, V^{k+1−j})) ,

and if ∇^{V,∗} denotes the adjoint of ∇^{V} and A′′ the adjoint of A′ with respect to g^{V}, then

(2.24)  A′′ = ∇^{V,∗} + ∑_{j} a′′_{j}  with  a′′_{j} = (a′_{j})^{∗} ∈ ⨁_{k} Ω^{j}(B; Hom(V^{k}, V^{k+1−j})) .

Flatness of A′ implies in particular

(2.25)  (a′_{0})^{2} = 0 ,  [∇^{V} + a′_{1}, a′_{0}] = 0 ,  and  (∇^{V} + a′_{1})^{2} + [a′_{0}, a′_{2}] = 0 .

This says first of all that (V^{∗}, a′_{0}) is a bundle of cochain complexes. Next, the coboundary operator a′_{0} is parallel with respect to ∇^{V} + a′_{1}, so the fibre-wise cohomology H^{∗}(V^{∗}, a′_{0}) forms a bundle over B denoted H → B, and ∇^{V} +a′_{1} induces a connection ∇^{H} on H. Finally, the term a′_{2} provides a cochain homotopy between the curvature of ∇^{V} + a′_{1} and 0, so the induced connection ∇^{H} is flat. The connection ∇^{H} is called the Gauß-Manin connection on H.

If (V, g^{V}) → B is a ℤ-graded Hermitian vector bundle as in Remark 2.22, let a′′_{0} denote the adjoint of a′_{0}. By finite dimensional Hodge theory, we may identify H_{b} with ker(a′′_{0} + a′_{0})_{b} = ker(a′′_{0} − a′_{0})_{b} for b ∈ B, so we get an identification of vector bundles

H = ker(a′′_{0} + a′_{0}) = ker(a′′_{0} − a′_{0}) ⊂ V ⟶ B .

Let P^{0} ∈ Γ(Hom(V, H)) denote the orthogonal projection onto H, and let g^{H} denote the restriction of g^{V} to H. If we take adjoints with respect to g^{V} and g^{H}, then by [BL], Proposition 2.6, we have

(2.26)  ∇^{H} = P^{0} (∇^{V} + a′_{1}) P^{0}  and  ∇^{H,∗} = P^{0} (∇^{V,∗} + a′′_{1}) P^{0} .

We replace B by \overline{B} = B ×ℝ, and let \overline{V} be the pullback of V to \overline{B}, with induced superconnection

\overline{A}′ = A′ + \frac{∂}{∂t} dt

as in (2.23). Let N^{V} ∈ Γ(End V) be the number operator that acts by multiplication with i on V^{i}, and consider the metric g\overline{V} on \overline{V} with

g\overline{V} ( · , ·)_{(b,t)} = g^{V} (t^{N^{V}} · , · )_{b} .

If A′′ is the adjoint of A′ with respect to g^{V} as in (2.24), then the adjoint of \overline{A}′ with respect to g\overline{V} is given by

\overline{A}′′ = t^{−N^{V}} (A′′ + \frac{∂}{∂t} dt) = ∇^{V,∗} + ∑_{j} t^{1−j}a_{j} + (\frac{∂}{∂t} + \frac{N^{V}}{t} ) dt .

We have an operator

\overline{X} = \frac{1}{2} (\overline{A}′′ − \overline{A}′) ∈ Ω^{∗}(\overline{B}; End \overline{V} ) ,

and we denote its restriction to B × {t} by X_{t}.

If we conjugate by t^{\frac{N^{V}}{2}} , we obtain superconnections

Ã′ = t^{\frac{N^{V}}{2}} \overline{A}′ t^{− \frac{N^{V}}{2}} = ∇^{V} + ∑_{j} t^{\frac{1−j}{2}} a′_{j} + (\frac{∂}{∂t} − \frac{N^{V}}{2t} ) dt ,

and  Ã′′ = t^{\frac{N^{V}}{2}} \overline{A}′′ t^{− \frac{N^{V}}{2}} = ∇^{V,∗} + ∑_{j} t^{\frac{1−j}{2}} a′′_{j} + (\frac{∂}{∂t} + \frac{N^{V}}{2t} ) dt ,

and Ã′′ and Ã′ are adjoint to each other with respect to the pull-back of g^{V} with to \overline{V} → \overline{B} that we still denote by g^{V}. Let

(2.27)  ω^{V} = ∇^{V,∗} − ∇^{V} = (g^{V} )^{−1} [∇^{V} , g^{V} ] ∈ Ω^{1}(B; End V ) ,

and as in (2.13), set

X̃ = \frac{1}{2} (Ã′′ − Ã′) = \frac{ω^{V}}{2} + ∑_{j} t^{\frac{1−j}{2}} \frac{a′′_{j} − a′_{j}}{2} + N^{V} \frac{dt}{2t} ,

and we write X̃_{t} for the restriction of X̃ to the submanifold B × {t}.

We define f̂ to be twice the form with the same name in [BL],

(2.28)  f̂(A′, g^{V} ) = (2πi)^{− \frac{N^{B}}{2}} str\overline{V} (N^{V} f′(X̃)) .

If g_{t}^{V} = g^{V} (t^{N^{V}} · , ·) denotes the restriction of g\overline{V} to B × \{t\}, then clearly

(2.29)  f(\overline{A}′, g\overline{V} ) = f(Ã′, g^{V} ) = f(Ã′_{t}, g^{V} ) + f̂(Ã′_{t}, g^{V} ) \frac{dt}{2t} = f(A′, g_{t}^{V} ) + f̂(A′, g_{t}^{V} ) \frac{dt}{2t} ,

and this form is closed on B × ℝ.

Again differing from [BL], we define

χ(V) = str_{V} (1) = ∑_{j}(−1)^{j} rk(V^{i})  and  χ′(V) = str_{V} (N^{V} ) = ∑_{j}(−1)^{j} j rk(V^{i}) ,

so χ(V) = χ(H). Then by [BL], Theorem 2.9, as t → ∞,

\begin{aligned}f(A′, g_{t}^{V} ) & = f(∇^{H}, g^{H}) + O(t^{− \frac{1}{2}} ) , \\ and  f̂(A′, g_{t}^{V} ) & = f′(0) χ′(H) + O(t^{− \frac{1}{2}} ) . \end{aligned} (2.30)

We define S_{f}(A′, g^{V}) as in [BL], [BG1], where convergence is granted by (2.30).

2.31. Definition. Set

S_{f}(A′, g^{V} ) = − ∫_{1}^{∞}(f̂(A′, g_{t}^{V} ) − f′(0) χ′(H)) \frac{dt}{2t} ∈ Ω^{∗}(B) .

Then we have the following important result of [BL], Theorems 2.16, and [BG1], Theorem 1.25.

2.32. Theorem. The form S_{f}(A′, g^{V}) is even and real, and satisfies

d^{B} S_{f}(A′, g^{V} ) = f(A′, g^{V} ) − f(∇^{H}, g^{H}) .

Proof. This follows from (2.29) and (2.30). □

Let now g^{V}, g^{′V} be two metrics on V → B, and let g^{H}, g^{′H} be the corresponding metrics on H → B. From Theorem 2.32 and Definition 2.16, we easily deduce that

(2.33)  S_{f}(A′, g^{′V} ) − S_{f}(A′, g^{V} ) = f̃(A′, g^{V} , g^{′V} ) − f̃(A′, g^{H}, g^{′H}) .

In order to control the behaviour of S_{f}(A′, g^{′V}) under variation of A′, we assume for a moment that f is an odd homolomorphic function of degree ≥ 3 that satisfies (2.12). Then k(z) = \frac{f′(z)}{2z} is also holomorphic. Let (A′_{s}, g_{s}^{V})_{s∈[0,1]} be a family of flat superconnections and ℤ-graded metrics on V. We assume that the dimension of the cohomology bundles H_{s} is independent of s, then these bundle combine to form a family over B × [0, 1], and we get a family (∇^{H,s}, g_{s}^{H})_{s∈[0,1]} of flat connections and metrics. We recall Theorem 2.15 of [BG1], again without proof.

2.34. Theorem. Modulo exact forms on B, we have

S_{f}(A′_{1}, g_{1}^{V} ) − S_{f}(A′_{0}, g_{0}^{V} ) = L_{k}(A′_{s}, g_{0}^{V} , g_{1}^{V} ) − L_{k}(∇^{H,s}, g_{0}^{H}, g_{1}^{H}) .

2.d. Family Thom-Smale complexes and their torsion forms. We define a family Thom-Smale complex to be a ℤ-graded complex vector bundle V → B with a flat superconnection A′ = ∇^{V} + a′ of total degree 1 such that the connection ∇^{V} is diagonal and the coefficients a′ are upper triangular with respect to the eigenspace decomposition of an endomorphism h^{V}, in a sense to be made precise. For such a vector bundle, together with an adapted metric g^{V}, we define a torsion form T_{f}(A′, g^{V} , h^{V}) relating f(∇^{H}, g^{H}) to f(∇^{V} , g^{V}). In the special case that a′_{j} = 0 for j ≠ 0, we may take h^{V} = N^{V}, and our torsion forms equal those of [BL], Chapter 2, that appeared in Theorem 0.1 of [BG1].

2.35. Definition. Let V be a ℤ-graded complex vector bundle as in Remark 2.22, let A′ = ∇^{V} +a′ be a flat superconnection of total degree 1, and let h^{V} ∈ Γ(End^{0} V) be an endomorphism of V of degree 0. Then (V, A′, h^{V}) is called a family Thom-Smale complex iff for each compact subset K ⊂ B there exists ε > 0, and for each point b ∈ K there exists a neighbourhood U of b, such that

(1) for each j, there is a splitting V^{j}|_{U} = ⨁_{α} V_{α}^{j}, which is preserved by ∇^{V};

(2) the endomorphism h^{V} acts on V_{α}^{j} by a function h_{α}^{j}: U → ℝ;

(3) for each j and each α,

a′(V_{α}^{j}) ⊂ ⨁_{{k,β \atop h_{α}^{j}+ε≤h_{β}^{k}}} Ω^{∗}(B; V_{β}^{k}) .

If F: M → B is a smooth map, then the pullback F^{∗}(V, A′, h^{V}) = (F^{∗}V, F^{∗}A′, F^{∗}h^{V}) is defined in the obvious way.

It seems that we should include ∇^{V} in the notation for a family Thom-Smale complex. However, the following statement shows that this is not necessary.

2.36. Proposition. The connection ∇^{V} in Definition 2.35 is uniquely determined by A′ and h^{V}. Moreover, it is flat.

Proof. Fix a compact subset K ⊂ B, a point b ∈ K and a small open neighbourhood U of b in B as above. Let V |_{U} = ⨁_{j} ⨁_{α} V_{α}^{j} be the local splitting of (1) above, and let P_{α}^{j}: V |_{U} → V_{α}^{j} be the corresponding product projections. By (3), we clearly have

P_{α}^{j} A′ P_{α}^{j} = P_{α}^{j} ∇^{V} P_{α}^{j} = ∇^{V_{α}^{j}}

over U, and since ∇^{V} preserves V_{α}^{j}, the connection ∇^{V_{α}^{j}} is simply the restriction of ∇^{V} over U to V_{α}^{j}. In particular, ∇^{V} |_{U} = ⨁_{j} ⨁_{α} ∇^{V_{α}^{j}} is uniquely determined by A′.

Similarly, we can calculate the curvature of ∇^{V_{α}^{j}} by

(∇^{V_{α}^{j}})^{2} = P_{α}^{j} (∇^{V} )^{2} P_{α}^{j} = P_{α}^{j} (A′)^{2} P_{α}^{j} = 0 ,

again using (3) to show that the derivatives of a′ do not contribute to (∇^{V_{α}^{j}})^{2}. □

For example, if A′ = ∇^{V} + a′_{0} is a flat family of complexes as in [BL], section 2.f, that is, a′_{0} ∈ Ω^{0}(B; End^{1} V), we may take h^{V} = N^{V} in order to obtain a family Thom-Smale complex. If we constructed V, ∇^{V} and A′ as in Section 1, then we may choose h^{V} to act on V_{α} ⊂ V by h_{α}.

2.37. Definition. Let (V, A′, h^{V}) be a family Thom-Smale complex. An adapted metric on V is a metric

g^{V} = ⨁_{i} g^{V^{i}} on V

such that the splitting of Definition 2.35 (1) is g^{V} -orthogonal.

2.38. Remark. If (V, A′, h^{V}) is a family Thom-Smale complex and g^{V} is an adapted metric, then the endomorphisms h^{V} and N^{V} are self-adjoint with respect to g^{V}. Because ∇^{V} and g^{V} respect the splitting of V in Definition 2.35 (1), so do ∇^{V,∗} and ω^{V}. If we set

∇^{V,u} = \frac{1}{2} (∇^{V,∗} + ∇^{V} ) ,

then ∇^{V,u} also preserves that splitting. If we write

dh^{V} = [∇^{V} , h^{V} ] = [∇^{V,∗}, h^{V} ] = [∇^{V,u}, h^{V} ] ,

then dh^{V} acts on V_{α}^{j} by dh_{α}^{j}. We have the obvious relations

[∇^{V,u}, dh^{V} ] = [∇^{V,u}, ω^{V} ] = [h^{V} , ω^{V} ] = [dh^{V} , ω^{V} ] = 0 .

Let \overline{A}′ = A′+ \frac{∂}{∂t} dt+ \frac{∂}{∂T} dT denote the trivial extension of A′ to the pullback \overline{V} of V to B×ℝ_{>}×ℝ. Define a metric g\overline{V} on \overline{V} by

(2.39)  g\overline{V} = g^{V} (t^{N^{V}} T^{h^{V}} · , · ) .

If we take the adjoint of A′ with respect to g\overline{V}, we obtain a flat superconnection

\overline{A}′′ = ∇^{V,∗} + ∑_{j} t^{1−j} T^{−h^{V}} a′′_{j} T^{h^{V}} + log T dh^{V} + (\frac{∂}{∂t} + \frac{N^{V}}{t} ) dt+(\frac{∂}{∂T} + \frac{h^{V}}{T} ) dT

and an operator

\overline{X} = \frac{1}{2} (\overline{A}′′ − \overline{A}′) .

We denote the restrictions of g\overline{V}, \overline{A}′, \overline{A}′′, \overline{X} to B × \{(t, T)\} by g_{t,T}^{V}, A′_{t,T}, A′′_{t,T} and X_{t,T}.

If we conjugate everything with t^{\frac{N^{V}}{2}} T^{\frac{h^{V}}{2}} , we obtain two superconnections

Ã′ = ∇^{V} + ∑_{j} t^{\frac{1−j}{2}} T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} − \frac{log T}{2} dh^{V} + (\frac{∂}{∂t} − \frac{N^{V}}{2t} ) dt+(\frac{∂}{∂T} − \frac{h^{V}}{2T} ) dT

and  Ã′′ = ∇^{V,∗} + ∑_{j} t^{\frac{1−j}{2}} T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} + \frac{log T}{2} dh^{V} + (\frac{∂}{∂t} + \frac{N^{V}}{2t} ) dt+(\frac{∂}{∂T} + \frac{h^{V}}{2T} ) dT

(2.40)

that are adjoints of each other with respect to g^{V}, and a g^{V} -skew-adjoint operator

(2.41)  X̃ = \frac{1}{2} (Ã′′ − Ã′) .

Fix a compact subset K of B and let ε be the constant of Definition 2.35. From now on, we keep t fixed. Then

\begin{aligned}X̃_{t,T} & = \frac{ω^{V}}{2} + \frac{1}{2} ∑_{j} t^{\frac{1−j}{2}} (T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} − T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} ) + \frac{log T}{2} dh^{V} \\ & = \frac{ω^{V}}{2} + \frac{log T}{2} dh^{V} + O(T^{\frac{ε}{2}} ) \end{aligned} (2.42)

uniformly on K as T → 0.

Define ∇^{V,u} as in Remark 2.38, then since f and ω^{V} + log T dh^{V} are odd with [ω^{V} , dh^{V}] = (dh^{V})^{2} = 0, and ω^{V} + log T dh^{V} commutes with h^{V}, we have

str_{V} (h^{V} f′(\frac{ω^{V} + log T dh^{V}}{2} )) = str_{V} (h^{V} ) f′(0) ,
and str_{V} (f(\frac{ω^{V} + log T dh^{V}}{2} )) = str_{V} (f(\frac{ω^{V}}{2} )) + \frac{log T}{2} str_{V} (dh^{V} f′(\frac{ω^{V}}{2} )) ,
with  str_{V} (dh^{V} f′(\frac{ω^{V}}{2} )) = str_{V} ([∇^{V,u}, h^{V} f′(\frac{ω^{V}}{2} )])

= d^{B} str_{V} (h^{V} f′(\frac{ω^{V}}{2} )) = str_{V} (dh^{V} ) f′(0)

by Remark 2.38, so

f(∇^{V} + \frac{∂}{∂T} dT, g\overline{V} ) = f(∇^{V} − \frac{log T}{2} dh^{V} + (\frac{∂}{∂T} − \frac{h^{V}}{2T} ) dT, g^{V} )

= (2πi)^{\frac{1−N^{B}}{2}} str_{V} ( f(\frac{ω^{V} + log T dh^{V}}{2} ))

+ (2πi)^{− \frac{N^{B}}{2}} str_{V} (h^{V} f′(\frac{ω^{V} + log T dh^{V}}{2} )) \frac{dT}{2T}

= f(∇^{V} , g^{V} ) + d\overline{B}(\frac{log T}{2} str_{V} (h^{V} )) f′(0) .

Define an odd closed and real form

(2.43)  α = f(\overline{A}′, g\overline{V} ) − f(∇^{V} + \frac{∂}{∂t} dt + \frac{∂}{∂T} dT, g\overline{V} ) ∈ Ω^{∗}(B × (0, ∞) × ℝ) ,

then for fixed t,

α_{t} = f(Ã′_{t,T} , g^{V} ) − f(∇^{V} , g^{V} )

+ (2πi)^{− \frac{N^{B}}{2}} str_{V} (h^{V} f′(X̃_{t,T} )) \frac{dT}{2T} − d\overline{B}(\frac{log T}{2} str_{V} (h^{V} )) f′(0) .

(2.44)

For a fixed compact K ⊂ B, let ε be as in Definition 2.35, and choose 0 < ε′ < \frac{ε}{2}. Using (2.42), we compute

α_{t} = (2πi)^{\frac{1−N^{B}}{2}} str_{V} (f(\frac{ω^{V} + log T dh^{V}}{2} + O(T^{\frac{ε}{2}} ))) − f(∇^{V} , g^{V} )

+ (2πi)^{− \frac{N^{B}}{2}} str_{V} (h^{V} f′(\frac{ω^{V} + log T dh^{V}}{2} + O(T^{\frac{ε}{2}} ))) \frac{dT}{2T}

− d\overline{B}(\frac{log T}{2} str_{V} (h^{V} )) f′(0)

= O(T^{ε′}) + O(T^{ε′}) \frac{dT}{2T}

(2.45)

as T → 0, where the terms O(T^{ε′}) do not involve the exterior variable dT. Using this formula and (2.30), we find that the integrals in the following definition converge.

2.46. Definition. Let (V, A′, h^{V}) be a family Thom-Smale complex with an adapted metric g^{V}, then we define its torsion form in Ω^{∗}(B)/d^{B}Ω^{∗}(B) by

T_{f}(A′, g^{V} , h^{V} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str_{V} (N^{V} f′(X_{t,1})) − χ′(H)f′(0)

− (χ′(V) − χ′(H)) f′(\sqrt{−t}/2)) \frac{dt}{2t}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{1}(str_{V} (h^{V} f′(X_{1,T})) − str_{V} (h^{V} ) f′(0)

− (χ′(V) − χ′(H)) (f′(\sqrt{−T} /2) − f′(0))) \frac{dT}{2T} .

2.47. Remark. Suppose that A′ = ∇^{V} +a_{0} is a flat superconnection on a ℤ-graded Hermitian vector bundle (V, g^{V}), then the form T_{f}(A′, g^{V} , N^{V}) clearly equals the form

((3.30)) T_{f}(A′, g^{V} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{∞}(str_{V} (N^{V} f′(X_{t,1}))

− χ′(H)f′(0) − (χ′(V) − χ′(H)) f′(\sqrt{−t}/2)) \frac{dt}{2t}

of [BL], [BG1]. This motivates some of the correction terms in Definition 2.46. In particular, the component of degree 0 can be written as

T_{f}(A′, g^{V} , N^{V} )^{[0]} = \frac{1}{2} ∑_{j} str_{H^{⊥}}(N^{V} log(a′_{0} + a′′_{0})^{2}) .

2.48. Theorem. The form T_{f}(A′, g^{V} , h^{V}) ∈ Ω^{∗}(B) is even and real, and if F: M → B is a smooth map, then

T_{f}(F^{∗}A′, F^{∗}g^{V} , F^{∗}h^{V} ) = F^{∗}T_{f}(A′, g^{V} , h^{V} ) .

We have

d^{B}T_{f}(A′, g^{V} , h^{V} ) = f(∇, g^{V} ) − f(∇^{H}, g^{H}) .

The component of T_{f}(A′, g^{V} , h^{V}) in degree zero is given by

T_{f}(A′, g^{V} , h^{V} )^{[0]} = \frac{1}{2} str(N^{V} log(a′_{0} + a′′_{0})^{2}|_{ker(a′_{0}+a′′_{0})^{⊥}}) .

Proof. The form T_{f}(A′, g^{V} , h^{V}) is even and real because f(\overline{A}′, g\overline{V} ) is odd on B × ℝ_{>} × ℝ and real by Theorem 2.15. The first equation of Theorem 2.48 is clear from Definition 2.46.

It follows from Definition 2.31 and Theorem 2.32 that

(2.49) − d^{B}((2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str_{V} (N^{V} f′(X_{t,1})) − χ′(H)f′(0)

− (χ′(V) − χ′(H)) f′(\sqrt{−t}/2)) \frac{dt}{2t})

= d^{B} S_{f}(A′, g^{V} ) = f(A′, g^{V} ) − f(∇^{H}, g^{H}) .

Fix K ⊂ B compact and ε′ > 0 as above, then equations (2.44) and (2.45) imply that similarly,

(2.50) − d^{B}((2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{1}(str_{V} (h^{V} f′(X_{1,T})) − str_{V} (h^{V} ) f′(0)

− (χ′(V) − χ′(H)) (f′(\sqrt{−T} /2) − f′(0))) \frac{dT}{2T} )

= −d^{B} ∫_{B×[0,1]/B} α_{1} = −α_{1,1} = f(∇^{V} , g^{V} ) − f(A′, g^{V} ) .

The second equation in Theorem 2.48 follows from (2.49) and (2.50).

Now fix b ∈ B and consider the ℤ-graded Hermitian vector bundle (V_{b} × [0, 1], g^{V_{b}}) → [0, 1] with the trivial flat connection ∇^{0}. This bundle admits a flat superconnection A′_{b} = a′_{0} + ∇^{0}. Let u denote the coordinate of [0, 1], then (V_{b} × [0, 1], A′_{b}, u h^{V} + (1 − u) N^{V}) is a family Thom-Smale complex with adapted metric g^{V_{b}}. We have

dT_{f}(A′_{b}, g^{V_{b}}, u h^{V} + (1 − u) N^{V} ) = f(∇^{0}, g^{V_{b}}) − f(∇^{H}, g^{H}) = 0 ∈ Ω^{∗}([0, 1]) .

On the other hand, Theorem 2.25 in [BL] shows that at u = 0, we obtain the classical Reidemeister torsion of the complex (V, a′_{0}), which proves the last assertion. □

2.51. Remark. Suppose that γ ∈ Γ(End V) acts by fibre-wise isometries on V, leaving A′ and h^{V} invariant. Then there is an obvious equivariant analogue T_{f,γ}(A′, g^{V} , h^{V}) satisfying an equivariant version of Theorem 2.48.

2.e. Rigidity of the forms T_{f}(A′, g^{V} , h^{V}). We analyse the behaviour of T_{f}(A′, g^{V} , h^{V}) under smooth variation of A′, g^{V} and h^{V}.

2.52. Definition. Let V → B be a ℤ-graded complex vector bundle, and let (A′_{s}, g_{s}^{V} , h_{s}^{V})_{s∈[0,1]} be family of flat superconnections of total degree 1 with A′_{s} = ∇^{V,s} + a′_{s}, ℤ-graded Hermitian metrics, and ℤ-graded endomorphisms on V. We call (A′_{s}, g_{s}^{V} , h_{s}^{V})_{s∈[0,1]} a family of family Thom-Smale complexes iff for each compactum K ⊂ B × [0, 1] there exists ε > 0, and for each point b ∈ K there exists a neighbourhood U of b in B × [0, 1], such that Definition 2.35 (1)–(3) hold over U.

Note that both in Definition 2.35 and here, the splittings of V^{j}|_{U} = ⨁_{α} V_{α}^{j} are allowed to depend on U. In particular, it may happen that several eigenvalues of h^{V} run together over a certain subset of B × [0, 1].

Let us assume for the moment that f is an odd holomorphic function of degree ≥ 3 that satisfies (2.12), cf. Theorem 2.34, and set k(z) = \frac{f′(z)}{2z} as before. We also assume that the dimension of the cohomology bundles H_{s} is independent of s ∈ [0, 1], so that they combine to form a bundle on B ×[0, 1]. Let (∇^{H,s}, g_{s}^{H})_{s∈[0,1]} be the induced family of flat connections and Hermitian metrics.

2.53. Proposition. Modulo exact forms on B, we have

T_{f}(A′_{1}, g_{1}^{V} , h_{1}^{V} ) − T_{f}(A′_{0}, g_{0}^{V} , h_{0}^{V} ) = L_{k}(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) − L_{k}(∇^{H,s}, g_{0}^{H}, g_{1}^{H}) .

Proof. Because we already know Theorem 2.34, we only have to care about the integral over T in Definition 2.46. For fixed t, define a form

ℓ_{t} = (2πi)^{− \frac{N^{B}}{2}} (str_{V} (k(\overline{X}_{s}) \frac{∂\overline{A}_{s}}{∂s} ) − str_{V} (k(\frac{ω^{V} + log T dh_{s}^{V}}{2} + \frac{h_{s}^{V}}{2T} dT ) \frac{∂∇^{V,s,u}}{∂s} )) ds

on B × [0, 1] × {t} × ℝ. As in (2.18), we have

d^{B×[0,1]×\{t\}×ℝ}ℓ_{t} = f(\overline{A}′_{s}, g\overline{V} ; s ) − f(∇^{V,s} + \frac{∂}{∂T} dT, g\overline{V} ; s ) = α_{s,t} ds ,

where α_{s,t} is the form defined in (2.43). In particular, the form ℓ_{t}+α_{t} is closed on B×[0, 1]×{t}×ℝ. As T → 0, we have

\frac{∂Ã_{s,t}}{∂s} = ∇^{V,s,u} + O(T^{\frac{ε}{2}} )
and  X̃_{s,t} = \frac{ω^{V,s} + log T dh_{s}^{V}}{2} + \frac{h_{s}^{V}}{2T} dT + O(T^{\frac{ε}{2}} ) .

This implies that

(2.54)  ℓ_{t} = O(T^{\frac{ε}{2}} ) + O(T^{\frac{ε}{2}} ) \frac{dT}{2T}

uniformly in s as T → 0, where the forms O(T^{\frac{ε}{2}} ) do not contain the exterior variable dT. By (2.45), (2.54) and Stokes’ theorem,

d^{B} ∫_{B×[0,1]×[0,1]/B}(ℓ_{1} + α_{1}) = ∫_{B×[0,1]/B}(α_{1,1} − α_{0,1}) − ∫_{B×[0,1]/B} ℓ_{1,1}

for t = 1 fixed, where the first integration on the right hand side is in the direction of T, and the second in the direction of s. Together with Theorem 2.34, this proves our proposition. □

The next step towards the general case is an analogue of Theorem 2.18 in [BG1]. Let f be an odd holomorphic function satisfying (2.12). Let a > 0, let R be a polynomial, and set

(2.55)  f_{a}(z) = R(\frac{∂}{∂a})|_{a=1} \frac{h(\sqrt{a} z)}{\sqrt{a}} .

Then f_{a} also satisfies (2.12).

42 SEBASTIAN GOETTE

2.56. Proposition. The torsion with respect to the generating function f_{a} is up to exact forms on B given by

T_{f_{a}}(A′, g^{V} , h^{V} ) = R(\frac{∂}{∂a})|_{a=1}(a^{\frac{N^{B}}{2}} T_{f}(A′, g^{V} , h^{V} )) .

Proof. Let us regard the form α of (2.43) for variable t, but T = 1. Because

f(∇^{V} + \frac{∂}{∂t} dt + \frac{∂}{∂T} dT, g\overline{V} )|_{T =1} = f(∇^{V} , g_{t}^{V} ) + χ′(V) \frac{dt}{2t} ,

we have

α|_{T =1} = f(\overline{A}′, g\overline{V} ) − f(∇^{V} , g_{t}^{V} ) − χ′(V) f′(0) \frac{dt}{2t} .

Setting

β = (χ′(V) − χ′(H)) (f′(\sqrt{−t}/2) − f′(0)) (\frac{dt}{2t} + \frac{dT}{2T} ) ,

we may therefore rewrite Definition 2.46 as T_{f}(A′, g^{V} , h^{V} ) = − ∫_{B×({1}×[0,1] ∪ (1,∞)×{1})/B}(α − β) .

Rescaling t to at gives

\begin{aligned}X̃_{at} & = \frac{ω^{V}}{2} + ∑_{j}(at)^{\frac{1−j}{2}} (T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} − T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} ) + \frac{log T}{2} dh^{V} + \frac{h^{V}}{2T} dT \\ & = a^{\frac{1−N^{B}}{2}} X̃_{t} a^{\frac{N^{B}}{2}} . \end{aligned}

This implies that

(\frac{f(\sqrt{a} · )}{\sqrt{a}})′(X_{t,T}) = f′(\sqrt{a} X_{t,T} ) = a^{\frac{N^{B}}{2}} f′(X_{at,T}) a^{− \frac{N^{B}}{2}} .

We may thus express the torsion form with respect to \frac{f(\sqrt{a} ·)}{\sqrt{a}} as T_{\frac{f(\sqrt{a} ·)}{\sqrt{a}}} (A′, g^{V} , h^{V} )

= −a^{\frac{N^{B}}{2}} ∫_{B×(1,∞)×\{1\}/B}(str_{V} (f′(X_{at,T})) − χ′(H) f′(0)

− (χ′(V) − χ′(H)) f′(\sqrt{−at}/2)) \frac{dt}{2t}

− a^{\frac{N^{B}}{2}} ∫_{B×\{a\}×[0,1]/B}(α − (χ′(V) − χ′(H)) (f′(\sqrt{−aT} /2) − f′(0)) \frac{dT}{2T} )

= −a^{\frac{N^{B}}{2}} ∫_{B×([1,a]×\{0\} ∪ \{a\}×[0,1] ∪ (a,∞)×\{1\})/B}(α − β) .

00 1  1  a  t T  ∞ {1}×[0,1] ∪ (1,∞)×{1} [1,a]×{0} ∪ {a}×[0,1] ∪ (a,∞)×{1}

Figure 2.57. Paths in the construction of T_{f} and T_{f_{α}}.

Here we have used that α vanishes on B × [1, a] × {0} by (2.45), and that β is symmetric in t and T. Since the integrand above is closed, we may integrate along any path from (∞, 1) to (1, 0), changing the result only by an exact form, so

T_{\frac{f(\sqrt{a} ·)}{\sqrt{a}}} (A′, g^{V} , h^{V} ) = a^{\frac{N^{B}}{2}} T_{f}(A′, g^{V} , h^{V} ) + a^{\frac{N^{B}}{2}} d^{B} ∫_{B×[1,a]×[0,1]/B}(α − β) .

Our theorem now follows, since

T_{f_{a}}(A′, g^{V} , h^{V} ) = R(\frac{∂}{∂a})|_{a=1} T_{\frac{f(\sqrt{a} ·)}{\sqrt{a}}} (A′, g^{V} , h^{V} )

= R(\frac{∂}{∂a})|_{a=1}(a^{\frac{N^{B}}{2}} T_{f}(A′, g^{V} , h^{V} )) . □

We can now finally prove the analogue of [BG1], Theorem 2.20. We return to the convention (2.11) that f(z) = z e^{z^{2}}, and we set k(z) = \frac{f′(z)−f′(0)}{2z}.

2.58. Theorem. Modulo exact forms on B, we have

(T_{f}(A′_{1}, g_{1}^{V} , h_{1}^{V} ) − T_{f}(A′_{0}, g_{0}^{V} , h_{0}^{V} ))^{[≥2]} = L_{k}(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) − L_{k}(∇^{H,s}, g_{0}^{H}, g_{1}^{H}) .

2.59. Remark. Note that the right hand side is independend modulo dΩ^{∗}(B) of the path connecting ∇^{V,0} with ∇^{V,1} and ∇^{H,0} with ∇^{H,1}, even though both L_{k}-terms may depend on the homotopy class of these paths in degree ≥ 4, and on the paths themselves in degree 2.

Proof. Choose f(z) = z e^{z^{2}} as in (2.11) and set R(X) = X in (2.55), so that f_{a}(z) = z^{3} e^{z^{2}} is now of degree 3, and let k_{a}(z) = \frac{f′_{a}(z)}{2z}. By Proposition 2.53 and Proposition 2.56,

L_{k_{a}}(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) − L_{k_{a}}(∇^{H,s}, g_{0}^{H}, g_{1}^{H})

= \frac{∂}{∂a}|_{a=1}(a^{\frac{N^{B}}{2}} (T_{f}(A′_{1}, g_{1}^{V} , h_{1}^{V} ) − T_{f}(A′_{0}, g_{0}^{V} , h_{0}^{V} )))

= \frac{N^{B}}{2} (T_{f}(A′_{1}, g_{1}^{V} , h_{1}^{V} ) − T_{f}(A′_{0}, g_{0}^{V} , h_{0}^{V} )) .

On the other hand, we have

k_{a}(z) = \frac{3z + z^{3}}{2} e^{z^{2}} = \frac{∂}{∂a}|_{a=1}(\frac{(1 + 2az^{2}) e^{az^{2}} − 1}{2z}) = \frac{∂}{∂a}|_{a=1}(\sqrt{a} k(\sqrt{a} z)) .

It is now easy to check that

L_{k_{a}}(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) = \frac{∂}{∂a}|_{a=1} ∫_{0}^{1} str_{V} (k(\sqrt{a} \frac{ω^{V,s}}{2} ) \sqrt{a} \frac{∂∇^{V,s,u}}{∂s} ) ds

= \frac{∂}{∂a}|_{a=1} ∫_{0}^{1} a^{\frac{N^{B}}{2}} str_{V} (k(\frac{ω^{V,s}}{2} ) \frac{∂∇^{V,s,u}}{∂s} ) ds

= \frac{N^{B}}{2} L_{k}(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) .

Together, these equations give

\frac{N^{B}}{2} (T_{f}(A′_{1}, g_{1}^{V} , h_{1}^{V} ) − T_{f}(A′_{0}, g_{0}^{V} , h_{0}^{V} )) = \frac{N^{B}}{2} (L_{k}(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) − L_{k}(∇^{H,s}, g_{0}^{H}, g_{1}^{H})) ,

from which our theorem finally follows. □

2.f. Fibre-wise Morse functions and torsion classes. We show how to associate torsion classes to families of compact manifolds with fibre-wise Morse functions. If we start with a flat bundle (F, ∇^{F}) → M that admits a parallel metric g^{F}, such that the fibre-wise cohomology (H, ∇^{H}) → B also admits a parallel metric, then we obtain a well-defined cohomology class T_{f}(∇^{F} , h, g^{F} , g^{H}) ∈ H^{even,≥2}(B) that is invariant under smooth deformations of h in the space of all fibre-wise Morsefunctions on M.

Let p: M → B be a bundle with compact fibres, and assume that (F, ∇^{F}) → M is a flat bundle and h: M → ℝ is a Morse function on each fibre of p. If p̂: C → B denotes the covering of B by the fibre-wise critical points of h, we define the bundle V = p̂_{∗}(Λ^{max}T^{u}X ⊗F |_{C}), with the ℤ-grading by the fibre-wise index of h as in Definition 1.1. Then h acts by multiplication on Λ^{max}T^{u}X ⊗F |_{C}, and we let h^{V} ∈ Γ(End^{0} V) be the pushdown of this action. Let A′ = ∇^{V} + a be the superconnection of Definition 1.65, then a ∈ Ω^{∗}(B; End_{+}^{∗} V) is of total degree 1 by Theorem 1.66.

2.60. Remark. The bundle (V, A′, h^{V}) is a family Thom-Smale complex in the sense of Definition 2.35, and any Hermitian metric g^{F} on F → M induces by restriction to F_{C} and push-down to V → B an adapted metric g^{V} on V. In fact, if we denote by C_{α}^{j} the various local leaves of p̂^{j} where h has index j, we may take the ∇^{V} -parallel splitting

V^{j} = ⨁_{α} F |_{C_{α}^{j}} ,

such that h^{V} acts on C_{α}^{j} by multiplication with h_{α}^{j} = h|_{C_{α}^{j}}, so points (1) and (2) of Definition 2.35 are satisfied. We have A′ = ∇^{V} + a′ with a ∈ Ω^{∗}(B; End_{+}^{∗} V) by Definition 1.3 and Theorem 1.66, so (3) also holds.

Suppose that A′_{0} and A′_{1} are two flat superconnections of total degree one on V → B, both constructed as in Section 1. In particular, both (V, A′_{0}, h^{V}) and (V, A′_{1}, h^{V}) are family Thom-Smale complexes. Then we know by Proposition 1.70 that there exists a flat superconnection

\overline{A}′ = ∇\overline{V} + ∑_{j} ā′_{j}

of total degree one on the pullback \overline{V} of V to \overline{B} = B × [0, 1], also constructed as in Section 1, such that \overline{A}′|_{B×\{s\}} = A′_{s} for s = 0, 1.

We fix a metric g^{F} on F → M, let g^{V} be the induced metric o V, and let g\overline{V} be its pull-back to \overline{V} → \overline{B}. Similarly, h^{V} pulls back to an endomorphism h\overline{V} of \overline{V}, and (\overline{V} , \overline{A}′, h\overline{V} ) is a family Thom-Smale complex with adapted metric g\overline{V}.

Let \overline{H} = H^{∗}(\overline{V} , ā′_{0}) → \overline{B} denote the fibre-wise cohomology with its Gauß-Manin connection ∇\overline{H}, and let g\overline{H} denote the metric induced on \overline{H} by g\overline{V} using finite-dimensional Hodge theory. Then for s = 0, 1, the restriction of (\overline{H}, ∇\overline{H}, g\overline{H}) to B × \{s\} is naturally isomorphic to the cohomology bundle H → B with Gauß-Manin connection ∇^{H} and metric g_{s}^{H} induced by g^{V}. Thus, we have

f̃(∇^{H}, g_{0}^{H}, g_{1}^{H}) = ∫_{0}^{1} f(∇\overline{H}, g\overline{H})

as in Definition 2.16.

2.61. Proposition. Let A′_{0} and A′_{1} be two superconnections on V that are constructed as in Definition 1.65, and let g^{V} be induced by a metric g^{F} on F → M. Then

T_{f}(A′_{1}, g^{V} , h^{V} ) − T_{f}(A′_{0}, g^{V} , h^{V} ) = −f̃(∇^{H}, g_{0}^{H}, g_{1}^{H}) modulo d^{B}Ω^{∗}(B).

Proof. By Theorem 2.48 and Stokes’ Theorem,

T_{f}(A′_{1}, g^{V} , h^{V} ) − T_{f}(A′_{0}, g^{V} , h^{V} ) = d ∫_{\overline{B}/B} T_{f}(\overline{A}′, g\overline{V} , h\overline{V} )
+ ∫_{\overline{B}/B}(f(∇\overline{V} , g\overline{V} ) − f(∇\overline{H}, g\overline{H}))
= −f̃(∇^{H}, g_{0}^{H}, g_{1}^{H}) modulo d^{B}Ω^{∗}(B),

because f(∇\overline{V} , g\overline{V} ) is just the pullback of f(∇^{V} , g^{V}) to \overline{B}, so ∫_{\overline{B}/B} f(∇\overline{V} , g\overline{V}) vanishes. □

In particular, if we fix metrics g^{F} on F and g^{H} on H, we can define a torsion form for the bundle M → B, equipped with a flat vector bundle (F, ∇^{F}) → M and a fibre-wise Morse function h that is independent of all choices made during the constructions in Section 1, by setting

(2.62)  T_{f}(∇^{F} , h, g^{F} , g^{H}) = T_{f}(A′, g^{V} , h^{V} ) + f̃(∇^{H}, g^{H}, g_{V}^{H}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

It follows from Proposition 2.61 that T_{f}(∇^{F} , h, g^{F} , g^{H}) is indeed well-defined.

2.63. Corollary. Assume that the metrics g^{F} |_{C} and g^{H} are parallel, then T_{f}(∇^{F} , h, g^{F} , g^{H}) defines a cohomology class in H_{dR}^{even}(B). Moreover, only the homogeneous part of degree 0 depends on the particular choice of g^{F} |_{C} and g^{H}.

Proof. The first assertion is immediate from Theorem 2.48 and Definition 2.64. The second claim follows because any two parallel metrics g_{0}^{F}, g_{1}^{F} on a flat bundle are related by family of flat metrics g_{s}^{F} = (1 − s) g_{0}^{F} + s g_{1}^{F}. Let g\overline{F} be the metric induced by this family on the pullback \overline{F} → \overline{B} = B × [0, 1]. Let ∇^{\overline{F} ,∗} be the adjoint of the pullback ∇\overline{F} of ∇^{F} to \overline{F} with respect to g\overline{F}. Then

ω\overline{F} = ∇^{\overline{F} ,∗} − ∇^{\overline{F} ,∗} ∈ Ω^{1}(\overline{B}, End \overline{F} )

vanishes on all vectors in T \overline{B} that are tangent to the factor B. In particular, (ω\overline{F} )^{k} = 0 for k > 1, so

f̃(∇^{F} , g_{0}^{F} , g_{1}^{F} ) = f̃(∇^{F} , g_{0}^{F} , g_{1}^{F} )^{[0]} . □

2.64. Definition. Assuming that the metrics g^{F} |_{C} on F |_{C} → C and g^{H} on H → B are parallel, we define the higher Morse torsion

T_{f}(M/B; F, h) = T_{f}(A′, g^{V} , h^{V} )^{[≥2]} + f̃(∇^{H}, g^{H}, g_{V}^{H})^{[≥2]} ∈ H_{dR}^{even,≥2}(B) .

2.g. Generalised metrics and the classes S_{f}(A′, g^{V} , g^{H}). We recall the notion of a generalised metric following [BG1], Section 2.9. Our presentation will follow the lines of Section 2.a above. We also recall the definition of S_{f} for generalised metrics. The content of this subsection will be needed only in Section 4, where generalised metrics appear naturally as restrictions of “classical” metrics to certain subbundles of Λ^{∗}T B ˆ⊗ Ω(M/B; F) formed by “generalised eigenspaces” of the Witten-deformed operator 𝕏_{t,T}.

Recall that in Section 2.a, we extended a metric g^{V} on a vector bundle V → B to an ℝ-bilinear map g^{V}: (Λ^{∗}T B ˆ⊗ V)^{2} → Λ^{∗}T B with the properties (2.2) and (2.3). A natural generalisation of metrics is the following.

2.65. Definition. A generalised Hermitian metric g^{V} on a ℤ_{2}-graded complex vector bundle V → B is an ℝ-bilinear map g^{V}: (Λ^{∗}T B ˆ⊗ V)^{2} → Λ^{∗}T B, such that

(1) the component (g^{V})^{[0]}: (g^{V})^{2} → ℂ is a Hermitian metric;

(2) g^{V} is even with respect to the total grading of Λ^{∗}T B ˆ⊗ V;

(3) for v, w ∈ V, we have

g^{V} (w, v) = (−1)^{[\frac{deg(g^{V} (w,v))}{2}]} \overline{g^{V} (v, w)} ;

(4) for α, β ∈ Λ^{∗}T B and v, w ∈ V, we have

g^{V} (α ˆ⊗ v, β ˆ⊗ w) = (−1)^{[\frac{deg(β)}{2}]} α ∧ g^{V} (v, w) ∧ \overline{β} .

Properties (3) and (4) are the natural generalisations of (2.2) and (2.3). Together, they imply that (3) still holds for v, w ∈ Λ^{∗}T B ˆ⊗V. Note that as in Remark 2.9, we may regard g^{V} as an even element of Ω^{∗}(B; Hom(V, \overline{V}^{∗})). Clearly, there is an analogous notion of a generalised Euclidean metric. Note that by (2), the part (g^{V})^{[0]} of horizontal degree 0 respects the ℤ^{2}-grading of V.

2.66. Remark. Since any convex combination of generalised metrics is again a generalised metric, the space of generalised metrics on a given vector bundle is contractible. In particular, any generalised metric can be deformed onto its component (g^{V})^{[0]}.

When we define adjoints of endomorphisms and superconnections with respect to general metrics, we have to take care of the fact that g^{V} (v, w) may be an odd form for v, w ∈ V.

2.67. Definition. For an endomorphism E ∈ Ω^{∗}(B; End V), we define the adjoint endomorphism by

E^{∗} = (g^{V})^{−1} \overline{E}^{∗} g^{V} ,

and for a superconnection A on V, we set

A^{∗} = (g^{V})^{−1} \overline{A}^{∗} g^{V} .

2.68. Remark. For an endomorphism E ∈ End V, Definition 2.67 is equivalent to

g^{V} (E^{∗}w, v) = (−1)^{|E| (|E|+|w|+|v|)} g^{V} (w, Ev)

for all v, w ∈ V. Note that |E| + |w| + |v| has the same parity as the exterior degree of the form g^{V} (w, Ev).

For a connection ∇^{V}, we get

∇^{V,∗} = ∇^{V} + ω^{V}  with  ω^{V} = (g^{V} )^{−1} [∇^{V} , g^{V} ] ∈ Ω^{∗}(B; End V) .

Definition 2.65 (3), (4) and Definition 2.67 imply that (E^{∗})^{∗} = E and (A^{∗})^{∗} = A for all superconnections A and all E ∈ Ω^{∗}(M; End V).

For odd elements α ˆ⊗E ∈ Ω^{∗}(M; End V)^{odd} and for superconnections A on V, Definition 2.65 (4) and Definition 2.67 imply that

0 = ⟨E^{∗}v, w⟩ + (−1)^{deg E (deg v+deg w)} ⟨v, Ew⟩
and  d⟨v, w⟩ = ⟨A^{∗}v, w⟩ + (−1)^{deg E (deg v+deg w)} ⟨v, Aw⟩

for all v, w ∈ Ω^{∗}(M; V). Thus, most of the properties of metrics, superconnections and their adjoints generalise to the present setting.

Let f: ℂ → ℂ be a holomorphic function satisfying (2.12), e.g., f(z) = z e^{z^{2}} as in (2.11). If g^{V} is a generalised Hermitian metric and A′ is a superconnection, let A′′ be the adjoint of A′ with respect to g^{V}, and set X = \frac{1}{2} (A′′ − A′) as in (2.13). We have a natural generalisation of Definition 2.14.

2.69. Definition. Let f(A′, g^{V}) ∈ Ω^{∗}(B) be defined by

f(A′, g^{V} ) = (2πi)^{\frac{1−N^{B}}{2}} str_{V} (f(X)) .

Theorem 2.15 extends literally to this generalised setting, so in particular, f(A′, g^{V}) is odd, closed, and real. Similarly, we define f̃(A′, g_{0}^{V} , g_{1}^{V}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) as in Definition 2.16. Again, Corollary 2.17 extends literally to this situation and gives in particular

d^{B}f̃(A′, g_{0}^{V} , g_{1}^{V} ) = f(A′, g_{1}^{V} ) − f(A′, g_{0}^{V} ) .

We also need a generalisation of Definition 2.31 to this setting. Therefore, let (V, g^{V}) → B be a ℤ-graded Hermitian vector bundle, let (g_{t}^{V})_{t∈(0,∞)} be a smooth family of generalised metrics on V, and let g′ ∈ Ω^{∗}(B; End V) be even, such that as t → ∞,

t^{\frac{n−N^{V}}{2}} g_{t}^{V} t^{− \frac{N^{V}}{2}} = g^{V} ((id_{V} +\frac{g′}{\sqrt{t}} + O(\frac{1}{t})) · , · ) ,

and  t^{\frac{N^{V}}{2}} (g_{t}^{V} )^{−1} \frac{∂g_{t}^{V}}{∂t} t^{− \frac{N^{V}}{2}} = (N^{V} − \frac{n}{2}) \frac{1}{t} + O(t^{− \frac{3}{2}} ) ,

(2.70)

where both O( ·) are even elements of Ω^{∗}(B; End V). Let ḡ^{V} be the generalised metric on \overline{V} = V × (0, ∞) → \overline{B} = B × (0, ∞) induced by g_{t}^{V}, in particular, ḡ^{V} does not contain the exterior variable dt.

Let A′ be a flat superconnection of total degree 1 as in Definition 2.21, so

A′ = ∇^{V} + ∑_{i} a_{i}  with  a_{i} ∈ Ω^{i}(B; End^{1−i} V ) ,

and let \overline{A}′ = A′ + \frac{∂}{∂t} dt be the trivial extension of A′ to \overline{V}. Let \overline{A}′′ be the adjoint of \overline{A}′ with respect to ḡ^{V}. Note that \overline{A}′′ is not necessarily of total degree −1, since ḡ^{V} can shuffle degrees. Nevertheless, using that \overline{A}′ is of total degree 1 together with (2.70), we establish the following properties of A′ and A′′_{t}.

2.71. Proposition ([BG1], Proposition 2.46). As t → ∞,

t^{\frac{N^{V}}{2}} A′ t^{− \frac{N^{V}}{2}} = \sqrt{t} a_{0} + ∇^{V} + a_{1} + O(t^{− \frac{1}{2}} ) ,
and  t^{\frac{N^{V}}{2}} A′′_{t} t^{− \frac{N^{V}}{2}} = \sqrt{t} a_{0}^{∗} + ∇^{V,∗} + a_{1}^{∗} + [a_{0}^{∗}, g′] + O(t^{− \frac{1}{2}} ) ,

where adjoints are taken with respect to g^{V}. □

As in (2.13), we define

\overline{X} = \frac{1}{2} (\overline{A}′′ − \overline{A}′) = \frac{1}{2} (A′′_{t} − A′) + \frac{1}{2} (g_{t}^{V} )^{−1} \frac{∂g_{t}^{V}}{∂t} dt .

We define an odd closed differential form on \overline{B} by

(2.72) \begin{aligned}β & = f(\overline{A}′, ḡ^{V} ) − (χ′(H) − \frac{n}{2} χ(V)) f′(0) \frac{dt}{2t} \\ & = f(A′, g_{t}^{V} ) + (2πi)^{− \frac{N^{B}}{2}} ( str_{V} (\frac{1}{2} (g_{t}^{V} )^{−1} \frac{∂g_{t}^{V}}{∂t} f′(X_{t})) \end{aligned}

− (χ′(H) − \frac{n}{2} χ(V)) \frac{f′(0)}{2t}) dt .

Let g^{H} be the metric induced by g^{V} on H = H^{∗}(V, a_{1}) → B, and let ∇^{H} be the Gauß-Manin connection on H induced by A′. If P^{H}: V → H denotes the g^{V} -orthogonal projection onto the (a_{0}+ a_{0}^{∗})-harmonic elements of V, then clearly P^{H} [a_{0}^{∗}, g′] P^{H} = 0. Now the proof of [BL], Theorem 2.9, translates to this situation and gives an estimate for β as t → ∞, cf. [BG1], Proposition 2.48.

2.73. Proposition. As t → ∞,

β = f(∇^{H}, g^{H}) + O(t^{− \frac{1}{2}} ) + O(t^{− \frac{3}{2}} ) dt ,

where the O( ·) ∈ Ω^{∗}(\overline{B}; End \overline{V} ) do not contain dt. □

Let g_{s,t}^{V} be two generalised metrics on V, together with g_{s}^{V} and g′_{s}, such that (2.70) holds for s = 0, 1. Then there exist families g_{s,t}^{V}, g_{s}^{V} and g′_{s} for s ∈ [0, 1] that restrict to the given metrics for s = 0, 1, such that (2.70) holds uniformly in s. We define a form \overline{β} on B × [0, 1] × (0, ∞), and we let β_{s} denote its restriction to B × {s} × (0, ∞). Let g_{s}^{H} denote the metric induced on H by g_{s}^{V}, then the obvious generalisation of (2.33) is

(2.74)  ∫_{(B×(1,∞))/B}(β_{1} − β_{0}) = f̃(A′, g_{0,1}^{V}, g_{1,1}^{V}) − f̃(∇^{H}, g_{0}^{H}, g_{1}^{H}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

Writing g^{V} = g_{1}^{V}, me can now define a generalisation of the class S_{f}(A′, g^{V}).

2.75. Definition. Set

S_{f}(A′, g^{V} , g^{H}) = ∫_{(B×(1,∞))/B} β ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

Note that the class S_{f}(A′, g^{V} , g^{H}) was called U_{f}(A′, g_{t}^{V}) in [BG1].

2.76. Theorem ([BG1], Theorem 2.50). The form S_{f}(A′, g^{V} , g^{H}) is even and satisfies

d^{B}S_{f}(A′, g^{V} , g^{H}) = f(A′, g^{V} ) − f(∇^{H}, g^{H}) .

If g^{′V} is another generalised metric on V and g^{′H} is another metric on H, them

S_{f}(A′, g^{′V} , g^{′H}) − S_{f}(A′, g^{V} , g^{H}) = f̃(A′, g^{V} , g^{′V} ) − f̃(∇^{H}, g^{H}, g^{′H}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

Let g^{V} be a fixed metric on V that respects the ℤ-grading. Note that the family g^{V} t^{N^{V}} does not quite satisfy (2.70). Me may however set

g_{t}^{V} = g^{V} t^{N^{V} − \frac{n}{2}} ,

and since t^{− \frac{n}{2}} commutes with all operators involved, we have

A′′_{t} = t^{−N^{V} + \frac{n}{2}} A′′ t^{N^{V} − \frac{n}{2}} = t^{−N^{V}} A′′ t^{N^{V}} .

Because of this, our form β of (2.72) becomes

β = f(A′, g^{V} t^{N^{V}} ) + (2πi)^{− \frac{N^{B}}{2}} (str_{V} ((N^{V} − \frac{n}{2}) f′(X_{t}))

− (χ′(H) − \frac{n}{2} χ(V)) f′(0)) \frac{dt}{2t}

= f(A′, g^{V} t^{N^{V}} ) + (2πi)^{− \frac{N^{B}}{2}} (str_{V} (N^{V} f′(X_{t})) − χ′(H) f′(0)) \frac{dt}{2t} .

Let again g^{H} be the metric induced on H by g^{V}. A comparison with our definition of S_{f}(A′, g^{V}) in Definition 2.31 shows that

(2.77)  S_{f}(A′, g^{V} , g^{H}) = S_{f}(A′, g^{V} ) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) ,

which justifies our notation.

2.h. Analytic torsion forms. We recall the construction of analytic torsion forms in [BL].

Let p: M → B be a smooth fibre bundle, and let (F, ∇^{F}) → M be a flat complex vector bundle. Let Ω^{∗}(M; F) denote the space of F -valued differential forms on M. If we fix a horizontal subbundle T^{H}M ⊂ T M with T^{H}M ⊕ T X = T M, then T^{H}M ≅ p^{∗}T B, and we may split Λ^{∗}T M = p^{∗}Λ^{∗}T B ˆ⊗ Λ^{∗}T X. We denote the space of vertical forms by Ω^{∗}(M/B; F) = p_{∗}(Λ^{∗}T X ⊗ F) → B, then we have

(2.78)  Ω^{∗}(M; F) = Ω^{∗}(B; Ω^{∗}(M/B; F)) .

Let Ω ∈ Γ(p^{∗}Λ^{2}T B ⊗ T X) denote the fibre-bundle curvature of T^{H}M, given by

Ω(v, w) = −[v̄, w̄]^{⊥} ∈ T X ,

where v̄, w̄ ∈ Γ(T^{H}M) are horizontal lifts of vectorfields v, w on B and ( ·)^{⊥} denote the projection T M → T X along T^{H}M. With respect to the splitting (2.78), we may interpret the exterior differential d^{M} induced by ∇^{F} on Ω^{∗}(M; F) as a flat superconnection 𝔸′ of total degree 1 on the infinite-dimensional vector bundle Ω^{∗}(M/B; F) → B. Classically, this superconnection decomposes as

(2.79)  𝔸′ = d^{M} = d^{X} + ∇^{Ω^{∗}(M/B;F)} + ι_{Ω}

with respect to horizontal degree, cf. [BGV].

Now let us fix a vertical metric g^{T X}, then g^{T X} induces a metric on Λ^{∗}T X that we still denote by g^{T X}. We obtain a natural L_{2}-metric on Ω^{∗}(M/B; F), which we write as

(2.80)  g^{T X,F} (ω, ω′)_{b} = ∫_{p^{−1}(b)}(g^{T X} ⊗ g^{F} )(ω, ω′)_{x} d Vol(x) ,

where d Vol is the fibre-wise volume element induced by g^{T X}. Let 𝔸′′ be the adjoint of 𝔸′ with respect to g^{T X,F} in the sense of Definition 2.7. Then

(2.81) 𝕏 = \frac{1}{2} (𝔸′′ − 𝔸′) = \frac{1}{2} ((d^{X,∗} − d^{X}) + ω^{Ω^{∗}(M/B;F)} − ε_{Ω} − ι_{Ω})

∈ Ω^{∗}(B; End Ω^{∗}(M/B; F))

is a fibre-wise elliptic differential operator on Ω^{∗}(M; F) with coefficients in Ω^{∗}(B). Moreover, 𝕏 is skew-adjoint in the sense of Definition 2.7.

By Hodge theory, we can identify the fibre-wise cohomology H = H_{dR}^{∗}(M/B; F) with the bundle of (d^{X,∗} − d^{X})-harmonic forms in Ω^{∗}(M/B; F). Then ∇^{Ω^{∗}(M/B;F)} induces the Gauß-Manin connection ∇^{H} on H. We let g_{L_{2}}^{H} denote the restriction of g^{T X,F} to H.

Because −𝕏^{2} has the same spectrum as the Hodge Laplacian −(𝕏^{[0]})^{2} = \frac{1}{4} (d^{X,∗} +d^{X})^{2}, the odd heat kernel 𝕏 e^{𝕏^{2}} is of trace-class. In particular, as in Definition 2.14, we obtain an odd, closed and real form

(2.82)  f(𝔸′, g^{T X,F} ) = str(𝕏 e^{𝕏^{2}}) .

Let N^{X} denote the fibre-wise number operator on Ω^{∗}(M; F) and Ω^{∗}(M/B; F). We set g_{t}^{T X} = g^{T X}(t^{−N^{X}} · , ·). Let g_{t}^{T X,F} denote the induced L_{2}-metric on Ω^{∗}(M/B; F), then

(2.83)  g_{t}^{T X,F} = g^{T X,F} (t^{N^{X}− \frac{n}{2}} · , · ) ,

where the additional t^{\frac{n}{2}} is due to the change of the fibre-wise volume element. Let ḡ^{T X,F} denote the metric on Ω^{∗}(M/B; F) × (0, ∞) such that ḡ^{T X,F} |_{B×\{t\}} = g_{t}^{T X,F}. Let \overline{𝔸}′′ be the adjoint of the obvious extension \overline{𝔸}′ of the superconnection 𝔸′ to the bundle Ω^{∗}(M/B; F) × (0, ∞) → B × (0, ∞), and set \overline{𝕏} = \frac{1}{2} (\overline{𝔸}′′ − \overline{𝔸}′). As before, restriction to B × \{t\} will be written ( ·)_{t}.

The form

f(\overline{𝔸}′, ḡ^{T X,F} ) − χ′(H) f′(0) \frac{dt}{2t}

= f(𝔸′_{t}, g^{T X,F} ) + (2πi)^{− \frac{N^{B}}{2}} (str(N^{X} f′(𝕏_{t})) − χ′(H) f′(0)) \frac{dt}{2t}

is closed and satisfies

f(\overline{𝔸}′, ḡ^{T X,F} ) − χ′(H) f′(0) \frac{dt}{2t} = f(∇^{H}, g_{L_{2}}^{H}) + O(t^{− \frac{1}{2}} ) + +O(t^{− \frac{3}{2}} ) dt ,

as t → ∞ by [BL], Theorem 3.21, where the O( ·)-terms do not involve dt.

2.84. Definition. Set

S_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str(N^{X} f′(𝕏_{t})) − χ′(H) f′(0)) \frac{dt}{2t} .

Then clearly, S_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F}) is even and real, and satisfies

d^{B}S_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = ch^{o}(𝔸′, g^{T X,F} ) − ch^{o}(∇^{H}, g^{H}) .

Let ∇^{T X} be the natural connection on T X → M that is induced by T^{H}M and g^{T X}, and let e(∇^{T X}) be its Euler form. As t → 0, again by [BL], Theorem 3.21,

f(\overline{𝔸}′, ḡ^{T X,F} ) − χ′(H) \frac{dt}{2t}

= e(∇^{T X}) f(∇^{F} , g^{F} ) + (\frac{n}{2} χ(H) − χ′(H)) f′(0) \frac{dt}{2t} + O(t^{\frac{1}{2}} ) + O(t^{− \frac{1}{2}} ) dt ,

where the O( ·)-terms do not contain dt.

2.85. Definition. Set

T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{∞}(str(N^{X} f′(𝕏_{t})) − χ′(H) f′(0)

− (\frac{n}{2} χ(H) − χ′(H)) f′(\frac{\sqrt{−t}}{2})) \frac{dt}{2t} .

Let T^{H,s}M, g_{s}^{T X}, ∇^{F,s} and g_{s}^{F} denote families of horizontal subbundles and vertical metrics on T M, and of flat connections and Hermitian metrics on F for s ∈ [0, 1], where we assume that the rank of H_{s} = H_{dR}^{∗}(M/B; (F, ∇^{F,s})) does not depend on s. Let ∇^{H,s} and g_{L_{2}}^{H,s} denote the families of induced flat connections and L_{2}-metrics. Let ∇^{T X,s} denote the induced connections on the vertical tangent bundle, and let ẽ(∇^{T X,0}, ∇^{T X,1}) be the Chern-Simons class associated to the Euler class. Let k(z) = \frac{f′(z)−f′(0)}{2z}. We now summarise [BL], Theorems 3.23, 3.24, 3.29 and [BG1], Theorem 3.45.

2.86. Theorem. The form T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F}) is even and real, and satisfies

(1)  d^{B}T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = ∫_{M/B} e(∇^{T X}) f(∇^{F} , g^{F} ) − f(∇^{H}, g_{L_{2}}^{H}) .

The component T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F})^{[0]} of degree 0 is equal to the Ray-Singer analytic torsion T_{RS}(g^{T X}, ∇^{F} , g^{F}) of the fibre. Moreover, modulo exact forms on B,

\begin{matrix}T_{f}(T^{H,1}M, g_{1}^{T X}, ∇^{F} , g_{1}^{F} ) & − T_{f}(T^{H,0}M, g_{0}^{T X}, ∇^{F} , g_{0}^{F} ) \\ = ∫_{M/B} ẽ(∇^{T X,0}, ∇^{T X,1}) f(∇^{F} , g_{0}^{V} ) \\ + ∫_{M/B} e(∇^{T X,1}) f̃(∇^{F} , g_{0}^{V} , g_{1}^{V} ) − f̃(∇^{H}, g_{L_{2}}^{H,0}, g_{L_{2}}^{H,1}) , \end{matrix} (2)

\begin{matrix}and T_{f}(T^{H}M, g^{T X}, ∇^{F,1}, g_{1}^{F} )^{[≥2]} & − T_{f}(T^{H}M, g^{T X}, ∇^{F,0}, g_{0}^{F} )^{[≥2]} \\ = ∫_{M/B} e(∇_{0}^{T X}) L_{k}(∇^{F,s}, g_{0}^{V} , g_{1}^{V} ) − L_{k}(∇^{H,s}, g_{L_{2}}^{H,0}, g_{L_{2}}^{H,1}) . \end{matrix} (3)

2.87. Remark. Note again that the right hand side of (3) is independend modulo dΩ^{∗}(B) of the paths connecting ∇^{F,0} with ∇^{F,1} and ∇^{H,0} with ∇^{H,1}, even though both L_{k}-terms may depend on the homotopy class of these paths in degree ≥ 4, and on the paths themselves in degree 2.

Suppose now that both the bundles F → M and H → B carry parallel metrics g^{F} and g^{H}. Then as in Corollary 2.63, the form

T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) + f̃(∇^{H}, g^{H}, g_{L_{2}}^{H})

is closed, and its cohomology class is independend of the choice of the parallel metrics g^{F} and g^{H} in degree ≥ 2.

2.88. Definition. If the metrics g^{F} and g^{H} are parallel, define the higher analytic torsion

T_{f}(M/B; F ) = T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} )^{[≥2]} + f̃(∇^{H}, g^{H}, g_{L_{2}}^{H})^{[≥2]} ∈ H_{dR}^{even,≥2}(B) .

2.i. The Chern normalisation. We recall the Chern normalisation of the odd characteristic classes of Definition 2.14 and the torsion forms of Definition 2.46 and Definition 2.85 following [BG1]. Note that “normalisation” means the choice of a particular generating function f. Theorem 0.1 does not depend on f, and the construction of torsion forms using the function f of (2.11) is very convenient for proving it. Nevertheless, results like Theorem 0.1 of [BG2] depend on the choice of normalisation. This suggests that the most natural normalisation of torsion forms is the one we are introducing now.

Let (F, ∇^{F}) → M be a flat vector bundle. A metric g^{F} on F determines an adjoint connection ∇^{F,∗} which is also flat, so

ch(∇^{F} ) = tr(e^{− \frac{(∇^{F})^{2}}{2πi}} ) = ch(∇^{F,∗}) = rk F

as differential forms. Let c̃h(∇^{F} , ∇^{F,∗}) be the natural Chern-Simons class in Ω^{∗}(M)/d^{M}Ω^{∗}(M), then

d^{M} c̃h(∇^{F} , ∇^{F,∗}) = ch(∇^{F,∗}) − ch(∇^{F} ) = 0 ,

so c̃h(∇^{F} , ∇^{F,∗}) is in fact a cohomology class.

In order to construct a natural representative for this class, we choose a connection

∇\overline{F} = (1 − s) ∇^{F} + s ∇^{F,∗} + \frac{∂}{∂s} ds

on \overline{F} = F × [0, 1] → \overline{B} = B × [0, 1]. Set ω^{F} = ∇^{F,∗} − ∇^{F} as before. Because ∇^{F} and ∇^{F,∗} are flat, the curvature of ∇\overline{F} is given by

(∇\overline{F} )^{2} = s(1 − s) [∇^{F} , ∇^{F,∗}] − (∇^{F^{∗}} − ∇^{F} ) ds = −s(1 − s) (ω^{F})^{2} − ω^{F} ds .

2.89. Definition. Set

ch^{o}(∇^{F} , g^{F} ) = ∫_{0}^{1} tr(\frac{ω^{V}}{2} e^{\frac{s(1−s) (ω^{V} )^{2}}{2πi}} ) ds .

2.90. Theorem. The form ch^{o}(∇^{F} , g^{F}) is real, odd and closed and represents the Chern-Simons class πi c̃h(∇^{F} , ∇^{F,∗}) in H_{dR}^{∗}(M). Moreover, for f(z) = z e^{z^{2}},

ch^{o}(∇^{F} , g^{F} ) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}−1}{2}} f(∇^{F} , g^{F} ) ds .

Proof. It is clear that ch^{o}(∇^{F} , g^{F}) represents πi c̃h(∇^{F} , ∇^{F,∗}) by the definition of Chern-Simons classes. Moreover,

ch^{o}(∇^{F} , g^{F} ) = ∫_{0}^{1} tr(\frac{ω^{V}}{2} e^{\frac{s(1−s) (ω^{V} )^{2}}{2πi}} ) ds

= ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}−1}{2}} tr(\frac{ω^{V}}{2} e^{\frac{(ω^{V} /2)^{2}}{2πi}} ) ds

= ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}−1}{2}} f(∇^{F} , g^{F} ) ds .

Finally, ch^{o}(∇^{F} , g^{F}) is real, odd and closed because f(∇^{F} , g^{F}) is. □

This motivates the following definitions, where we make the same assumptions as before. In particular, let (V, A′, h^{V}) → B be a family Thom-Smale complex with an adapted metric g^{V}. Let p: M → B be a smooth bundle with compact fibres, with horizontal subbundle T^{H}M and vertical metric g^{V}. Let (F, ∇^{F}) → M be a flat complex vector bundle, and let g^{F}, g_{0}^{F} and g_{1}^{F} be metrics on F. Let (V, A′_{s}, h_{s}^{V}) be a family of family Thom-Smale complexes with adapted metrics g_{s}^{V} for s ∈ [0, 1], and let ∇^{F,s} be a family of flat connections on F → M, such that the fibre-wise cohomology H_{s} → B has constant rank, with ∇^{H,s}, g_{V}^{H,s} and g_{L_{2}}^{H,s} in both cases denoting the respective induced family of metrics and connections on H_{s}, respectively.

2.91. Definition. Let f̃(∇^{F} , g_{0}^{F} , g_{1}^{F}), L_{k}(A′_{s}, g_{0}^{V} , g_{1}^{V}), T_{f}(A′, g^{V} , h^{V}), and T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F}) be defined as in Definition 2.16, Definition 2.19, Definition 2.46 and Definition 2.85, and set

c̃h^{o}(∇^{F} , g_{0}^{F} , g_{1}^{F} ) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} f̃(∇^{F} , g_{0}^{F} , g_{1}^{F} ) ds  ∈ Ω^{∗}(M) ,
L(A′_{s}, g_{0}^{V} , g_{1}^{V} ) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} L_{k}(A′_{s}, g_{0}^{V} , g_{1}^{V} ) ds ,
T (A′, g^{V} , h^{V} ) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} T_{f}(A′, g^{V} , h^{V} ) ds ,
and  T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) ds ∈ Ω^{∗}(B) .

Let us assume that (H, ∇^{H}) is the fibre-wise cohomology of (V, A′) (or (Ω^{∗}(M/B; F), d^{M}) respectively), with induced metric g_{V}^{H} (or g_{L_{2}}^{H}). Then we have the obvious analogues of Corollary 2.17, Theorem 2.48, Theorem 2.58 and Theorem 2.86.

2.92. Corollary. The forms c̃h^{o}(∇^{F} , g^{F} , g^{F ′}), L(A′_{s}, g_{0}^{V} , g_{1}^{V}), T (A′, g^{V} , h^{V}) and T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) are even and real, and satisfy

(1)  d^{M} c̃h^{o}(∇^{F} , g_{0}^{F} , g_{1}^{F} ) = ch^{o}(∇^{F} , g_{1}^{F} ) − ch^{o}(∇^{F} , g_{0}^{F} ) ,
(2)  d^{B}L(A′_{s}, g_{0}^{V} , g_{1}^{V} ) = ch^{o}(A′_{1}, g_{1}^{V} )^{[≥3]} − ch^{o}(A′_{0}, g_{0}^{V} )^{[≥3]} ,
(3)  d^{B}T (A′, g^{V} , h^{V} ) = ch^{o}(∇^{V} , g^{V} ) − ch^{o}(∇^{H}, g_{V}^{H}) ,
(4)  and  d^{B}T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = ∫_{M/B} e(∇^{T X}) ch^{o}(∇^{F} , g^{F} ) − ch^{o}(∇^{H}, g_{L_{2}}^{H}) .

Modulo exact forms on B,

\begin{matrix}T (A′, g_{1}^{V} , h^{V} ) & − T (A′, g_{0}^{V} , h^{V} ) \\ = c̃h^{o}(∇^{V} , g_{0}^{V} , g_{1}^{V} ) − c̃h^{o}(∇^{H}, g_{V}^{H,0}, g_{V}^{H,1}) , \end{matrix} (5)

T (T_{1}^{H}M, g_{1}^{T X}, ∇^{F} , g_{1}^{F} ) − T (T_{0}^{H}M, g_{0}^{T X}, ∇^{F} , g_{0}^{F} )
= ∫_{M/B} ẽ(∇^{T X,0}, ∇^{T X,1}) ch^{o}(∇^{F} , g_{1}^{F} )
+ ∫_{M/B} e(∇^{T X,0}) c̃h^{o}(∇^{F} , g_{0}^{F} , g_{1}^{F} ) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H,0}, g_{L_{2}}^{H,1}) ,

(6)

\begin{matrix}T (A′_{1}, g_{1}^{V} , h_{1}^{V} )^{[≥2]} & − T (A′_{0}, g_{0}^{V} , h_{0}^{V} )^{[≥2]} \\ = L(∇^{V,s}, g_{0}^{V} , g_{1}^{V} ) − L(∇^{H,s}, g_{V}^{H,0}, g_{V}^{H,1}) , \end{matrix} (7)

\begin{matrix}and T (T^{H}M, g^{T X}, ∇^{F,1}, g_{1}^{F} )^{[≥2]} & − T (T^{H}M, g^{T X}, ∇^{F,0}, g_{0}^{F} )^{[≥2]} \\ = ∫_{M/B} e(∇^{T X})L(∇^{F,s}, g_{0}^{F} , g_{1}^{F} ) − L(∇^{H,s}, g_{L_{2}}^{H,0}, g_{L_{2}}^{H,1}) . \end{matrix} (8)

Theorem 2.48 and Corollary 2.92 (7) imply that T (A′, g^{V} , h^{V}) is independent of h^{V} modulo exact forms in all degrees. In [G], we will give a slightly more complicated definition of T (A′, g^{V} , h^{V}) that does not depend on h^{V} explicitly, but uses only the splitting A′ = ∇^{V} + a′ induced by h^{V}.

Assume that the metrics g^{F} on F → B and g^{H} on H → B are parallel. Then Corollary 2.92 implies in particular that we get higher torsion classes

\begin{aligned}T (M/B; F, h) & = T (A′, g^{V} , h^{V} )^{[≥2]} + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})^{[≥2]} \\ T (M/B; F) & = T (T^{H}M, g^{T X}, ∇^{F} , g^{F} )^{[≥2]} + c̃h^{o}(∇^{H}, g^{H}, g_{L_{2}}^{H})^{[≥2]} \end{aligned} (2.93)

in H_{dR}^{even,≥2}(B) as in Definition 2.64 and Definition 2.88. Moreover, the classes T (M/B; F, h) and T (M/B; F) are independent of the particular parallel metrics on F and H and of all other secondary data needed to define the respective analytic torsion forms.

3. Higher analytic torsion and Morse functions

We recall our main result, Theorem 0.1, and show that it is compatible with other known results on higher analytic torsion. We also name a few conclusions in [BG1] that do not hold without Smale’s transversality condition. Lott used the higher analytic torsion to define a pushdown in a secondary K-theory in [L]. In Section 3.c, we apply Theorem 0.1 to simplify this pushdown in the presence of a fibre-wise Morse function. In Section 3.d, we adapt the proof of Theorem 0.1 in [BG1] to the current situation. Section 4 contains a new proof for the only intermediate result of [BG1] that is affected by dropping the Smale condition.

3.a. Statement of the main result. Let p: M → B be a smooth fibre bundle with compact fibres, let (F, ∇^{F}) → M be a flat complex vector bundle, and assume that there exists a smooth function h: M → ℝ such that h is a Morse function on each fibre of p.

Let g^{F} be a Hermitian metric on F. Then we have an odd characteristic form ch^{o}(∇^{F} , g^{F}) of Definition 2.89 that vanishes if g^{F} is parallel with respect to ∇^{F}.

Let T^{H}M ⊂ T M be a horizontal subbundle complementary to the vertical tangent bundle T X, and let g^{T X} be vertical metric on T X. Let ∇^{T X} be the natural connection on T X induced by these data.

Let H = H^{∗}(M/B; F) → B denote the bundle of the fibre-wise cohomology with values in F, then H carries a natural Gauß-Manin connection ∇^{H}, such that (H, ∇^{H}) is the E_{1}-term in the Leray spectral sequence in de Rham theory for p with coefficients in F. Moreover, fibre-wise Hodge theory gives an L_{2}-metric g^{H} on H as in [BG1]. Let ch^{o}(∇^{H}, g^{H}) be the corresponding odd characteristic form.

Let T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) ∈ Ω^{∗}(B)/dΩ^{∗}(B) be the analytic torsion form of Definitions 2.85 and 2.91 in the Chern normalization of [BG1], such that

(3.1)  d^{B}T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) = ∫_{M_{g}/B} e(∇^{T X}) ch^{o}(∇^{F} , g^{F} ) − ch^{o}(∇^{H}, g^{H}) ,

as explained in Theorem 2.86 and Corollary 2.92.

Let h: M → ℝ be a smooth function that is Morse on each fibre of p, and let ∇^{T X}h be the fibre-wise gradient of h with respect to g^{T X}. Let C ⊂ M denote the set of fibre-wise critical points of h, then p̂ = p|_{C}: C → B is a finite covering of B. We denote the stable and unstable part of the vertical tangent bundle T X|_{C} of C by T^{s}X, T^{u}X → C.

Let V = (p|_{C})_{∗}o(T^{u}X) ⊗ F |_{C} → B be the bundle constructed in Definition 1.1, equipped with a flat superconnection A′ constructed as in Theorem 1.66. The component a_{0} of A′ of horizontal degree 0 in Ω^{∗}(B) defines a cochain complex on V. Let h^{V} be the endomorphism of V induced by h|_{C}. Let g^{V} denote the Hermitian metric on V induced by g^{F}, and let g_{V}^{H} denote the L_{2}-metric on H that is constructed as the restriction of g^{V} to the (a_{0} + a_{0}^{∗})-harmonic elements of V. Then we have the Chern-Simons class c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) of Definitions 2.16 and 2.91, which satisfies

(3.2)  d^{B}c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H}) = ch^{o}(∇^{H}, g_{V}^{H}) − ch^{o}(∇^{H}, g_{L_{2}}^{H}) .

Let T (A′, g^{V} , h^{V}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) be the torsion form of Definitions 2.46 and 2.91, such that

(3.3)  d^{B}T (A′g^{V} , h^{V} ) = ch^{o}(∇^{V} , g^{V} ) − ch^{o}(∇^{H}, g_{V}^{H}) ∈ Ω^{∗}(B)_{h} .

Let δ_{0} be the π^{∗}o(T X)-valued current on the total space of the vertical tangent bundle π: T X → M, given by integration over the zero section M of T X. Let Ω^{∗}(T X)_{0} denote currents that are smooth away from the zero section of T X, and whose wave front set is contained in T^{∗}X|_{M} ⊂ T^{∗}T X. Let ψ(∇^{T X}, g^{T X}) denote the π^{∗}o(T X)-valued Mathai-Quillen current on T^{∗}X of [MQ], see also [BZ1], such that

(3.4)  d^{T X}ψ(∇^{T X}, g^{T X}) = π^{∗}e(∇^{T X}) − δ_{0} ∈ Ω^{∗}(T X)_{0} .

Let δ_{C} be the o(T X)-valued current on M, given by integration over C. Then the wave front sets of δ_{C} and (∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) are contained in T^{∗}X|_{C}, and we have

(3.5)  d^{M}((∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X})) = e(∇^{T X}) − (−1)^{ind^{h}} δ_{C} .

Finally, let ^{0}J denote the additive genus introduced in [BG1]. Let ζ denote the Riemann ζfunction, and for a real vector bundle E → M, let ch(E)^{[4k]} denote the component of the Chern character of E in H^{4k}(M). Then

(3.6)  ^{0}J(E) = \frac{1}{2} ∑_{k=1}^{∞} ζ′(−2k) ch(E)^{[4k]} .

For convenience, we recall that Theorem 0.1 states that

(3.7) T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) − T (A′, g^{V} , h^{V} ) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H})

= − ∫_{M/B} ch^{o}(∇^{F} , g^{F} ) (∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) + p̂_{∗}((−1)^{ind^{h}} ^{0}J(T^{s}X − T^{u}X)) rk F .

modulo exact forms on B. Theorem 0.1 will be proved at the end of this section.

3.8. Remark. The formulation of Theorem 0.1 is the same as that of Theorem 0.1 in [BG1]. The only difference is in the definition of the “combinatorial torsion form” T.

We discuss an important special case of Theorem 0.1. Assume therefore that there are parallel metrics g^{F} on F → M and g^{H} on H → B, and recall the definition of the higher torsion classes T (M/B; F, h) and T (M/B; F) in (2.93), cf. Definition 2.64 and Definition 2.88. Theorem 0.1 immediately implies

3.9. Corollary. Assume that the vector bundles F → M and H → B carry parallel metrics. Then

T (M/B; F) − T (M/B; F, h) = p̂_{∗}((−1)^{ind^{h}} ^{0}J(T^{s}X − T^{u}X)) rk F .

We will give an application of Corollary 3.9 in Section 5.

3.b. Compatibility with known results on analytic torsion forms. We check that Theorem 0.1 is compatible with the transgression and variation formulas for higher analytic torsion forms of [BL] and [BG1], and with Poincaré duality, cf. [BG1], sections 7.4–7.6. On the other hand, we remark that most of the results of sections 7.8–7.10 in [BG1] fail if h does not admit a fibre-wise Morse-Smale gradient field.

3.10. Remark. From equations (3.1)–(3.5) it is clear that applying the exterior differential d^{B} on B to both sides of (3.7) gives the trivial identity

∫_{M/B} e(∇^{T X}) ch^{o}(∇^{F} , g^{F} ) − ch^{o}(∇^{V} , g^{V} ) = ∫_{M/B} ch^{o}(∇^{F} , g^{F} ) e(∇^{T X}) − p̂_{∗} ch^{o}(∇^{F} , g^{F} ) .

Thus, Theorem 0.1 is compatible with the transgression formulas satisfied by T , T, c̃h^{o} and ψ. This automatically implies that Theorem 0.1 is also compatible with the variation formulas for T , T, ch^{o}, c̃h^{o} and ψ under smooth variations of T^{H}M, g^{T X}, and g^{F}. In particular, we may choose these data such that the simplifying assumptions of Proposition 1.10 are satisfied when we prove Theorem 0.1.

Let ∇^{F,s} be a smooth one-parameter family of flat connections on F, and let g^{F,s} be a smooth one-parameter family of Hermitian metrics, with s ∈ [0, 1]. Let (H_{s}, ∇^{H,s}, g_{L_{2}}^{H,s}) be the fibre-wise cohomology with values in (F, ∇^{F,s}) with its induced L_{2}-metric. We assume that dim H_{s} is constant, so that (H_{s}, ∇^{H,s}) becomes a smooth vector bundle on B × [0, 1].

For k ≥ 0, let ( ·)^{[≥k]} denote the sum of the homogeneous parts of degree ≥ k of a differential form. In [BG1], Definition 2.4, we defined the so-called Bott-Chern classes L(∇^{F,s}, g^{F,s}) ∈ Ω^{≥2}(M)/dΩ^{∗}(M) and L(∇^{H,s}, g_{L_{2}}^{H,s}) ∈ Ω^{≥2}(B)/dΩ^{∗}(B), see Definitions 2.19 and 2.91, such that

d^{M}L(∇^{F,s}, g^{F,s}) = ch^{o}(∇^{F,1}, g^{F,1})^{[≥3]} − ch^{o}(∇^{F,0}, g^{F,0})^{[≥3]}
and  d^{B}L(∇^{H,s}, g^{H,s}) = ch^{o}(∇^{H,1}, g^{H,1})^{[≥3]} − ch^{o}(∇^{H,0}, g^{H,0})^{[≥3]} ,

(3.11)

see Theorem 2.20 and Corollary 2.92 (2). In [BG1], Theorem 3.45, it was shown that

(3.12) T (T^{H}M, g^{T X}, ∇^{F,1}, g^{F,1})^{[≥2]} − T (T^{H}M, g^{T X}, ∇^{F,0}, g^{F,0})^{[≥2]}

= ∫_{M/B} e(∇^{T X}) L(∇^{F,s}, g^{F,s}) − L(∇^{H,s}, g^{H,s}) ,

see Theorem 2.86 (3) and Corollary 2.92 (8). Similarly, we have shown in Theorem 2.58 and Corollary 2.92 (7) that

(3.13)  T (A^{′1}, g^{V,1}, h^{V} )^{[≥2]} − T (A^{′0}, g^{V,0}, h^{V} )^{[≥2]} = L(∇^{V,s}, g^{V,s}) − L(∇^{H,s}, g_{V}^{H,s}) .

3.14. Remark. By equations (3.11)–(3.13), subtracting (3.7) for \overline{F}_{0} from (3.7) for \overline{F}_{1} in degree ≥ 2 after a straightforward partial integration gives the trivial identity

∫_{M/B} e(∇^{T X}) L(∇^{F,s}, g^{F,s}) − L(∇^{V,s}, g^{V,s})

= ∫_{M/B} L(∇^{F,s}, g^{F,s}) e(∇^{T X}) − p̂_{∗}L(∇^{F,s}, g^{F,s}) .

In particular, Theorem 0.1 is compatible with the known variation formulas for analytic torsion forms in dependence of the flat connection on F.

Let o(T X) → M denote the orientation line bundle of T X, and let n denote the dimension of the fibres of p: M → B. Fibre-wise Poincaré duality implies that there exists a canonical isomorphism

\overline{H^{k}(M/B; F)}^{∗} ≅ H^{n−k}(M/B; \overline{F}^{∗} ⊗ o(T X)) .

As in [BG1], one verifies that the analytic torsion form satisfies

(3.15) T (T^{H}M, g^{T X}, ∇^{\overline{F}^{∗}⊗o(T X)}, g^{\overline{F}^{∗}⊗o(T X)})

= (−1)^{n+1} T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) ∈ Ω^{∗}(B)/dΩ^{∗}(B) .

If we replace the fibrewise Morse function h by −h and F by \overline{F}^{∗}⊗o(T X), then the bundles V^{k} → B get replaced by \overline{V^{n−k}}^{∗} → B. The superconnections A′ and A′′ on V get replaced by \overline{A′}^{∗} and \overline{A′′}^{∗}, which are conjugates of A′′ and A′ by the metric g^{V}. As in [BG1], Definition 2.46 implies that

(3.16)  T (\overline{A′}^{∗}, g^{\overline{V}^{∗}}, −h^{\overline{V}^{∗}}) = (−1)^{n+1} T (A′, g^{V} , h^{V} ) ∈ Ω^{∗}(B)/dΩ^{∗}(B) .

3.17. Remark. It has been checked in [BG1] that the remaining terms in (3.7) satisfy equations analogous to (3.15) and (3.16) if one replaces h by −h and F by \overline{F}^{∗} ⊗ o(T X). Thus, Theorem 0.1 is compatible with Poincaré duality.

3.18. Remark. In [BG1], sections 7.8 and 7.9, we drew some conclusions in the special case that the metric g^{F} |_{C} is parallel. We found that then the combinatorial torsion vanishes in higher degrees on B, and that the metric g_{V}^{H} induced by g^{V} on H → B is parallel; both due to the fact that A′ takes the special form a_{0} + ∇^{V} with a_{0} of degree 0 on B and ∇^{V} unitary with respect to g^{V}.

If ∇^{T X}h does not satisfy Smale’s transversality condition, then A′ is more complicated, and we will give an example in [G] where T (A′, g^{V} , h^{V}) carries cohomological information even though ∇^{V} is unitarily flat and H = 0.

Moreover in our situation, ∇^{H} is induced by ∇^{V} + a_{1}, which is not necessarily unitary, so we cannot conclude any longer that g^{H} is a parallel metric. Thus, the arguments in [BG1], sections 7.8 and 7.9 fail in our context.

Note also that in [BG1], Thm 7.11, we deduced that if the fibres are even-dimensional, then the bundle

p̂_{∗}((−1)^{ind^{h}} T X|_{C})

becomes trivial in KR_{0}(B) ⊗ ℚ. By Theorem 0.3 that will be proved in [G], the above bundle gives the difference between the higher analytic torsion and the higher Franz-Reidemeister torsion

∫_{M/B} e(T X) ^{0}J(T X) rk F = p̂_{∗}((−1)^{ind^{h}} ^{0}J(T X|_{C})) rk F ,

which may be nonzero.

3.19. Remark. In [BG1], section 7.10, we used Smale’s transversality condition to show that for every subcovering C′ ⊂ C → B that leads to an acyclic direct summand V′ ⊂ V, we have

(p̂|_{C′})_{∗}((−1)^{ind^{h}} ^{0}J(T^{s}X − T^{u}X)) = 0 .

We will see in Remark 5.14 that this is not the case in Hatcher’s example, which shows that the Morse function in Hatcher’s example does not admit a global Smale gradient field, and that Theorem 7.12 in [BG1] becomes false without the transversality condition.

3.c. Morse functions and Lott’s pushdown. In this subsection, we give a short review of Lott’s \overline{K}_{R}^{0}-groups and describe the pushdown map p_{!}: \overline{K}_{R}^{0}(M) → \overline{K}_{R}^{0}(B) associated to a proper submersion p: M → B in the special case that M carries a fibre-wise Morse function. In Section 5.f, we will apply the \overline{K}_{R}^{0}-functor to Hatcher’s example and discuss a conjectural relation of p_{!} to a Becker-Gottlieb transfer.

Let R be a right-regular, right-Noetherian ring, e.g., R = ℤ, R = ℝ or R = ℂ. Let ρ: R → End(ℂ^{n}) be a complex representation of R such that ℂ^{n} becomes a flat R-module. If M is a manifold and F → M is a local system of right-R-modules, let F_{ℂ} = F ⊗_{ρ} ℂ^{n} denote the complex flat vector bundle associated to F with flat connection ∇^{F}.

Lott first defines a Grothendieck group K̂_{R}^{0}(M) with generators (F, g^{F} , η), where F is a local system of right-R-modules, g^{F} is a Hermitian metric on F_{ℂ}, and η ∈ Ω^{even}(M)/dΩ^{odd}(M) is a class of even differential forms modulo exact forms on M.

To define the relations in K̂_{R}^{0}(M), let

(3.20)  0 ⟶ F_{1} ⟶ F_{2} ⟶ F_{3} ⟶ 0

be a short exact sequence, choose metrics g^{F_{i}} on F_{i,ℂ} and forms η_{i} ∈ Ω^{even}(M)/dΩ^{odd}(M), and let T denote the Bismut-Lott analytic torsion of the finite-dimensional flat family of acyclic complexes

(3.21)  0 ⟶ (F_{1,ℂ}, g^{F_{1}}) ⟶ (F_{2,ℂ}, g^{F_{2}}) ⟶ (F_{3,ℂ}, g^{F_{3}}) ⟶ 0

over M. Then in K̂_{R}^{0}(M),

(3.22)  [F_{2}, g^{F_{2}}, η_{2}] = [F_{1}, g^{F_{1}}, η_{1}] ⊕ [F_{3}, g^{F_{3}}, η_{3}]  if  η_{2} = η_{1} + η_{3} + T .

There is a well-defined map ch^{o}: K̂_{R}^{0}(M) → Ω^{odd}(M) given on generators by

ch^{o}[F, g^{F} , η] = ch^{o}(∇^{F} , g^{F} ) − dη ,

and \overline{K}_{R}^{0}(M) ⊂ K̂_{R}^{0}(M) is defined as the kernel of ch^{o}. Then \overline{K}_{R}^{0}(M) fits into an exact sequence

(3.23)  H^{even}(M, ℝ) ⟶ \overline{K}_{R}^{0}(M) ⟶ K_{R}^{0}(M) ⟶ H^{odd}(M, ℝ) ,

where K_{R}^{0}(M) is simply the Grothendieck group of local systems of right-R-modules on M, and the rightmost arrow is the characteristic class ch^{o}. For any smooth map f: M → N there is a natural pullback f^{∗}: K̂_{R}^{0}(N) → K̂_{R}^{0}(M) that restricts to a pullback f^{∗}: \overline{K}_{R}^{0}(N) → \overline{K}_{R}^{0}(M) and is compatible with (3.23).

Now let p: M → B be a smooth proper submersion, and fix a horizontal subbundle T^{H}M and a vertical metric g^{T X}. Let H = H^{∗}(M/B; F) denote the fibre-wise cohomology of F, which is again a local system of right-R-modules over B, and let g_{L_{2}}^{H} denote the L_{2}-metric on H_{ℂ} → B. Then a pushdown in K̂_{R}^{0}(M) is given on generators by

(3.24)  p_{!}[F, g^{F} , η] = [H, g_{L_{2}}^{H}, ∫_{M/B} e(T X, ∇^{T X}) η − T (T^{H}M, g^{T X}, ∇^{F} , g^{F} )] .

It is proved in [L] that p_{!} is compatible with (3.22). Moreover, the pushdown restricts to a map p_{!}: \overline{K}_{R}^{0}(M) → \overline{K}_{R}^{0}(B) that is independent of T^{H}M and g^{T X}.

If h: M → B is a fibre-wise Morse-function, we define C, (V, ∇^{V} , A′, g^{V}) → B and g_{V}^{H} as above. We define another pushdown on K̂_{R}^{0}(M) by

(3.25) p_{!h}[F, g^{F} , η] = [H, g_{V}^{H}, p̂_{∗}((−1)^{ind^{h}} (^{0}J(T^{s}X − T^{u}X) rk F_{ℂ} + η)) − T (A′, g^{V} , h^{V} )] .

As in [L], one can show that this pushdown is compatible with (3.22).

3.26. Theorem. Suppose that h: M → ℝ is a fibre-wise Morse function. Then p_{!h} restricts to the pushdown p_{!}: \overline{K}_{R}^{0}(M) → \overline{K}_{R}^{0}(B).

Proof. The relations (3.22)imply that

(3.27)  [F, g^{′F} , η′] = [F, g^{F} , η′ − c̃h^{o}(∇^{F} , g^{F} , g^{′F} )] ∈ K̂_{R}^{0}(M) .

Also, a straightforward partial integration shows that

∫_{M/B} e(T X, ∇^{T X}) η − p̂_{∗}η = ∫_{M/B}(∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) dη

modulo exact forms on B. Using (3.7) and the two observations above, we may rewrite (3.24) as

p_{!}[F, g^{F} , η] = [H, g_{V}^{H}, p̂_{∗}η + ∫_{M/B}(∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) dη
− T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H})]

= p_{!h}[F, g^{F} , η] + [0, 0, ∫_{M/B}(∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) (dη − ch^{o}(∇^{F} , g^{F} ))] .

The theorem follows because in \overline{K}_{R}^{0}(M), we have

dη − ch^{o}(∇^{F} , g^{F} ) = − ch^{o}[F, g^{F} , η] = 0 . □

3.d. A proof of Theorem 0.1. In order to prove Theorem 0.1, we refer to the proof of Theorem 0.1 of [BG1]. However, we have to replace all arguments that rely on the Smale transversality condition for the negative gradient flow, or on its consequences. We start by summarizing the places in [BG1] that depend on the Smale condition, then we indicate how to extend [BG1] to the more general case of fibre-wise Morse functions that do not necessarily admit a fibre-wise Morse-Smale gradient field, and finally, we show how to fit the contents of this paper into the general framework of [BG1], thereby proving Theorem 0.1.

In Chapter 5 of [BG1], we used the transversality condition to prove a families’ version of the result of Laudenbach in the appendix of [BZ1]. In particular, we constructed a vector bundle V = C^{∗}(W^{u}; F) → B with a flat superconnection

(3.28)  A′ = ∇^{V} + v  with  v ∈ Ω^{0}(B; End^{1} V )

of total degree one, and an Ω^{∗}(B)-linear quasi-isomorphism

(3.29)  P^{∞}: Ω^{∗}(M; F) ⟶ Ω^{∗}(B; V ) ,

given as integration over the fibres of the unstable cells.

In Chapters 1 and 2 of [BG1], we recalled and extended the construction of torsion forms from Section 2 (f) in [BL]. The torsion forms

(3.30)  T_{f}(A′, g^{V} ) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B)

are only defined for flat superconnections of the type (3.28), in particular, they converge only if there is no component in Ω^{≥2}(B; End V). Moreover, the identity

(3.31)  dT_{f}(A′, g^{V} ) = f(∇^{V} , g^{V} ) − f(∇^{H^{∗}(V,v)}, g^{H^{∗}(V,v)})

only holds if the connection part of A′ equals ∇^{V}.

In Theorem 9.8 of [BG1], we proved that up to a few divergences and correction terms, the analytic torsion form of [BL] converges to the form T_{f}(A′, g^{V}) of (3.30). We gave two proofs of this theorem in Chapters 10 and 11 of [BG1]. While the proof in Chapter 11 needs the transversality condition in its full strength, the proof in Chapter 10 uses only that A′ is of the form (3.28), and this assumption is needed only in very few places.

We now indicate the changes that lead to Theorem 0.1. In Section 1 of this paper, we constructed a flat superconnection

A′ = ∇^{V} + ∑_{j} a_{j}  with  a_{j} ∈ Ω^{j}(B; End^{1−j} V )

of total degree one on V, which replaces the one of (3.28). If we view h|_{C} as the “height” of a critical point, then the coefficients a_{j} “move” elements of V “up” by at least ε^{2} > 0. We also constructed an Ω^{∗}(B)-linear quasi-isomorphism

I: Ω^{∗}(M; F) ⟶ Ω^{∗}(B; V)

that takes the role of P^{∞} of (3.29). The main properties of I and A′ are summarised in Theorem 1.66.

In Section 2, we generalised the torsion forms of [BL], [BG1] to general flat superconnections of total degree one that are compatible with an endomorphism h^{V} given by the Morse function h|_{C}.

The necessary condition is that (V, A′, h^{V}) is a family Thom-Smale complex in the sense of Definition 2.35 with an adapted metric g^{V}, and the torsion forms T_{f}(A′, g^{V} , h^{V}) are constructed in Definition 2.46.

In Section 4, we summarise Chapter 10 of [BG1] and prove our Theorem 4.31, which replaces Theorem 9.8 of [BG1]. We still use the Witten deformation as in [BZ2] and [BG1]. In fact, our proof is a bit more direct because our definition of T_{f}(A′, g^{V} , h^{V}) is modelled itself on a “Witten deformation” of the complex (V, A′) with metric g^{V}.

Let us recall Theorem 9.8 of [BG1]. The definition of T_{f}(A′, g^{V}) in [BL], Chapter 2 and [BG1], Definition 1.29 was recalled in (3.30). It implies in particular that

T_{f}(A′, g^{V} ) − (χ′(V) − χ′(H)) ∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t}

− (χ′(V) − χ′(H)) ∫_{0}^{1}(f′(\sqrt{−t}/2) − f′(0)) \frac{dt}{2t}

= −(2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str_{V} (N^{V} γ f′(X_{t,1})) − χ′(H)f′(0)) \frac{dt}{2t}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{1}(str_{V} (N^{V} γ f′(X_{t,1})) − χ′(V)f′(0)) \frac{dt}{2t} ,

where f(z) = z e^{z^{2}}, so in particular, f′(0) = 1.

We must also recall the definition of χ̃^{′±}(F) in [BG1]. Starting from (9.6) in [BG1], an elementary calculation shows that

χ̃^{′+}(F) = ∑_{c∈C/B}(−1)^{ind_{h}(c)} (n − ind_{h}(c)) rk F
= str_{V} ((n − N^{V})) = n χ(V) − χ′(V) ,
and  χ̃^{′−}(F) = ∑_{c∈C/B}(−1)^{ind_{h}(c)} ind_{h}(c) rk F
= str_{V} (N^{V} ) = χ′(V) .

With this preliminaries, we can now reformulate Theorem 9.8 of [BG1] in our notation.

3.32. Theorem. In the limit T → ∞, we have

f̃(∇^{H}, g^{H}, g_{T}^{H}) + S_{f}(𝔸′, g_{T}^{T X,F} ) − (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) log T + T str_{V} (h^{V} γ)

⟶ f̃(∇^{H}, g^{H}, g_{V}^{H}) − (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) log π + T_{f}(A′, g^{V} )

− (χ′(V) − χ′(H)) ∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t}

− (χ′(V) − χ′(H)) ∫_{0}^{1}(f′(\sqrt{−t}/2) − 1) \frac{dt}{2t}

modulo exact forms on B. □

In Section 4, we prove Theorem 4.31, which equals the theorem above, except that we do no longer require the existence of a Smale gradient field (which is used implicitly in [BG1], Theorem 9.8), and that T_{f}(A′, g^{V}) gets replaced by T_{f}(A′, g^{V} , h^{V}).

Proof of Theorem 0.1. By Remark 3.10, we may assume that the simplifying assumptions of Proposition 1.10 are satisfied. By the definitions and arguments of Section 2.i, we may replace the Chern normalization in Theorem 0.1 by the normalisation f(z) = z e^{z^{2}} of (2.11). If we define

(3.33)  ^{0}I_{f}(E) = \frac{1}{2} ∑_{k=1}^{∞} \frac{(4k + 1)!}{2^{4k} (2k)!^{2}} ζ′(−2k) ch(E)^{[4k]}

in analogy with (3.6), then an elementary calculation shows that

0J(E) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} ^{0}I_{f}(E) ds .

It remains to establish

(3.34) T_{f}(T^{H}M, g^{T X}, ∇^{F} , g^{F} ) − T_{f}(A′, g^{V} , h^{V} ) − f̃(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H})

= − ∫_{M/B} f(∇^{F} , g^{F} ) (∇^{T X}h)^{∗}ψ(∇^{T X}, g^{T X}) + p̂_{∗}((−1)^{ind^{h}} ^{0}I_{f}(T^{s}X − T^{u}X)) rk F .

As indicated above, this equation now follows from the proof given in Chapter 9 of [BG1], where Theorem 4.31 replaces Theorem 9.8. □

4. The Witten deformation

We consider the Witten deformation of the fibre-wise de Rham complex with parameter T > 0. Proceeding as in [BG1], we show that for T sufficiently large, the de Rham complex splits naturally into two sub-complexes. One of these sub-complexes represents all eigenvalues that grow at least linearly in T (this will be made precise), so that it does not contribute to the torsion integral in the limit. The other one can be identified with the sections of a finite-dimensional vector bundle V → B, generalising the families’ Thom-Smale complex of [BG1]. While in [BG1], the identification was provided by the integration over the unstable bundles of the fibre-wise gradient flow, we now use the integration map I: Ω^{∗}(M; F) → Ω^{∗}(B; V) constructed in Section 1.

In Sections 4.a–4.c, we recall several results of [BG1] related to the model operator near the fibre-wise critical points, the space spanned by the generalised eigenforms with small eigenvalues, and the behaviour of the torsion integrand for large values of t, always using I instead of the de Rham map employed in [BG1]. In Section 4.d we show how the definition of T (A′, g^{V} , h^{V}) via finite-dimensional Witten deformation on Ω^{∗}(B; V) fits into the picture of the infinite-dimensional Witten deformation on Ω^{∗}(M; F), thereby establishing Theorem 4.31, which was used in Section 3.d to prove Theorem 0.1.

4.a. The model operator near the critical points. We recall constructions from [BG1], sections 4 and 10 related to a model operator 𝕏̂_{T} acting on forms on the total space of a ℤ_{2}-graded Euclidean vector bundle E = E^{+} ⊕ E^{−} over a smooth manifold C, with values in a flat vector bundle F → E that is pulled back from C. This will be applied as follows. Assume that p: M → B carries a fibre-wise Morse function h: M → ℝ with fibre-wise critical set C, and there exists a vertical metric g^{T X} on the vertical tangent bundle T X → M, such that h and g^{T X} satisfy the simplifying assumptions of Proposition 1.10. Then the restriction of the vertical tangent bundle to C splits orthogonally as T X|_{C} = T^{s}X ⊕ T^{u}X. The Witten deformed operator 𝕏̃_{T} acting on Ω^{∗}(M; F) coincides with 𝕏̂_{T} on E = T X|_{C} in a tubular neighbourhood of C. It has been shown in [BG1] that for T → ∞, the operator e𝕏̂_{T}^{2} provides us with a good approximation for the heat kernel e𝕏̃_{T}^{2} , in a sense to be made precise later.

Let π: (E, g^{E}) = (E^{+}, g^{E^{+}}) ⊕ (E^{−}, g^{E^{−}}) → C be a ℤ_{2}-graded Euclidean vector bundle of dimension n = n^{+} + n^{−}. Let ∇^{E} = ∇^{E^{+}} ⊕ ∇^{E^{−}} be a Euclidean connection that respects the grading, with curvature F^{E}. If we identify π^{∗}E → E with the vertical tangent bundle of E → C, we can define a vertical radial vector field on the total space E by

R = R^{+} + R^{−}  with  R^{±}(p, e^{+}, e^{−}) = e^{±} ∈ π^{∗}E ⊂ T E ,

for all p ∈ C and all (e^{+}, e^{−}) ∈ E_{p}.

The connection ∇^{E} induces a bundle connection on T E, i.e., a projection T E → π^{∗}E with kernel T^{H}E, and g^{E} becomes a vertical metric. These data induce a connection on the vertical tangent bundle as in [B], Theorem 1.1, which coincides with the pull back ∇^{π^{∗}E} of the connection ∇^{E}. The fibre bundle curvature of T^{H}E ⊂ T E is given by Ω^{E} = F^{E}R, see (1.9).

Let Ω^{∗}(E/C) → C denote the bundle of differential forms on the fibres of E, equipped with the L_{2}-metric

g^{Ω^{∗}(E/C)}(s, s′)_{p} = ∫_{E_{p}} g^{E}(s(e), s′(e)) d Vol_{E}(e) ,

where d Vol_{E} is the volume element induced by the vertical metric g^{E}. Then ∇^{E} induces a natural connection ∇^{Ω^{∗}(E/C)} on Ω^{∗}(E/C), and ∇^{Ω^{∗}(E/C)} is compatible with the L_{2}-metric g^{Ω^{∗}(E/C)}.

If we identify E with E^{∗}, then E acts on Ω^{∗}(E/C) by exterior multiplication ε_{e}, interior multiplication ι_{e}, and two types of Clifford multiplication

c_{e} = ε_{e} − ι_{e}  and  ĉ_{e} = ε_{e} + ι_{e} ,

satisfying

[c_{e}, c_{e′}] = −g^{E}(e, e′) ,  [c_{e}, ĉ_{e′}] = 0 ,  and  [ĉ_{e}, ĉ_{e′}] = g^{E}(e, e′) ,

where we use [A, B] = AB − (−1)^{deg A·deg B}BA to denote the supercommutator of A and B. Fixing a local orthonormal basis e_{1}, …, e_{n} of E, we abbreviate ε_{j} = ε_{e_{j}}, ι_{j} = ι_{e_{j}} etc.

Finally, let (F, ∇^{F}) → C be a flat complex vector bundle with a Hermitian metric g^{F} that is not necessarily parallel. Let ∇^{F,∗} be the adjoint of ∇^{F} with respect to g^{F}, and set

ω^{F} = ∇^{F,∗} − ∇^{F} = (g^{F} )^{−1}[∇^{F} , g^{F} ] ,  and  ∇^{F,u} = \frac{1}{2} (∇^{F,∗} + ∇^{F} ) = ∇^{F} + \frac{1}{2} ω^{F} .

Then ∇^{F,u} is unitary with respect to g^{F}. By an abuse of notation, we write (F, ∇^{F}) → E and g^{F} for the pull-backs (π^{∗}F, π^{∗}∇^{F}) and π^{∗}g^{F}. Then g^{E} and g^{F} induce an L_{2}-metric

g^{Ω^{∗}(E/C;F)}(s, s′)_{p} = ∫_{E_{p}}(g^{E} ⊗ g^{F} )(s(e), s′(e)) d Vol_{E}(e)

on the bundle Ω^{∗}(E/C; F) = Ω^{∗}(E/C) ⊗ F → C. The connections ∇^{F}, ∇^{F,∗} and ∇^{F,u} induce connections on Ω^{∗}(E/C; F) denoted by

∇^{Ω^{∗}(E/C;F)} , ∇^{Ω^{∗}(E/C;F),∗} = ∇^{Ω^{∗}(E/C;F)} + ω^{F} ,

and ∇^{Ω^{∗}(E/C;F),u} = \frac{1}{2} (∇^{Ω^{∗}(E/C;F),∗} + ∇^{Ω^{∗}(E/C;F)}) .

We are now ready to construct two superconnections on the bundle Ω^{∗}(E/C; F) → C by

\begin{aligned}𝔸̂′_{0} & = d^{E} = ε_{i} ∇_{e_{i}}^{F} + ∇^{Ω^{∗}(E/C;F)} + ι_{F^{E}R} , \\ and  𝔸̂′′_{0} & = −ι_{i} ∇_{e_{i}}^{F} + ∇^{Ω^{∗}(E/C;F),∗} − ε_{F^{E}R} . \end{aligned} (4.1)

Then 𝔸̂′′ is the adjoint of 𝔸̂′ with respect to g^{Ω^{∗}(E/C;F)}.

Now suppose that h^{C}: C → ℝ is any smooth function, then the function h^{E}: E → ℝ given by

h^{E} = h^{C} ∘ π + \frac{1}{2} (|R^{+}|^{2} − |R^{−}|^{2})

is a fibre-wise Morse function. The critical set is the zero section C ⊂ E of E → C, and its index is n^{−} = dim E^{−}. We extend the superconnections 𝔸̂′_{0} and 𝔸̂′′_{0} to superconnection on the pullback Ω^{∗}(Ê/Ĉ; F) of Ω^{∗}(E/C; F) to Ĉ = C × (0, ∞) by setting

𝔸̂′ = e^{−T h^{E}} (𝔸̂′_{0} + \frac{∂}{∂T} dT ) e^{T h^{E}} ,  and  𝔸̂′′ = e^{T h^{E}} (𝔸̂′′_{0} + \frac{∂}{∂T} dT ) e^{−T h^{E}} ,

where T is the standard coordinate on (0, ∞). Then 𝔸̂′′ is the adjoint of 𝔸̂′′ with respect to the pullback of the L_{2}-metric g^{Ω^{∗}(E/C;F)}. We denote the restriction of any of these connections to the submanifold C × {T } by adding a subscript ( ·)_{T}.

Define an operator 𝕏̂ on Ω^{∗}(Ê/Ĉ; F) by

𝕏̂ = \frac{1}{2} (𝔸̂′′ − 𝔸̂′) = −\frac{1}{2}(ĉ_{j} ∂_{j} + T c_{R^{+}−R^{−}}) + \frac{1}{2} ω^{F} − \frac{1}{2} ĉ_{F^{E}R}

− T dh^{E} − \frac{|R^{s}|^{2} − |R^{u}|^{2}}{2} dT − h^{C} dT = 𝕏̂_{T} − h^{E} dT .

The square of this operator is given by

\begin{aligned}𝕏̂^{2} & = −\frac{1}{4}(Δ^{E} + T^{2} |R|^{2} − ⟨F^{E}R, F^{E}R⟩) + \frac{1}{2}∇_{F^{E}R} \\ & − \frac{T}{2} (N^{+} − N^{−} + \frac{n^{−} − n^{+}}{2} ) + \frac{1}{4}⟨F^{E}e_{j}, e_{k}⟩ĉ_{j}ĉ_{k} \\ & + \frac{1}{4}(ω^{F} )^{2} + \frac{1}{2} ĉ_{R^{+}−R^{−}} dT \\ & = 𝕏̂_{T}^{2} + \frac{1}{2} ĉ_{R^{+}−R^{−}} dT , \end{aligned} (4.2)

where Δ^{E} denotes the fibre-wise Laplacian on E, and N^{+}, N^{−} are number operators that act on Λ^{p}E^{+} ⊗ Λ^{q}E^{−} ⊂ Λ^{∗}E by p and q respectively.

The model operator 𝕏̂ is invariant under an ℝ_{+}^{∗}-action that we now define. Let o(E^{−}) = λ^{max}Ê^{−} → Ĉ be the orientation bundle of Ê^{−} → Ĉ. Let N^{dT} denote the number operator on Λ^{∗}T (0, ∞) = ℝ ⊕ ℝ dT.

4.3. Definition. For λ > 0, set

(1)  r_{λ}(p̂) = (p, T/λ)  for p̂ = (p, T) ∈ Ĉ,
(2)  r_{λ}^{o(E^{−})}(o) = o  ∈ o(E^{−})_{r_{λ}p̂} for o ∈ o(E^{−})p̂,
(3)  r_{λ}^{F} (f) = f  ∈ F_{r_{λ}p̂}  for f ∈ Fp̂,
(4)  r_{λ}^{Λ^{∗}T Ĉ}(α) = λ^{\frac{N_{C}}{2}} α ∈ Λ^{∗}T_{r_{λ}p̂}Ĉ  for α ∈ Λ^{∗}Tp̂Ĉ,
(5)  (r_{λ}^{Ω^{∗}(Ê/Ĉ)}(ω))(r_{λ}p̂, e) = λ^{− \frac{n}{4}} ω(p̂, e/\sqrt{λ})  for ω ∈ Ω^{∗}(Ê/Ĉ)p̂,

where we agree that the rescaling of the coordinates T and e affects dT, but not the vertical forms. For ω ∈ Ω^{∗}(Ê; F ) = Ω^{∗}(Ĉ; Ω^{∗}(Ê/Ĉ; F)), set

(6)  (r_{λ}^{Ω^{∗}(Ê;F)}ω)(p̂, e) = (r_{λ}^{Λ^{∗}T Ĉ⊗Ω^{∗}(Ê/Ĉ;F)} ω_{r_{λ}^{−1}p̂})(e) = λ\frac{N_{C}}{2} − \frac{n}{4} ω(p, λT, e/\sqrt{λ}) .

Note that r_{λ}^{o(E^{−})}, r_{λ}^{F}, and r_{λ}^{Ω^{∗}(Ê/Ĉ)} are isometries with respect to the pull-backs to Ĉ of the bundle metrics g^{o(E^{−})}, g^{F} and g^{Ω^{∗}(E/C;F)}. When no confusion can arise, we will drop the superscript and write simply r_{λ}.

4.4. Proposition. The operator 𝕏̂^{2} satisfies

𝕏̂^{2} = λ r_{λ} 𝕏̂^{2} r_{λ}^{−1} .

Proof. This is clear from (4.2) and Definition 4.3. Note that dT is multiplied by λ^{\frac{3}{2}} by Definition 4.3 (1) and (4). □

Let

𝕏̂_{1}^{[0]} = −\frac{1}{2} (ĉ_{j}∇_{e_{j}}^{F} + c_{R^{+}−R^{−}})

be the homogeneous part of horizontal degree 0 with respect to Ω^{∗}(Ĉ). Then

−(𝕏̂_{1}^{[0]})^{2} = \frac{1}{4}(Δ^{E} + |R|^{2}) + \frac{N^{+} − N}{2} − \frac{n^{+} − n^{−}}{4}

is a fibre-wise harmonic oscillator with spectrum \{0, \frac{1}{2}, 1, … \}. Let o(E^{−}) = Λ^{max}E^{−} denote the orientation bundle of E^{−}. Then the map

Ĵ_{1}^{[0]} : o(E^{−}) ⊗ F ⟶ ker(𝕏̂_{1}^{[0]})^{2} ⊂ Ω^{∗}(E/C; F )
with  Ĵ_{1}^{[0]}(v) = π^{− \frac{n}{4}} e^{− \frac{| · |^{2}}{2}} π^{∗}v

(4.5)

is an isometry with respect to g^{F} and g^{Ω^{∗}(E/C;F)}.

Because 𝕏̂_{1}^{2} − (𝕏̂_{1}^{[0]})^{2} is nilpotent, the operators 𝕏̂_{1}^{[0]} and 𝕏̂_{1} have the same spectrum. By Proposition 4.4, the operator 𝕏̂^{2} has spectrum Spec(𝕏̂^{2})_{C×\{T \}} = \{0, \frac{T}{2} , T, … \}. Let δ ⊂ ℂ be circle of radius 1 around 0 and consider the map

ℙ̂ = \frac{1}{2πi} ∫_{\frac{T}{4} δ} \frac{dλ}{λ − 𝕏̂^{2}}
= ℙ̂_{T} + ℚ̂_{T} dT ∈ Ω^{∗}(Ĉ; End Ω^{∗}(Ê/Ĉ; F )) .

We define another map Ĵ: Λ^{∗}T Ĉ ⊗ o(E^{−}) ⊗ F → Λ^{∗}T Ĉ ⊗ Ω^{∗}(Ê/Ĉ; F ) by

(4.6)  Ĵ = ℙ̂ ∘ ((r_{T}^{Λ^{∗}T Ĉ⊗Ω^{∗}(Ê/Ĉ;F)})^{−1} ∘ J_{1}^{[0]} ∘ r_{T}^{Λ^{∗}T Ĉ⊗o(E^{−})⊗F} ) .

Then we may rewrite Theorem 10.53 in [BG1] as follows, where for any vector bundle V → Ĉ, we set

Ω^{∗}(Ĉ/(0, ∞); V ) = \{ ω ∈ Ω^{∗}(Ĉ; V ) | ι_{\frac{∂}{∂T}} ω = 0 \} .

4.7. Theorem. The map ℙ̂ is a spectral projection that commutes with 𝔸̂′ and 𝔸̂′′, is orthogonal with respect to g^{Ω^{∗}(E/C;F)}, and we have

(1)  ℙ̂ ∈ Ω^{even}(Ĉ/(0, ∞); End^{even} Ω^{∗}(Ê/Ĉ) ⊕ End^{odd} Ω^{∗}(Ê/Ĉ) dT ) .

For any positive integer k,

(2)  ℙ̂ = \frac{1}{2πi} ∫_{(\frac{T}{4} )^{\frac{k}{2}} δ} \frac{dλ}{λ − 𝕏̂^{k}} .

Let 𝔽̂ be the range of ℙ̂, then 𝔽̂ is a finite dimensional bundle over Ĉ which is preserved by the action of r_{λ}, and 𝔽̂ = ker(𝕏̂^{k}) for k sufficiently large.

The map Ĵ maps Λ^{∗}T Ĉ ⊗ o(E^{−}) ⊗ F isomorphically to 𝔽̂, and Ĵ^{[0]}: o(E^{−}) ⊗ F → 𝔽̂^{[0]} = 𝔽̂ ∩ Ω^{∗}(Ê/Ĉ; F ) is an isometry. We may write

(3)  Ĵ(α ⊗ o ⊗ f) = α ⊗ ((\frac{T}{π} )^{\frac{n}{4}} e^{− \frac{T |R|^{2}}{2}} (1 + RÊ)(π^{∗}o)) ⊗ π^{∗}f

for α ∈ Λ^{even}T Ĉ, o ∈ o(E^{−}), and f ∈ F, where

(4)  RÊ ∈ Ω^{even}(Ĉ/(0, ∞); End^{even} Ω^{∗}(Ê/Ĉ) ⊕ End^{odd} Ω^{∗}(Ê/Ĉ) dT )

is an r_{λ}-invariant polynomial in R^{±} with coefficients depending only on F^{E^{±}} and T. □

Proof. It follows from Proposition 4.4 that ℙ̂ and 𝔽̂ are r_{λ}-invariant. The other assertions on ℙ̂ and 𝔽̂ follow as in Theorem 10.53 in [BG1]. The fact that the only term in (4.2) containing dT is

\frac{1}{2} ĉ_{R^{+}−R^{−}} dT ∈ End^{odd} Ω^{∗}(Ê/Ĉ) dT

accounts for the form of (1) compared to the corresponding statement in Theorem 10.53 in [BG1]. From [BZ2], we know that Ĵ^{[0]} takes the form

Ĵ^{[0]}(v) = (\frac{T}{π} )^{\frac{n}{4}} e^{− \frac{T |R|^{2}}{2}} π^{∗}(v) .

Then (3) follows immediately from (1), and RÊ is a polynomial by (10.191) in [BG1], which is r_{λ}-invariant since 𝔽̂ and ℙ̂ are r_{λ}-invariant, and of the form (4) by (1). □

Let Ĵ^{∗} be the adjoint of Ĵ with respect to g^{o(E^{−})⊗F} and g^{Ω^{∗}(Ê/Ĉ;F)} in the sense of Definition 2.7. Integration over the fibres of Ê^{−} → Ĉ defines a map

Î: Ω^{∗}(Ê; F ) ⟶ Ω^{∗}(Ĉ; o(E^{−}) ⊗ F )  with  Î(α) = ∫_{Ê^{−}/Ĉ} α .

Then we have the following version of [BG1], Proposition 10.58 and Theorem 10.59 for our model situation.

4.8. Proposition. There exist r_{λ}-invariant characteristic differential forms R_{J^{∗}J} and R_{IJ} ∈ Ω^{even,≥2}(Ĉ/(0, ∞)) of R^{E^{±}}, such that

(1)  Ĵ^{∗} ∘ Ĵ = 1 + R_{J^{∗}J}

(2)  and  Î∘ e^{T h^{E}} ∘ Ĵ = e^{T h^{C}} (\frac{T}{π} )^{\frac{n^{+}−n^{−}}{4}} (1 + R_{IJ}) .

In particular, R_{J^{∗}J}, R_{IJ} are polynomials in \frac{1}{T} without constant term.

Note in particular that the right hand sides of both formulas are closed differential forms that do not contain any multiple of dT.

Proof. These claims follow by elementary calculations using r_{λ}-invariance, see also [BG1], Section 10.15. Since RÊ is r_{λ}-invariant,

RÊ|_{(p,T,e)} = T^{− \frac{N_{C}}{2}} RÊ|_{(p,1,\sqrt{T} e)} .

We will use the symbol | · |^{2} in the sense of (2.2). Let v ∈ (o(E^{−}) ⊗ F)_{p,T}, then

(Ĵ^{∗} ∘ Ĵ)(v) = ∫_{E_{p}}(\frac{T}{π} )^{\frac{n}{2}} e^{−T |e|^{2}} |(1 + RÊ|_{(p,T,e)})(π^{∗}v)|^{2} d Vol^{E}(e)

= T^{− \frac{N_{C}}{2}} ∫_{E_{p}} π^{− \frac{n}{2}} e^{−|e|^{2}} |(1 + RÊ|_{(p,1,e)})(π^{∗}v)|^{2} d Vol^{E}(e) ,

which proves (1). Note that the part of RÊ that contains dT does not contribute because it is odd as an endomorphism of Ω^{∗}(Ê/Ĉ).

Similarly, we get (2) because

(Î∘ e^{T h^{E}} ∘ Ĵ) = e^{T h^{C}} (\frac{T}{π} )^{\frac{n}{4}}

∫_{Ê^{−}/Ĉ|_{p}} e^{T (h^{E}−h^{C})(e^{−})} e^{− \frac{T|e^{−}|^{2}}{2}} (1 + RÊ|_{(p,T,e^{−})})(π^{∗}v)

= e^{T h^{C}} (\frac{T}{π} )^{\frac{n^{+}−n^{−}}{4}} π^{− \frac{n}{2}}

T^{− \frac{N_{C}}{2}} ∫_{Ê^{−}/Ĉ|_{p}} e^{−|e^{−}|^{2}} (1 + RÊ|_{(p,1,e^{−})})(π^{∗}v) .

Note that if v = o ⊗ f ∈ o(E^{−}) ⊗ F, then π^{∗}o is a volume form on the fibres of Ê^{−}, so only the component of RÊ in Ω^{∗}(Ĉ; {id_{Ω^{∗}(Ê/Ĉ;F)}}) contributes, and this component again does not contain dT since it is even.

Then both R_{J^{∗}J} and R_{IJ} are r_{λ}-invariant, even forms of degree ≥ 2, so they are polynomials in \frac{1}{T} without constant terms. □

4.b. Small eigenvalues and spectral projections. Let

\overline{M} = M × (0, ∞) × ℝ ⟶ \overline{B} = B × (0, ∞) × ℝ

be the pull-back of M → B to \overline{B}. By an abuse of notation, let T X → \overline{M} and F → \overline{M} denote the pull-backs of the vector bundles T X → M and F → M to \overline{M}. Let g^{T X} and g^{F} denote the pull-back metrics on these bundles, then these metrics induce a fibre-wise L_{2}-metric g^{T X,F} on the infinite dimensional vector bundle Ω^{∗}(M/B; F) → B. For t ∈ (0, ∞), T ∈ ℝ consider the metrics

g_{t}^{T X} = \frac{1}{t} g^{T X}  and  g_{T}^{F} = e^{−2T h} g^{F} .

These metrics induce a fibre-wise L_{2}-metric ḡ^{T X,F} on Ω^{∗}(\overline{M}/\overline{B}; F ) such that

ḡ^{T X,F} = g^{T X,F} (t^{N^{X}− \frac{n}{2}} e^{−2T h} · , · ) .

Let 𝔸′ = d^{M} denote the total derivative on Ω^{∗}(M; F). We extend 𝔸′ to a superconnection \overline{𝔸}′ on Ω^{∗}(\overline{M}; F ) by

\overline{𝔸}′ = 𝔸′ + \frac{∂}{∂t} dt + \frac{∂}{∂T} dT .

Let \overline{𝔸}′′ be the adjoint superconnection of \overline{𝔸}′ with respect to ḡ^{T X,F}. Then we have

(4.9)  \overline{𝔸}′′ = t^{−N^{X}} e^{2T h} (𝔸′′ + \frac{∂}{∂t} dt + \frac{∂}{∂T} dT ) e^{−2T h} t^{N^{X}}

Then we define a self-adjoint superconnection \overline{𝔸} on Ω^{∗}(M/B; F) and a skew-adjoint operator \overline{𝕏} ∈ Ω^{∗}(B; End Ω(M/B); F)—both with respect to ḡ^{T X,F} —by

\overline{𝔸} = \frac{1}{2} (\overline{𝔸}′′ + \overline{𝔸}′)  and  \overline{𝕏} = \frac{1}{2} (\overline{𝔸}′′ − \overline{𝔸}′) .

We will denote the restrictions of \overline{𝔸}′, \overline{𝔸}′′ and ḡ^{T X,F} to M × \{t\} × \{T \} by 𝔸′_{t,T}, 𝔸′′_{t,T} and g_{t,T}^{T X,F}, respectively. We have

\overline{𝕏} = 𝕏_{t,T} + N^{X} \frac{dt}{2t} − h dT .

If 𝔸_{t,T}^{[0]} and 𝕏_{t,T}^{[0]} ∈ Ω^{0}(B; End Ω^{∗}(M/B; F)) denote the homogeneous part of horizontal degree 0,

then \overline{𝔸}^{2}|_{t,T} − (𝔸_{t,T}^{[0]} )^{2} and \overline{𝕏}^{2}|_{t,T} − (𝔸_{t,T}^{[0]} )^{2} are elements of Ω^{>0}(B; End Ω^{∗}(M/B; F)) and thus nilpotent in Ω^{∗}(B; End Ω^{∗}(M/B; F)). This implies that

Spec \overline{𝔸}^{2}|_{t,T} = Spec 𝔸_{t,T}^{2} = Spec(𝔸_{t,T}^{[0]} )^{2}
= − Spec \overline{𝕏}^{2}|_{t,T} = − Spec 𝕏_{t,T}^{2} = − Spec(𝕏_{t,T}^{[0]} )^{2} .

Recall that a vector bundle V → B was defined in Definition 1.1 by

V^{j} = p̂_{∗}^{j}F |_{C^{j}} ⊗ o(N^{u}) ,
and  V = p̂_{∗}F |_{C} ⊗ o(N^{u}) = ⨁_{j} V^{j} ,

and that we have constructed a map I: Ω^{∗}(M; F) → Ω^{∗}(B; V) in Definition 1.65, which is an Ω^{∗}(B)-linear combination of maps that arise by integration over the fibres of certain subbundles of M → B. Let I^{[0]}: Ω^{∗}(M/B; F) → V denote the homogeneous part of horizontal degree 0. Then we have the following result of [BZ1], Theorems 7.8 and 7.9, and [BZ2], Theorem 6.11, cf. [BG1], Theorems 10.2 and 10.59.

4.10. Theorem. There exists T_{0} ≥ 0 such that for all T ≥ T_{0},

Spec(𝔸_{t,T}^{2} ) = − Spec(𝕏_{t,T}^{2} ) ⊂ [0, \frac{t}{4}] ∪ (4t, ∞) ,

and the direct sums of eigenspaces associated to eigenvalues in [0, \frac{t}{4}] form a vector bundle W → B × (0, ∞) × [T_{0}, ∞) ⊂ \overline{B}, such that

I^{[0]}|_{W}: W ⟶ V

is a linear isomorphism.

Proof. We can copy the proof of [BZ2], Theorem 6.11. However, since I^{[0]} is not simply integration over the unstable cell of the gradient flow, we have to use Theorem 1.66. □

Let S^{1}(t) ⊂ ℂ denote a circle around the origin of radius t, and note that if \frac{t}{4} < t_{0}, t_{1} < 4t, then

\frac{1}{2πi} ∫_{S^{1}(t_{0})} \frac{dλ}{λ − \overline{𝔸}^{2}} = \frac{1}{2πi} ∫_{S^{1}(t_{1})} \frac{dλ}{λ − \overline{𝔸}^{2}} .

This allows us to patch spectral projectors that are defined on B × (\frac{t_{0}}{4} , 4t_{0}) × [T_{0}, ∞) together as in the following definition, cf. Definition 10.33 in [BG1].

4.11. Definition. Define an endomorphism \overline{ℙ} of the bundle Λ^{∗}T (\overline{B}) ⊗ Ω^{∗}(\overline{M}/\overline{B}; F ) for T ≥ T_{0} in a neighbourhood of B × {(t_{0}, T)} in B × (0, ∞) × [T_{0}, ∞) by

\overline{ℙ} = \frac{1}{2πi} ∫_{S^{1}(t_{0})} \frac{dλ}{λ − \overline{𝔸}^{2}} .

We recall parts of Theorem 10.9 from [BG1].

4.12. Theorem. For T ≥ T_{0}, the operator \overline{ℙ} ∈ Ω^{∗}(\overline{B}; End Ω^{∗}(M/B; F)) is an even self-adjoint projector with range a finite dimensional vector bundle \overline{𝕎} → B ×(0, ∞)×[T_{0}, ∞), which commutes with the action of Λ^{∗}T (\overline{B}) and with \overline{𝔸}′ and \overline{𝔸}′′. For any integer k > 0 and t ∈ (\frac{t_{0}}{4} , 4t_{0}), we have

\overline{ℙ} = \frac{1}{2πi} ∫_{S^{1}(t \frac{k}{2} ; 0 )} \frac{dλ}{λ − \overline{𝕏}^{k}} .

Let g_{T}^{𝕎} denote the restriction of the L_{2}-metric g_{1,T}^{T X,F} on Ω^{∗}(\overline{M}/\overline{B}; F ) to 𝕎_{T} = \overline{𝕎}|_{B×\{(1,T)\}}, which is a generalized metric on 𝕎 in the sense of Definition 2.65. Similarly, let g_{T}^{H} denote the L_{2}-metric on H induced by g_{1,T}^{T X,T}.

4.13. Theorem. There exists a constant δ ∈ (0, 1) such that for T ≥ T_{0},

(1)  ∫_{1}^{∞}(str(N^{X} f′(𝕏_{t,T} )) − str(N^{X} ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} ))) \frac{dt}{2t} = O(T^{−δ}) ,

and for all T_{1} ≥ T_{0},

(2) f̃(∇^{H}, g^{H}, g_{T_{1}}^{H}) − (2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str(N^{X} f′(𝕏_{t,T_{1}})) − χ′(H) f′(0)) \frac{dt}{2t}

= f̃(∇^{H}, g^{H}, g_{T_{0}}^{H}) − (2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str(N^{X} ℙ_{t,T_{0}} f′(𝕏_{t,T_{0}} ℙ_{t,T_{0}})) − χ′(H) f′(0)) \frac{dt}{2t}

+ f̃(𝔸′_{1, ·} , g_{T_{0}}^{𝕎}, g_{T_{1}}^{𝕎}) + O(T_{1}^{−δ}) modulo an exact form on B. Proof. Equation (1) is a consequence of Theorem 4.10, cf. [BG1], Theorem 10.5.

We recall that

\overline{𝕏}_{T} = 𝕏_{t,T} + (g_{t,T}^{T X,F} )^{−1} \frac{∂g_{t,T}^{T X,F}}{∂t} dt = 𝕏_{t,T} + \frac{N^{X}}{2t} dt .

Let \overline{ℙ}_{T} = ℙ_{t,T} + ℙ′_{t,T} dt and note that \overline{ℙ}_{T}^{2} = \overline{ℙ}_{T} implies

ℙ′_{t,T} = ℙ_{t,T} ℙ′_{t,T} + ℙ′_{t,T} ℙ_{t,T} ,

so ℙ′_{t,T} interchanges the images of ℙ_{t,T} and id −ℙ_{t,T}. This implies that

str(\overline{ℙ}_{T} f(\overline{𝕏}_{T} \overline{ℙ}_{T} )) = str(ℙ_{t,T} f(𝕏_{t,T} ℙ_{t,T} ))

+ str((−ℙ′_{t,T} 𝕏_{t,T} ℙ_{t,T} + ℙ_{t,T} \frac{N^{X}}{2t} ℙ_{t,T} + ℙ_{t,T} 𝕏_{t,T} ℙ′_{t,T} )

f′(ℙ_{t,T} 𝕏_{t,T} ℙ_{t,T} )) dt

= str(ℙ_{t,T} f(𝕏_{t,T} ℙ_{t,T} )) + str(N^{X} ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} )) \frac{dt}{2t} .

(4.14)

A similar argument with fixed t and variable T shows that

(4.15)  str(\overline{ℙ}_{t} f(\overline{𝕏}_{t} \overline{ℙ}_{t})) = str(ℙ_{t,T} f(𝕏_{t,T} ℙ_{t,T} )) − str(h ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} )) dT .

Now consider the closed differential form on B × (0, ∞) × {T }, which is given by

str(f(\overline{ℙ} \overline{𝕏} \overline{ℙ})) − χ′(H) f′(0) \frac{dt}{t}

= str(ℙ_{t,T} f(𝕏_{t,T} ℙ_{t,T} )) + (str(N^{X} ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} )) − χ′(H) f′(0)) \frac{dt}{2t}

− str(h ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} )) dT + R dt dT ,

with R ∈ Ω^{∗}(B × (0, ∞) × [T_{0}, ∞)).

Let ḡ^{H} be the metric on the pull-back of H to B × [T_{0}, ∞), let \overline{∇}^{H,∗} denote the adjoint of the pull-back \overline{∇}^{H} of \overline{∇}^{H} with respect to ḡ^{H}, and let

\overline{ω}^{H} = \overline{∇}^{H,∗} − \overline{∇}^{H} .

Then we know that as t → ∞,

str(f(\overline{ℙ} \overline{𝕏} \overline{ℙ})) − χ′(H) f′(0) \frac{dt}{t} = str(f(\overline{ω}^{H}/2)) + O(t^{− \frac{1}{2}} ) + O(t^{− \frac{3}{2}} ) dt ,

cf. [BL], Theorems 3.16 and 3.21. Now, equation (2) follows easily. □

4.c. The limit t → ∞ and the class S_{f}(A′, g^{V}). We identify the bundle 𝕎 → B with Λ^{∗}T B⊗V. This allows us to compare the term

−(2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str(N^{X} ℙ_{t,T_{0}} f′(𝕏_{t,T_{0}} ℙ_{t,T_{0}})) − χ′(H) f′(0)) \frac{dt}{2t}

of Theorem 4.13 (2) with a class S_{f}(A′, g^{V} , g^{H}) of the bundle V → B.

Let I: Ω^{∗}(M; F) → Ω^{∗}(B; V) be the integration map of Definition 1.65, then we have an obvious extension

\overline{I}: Ω^{∗}(\overline{M}; F ) ⟶ Ω^{∗}(\overline{B}; V ) ,

for which the analogue of Theorem 1.66 still holds. By Theorem 4.10, I^{[0]} provides us with an isomorphism of the bundle W_{t,T} → B of the small eigenspaces with the bundle V of Definition 1.1. By Ω^{∗}(B)-linearity, this implies that

(4.16)  \overline{𝕀} = \overline{I}|\overline{𝕎}: \overline{𝕎} ⟶ Λ^{∗}T \overline{B} ⊗ V

is an isomorphism as well. Define

(4.17)  ḡ_{t,T}^{V} = (\overline{𝕀}^{−1})^{∗}(g\overline{𝕎}) = g^{T X,F} (\overline{𝕀}^{−1} · , \overline{𝕀}^{−1} · )

Let N^{M} denote the total number operator on Ω^{∗}(M; F). Recall that we used the subscript ( ·)_{T} as an abbreviation for ( ·)_{1,T}. We collect a few properties of the maps \overline{ℙ} and \overline{𝕀} and of ḡ^{V}.

4.18. Proposition. For T ≥ T_{0}, the map ℙ_{t,T} satisfies

(1)  ℙ_{t,T} = t^{− \frac{N^{M}}{2}} ℙ_{T} t^{\frac{N^{M}}{2}} .

Moreover, ℙ_{t,T} is even in Ω^{∗}(B; End Ω^{∗}(M/B; F)) and polynomial in \frac{1}{t}, and ℙ_{t,T}^{[0]} = ℙ_{T}^{[0]} independent of t.

The inverse of 𝕀_{t,T} is even in Ω^{∗}(B; Hom(V, Ω^{∗}(M/B; F))) and polynomial in \frac{1}{t} with (𝕀_{t,T}^{−1} )^{[0]} = (I^{[0]}|_{W_{T}})^{−1} independent of t, and satisfies

(2)  𝕀_{t,T}^{−1} = t^{− \frac{N^{M}}{2}} 𝕀_{T}^{−1} t^{\frac{N^{B}+N^{V}}{2}} .

The map g_{t,T}^{V} is a generalized metric on V that depends polynomially on \frac{1}{t}. If we set g_{T}^{V} = (g_{T}^{V})^{[0]}, then there exists an even g′_{T} ∈ Ω^{∗}(B; End V) such that (2.70) holds for each T ≥ T_{0}.

Let A′′_{t,T} denote the adjoint of A′ with respect to g_{t,T}^{V} and set X_{t,T} = \frac{1}{2} (A′′_{t,T} − A′), then

(3)  𝕀_{t,T} 𝕏_{t,T} = X_{t,T} 𝕀_{t,T} .

Proof. Since 𝔸′ is of total degree 1 and 𝔸′′ is of total degree −1, we have

t^{\frac{1−N^{M}}{2}} 𝔸′ t^{\frac{N^{M}}{2}} = 𝔸′  and  t^{\frac{1−N^{M}}{2}} 𝔸′′_{T} t^{\frac{N^{M}}{2}} = 𝔸′′_{t,T} .

By Definition 4.11, this implies (1).

Next, we note that 𝔸′_{t,T} raises the total degree by 1, while 𝔸_{t,T}^{′′[k]} raises it by 2k − 1. This implies that

𝔸_{t,T}^{2} = 𝔸′_{t,T} 𝔸′′_{t,T} + 𝔸′′_{t,T} 𝔸′_{t,T}

does not lower the total degree. Moreover

(𝔸_{t,T}^{[0]} )^{2} = 𝔸_{t,T}^{′[0]}𝔸_{t,T}^{′′[0]} + 𝔸_{t,T}^{′′[0]}𝔸_{t,T}^{′[0]} = t (𝔸_{T}^{[0]})^{2}

preserves the total degree, and 𝔸_{t,T}^{2} − (𝔸_{t,T}^{[0]} )^{2} is nilpotent. In particular,

\begin{aligned}ℙ_{t,T} & = \frac{1}{2πi} ∫_{S^{1}(t)} \frac{1}{λ − (𝔸_{t,T}^{[0]} )^{2}} ∑_{k≥0}((𝔸_{t,T}^{2} − (𝔸_{t,T}^{[0]} )^{2}) \frac{1}{λ − (𝔸_{t,T}^{[0]} )^{2}} )^{k} dλ \\ & = \frac{1}{2πi} ∫_{S^{1}(1)} \frac{1}{λ − (𝔸_{T}^{[0]})^{2}} ∑_{k≥0}((\frac{1}{t} 𝔸_{t,T}^{2} − (𝔸_{T}^{[0]})^{2}) \frac{1}{λ − (𝔸_{T}^{[0]})^{2}} )^{k} dλ , \end{aligned}

where the summation index k is bounded by dim B. Since \frac{1}{t} 𝔸_{t,T}^{2} − (𝔸_{T}^{[0]})^{2} is even, it is polynomial in \frac{1}{t} by (1), which proves the remaining statements on ℙ_{t,T}.

Taking the inverse of I^{[0]}|_{W_{T}}, and noting that 𝕀_{t,T} = I ℙ_{t,T} and (I ℙ_{t,T} (I^{[0]}|_{W_{T}})^{−1})^{[0]} = id, we may write

𝕀_{t,T}^{−1} = ℙ_{t,T} (I^{[0]}|_{W_{T}} )^{−1} (I ℙ_{t,T} (I^{[0]}|_{W_{T}} )^{−1})^{−1} .

This explains why 𝕀_{t,T} is even and polynomial in \frac{1}{t}. Claim (2) follows from the analogous statement for ℙ_{t,T}, because I^{[0]} preserves the total degree.

Since (I^{[0]}|_{W})^{−1}: V → Ω^{(}M/B; F) is clearly injective and preserves the total degree, we know that (g_{t,T}^{V})^{[0]} = ((I^{[0]}|_{W})^{−1})^{∗}g_{t,T}^{T X,F} is a Hermitian metric on V that respects the ℤ-grading. Moreover, g_{t,T}^{V} is even because 𝕀_{t,T}^{−1} is. Finally, Definition 2.65 (3), (4) are clearly satisfied by (2.2), so g_{t,T}^{V} is a generalised metric. It is also clear that g_{t,T}^{V} is polynomial in \frac{1}{t}.

In order to check (2.70), note that

g_{t,T}^{V} = t^{− \frac{n}{2}} g_{T}^{T X,F} (t^{\frac{N^{X}}{2}} 𝕀_{t,T}^{−1} · , t^{\frac{N^{X}}{2}} 𝕀_{t,T}^{−1} · )
= t^{−N^{B}− \frac{n}{2}} g_{T}^{T X,F} (𝕀_{T}^{−1} t^{\frac{N^{B}+N^{V}}{2}} · , 𝕀_{T}^{−1} t^{\frac{N^{B}+N^{V}}{2}} · )
= t^{−N^{B}− \frac{n}{2}} g_{T}^{V} (t^{\frac{N^{B}+N^{V}}{2}} · , t^{\frac{N^{B}+N^{V}}{2}} · ) .

Since 𝕀_{T}^{−1} is Ω^{∗}(B)-linear, it cannot lower horizontal degrees, which proves that t^{\frac{n−N^{V}}{2}} g_{t,T}^{V} t^{− \frac{N^{V}}{2}} is a polynomial in \frac{1}{\sqrt{t}} with leading term g_{T}^{V}. This implies the first equation of (2.70). It also implies that

t^{\frac{N^{V}}{2}} (g_{t,T}^{V} )^{−1} \frac{∂g_{t,T}^{V}}{∂t} t^{− \frac{N^{V}}{2}} = (t^{\frac{n−N^{V}}{2}} g_{t,T}^{V} t^{− \frac{N^{V}}{2}} )^{−1} \frac{∂(t^{\frac{n−N^{V}}{2}} g_{t,T}^{V} t^{− \frac{N^{V}}{2}} )}{∂t}

+ \frac{N^{V}}{2t} + (t^{\frac{n−N^{V}}{2}} g_{t,T}^{V} t^{− \frac{N^{V}}{2}} )^{−1} \frac{N^{V} − n}{2t} (t^{\frac{n−N^{V}}{2}} g_{t,T}^{V} t^{− \frac{N^{V}}{2}} )

MORSE THEORY AND HIGHER TORSION INVARIANTS I 73

is a polynomial in \frac{1}{\sqrt{t}} with leading term \frac{N^{V}}{t} − \frac{n}{2t}, which implies the second equation of (2.70).

Finally, the adjoint of A′ with respect to g_{t,T}^{V} can be written as

A′′_{t,T} = (\overline{𝕀_{t,T}^{−1}}^{∗} g_{t,T}^{T X,F} 𝕀_{t,T}^{−1} )^{−1} \overline{A′}^{∗} (\overline{𝕀_{t,T}^{−1}}^{∗} g_{t,T}^{T X,F} 𝕀_{t,T}^{−1} )
= 𝕀_{t,T} (g_{t,T}^{T X,F} )^{−1} \overline{𝔸′}^{∗} g_{t,T}^{T X,F} 𝕀_{t,T}^{−1}
= 𝕀_{t,T} 𝔸′′_{t,T} 𝕀_{t,T}^{−1} ,

which implies (3). □

Let \overline{A}′, \overline{A}′′_{T} and \overline{X}_{T} denote the obvious extensions of A′, A′′_{t,T} and X_{t,T} to the pullback of V to B × (0, ∞) × {T } . Note that

str(f′(ℙ_{t,T} 𝕏_{t,T} ℙ_{t,T} )) = χ(V) f′(0)

because f′ is even and ℙ_{t,T} 𝕏_{t,T} ℙ_{t,T} is odd. With this observed, we can now calculate

− (2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str(N^{X} ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} )) − χ′(H) f′(0)) \frac{dt}{2t}

= −(2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str((N^{X} − \frac{n}{2}) ℙ_{t,T} f′(𝕏_{t,T} ℙ_{t,T} ))

− (χ′(H) − \frac{n}{2} χ(V)) f′(0)) \frac{dt}{2t}

= −(2πi)^{− \frac{N^{B}}{2}} ∫_{(B×(1,∞)×\{T \})/B×\{T \}}(str(f(\overline{ℙ}_{T} \overline{𝕏}_{T} \overline{ℙ}_{T} ))

− (χ′(H) − \frac{n}{2} χ(V)) f′(0) \frac{dt}{2t})

= −(2πi)^{− \frac{N^{B}}{2}} ∫_{(B×(1,∞)×\{T \})/B×\{T \}}(str(f(\overline{X}_{T} ))

− (χ′(H) − \frac{n}{2} χ(V)) f′(0) \frac{dt}{2t})

= −(2πi)^{− \frac{N^{B}}{2}} ∫_{1}^{∞}(str_{V} ((g_{t,T}^{V} )^{−1} \frac{∂g_{t,T}^{V}}{∂t} f′(X_{t,T} ))

− (χ′(H) − \frac{n}{2} χ(V)) f′(0)) \frac{dt}{2t}

= S_{f}(A′, g_{T}^{V} , g_{T}^{H}) .

Using Theorem 2.76 and (4.15), we can now rewrite the equation in Theorem 4.13 (2) as (4.19) f̃(∇^{H}, g^{H}, g_{T}^{H}) + S_{f}(T^{H}M, g^{T X}, ∇^{F} , g_{T}^{F} )

\begin{matrix}= f̃(∇^{H}, g^{H}, g_{T_{0}}^{H}) + S_{f}(A′, g_{T_{0}}^{V}, g_{T_{0}}^{H}) + f̃(A′, g_{T_{0}}^{V}, g_{T}^{V} ) + O(T^{−δ}) \\ = f̃(∇^{H}, g^{H}, g_{V}^{H}) + S_{f}(A′, g^{V} ) + f̃(A′, g^{V} , g_{T}^{V} ) + O(T^{−δ}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) . \end{matrix}

4.d. The limit T → ∞ and the class T_{f}(A′, g^{V} , h^{V}). We state some estimates for the generalised metric g_{T}^{V} as T → ∞ that were established in [BG1], and we use these estimates to identify the right hand side of (4.19) with the torsion form T_{f}(A′, g^{V} , h^{V}), up to divergences and correction terms.

Recall that we have defined an Ω^{∗}(C)-linear map Ĵ: Ω^{∗}(C; o(E^{−}) ⊗ F) → Ω^{∗}(Ê; F ) in (4.6) that maps Λ^{∗}T Ĉ ⊗ o(E^{−}) ⊗ F isomorphically onto the bundle 𝔽 → Ĉ of the fibre-wise generalised eigenspaces of the model operator 𝕏̂ for the generalised eigenvalue 0. We will apply this definition to E = T X|_{C} = T^{s}X ⊕ T^{u}X. Restriction to T ≥ T_{0} and push-down to B × [T_{0}, ∞) gives a map

(4.20)  \overline{J}: Ω^{∗}(B × [T_{0}, ∞), V ) ⟶ Ω^{∗}(T X|_{C} × [T_{0}, ∞), π^{∗}F ) .

Recall that by the simplifying assumptions of Proposition 1.10, the fibre-wise exponential map at C ⊂ M maps the 2ε-disc bundle of T X|_{C} fibre-wise isometrically onto a tubular neighbourhood of C in M. Let us fix a C^{∞} cut-off function φ: ℝ → [0, 1] with

φ(r) = \{ \begin{matrix}1 & if r ≤ ε, and \\ 0 & if r ≥ 2ε. \end{matrix}

Then we may regard φ(|R|) \overline{J} as a map

(4.21)  φ(|R|) \overline{J}: Ω^{∗}(B × [T_{0}, ∞), V ) ⟶ Ω^{∗}(M × [T_{0}, ∞), F ) .

This allows to define

(4.22)  e_{T}: Λ^{∗}T B ⊗ V ⟶ 𝕎_{T}  by  e_{T} = ℙ_{T} (e^{T h} φ(|R|) J_{T} ) .

We recall a fundamental result from [BG1].

4.23. Theorem ([BG1], Theorem 10.56, cf. [BZ2], Theorem 6.7). There exists a constant c > 0 such that as T → ∞ for any k ≥ 0 and any v ∈ Ω^{∗}(B; V),

(e_{T} − e^{T h} φ(|R|) J_{T} )(v) = e^{T h} O(e^{−cT} |v|)

uniformly in C^{k}-topology over compact subsets of B. In particular, e_{T} is an Ω^{∗}(B)-linear vector bundle isomorphism for sufficiently large T. □

This theorem was applied to prove an estimate for g_{T}^{V} in [BG1], Section 10.16, which we now recall.

4.24. Theorem. There exists an r_{λ}-invariant and g^{V} -self-adjoint R_{T} ∈ Ω^{even,≥2}(B; End^{even} V) commuting with ∇^{V}, h^{V} and N^{V}, which is a polynomial in \frac{1}{T} without constant term, and another g^{V} -self-adjoint R′_{T} ∈ Ω^{∗}(B; End V) with |R′_{T} | = O(e^{−cT}) uniformly in C^{k}-topology, such that

g_{T}^{V} = ((id_{V} +R_{T} + R′_{T} ) e−T h^{V} (\frac{T}{π} )\frac{N^{V}}{2} − \frac{n}{4} )^{∗}g^{V} .

Proof. By definition of g_{T}^{V},

\begin{aligned}g_{T}^{V} & = g_{T}^{T X,F} (𝕀_{T}^{−1} · , 𝕀_{T}^{−1} · ) \\ & = g_{T}^{T X,F} (e_{T} ∘ (I ∘ e_{T})^{−1} · , e_{T} ∘ (I ∘ e_{T})^{−1} · ) . \end{aligned} (4.25)

We will prove Theorem 4.24 by giving suitable approximations for I ∘ e_{T} and e_{T}^{∗} g_{T}^{T X,F} using Theorem 1.66 and Proposition 4.8.

Let c ∈ C_{β} be a fibre-wise critical point of h, and let v ∈ F_{c} ⊗ o(T_{c}^{u}X). Then

(4.26) \begin{aligned}(I ∘ e_{T} )_{α}(v) & = e^{T h_{α}} I(e^{−T h_{α}}e_{T} (v)) \\ & = e^{T h_{α}} I(e^{T (h−h_{α})} φ(|R|) J(v)) + e^{T h_{α}} I(e^{T (h−h_{α})} O(e^{−cT} |v|)) . \end{aligned}

Clearly,

e^{T (h−h_{α})}|_{h^{−1}(−∞,h_{α}−ε^{2})} = O(e^{−ε^{2}T} ) .

Together with Theorem 1.66 (2), this allows us to replace M near C by the total space of the vertical tangent bundle E = T X|_{C} → C, and thus to approximate e^{−T h_{α}} I_{T} ∘ e_{T} ∈ by Î∘ e^{T h^{E}} ∘ Ĵ considered in Proposition 4.8 (2). Here h^{E} replaces h − h_{α}, and we set h^{C} = 0. In particular, if α ≠ β, then (4.26) gives

(I_{T} ∘ e_{T} )_{α}(v) = e^{T h_{α}} O(e^{−cT} |v|)

uniformly in C^{k}-topology, possibly with a new constant c.

If we assume that β = α, then Proposition 4.8 (2) and (4.26) imply that

(I_{T} ∘ e_{T} )_{α}(v) = e^{T h_{α}} (Î∘ e^{T (h−h_{α})} ∘ Ĵ(v)) + O(e^{−cT} |v|))

= e^{T h_{α}} ((\frac{T}{π} )^{\frac{n^{+}−n^{−}}{4}} (1 + R_{IJ,T} ) v + O(e^{−cT} |v|)) ,

where R_{IJ} is an r_{λ}-invariant characteristic differential form of ∇^{T X} on C. We can summarize our results so far by

(4.27)  I_{T} ∘ e_{T} = eT h^{V} (\frac{T}{π} )\frac{n}{4} − \frac{N^{V}}{2} (1 + p̂_{∗}R_{IJ,T} + O(e^{−cT} ))

with O(e^{−cT}) ∈ Ω^{∗}(B; End V) for some c > 0, cf. [BG1], Theorem 10.59, and [BZ2], Theorem 6.11. Here, p̂_{∗}R_{IJ,T} ∈ Ω^{≥2}(B, End V) clearly commutes with ∇^{V}, h^{V}, and N^{V}. By Proposition 4.8, R_{IJ,T} is a polynomial in \frac{1}{T} without constant term.

We regard the generalised metric e_{T}^{∗} g_{T}^{T X,F} on V. Taking adjoints with respect to g^{T X,F} and g^{V}, Theorem 4.23 implies

\begin{aligned}e_{T}^{∗} g_{T}^{T X,F} & = g^{T X,F} (e^{−T h} e_{T} · , e^{−T h} e_{T} · ) \\ & = g^{V} ( · , (J_{T}^{∗} φ(R) + O(e^{−cT} )) (φ(R) J_{T} + O(e^{−cT} )) · ) . \end{aligned}

The presence of the cut-off function φ(R) allows us again to replace M by p̂_{∗}(T X|_{C}). By Theorem 4.7 and Proposition 4.8, we find

(4.28) e_{T}^{∗} g_{T}^{T X,F} = g^{V} ( · , (p̂_{∗}(Ĵ_{T}^{∗} Ĵ_{T} ) + O(e^{−cT} )) · ) = g^{V} ( · , (id_{V} +p̂_{∗}R_{J^{∗}J,T} + O(e^{−cT} )) · )

uniformly in C^{k}-topology, cf [BG1], Proposition 10.58. Moreover, p̂_{∗}R_{J^{∗}J,T} ∈ Ω^{≥2}(B, End V) again commutes with ∇^{V}, h^{V}, and N^{V}, and p̂_{∗}R_{J^{∗}J,T} is a polynomial in \frac{1}{T} without constant term.

Let (id_{V} +p̂_{∗}R_{J^{∗}J,T} + O(e^{−cT}))^{\frac{1}{2}} denote the square-root of (id_{V} +p̂_{∗}R_{J^{∗}J,T} + O(e^{−cT})), which is self adjoint in the sense of Definition 2.7. Combining (4.25), (4.27), and (4.28), we finally find that

g_{T}^{V} = ((id_{V} +p̂_{∗}R_{J^{∗}J,T} + O(e^{−cT} ))^{\frac{1}{2}}

(1 + p̂_{∗}R_{IJ,T} + O(e^{−cT} ))^{−1} (\frac{T}{π} )\frac{N^{V}}{2} − \frac{n}{4} e−T h^{V} )^{∗}g^{V}

= ((id_{V} +R_{T} + O(e^{−cT} )) (\frac{T}{π} )\frac{N^{V}}{2} − \frac{n}{4} e−T h^{V} )^{∗}g^{V} . □

We can now compare the generalised metric g_{T}^{V} with the deformed metric e^{−2T h} g^{V}.

4.29. Theorem. As T → ∞,

f̃(A′, e^{−2T h} g^{V} , g_{T}^{V} ) = (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) (log T − log π) f′(0) + O(\frac{1}{T} ) .

Proof. We set

E_{s,T} = (id_{V} +s (R_{T} + R′_{T} )) e−T h^{V} (\frac{T}{π} )s (\frac{N^{V}}{2} − \frac{n}{4}) ∈ Ω^{∗}(B; End V ) ,

and construct a family of metrics

g̃_{s,T}^{V} = E_{s,T}^{∗} g^{V} ,

so g^{V} = g̃_{0,T}^{V} and g_{T}^{V} = g̃_{1,T}^{V}. Let Ã′′_{s,T} denote the adjoint of A′ with respect to g̃_{s,T}^{V}, and let X̃_{s,T} = \frac{1}{2} (Ã′′_{s,T} − A′), then we have

(4.30)  f̃(A′, e^{−2T h} g^{V} , g_{T}^{V} ) = (2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{1} str_{V} (\frac{1}{2}(g̃_{s,T}^{V} )^{−1} \frac{∂g̃_{s,T}^{V}}{∂s} f′(X̃_{s,T} )) ds .

Using Theorem 4.24, we can estimate the various terms appearing in (4.30). First of all,

E_{s,T} (g̃_{s,T}^{V} )^{−1} \frac{∂g̃_{s,T}^{V}}{∂s} E_{s,T}^{−1} = (id_{V} +s(R_{T} + R′_{T} ))^{−1} ((R_{T} + R′_{T} )

+ (\frac{N^{V}}{2} − \frac{n}{4}) (log T − log π) (id_{V} +s(R_{T} + R′_{T} )))

+ ((\frac{N^{V}}{2} − \frac{n}{4}) (log T − log π) (id_{V} +s(R_{T} + R′_{T} ))

+ (R_{T} + R′_{T} )) (id_{V} +s(R_{T} + R′_{T} ))^{−1}

= (N^{V} − \frac{n}{2}) (log T − log π) + 2R_{T} (id +s R_{T} )^{−1} + O(e^{−cT} )

uniformly in s, since R_{T} and N^{V} commute.

Now, we investigate the superconnections A′ and Ã′′_{s,T}. Let A′′ be the adjoint of A′ with respect to g^{V}. Then we have

A′ = ∇^{V} + ∑_{j} a_{j}  and  A′′ = ∇^{V,∗} + ∑_{j} a_{j}^{∗} ,

where adjoints are taken with respect to g^{V}, and we know that a_{j} raises eigenvalues of h^{V} by at least ε^{2} because of Theorem 1.66, while a_{j}^{∗} lowers them by at least the same amount. Since the estimates in Theorem 4.23 and Theorem 4.24 include derivatives of R′_{T}, we find

E_{s,T} A′ E_{s,T}^{−1} = (id_{V} +s(R_{T} + R′_{T} )) (∇^{V} + T [∇^{V} , h^{V} ]

+ ∑_{j}(\frac{T}{π} )^{\frac{s(1−j)}{2}} e^{−T h^{V}} a_{j} e^{T h^{V}} ) (id_{V} +s(R_{T} + R′_{T} ))^{−1}

= ∇^{V} + T [∇^{V,∗}, h^{V} ] + O(e^{−cT} ) .

Similarly, we obtain

E_{s,T} Ã′′_{s,T} E_{s,T}^{−1} = E_{s,T}^{−1} A′′ E_{s,T} = ∇^{V,∗} − T [∇^{V} , h^{V} ] + O(e^{−cT} ) .

Now we note that h^{V} acts as a scalar function on each component V^{α} = V̂ |_{C_{α}} of V, and that R_{T}, ∇^{V} and ∇^{V,∗} preserve these components. Writing

ω^{V} = ∇^{V,∗} − ∇^{V}  and  dh^{V} = [∇^{V} , h^{V} ] = [∇^{V,∗}, h^{V} ] ,

we finally obtain

str_{V} (\frac{1}{2} (g̃_{s,T}^{V} )^{−1} \frac{∂g̃_{s,T}^{V}}{∂s} f′(X̃_{s,T} ))

= str_{V} (\frac{1}{2} ((N^{V} + \frac{n}{2}) (log T − log π) + 2R_{T} (id +R_{T} )^{−1}) f′(\frac{ω^{V}}{2} − T dh^{V} ) + O(e^{−cT} )

= (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) (log T − log π) f′(0) + O(\frac{1}{T} ) ,

because \frac{ω^{V}}{2} +T dh^{V} is odd and commutes with N^{V} +2R_{T} (id +R_{T})^{−1}, while f′ is an even function, and supertraces of even powers of odd operators vanish. By (4.30), this finishes our proof of Theorem 4.29. □

Using this result, we can summarize the results of this section, where we now make use of the fact that f(z) = ze^{z^{2}} by (2.11).

4.31. Theorem. As T → ∞, we have

f̃(∇^{H}, g^{H}, g_{T}^{H}) + S_{f}(T^{H}M, g^{T X}, ∇^{F} , g_{T}^{F} ) − (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) log T + T str_{V} (h^{V} )

⟶ f̃(∇^{H}, g^{H}, g_{V}^{H}) + T_{f}(A′, g^{V} , h^{V} )

− (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) log π − (χ′(V) − χ′(H)) ∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t}

− (χ′(V) − χ′(H)) ∫_{0}^{1}(f′(\sqrt{−t}/2) − 1) \frac{dt}{2t}

78 SEBASTIAN GOETTE

modulo exact forms on B.

This theorem is our analogue of Theorem 9.8 in [BG1]. Note however, that a different notion of torsion appears in [BG1]. Proof. From (4.19), Theorem 4.29 and the fact that f′(0) = 1, we get

(4.32) f̃(∇^{H}, g^{H}, g_{T}^{H}) + S_{f}(T^{H}M, g^{T X}, ∇^{F} , g_{T}^{F} ) − (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) (log T − log π)

= f̃(∇^{H}, g^{H}, g_{V}^{H}) + S_{f}(A′, g^{V} ) + f̃(A′, g^{V} , e^{−2T h^{V}} g^{V} ) + O(T^{−δ}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

We change back to the notation of Section 2.d. In particular, A′′_{1,τ} now refers to the adjoint of A′ with respect to the metric g^{V} (T^{h^{V}} · , · ), and of course X_{1,T} = \frac{1}{2} (A′′_{1,T} − A′). Then

\begin{aligned}f̃(A′, g^{V} , e^{−2T h^{V}} g^{V} ) & = −(2πi)^{− \frac{N^{B}}{2}} ∫_{0}^{T} str_{V} (h^{V} f′(X_{1,e^{−2τ}} )) dτ \\ & = −(2πi)^{− \frac{N^{B}}{2}} ∫_{e^{−2T}}^{1} str_{V} (h^{V} f′(X_{1,T′})) \frac{dT′}{2T′} . \end{aligned} (4.33)

By (4.32) and (4.33),

f̃(∇^{H}, g^{H}, g_{T}^{H}) + S_{f}(T^{H}M, g^{T X}, ∇^{F} , g_{T}^{F} ) − (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) (log T − log π) + T str_{V} (h^{V} )

= f̃(∇^{H}, g^{H}, g_{V}^{H}) + S_{f}(A′, g^{V} ) − (2πi)^{− \frac{N^{B}}{2}} ∫_{e^{−2T}}^{1} (str_{V} (h^{V} f′(X_{1,T′}))

− str_{V} (h^{V} )) \frac{dT′}{2T′} + O(T^{−δ}) ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) .

In the limit T → ∞, Definition 2.46 gives us

f̃(∇^{H}, g^{H}, g_{T}^{H}) + S_{f}(T^{H}M, g^{T X}, ∇^{F} , g_{T}^{F} ) − (\frac{χ′(V)}{2} − \frac{n χ(V)}{4}) (log T − log π) + T str_{V} (h^{V} )

⟶ f̃(∇^{H}, g^{H}, g_{V}^{H}) + T_{f}(A′, g^{V} , h^{V} )

− (χ′(V) − χ′(H)) ∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t}

− (χ′(V) − χ′(H)) ∫_{0}^{1}(f′(\sqrt{−t}/2) − 1) \frac{dt}{2t} ∈ Ω^{∗}(B)/d^{B}Ω^{∗}(B) ,

which finishes the proof. □ 4.34. Remark. Suppose that γ is a fibre-wise isometry of M that acts on F and preserves T^{H}M, g^{T X}, ∇^{F} and g^{F}. Suppose also that γ acts on the fibres of V → B preserving A′, g^{V} and h^{V}, and that the map I: Ω^{∗}(M; F) → Ω^{∗}(B; V) is γ-equivariant. Then there are forms f̃_{γ}(∇^{H}, g^{H}, g_{T}^{H}), S_{f,γ}(T^{H}M, g^{T X}, ∇^{F} , g_{T}^{F}), T_{f,γ}(A′, g^{V} , h^{V}) etc., and all considerations in this section still go through and provide us with a γ-equivariant version of Theorem 4.31, cf. [BG1], Theorem 9.8

5. Exotic smooth fibre bundles

We calculate the analytic torsion form of a sphere bundle p_{γ}: M_{γ} → S^{4k} that is fibre-wise homeomorphic to a trivial bundle, but not diffeomorphic. We then use this bundle to construct “exotic” bundle structures on a quite general class of compact fibre bundles.

The construction of the sphere bundle p_{γ} in Section 5.a goes back to Hatcher, and was kindly explained to us by Igusa, see also [I2], Section 6. The computation of T (M_{γ}, S^{4k}; ℂ) in Section 5.b and the extension to bundles with arbitrary fibres in Section 5.c are still closely related to Igusa’s work.

In Section 5.d, we use Hatcher’s example to exhibit a large class of fibre bundles M → B where we can construct infinitely many bundles that are homeomorphic but not diffeomorphic as bundles to M → B, see Corollary 5.23. In Section 5.e we finally prove a refinement of Theorem 0.2, see Theorem 5.31. We will also relate our results to [DWW].

5.a. Hatcher’s construction. We will construct a bundle N_{γ} → S^{4k}, with fibre

(5.1)  D^{m+n} = (S^{m} × D^{n}) ∪_{S^{m}×D^{n−1}} (D^{m+1} × D^{n−1}) ,

see Figure 5.2. Here, γ ∈ ker J, see (5.3) below. The bundle M_{γ} will then be given by the fibrewise double of N_{γ}. Over the “southern” hemisphere D_{−}^{4k} ⊂ S^{4k} of the base sphere, we take a fixed embedding of D^{n−1} into a fixed hemisphere of S^{n−1} = ∂D^{n} in (5.1), thus obtaining a fixed trivialisation of D^{m+n} × D_{−}^{4k}.

S^{m}×D^{n−1} S^{m}×D^{n}  D^{m+1}×D^{n−1}  D^{m+n}

Figure 5.2. A single fibre of N_{γ}.

We will now describe the construction over the “northern” hemisphere D_{+}^{4k} ⊂ S^{4k}. For fixed k and m, we have the J-homomorphism, which is given as

(5.3)  J: π_{4k−1}(O(m)) ⟶ π_{4k−1}(C^{0}((S^{m}, x_{0}), (S^{m}, x_{0}))) ≅ π_{4k−1+m}(S^{m}, x_{0}) ,

where x_{0} ∈ S^{m} denotes the north pole. The groups π_{4k−1}(O(m)) and π_{4k−1+m}(S^{m}, x_{0}) are independent of m for m sufficiently large, and π_{4k−1+m}(S^{m}, x_{0}) is finite, while π_{4k−1}(O(m)) is infinite cyclic. Thus, pick 0 ≠ γ ∈ ker J, which is represented by a map γ: S^{4k−1} → O(m) that extends continuously to γ: D_{+}^{4k} → C^{0}((S^{m}, x_{0}), (S^{m}, x_{0})).

For n large enough, using transversality, we can approximate the map

D_{+}^{4k} × (S^{m}, x_{0}) ⟶^{γ} (S^{m}, x_{0}) ⟶ (S^{m} × D^{n−1}, (x_{0}, 0))

by a smooth family of embeddings γ̃_{q}: (S^{m}, x_{0}) ↪ (S^{m} × D^{n−1}, (x_{0}, 0)) for q ∈ D_{+}^{4k}, which equals

(qr, x) ⟼ (γ_{q}(x), 0) for q ∈ S^{4k−1} = ∂D_{+}^{4k}, r ∈ [\frac{1}{2}, 1], and x ∈ S^{m} .

The normal bundle N_{q} → S^{m} of γ̃_{q} is stably trivial for all q ∈ D_{+}^{4k}, because the tangent bundles of S^{m} and S^{m} × D^{n−1} are stably trivial. Making n even larger if necessary, we may assume that the N_{q} are trivial. Then the bundle N → D_{+}^{4k} × S^{m} is of course also trivial. Fixing a trivialisation of N and using the normal exponential map, we can extend γ̃ to a family of embeddings

γ̃_{q}: S^{m} × D^{n−1} ⟶ S^{m} × D^{n−1} for q ∈ D_{+}^{4k} .

We may assume that there exists a map γ′: S^{4k−1} × S^{m} → O(n − 1) such that for all q ∈ S^{4k−1} = ∂D_{+}^{4k} and all r ∈ [\frac{1}{2}, 1], the embedding γ̃_{qr} becomes

γ̃_{qr}(x, y) = (γ_{q}(x), γ′_{q,x}(y)) .

5.4. Proposition. The family of embeddings γ̃_{q}: S^{m} ×D^{n−1} → S^{m} ×D^{n−1} for q ∈ D_{+}^{4k} is isotopic to a family of embeddings that restricts over S^{4k−1} = ∂D_{+}^{4k} to

(x, y) ⟼ (γ_{q}(x), (γ_{q}^{−1} ⊕ id_{ℝ^{n−1−m}})(y)) for all q ∈ S^{4k−1} .

With this result, we finally construct N_{γ}. We still identify D^{n−1} with a fixed hemisphere of S^{n−1} = ∂D^{n}. For q ∈ D_{+}^{4k}, we construct the fibre

D^{m+n} = (S^{m} × D^{n}) ∪_{S^{m}×D^{n−1}} (D^{m+1} × D^{n−1})

as in (5.1), identifying

(x, y) ∈ S^{m} × D^{n−1} ⊂ ∂(D^{m+1} × D^{n−1})  with  γ̃_{q}(x, y) ∈ S^{m} × D^{n−1} ⊂ ∂(S^{m} × D^{n}) .

Then we can glue N_{γ}|_{D_{+}^{4k}} with N_{γ}|_{D_{−}^{4k}} along S^{4k−1} using the identity on S^{m} × D^{n} and identifying

(q, x, y) ∈ N_{γ}|_{D_{+}^{4k}}  with  (q, γ_{q}(x), (γ_{q}^{−1} ⊕ id_{ℝ^{n−1−m}})(y)) ∈ N_{γ}|_{D_{−}^{4k}} .

Finally, we construct the sphere bundle

(5.5)  p: M_{γ} = N_{γ} ∪_{id_{∂Nγ}} N_{γ} ⟶ S^{4k} .

Proof of Proposition 5.4. Consider the family of rotations S^{4k−1} → O(2m) given by

q ⟼ (γ_{q} ⊕ γ_{q}^{−1}): S^{4k−1} ⟶ O(2m) .

By a well-known trick from K-theory, this family is homotopic to the trivial family q ↦ id_{ℝ^{2m}}: apply the Cartan projection GL(2m, ℝ) → O(2m) to the homotopy

(5.6)  (q, t) ⟼ ( \begin{matrix}1 & tγ_{q} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ −tγ_{q}^{−1} & 1 \end{matrix} ) ( \begin{matrix}1 & tγ_{q} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 − t & −t \\ t & 1 − t \end{matrix} )

in GL(2m, ℝ), where t ∈ [0, 1]. In particular, there exists a map G: D_{+}^{4k} → O(m + n) that for all q ∈ S^{4k−1} and all r ∈ [\frac{1}{2}, 1] takes the form

G_{rq} = (γ_{q} ⊕ γ_{q}^{−1} ⊕ id_{ℝ^{n−m}}) .

The map γ̃_{q}: S^{m} × D^{n−1} → S^{m} × D^{n−1} is determined by its 1-jet

Dγ̃_{q}|_{S^{m}×{0}}: T S^{m} × ℝ^{n−1} ⟶ T S^{m} × T D^{n−1} .

We add the trivial line ℝ on both sides, and identify T S^{m} × ℝ with ℝ^{m+1} using the standard embedding S^{m} ↪ ℝ^{m+1}. Then Dγ̃_{q}|_{S^{m}×{0}} acts on T S^{m} × ℝ by the matrix γ_{q} ⊕ id_{ℝ} ∈ O(m + 1) for q ∈ S^{4k−1}. This implies that G_{q}^{−1} ∘ Dγ̃_{q}|_{S^{m}×{0}} acts trivially on T S^{m} × ℝ for q ∈ S^{4k−1}.

Because the Stiefel manifold O(m + n + 1)/O(m + 1) is 4k-connected for n sufficiently large, we can deform G_{q}^{−1} ∘ Dγ̃_{q} on D_{+}^{4k} relative to S^{4k−1} such that T S^{m} × ℝ ≅ ℝ^{m+1} is trivially mapped to itself. Finally, we can deform the action on the factor ℝ^{n−1} to the trivial one over D_{+}^{4k}. This shows that the map G_{q}^{−1} ∘ Dγ̃_{q}|_{S^{m}×{0}} can be deformed to the identity, relative to S^{4k−1} × T S^{m} × ℝ. Then Dγ̃_{q}|_{S^{m}×{0}} can be deformed to G_{q} relative to S^{4k−1} × T S^{m} × ℝ. In particular, the embedding γ̃_{q} is isotopic to an embedding that takes the form given in the proposition. □

In the remainder of this chapter, we will use the example above to construct families of bundles which are pairwise homeomorphic as bundles, but not diffeomorphic in one the following two senses.

5.7. Definition. Two fibre bundles p: M → B and p′: M′ → B are homeomorphic (diffeomorphic) as bundles iff there exists a homeomorphism (diffeomorphism) Φ such that the left diagramme below commutes.

M ⟶_{∼}^{Φ} M′  M ⟶_{∼}^{Φ} M′
p↓  ↓^{p′}  ^{p}↓  ↓^{p′}
B ⟶^{id} B  B ⟶_{∼}^{Ψ} B

We say that p and p′ are homeomorphic (diffeomorphic) as maps iff there exist homeomorphisms (diffeomorphisms) Φ and Ψ such that the right diagramme above commutes.

5.8. Proposition. The bundle p: M_{γ} → S^{4k} is homeomorphic as a bundle to a trivial sphere bundle.

Proof. It is well known that the diffeomorphism group of D^{m+n} with the C^{0}-topology retracts to O(m+n), so M_{γ} → S^{4k} is homeomorphic to some sphere bundle with structure group O(m+n). Our claim then follows because the bundle p: M_{γ} → S^{4k} admits sections C_{1}, …, C_{6}, and the restriction of T X to each of these sections is trivial. □

5.9. Remark. In the construction above, we did not care at all about the dimension of the fibre of p: M_{γ} → S^{4k}. It should be clear that there are some numbers p, q such that one can always keep the dimension of the fibre at most pk + q, and it is interesting to know explicite values of p and q. This question is adressed in [I2], see Theorem 6.2.2 and Corollary 6.5.8.

5.b. The higher torsion invariants of Hatcher’s sphere bundle. We calculate Bismut-Lott’s analytic torsion form and Igusa’s higher Franz-Reidemeister torsion of p: M_{γ} → S^{4k} by specifying a fibre-wise Morse function on M_{γ}.

We first construct a fibre-wise Morse function h: N_{γ} → ℝ such that dh is positive on the outward normal vector to ∂N_{γ}. We start with a fixed Morse function on S^{m}, having exactly two critical points: a minimum of −3 at the south pole, and a maximum of −2 at the north pole. Adding a small positive function on D^{n} whose only critical point is a minimum 0 at the origin, we obtain a Morse function on the trivial bundle S^{m} × D^{n} × S^{4k} → S^{4k}.

We then choose a Morse function on D^{m+1} × D^{n−1} with only one critical point at (0, 0) of value −1, such that D^{m+1}×{0} is the descending cell, and {0}×D^{n−1} is the ascending cell. Clearly, these fibre-wise Morse functions can be patched together to give a fibre-wise Morse function h: N_{γ} → ℝ with three leaves of fibre-wise critical points of indices 0, m and m + 1, see Figure 5.10. We may assume that h^{−1}(0) = ∂N_{γ}.

We finally get a Morse function on M_{γ} = N_{γ} ∪_{id_{∂Nγ}} N_{γ} by taking h on one copy of N_{γ} and −h on the other. Assuming that n ≥ m + 3, this new Morse function has six fibre-wise critical leaves C_{1}, …, C_{6} of indices 0, m, m + 1, n − 1, n and m + n, with values −3, −2, −1, 1, 2 and 3, respectively. In particular, no critical leaf can “cross over” another critical leaf of higher or equal index. For this reason, the “simplicial superconnection” a of Proposition 1.33 has no terms of higher degree, i.e., a_{αβ}(σ) = 0 unless σ ∈ S_{0}.

Letting F = M_{γ} × ℂ be trivial, the bundle V = S^{4k} × ℂ^{6} is also trivial. We can arrange the superconnection A′ of Theorem 1.66 to equal a_{0} + ∇^{V}, with

a_{0} = ( \begin{matrix}0 \\ & 0 \\ & 1 & 0 \\ & & & 0 \\ & & & 1 & 0 \\ & & & & & 0 \end{matrix} ) ∈ Γ(End V)

C_{2}  C_{3}  C_{1}

Figure 5.10. The negative gradient flow of the fibre-wise Morse function h on N_{γ}.

parallel with respect to the trivial connection ∇^{V}. Thus

(Ω^{∗}(S^{4k}; V), A′) = (Ω^{∗}(S^{4k}), d) ⊗ (ℂ^{6}, a_{0})

is a product, and the bundle H = H^{∗}(M_{γ}/S^{4k}; ℂ) → S^{4k} is trivial, too.

We also find that T (A′, g^{V} , h^{V}) ∈ ℝ ⊂ Ω^{∗}(S^{4k}) is homogeneous of degree 0 (it actually vanishes). However, the term

p̂_{∗}^{0}J((−1)^{ind^{h}} (T^{s}X − T^{u}X))

in Theorem 0.1 does not vanish. The critical leaves C_{1}, C_{2}, C_{5} and C_{6} lie within the trivial bundle

((S^{m} × D^{n}) ∪_{id_{S^{m}×D^{n−1}}} (S^{m} × D^{n})) × S^{4k} ⟶ S^{4k}

and come with trivial bundles T^{s}X, T^{u}X, so they do not contribute. The bundles T X|_{C_{3}} and T X|_{C_{4}} are trivial, because they arise from trivial bundles over D_{+}^{4k} and D_{−}^{4k} using the clutching function q ↦ (γ_{q} ⊕ γ_{q}^{−1} ⊕ id), which is homotopic to the identity by (5.6). However, the bundles T^{s,u}X|_{C_{3}} and T^{s,u}X|_{C_{4}} are not trivial because γ_{q} was chosen non-trivial. The bundle T^{s}X|_{C_{4}} is isomorphic to T^{u}X|_{C_{3}}, and has clutching function γ^{−1}. If u is a generator of H^{4k}(S^{4k}; ℤ) ⊂ H_{dR}^{4k}(S^{4k}), if γ is the n_{γ}-th multiple of a generator of ker J, and if 0 ≠ b_{k} ∈ ℚ is the coefficient discussed in [I2], Remark 6.4.3, then

(5.11)  p_{j}(T^{u}X|_{C_{3}}) = p_{j}(T^{s}X|_{C_{4}}) = \{ \begin{matrix}n_{γ} b_{k} u & if j = k, and \\ 0 & otherwise. \end{matrix}

By triviality of T X|_{C_{3}} and T X|_{C_{4}} and multiplicativity of the total Pontrijagin class,

(5.12)  p_{j}(T^{u}X|_{C_{4}}) = p_{j}(T^{s}X|_{C_{3}}) = −p_{j}(T^{u}X|_{C_{3}}) = \{ \begin{matrix}−n_{γ} b_{k} u & if j = k, and \\ 0 & otherwise, \end{matrix}

Because the two flat bundles F → M_{γ} and H → S^{4k} are trivial, we have a higher analytic torsion class T (M_{γ}/S^{4k}; ℂ) ∈ H^{≥2}(S^{4k}) as in Definition 2.88. By Theorem 0.1 and Corollary 3.9, we get a parallel result to [I2], Theorem 6.4.2. Note that passing from Pontrijagin classes to the Chern character in the definition of ^{0}J gives an extra factor 2^{−k}.

5.13. Theorem. The analytic torsion class of the bundle M_{γ} → S^{4k} above is given by

T (M_{γ}/S^{4k}; ℂ) = (−1)^{m+1} ^{0}J(T^{s}X|_{C_{3}}) + (−1)^{m} ^{0}J(T^{u}X|_{C_{3}})
+ (−1)^{n−1} ^{0}J(T^{s}X|_{C_{4}}) + (−1)^{n} ^{0}J(T^{u}X|_{C_{4}})

= \{ \begin{matrix}(−1)^{m} 2^{2−k} n_{γ} b_{k} ζ′(−2k) u ∈ H^{4k}(S^{4k}; ℝ) & if n + m is odd, and \\ 0 & if n + m is even. \end{matrix}

In particular, neither the sphere bundle M_{γ} → S^{4k} nor the disc bundle N_{γ} → S^{4k} are diffeomorphic to a trivial bundle. Thus we recover a result by Bökstedt [Bo], cf. [I2], Chapter 6.

Note on the other hand that using Igusa’s framing principle (Theorem 6.1.1 in [I2]), the higher Franz-Reidemeister torsion of [I2] equals

τ(M_{γ}/S^{4k}; ℂ) = (−1)^{m} 2^{0}J(T^{u}X|_{C_{3}}) + (−1)^{n} 2^{0}J(T^{u}X|_{C_{4}}) = T (M_{γ}/S^{4k}; ℂ) .

This is consistent with Theorem 0.3, by which

T (M_{γ}/S^{4k}; ℂ) − τ(M_{γ}/S^{4k}; ℂ) = ∫_{M_{γ}/S^{4k}} e(T X)^{0}J(T X) = 0 ,

because the bundle T X|_{C} is trivial by construction.

5.14. Remark. Note that the superconnection A′ = ∇^{V} + a_{0} on V → S^{4k} looks like a superconnection of a fibre-wise Morse function with a fibre-wise gradient field satisfying the Smale transversality condition, cf. [BG1], section 5. On the other hand, if there was a fibre-wise Smale gradient field for h, we would have

T^{s}X|_{C_{2}} ≅ T^{s}X|_{C_{3}} ⊕ ℝ ,  T^{u}X|_{C_{2}} ⊕ ℝ ≅ T^{u}X|_{C_{3}} ,
T^{s}X|_{C_{4}} ≅ T^{s}X|_{C_{5}} ⊕ ℝ ,  and  T^{u}X|_{C_{4}} ⊕ ℝ ≅ T^{u}X|_{C_{5}}

by [BG1], section 7.10, contradicting (5.11), (5.12). This shows that the considerations in Section 1 are indeed needed to deal with this example.

5.15. Remark. Consider once more the (m + n)-disc bundle N_{γ} → S^{4k} above. If m + n is odd, the boundary ∂N_{γ} becomes a trivial smooth sphere bundle, possibly after replacing N_{γ} by N_{γ} × D^{2l} → S^{4k} for some l ≥ 0. This means that instead of doubling N_{γ}, we could glue N_{γ} and a trivial disc bundle to obtain another sphere bundle \overline{N}_{γ} → S^{4k} with a fibre-wise Morse function with only four fibre-wise critical points, namely C_{1}, C_{2} and C_{3} on N_{γ} and a maximum on the trivial disc bundle. Then clearly

T (\overline{N}_{γ}/S^{4k}; ℂ) = \frac{1}{2} T (M_{γ}/S^{4k}; ℂ) = (−1)^{m} 2^{1−k} n_{γ} b_{k} ζ′(−2k) u ∈ H^{4k}(S^{4k}; ℝ) .

In order to keep this chapter reasonably self-contained, we will stick to the slightly more complicated bundle p: M_{γ} → S^{4k} constructed above.

5.c. Exotic bundles over S^{4k} with general fibres. We extend Hatcher’s example to generate examples of bundles with arbitrary fibre. This involves a very simple kind of a vertical gluing formula. A more general kind of vertical gluing will be discussed in [G]. The following construction is closely related to Corollary 6.5.8 in [I2]. We need it as an intermediate step for the general construction in the following subsection.

Fix k > 0 and let X be a compact manifold of dimension dim X = 2l−1, such that l is sufficiently large with respect to k. Let p_{X}: X × S^{4k} → S^{4k} be the trivial bundle. We fix a Morse function on X, which gives rise to a trivial fibre-wise Morse function h_{X}: X × S^{4k} → ℝ. Any flat vector bundle F_{X} → X of rank r with a parallel metric pulls back to a flat bundle F_{X} → X × S^{4k}.

Let p_{γ}: M_{γ} → S^{4k} be the bundle with fibre S^{2l−1} constructed in Section 5.a. In particular, there exists a fibre-wise Morse function h_{γ}: M_{γ} → ℝ that has six critical points on each fibre. Let S^{4k} ≅ C_{6} ⊂ M_{γ} be the fibre-wise maximum, then C_{6} has a trivial neighbourhood U ≅ C_{0} × D^{2l−1} in M_{γ}.

We can now replace a small neighbourhood of one of the local minima of h_{X} by M_{γ}\U, obtaining a new bundle p′_{γ}: M′_{γ} → S^{4k} with fibre X that is still homeomorphic as a bundle to p_{X}: X×S^{4k} → S^{4k}. By gluing the bundle F_{X} → X × S^{4k} with the trivial bundle of rank r over M_{γ}, we get a flat vector bundle F′_{γ} → M′_{γ}, and F′_{γ} → M′_{γ} is the pull back of F by the homeomorphism M′_{γ} → X × S^{4k} above. In particular, the fibre-wise cohomology of M′_{γ} → S^{4k} is still a trivial bundle over S^{4k} with fibre H^{∗}(X; F_{X}).

5.16. Proposition (cf. Corollary 6.5.8 in [I2]). If F_{X} → X is trivial, we have

T (M′_{γ}/S^{4k}; ℂ) = 2^{2−k} n_{γ} b_{k} ζ′(−2k) u ∈ H^{4k}(S^{4k}; ℝ) .

Thus again, the higher analytic torsion is able to detect the “exotic” differentiable structure of the homeomorphically trivial bundle p′_{γ}: M′_{γ} → S^{4k}.

In [G], we will exhibit a spectral sequence argument which gives rise to a gluing formula for higher torsion invariants. In particular, we will find that

T (M′_{γ}/S^{4k}; F′_{γ}) = 2^{2−k} n_{γ} b_{k} ζ′(−2k) u rk F_{X} ∈ H^{4k}(S^{4k}; ℝ) .

Note that in the argument below, there are no spectral sequence contributions by (5.18).

Proof. After subtracting a suitable constant from h_{γ}, gluing with h_{X}, and smoothing along the common boundary, we obtain a new fibre-wise Morse function h′_{γ}: M′_{γ} → ℝ. Let C′_{γ} ⊂ M′_{γ} denote the critical points of h′_{γ}, then

C′_{γ} = C_{γ}^{′+} ˙∪ C_{γ}^{′−} ,

where C_{γ}^{′−} are the five remaining critical levels of h_{γ}, and C_{γ}^{′+} are the remaining critical points of h_{X}. This induces a splitting

(5.17)  V′_{γ} = V_{γ}^{′+} ⊕ V_{γ}^{′−} ⟶ B .

Since the descending cells of the critical levels C_{γ}^{′+} can meet the ascending cells of C_{γ}^{′−} in any possible way, we cannot conclude any more that the superconnection A′ = ∇^{V′_{γ}} +a′ of Definition 1.65 on the family Thom-Smale complex Ω^{∗}(S^{4k}; V) still takes the form A′ = ∇^{V′_{γ}} +a_{0} with a_{0} ∈ Ω^{0}(B, End V). We may however conclude that

a′ = ( \begin{matrix}b′ & d′ \\ 0 & c′ \end{matrix} )

with respect to (5.17), such that b′ ∈ Ω^{0}(B; End V_{γ}^{′+}) and c′ ∈ Ω^{0}(B; End V_{γ}^{′−}) are both of horizontal degree 0 and parallel with respect to ∇^{V′_{γ}}.

We note that the fibre-wise cohomologies of (V_{γ}^{′+}, b′) and (V_{γ}^{′−}, c′) are given by

H^{j}(V_{γ}^{′+}, b′) = \{ \begin{matrix}0 & j = 0 , \\ H^{j}(X; ℂ) & j > 0 , \end{matrix}  and  H^{j}(V_{γ}^{′−}, c′) = \{ \begin{matrix}ℂ & j = 0 , \\ 0 & j > 0 , \end{matrix}

because H^{∗}(V_{γ}^{′−}, c′) is the cohomology of a disc. This implies in particular that

(5.18)  H^{∗}(V′_{γ}, a′_{0}) = H^{∗}(X; ℂ) = H^{j}(V_{γ}^{′+}, b′) ⊕ H^{j}(V_{γ}^{′−}, c′) ,

so the spectral sequence associated with the filtration V′γ ⊃ V_{γ}^{′+} ⊃ 0 degenerates already at E_{1} = E_{∞}.

We now need the analytic analogue of Igusa’s splitting principle ([I2], Lemma 5.7.8). Multiplying the (parallel) metric g^{V} restricted to V_{γ}^{′−} by a small positive constant T^{−2} has the same effect

on T (A′, g^{V} , h^{V}) as conjugating a′ by ( \begin{matrix}1 & 0 \\ 0 & T \end{matrix} ). In the limit T → ∞, only terms in Ω^{0}(S^{4k}; End V)

remain, giving a torsion form in Ω^{0}(S^{4k}). The class T (M′_{γ}/S^{4k}; ℂ, h) ∈ H^{even,≥2}(S^{4k}) is unaffected by this deformation for all T > 0 according to Corollary 2.63. Because the rank of the fibre-wise cohomology does not jump in the limit T → ∞ by (5.18), we find that

T (M′_{γ}/S^{4k}; ℂ, h) = T (∇^{V_{γ}^{′+}} + b′, g^{V_{γ}^{′+}})^{[≥2]} + T (∇^{V_{γ}^{′−}} + c′, g^{V_{γ}^{′−}})^{[≥2]} = 0 .

Because the bundles T^{s}X and T^{u}X are trivial along C_{γ}^{′+}, and the ^{0}J-classes of (T^{s}X −T^{u}X)|_{C_{γ}^{′−}} are the same as in Theorem 5.13, Theorem 0.1 implies that

(5.19)  T (M′_{γ}/S^{4k}; ℂ) = 2^{2−k} n_{γ} b_{k} ζ′(−2k) u . □

5.d. Exotic structures on general fibre bundles. We show that we can alter the differentiable structure on a quite general fibre bundle by “horizontally” gluing with the bundle M′_{γ} of the previous subsection. In the following subsection, we use the bundles obtained this way to prove Theorem 0.2.

Let p = p_{0}: M_{0} → B be a smooth proper submersion of compact manifolds, and assume that the base is orientable with dim B = 4k, and that dim X = dim M_{0} − dim B = 2l − 1, such that l is sufficiently large with respect to k. Let F → M_{0} be a unitarily flat vector bundle. If H = H^{∗}(M_{0}/B; F) → B admits a parallel metric, the higher torsion T (M_{0}/B; F) ∈ H_{dR}^{even,≥2}(B) is well-defined. Otherwise, we fix an arbitrary metric g^{H} on H and regard the class

(5.20)  T (M_{0}/B; F, g^{H}) = (T (T^{H}M_{0}, g^{T X}, ∇^{F} , g^{F} ) + c̃h^{o}(∇^{H}, g^{H}, g_{L_{2}}^{H}))^{[≥2]}

in Ω^{even}(B)/dΩ^{odd}(B), which is independent of the choices of T^{H}M_{0}, g^{T X} and the parallel metric g^{F}, but depends on g^{H}.

Let X_{0} = p^{−1}(b_{0}) ⊂ M_{0} be a fixed fibre of p for a fixed b_{0} ∈ B, and let F_{0} be the restriction of F to X_{0}. Let γ ∈ π_{4k−1}(O) be a generator of ker J and let M′_{γ} → S^{4k} be constructed as in the previous subsection with fibre X = X_{0}. We may assume that the bundles M_{0} → B and M′_{γ} → S^{4k} are trivialised in small discs D_{0}^{4k} around b_{0} ∈ B and D_{−}^{4k} ⊂ S^{4k}, and we pick the horizontal subbundles T^{H}M_{0} and T^{H}M′_{γ} according to that trivialisation. Fixing a metric g^{X} on X and g_{0}^{H} on H(X; F), we define g^{T X} and g_{0}^{H} over D_{0}^{4k} and D_{−}^{4k} using the trivialisation above together with the given g^{X} and g_{0}^{H}. Then the data defining the classes T (M_{0}/B; F, g^{H}) and T (M′_{γ}/S^{4k}; E_{i}) coincide on ∂D_{0}^{4k} and ∂D_{−}^{4k}, which allows us to glue M_{0}|_{B\D_{0}^{4k}} and M′_{γ}|_{S^{4k}\D_{+}^{4k}} along their common boundary, obtaining a new bundle

p_{1}: M_{1} ⟶ B_{1} ,

where B_{1} is naturally diffeomorphic to B. In other words, we replace the restriction of M_{0} to D_{0}^{4k} by the restriction M′_{γ} to D_{+}^{4k}.

Then p_{1}: M_{1} → B_{1} ≅ B is homeomorphic to p_{0}: M_{0} → B as a bundle by Proposition 5.8. Because the bundle F |_{X} is diffeomorphic and isometric to E_{j}|_{p^{−1}(b_{j})}, we can also identify the flat vector bundles F and E_{j} along M|_{∂D}, obtaining a new flat vector bundle F_{1} → M_{1}.

We note that all secondary data T^{H}M, g^{T X}, g^{F} and g_{0}^{H} has carefully been chosen so as to be compatible with the gluing process. We now restrict our attention to the trivial bundle F = M_{0}×ℂ. Proposition 5.16 implies that

(5.21) T (T^{H}M_{1}, g^{T X_{1}}, d, g^{triv}) + c̃h^{o}(∇^{H}, g_{0,L_{2}}^{H}, g_{1,L_{2}}^{H})

= T (T^{H}M_{0}, g^{T X_{0}}, d, g^{triv}) + 2^{2−k} b_{k} ζ′(−2k) u ∈ Ω^{∗}(B)/dΩ^{∗}(B) ,

where g_{0,L_{2}}^{H} and g_{1,L_{2}}^{H} are the L_{2}-metrics on H ≅ H^{∗}(M_{0}/B) ≅ H^{∗}(M_{1}/B), and where u represents a generator of H^{4k}(B; ℤ) ⊂ H_{dR}^{4k}(B). Note that both c̃h^{o}(∇^{H}, g_{0,L_{2}}^{H}, g_{1,L_{2}}^{H}) and u are given by smooth forms supported on D_{0}^{4k} ⊂ B, which are well-defined modulo differentials of forms supported on D_{0}^{4k}.

Because the bundles p_{1} and p_{0} are naturally homeomorphic, we can identify the fibre-wise cohomology bundles H(M_{1}/B; ℂ) → B_{1} ≅ B and H(M_{0}/B; ℂ) → B as flat vector bundles, and this identification is canonical on B ⊂ D_{0}^{4k} = B_{1} ⊂ D_{+}^{4k}. In particular, the metric g^{H} can be regarded as a metric on H(M_{1}/B; ℂ) as well. Passing to classes, equation (5.21) becomes

T (M_{1}/B; ℂ, g^{H}) = T (M_{0}/B; ℂ, g^{H}) + 2^{2−k} b_{k} ζ′(−2k) u ∈ Ω^{∗}(B)/dΩ^{∗}(B) .

Iterating this process gives bundles p_{j}: M_{j} → B for all j ≥ 0. If we repeat the constructions above with γ replaced by γ^{−1} in (5.5), we also obtain bundles p_{j}: M_{j} → B for all j < 0, such that

(5.22)  T (M_{j}/B; ℂ, g^{H}) = T (M_{0}/B; ℂ, g^{H}) + 2^{2−k}j b_{k} ζ′(−2k) u ∈ Ω^{∗}(B)/dΩ^{∗}(B)

for all j ∈ ℤ. Let us summarize our examples so far.

5.23. Corollary. Let p: M → B be a smooth fibre bundle with compact fibre X of dimension 2l−1 and an orientable, 4k-dimensional, compact base space B, such that l is sufficiently large with respect to k, and let g^{H} be a fixed metric on the fibre-wise cohomology H(M/B; ℂ) → B. Then there exist infinitely many bundles p_{j}: M_{j} → B for j ∈, j > 0, with fibres diffeomorphic to X, that are all homeomorphic to p = p_{0} as bundles, but with pairwise different values of

∫_{B} T (M_{j}/B; ℂ, g^{H}) .

5.24. Remark. The result above is related to Proposition 6.5.7 in [I2] for trivial bundles over arbitrary compact orientable base spaces. Indeed, if the bundle H(M/B; ℂ) → B is trivial, then we could have stated Corollary 5.23 using Igusa’s higher Franz-Reidemeister torsion τ(M_{j}/B; ℂ). However, Igusa’s τ(M_{j}/B; ℂ) has so far only been defined if H(M/B; ℂ) → B is trivial, so we use T (M_{j}/B; F, g^{H}) to get a more general statement. Since we used Theorem 0.1 to relate the higher analytic torsion of Hatcher’s sphere bundles to τ(M_{γ}/B, ℂ), Corollary 5.23 is in fact an application of the methods developped in this paper.

5.25. Remark. At this point, it is natural to ask for those cohomology classes on B that can be realised as differences of higher torsion classes of fibre bundles that are homeomorphic to a given M → B as bundles. Corollary 5.23 only realises certain multiples of the fundamental class as such differences. However, the gluing techniques used in the proof immediately give a slightly more general result, which applies in particular to the trivial bundle M × B → B.

Let p: M → B be a smooth fibre bundle with compact fibre X of dimension 2l − 1 and base space B. Assume that B contains a properly immersed oriented submanifold A of codimension 4k > 0 with trivial normal bundle, which has a lift N ⊂ M, such that l is sufficiently large with respect to k, and the vertical tangent bundle T X|_{N} is trivial. Let α ∈ H^{4k}(B, ℝ) be the Poincaré dual of A. Let g^{H} be a fixed metric on the fibre-wise cohomology H(M/B; ℂ) → B. Then there exist infinitely many bundles p_{j}: M_{j} → B for j ∈, j > 0, with fibres diffeomorphic to X, that are all homeomorphic to p = p_{0} as bundles, but with

(5.26)  T (M_{1}/B; ℂ, g^{H}) = T (M_{0}/B; ℂ, g^{H}) + 2^{2−k} b_{k} ζ′(−2k) α ∈ Ω^{∗}(B)/dΩ^{∗}(B) .

Clearly, equation (5.22) is contained here as a special case where A is a single point.

To proof this claim, we may assume that N is embedded in M by transversality. Then let U denote a tubular neighbourhood of N, fix p ∈ N, and let D_{p} be a fibre of the bundle U → N. We may assume that U fibres over a dim B-dimensional submanifold V ⊂ M with fibre D^{2l−1}, where V is transversal to the fibres of M → B. By our assumptions, the bundle U → V is trivial, and we have a projection

π: V ⟶ D^{4k}  with  U = π^{∗}(D^{2l−1} × D^{4k}) ⟶ V .

Thus, we can replace U by the pullback of the bundle M_{γ}|_{D_{+}^{4k}} → D^{4k} with a neighbourhood of the critical level C_{1} removed as in Section 5.c. Then (5.26) follows by the same arguments as (5.19) and (5.22).

5.e. Homeomorphism classes and diffeomorphism classes of submersions, and a proof of Theorem 0.2. In the last section, we have exhibited a method to construct an infinite family of pairwise homeomorphic fibre-bundles with pairwise different higher analytic torsion classes, starting from an arbitrary fibre-bundle with compact fibre and compact orientable base of the right dimensions. To prove Theorem 0.2, we have to show that an infinite subfamily has diffeomorphic total spaces and that these bundles are pairwise not diffeomorphic as bundles, which is again a consequence of Corollary 5.23 above. In Theorem 5.31, we give a necessary condition for a diagramme

M ⟶_{∼}^{Φ} M
p_{i}↓  ↓^{p_{j}}
B ⟶_{∼}^{Ψ} B

as in Definition 5.7 to commute, where Φ and Ψ are now diffeomorphisms. This proves Theorem 0.2 and gives the refinement mentioned in the introduction.

Therefore, let p: M → B be a smooth fibre bundle with compact fibre X of dimension 2l − 1 and an orientable, 4k-dimensional, compact base space B, such that l is sufficiently large with respect to k. Let F → M be a unitarily flat vector bundle, and let p_{j}: M_{j} → B and F_{j} → M_{j} be constructed as in Corollary 5.23. We do not require that the pair-wise isomorphic flat vector bundles H_{j} = H^{∗}(M_{j}/B; ℂ) → B carry a parallel metric.

A careful investigation of the construction in Section 5.d shows that M_{j} arises from M by removing several subsets D_{i} diffeomorphic to D^{4k} × D^{2l−1} and gluing them back in a different way, such that before and after the gluing, the fibres of p respectively p_{j} intersect D^{4k} × D^{2l−1} either in {b} × D^{2l−1} for some b ∈ D^{4k} or not at all. After sliding the sets D_{i} around within M, we may assume that all D_{i} are contained in a larger set \overline{D} ⊂ M diffeomorphic to D^{4k} × D^{2l−1}, such that each inclusion D_{i} ↪ \overline{D} projects to the identity on D^{4k}. If we denote the corresponding subset of M_{j} by \overline{D}_{j}, then \overline{D}_{j} is still diffeomorphic to D^{4k} × D^{2l−1}, where the fibre of p_{j} intersect D^{4k} × D^{2l−1} either in {b} × D^{2l−1} for some b ∈ D^{4k} or not at all.

We construct M_{j} for j < 0 similarly by glueing in |j| copies of another exotic sphere bundle where we start with the inverse of γ in π_{4k−1}(O(m)). Then (5.22) still holds.

5.27. Proposition. There exists j_{0} ∈ ℤ such that there exists a diffeomorphism

Θ_{j}: M ⟶ M_{j}  with  Θ_{j}|_{M\\overline{D}} = id_{M\\overline{D}}: M \ \overline{D} ⟶ M_{j} \ \overline{D}_{j}

for all j ∈ j_{0}ℤ. In particular, Θ_{j} lifts to a parallel map

\overline{Θ}_{j}: F ⟶ F_{j}  with  \overline{Θ}_{j}|_{F |_{M\\overline{D}}} = id_{F |_{M\\overline{D}}}: F |_{M\\overline{D}} ⟶ F_{j}|_{M_{j}\\overline{D}_{j}} .

Proof. By the constructions in Subsections 5.c and 5.d, there exists a possibly exotic (4k + 2l − 1)- dimensional sphere Σ such that

M_{j} = M# \underbrace{Σ# ⋯ #Σ}_{j times}

for all j > 0, where the connected sum operation takes places inside \overline{D}. For j < 0, we have to take the connected sum with |j| copies of another possibly exotic sphere Σ′. Because the group of exotic spheres is finite in each dimension, the first claim now follows.

The second claim is then clear because F and F_{j} are up to isomorphism the unique extensions of F |_{M\\overline{D}} to M and M_{j}, respectively, thus Θ_{j}^{∗}F_{j} and F are isomorphic as flat bundles and can thus be identified. □

It is an immediate consequence of Corollary 5.23 that the maps p and p_{j} ∘ Θ_{j} differ somewhere on \overline{D} for j ≠ 0. On the other hand, recall that by Proposition 5.8, there always exists a homeomorphism Θ′_{j}: M → M_{j} supported on \overline{D} such that p = p_{j} ∘ Θ′_{j}. In the remainder of this chapter, we want to generalise and strengthen this observation. But first of all, we need some facts on Chern-Simons classes.

Let (F, ∇^{F}) → M be a flat vector bundle, and let Diff (M; F) denote the group of fibre preserving, fibre-wise linear, parallel diffeomorphisms \overline{Φ}: F → F. An element Φ ∈ Diff (M; F) is a parallel lift of a diffeomorphism of M, which by an abuse of notation will again be named Φ. If M is oriented, let Diff^{+}(M; F) ⊂ Diff (M; F) denote the subgroup of lifts of orientation preserving diffeomorphisms of M.

Let \overline{F} = F × [0, 1] denote the trivial extension of F to \overline{M} = M × [0, 1]. For Φ ∈ Diff (M; F), the bundle F naturally extends to a flat vector bundle

T_{Φ}F = \overline{F} / ∼ ⟶ T_{Φ}M = \overline{M}/ ∼

over the mapping torus T_{Φ}M, where “∼” is generated by (v, 1) ∼ (Φ(v), 0) for all v ∈ F, respectively (p, 1) ∼ (Φ(p), 0) for all p ∈ M. If g^{F} is any metric on F, let g_{t}^{F} be a family of metrics that interpolates between g_{0}^{F} = g^{F} and g_{1}^{F} = Φ^{∗}g^{F}. Let g\overline{F} denote the corresponding metric on \overline{F}, then g\overline{F} descends to a metric g^{T_{Φ}F} on T_{Φ}F.

Now we assume in addition that M is oriented and that Φ preserves the orientation. By the definition of the Chern-Simons class c̃h^{o}, we have

(5.28)  ∫_{M} c̃h^{o}(∇^{F} , g^{F} , Φ^{∗}g^{F} ) = ∫_{M×[0,1]} ch^{o}(∇\overline{F} , g\overline{F} ) = ch^{o}(∇^{T_{Φ}F} )[T_{Φ}M] ,

independent of the choice of g^{F}.

5.29. Proposition. Assume that M is oriented, then the map

Diff^{+}(M; F) ⟶ ℝ  with  Φ ⟼ ch^{o}(∇^{T_{Φ}F} )[T_{Φ}M]

is a group homomorphism.

Assume in addition that dim M > 0. Then ch^{o}(∇^{T_{Φ}F})[T_{Φ}M] depends only on the homotopy type of the action of Φ on M. In particular, if F carries a parallel metric or if Φ^{k} ∼ id_{M} for some k ≠ 0, then ch^{o}(∇^{T_{Φ}F})[T_{Φ}M] = 0.

To see that the invariant ch^{o}(∇^{T_{Φ}F})[T_{Φ}M] is nontrivial, one may consider a surface M of genus ≥ 2 and a diffeomorphism Φ such that T_{Φ}M = ℍ^{3}/Γ becomes hyperbolic with fundamental group Γ ⊂ SL(2, ℂ). We then choose F → M and a lift Φ to F such that T_{Φ}F becomes the flat bundle associated to the adjoint representation of SL(2, ℂ) restricted to Γ. In this case, ch^{o}(∇^{T_{Φ}F})[T_{Φ}M] is a nonzero multiple of the hyperbolic volume of T_{Φ}M.

Note that it is easy to see that if dim M > 0, the value of ch^{o}(∇^{T_{Φ}F})[T_{Φ}M] depends only on the homotopy type of the action of Φ on M and F, because a homotopy (Φ_{t})_{t∈[0,1]} then gives rise to a smooth family of flat vector bundles over the diffeomorphic spaces T_{Φ_{t}}M, and the higher odd Chern classes are locally constant on the space of flat vector bundles. This argument fails if the action of Φ_{0}, Φ_{1} on F are not homotopic.

Proof. The map c̃h^{o}(M; F) is a homomorphism by (5.28), because for Φ, Φ′ ∈ Diff^{+}(M; F), we have

c̃h^{o}(∇^{F} , g^{F} , (Φ′ ∘ Φ)^{∗}g^{F} ) = c̃h^{o}(∇^{F} , g^{F} , Φ^{∗}g^{F} ) + Φ^{∗}c̃h^{o}(∇^{F} , g^{F} , (Φ′)^{∗}g^{F} )

modulo exact forms.

We now assume that dim M > 0. Because of the homomorphism property, we may assume that the action of Φ on M is homotopic to the identity. Then T_{Φ}M is diffeomorphic to M × S^{1}. We momentarily assume that the bundle T_{Φ}F is of the form

(5.30)  π_{1}^{∗}F_{M} ⊗ π_{2}^{∗}F_{S^{1}} ,

where F_{M} → M and F_{S^{1}} → S^{1} are flat vector bundles; the general case will be discussed later. By Proposition 1.13 in [BL], we have

ch^{o}(∇^{π_{1}^{∗}F_{M} ⊗π_{2}^{∗}F_{S^{1}}} ) = π_{1}^{∗} ch^{o}(∇^{F_{M}} ) rk F_{S^{1}} + rk F_{M} π_{2}^{∗} ch^{o}(∇^{F_{S^{1}}} ) ,

thus in particular

ch^{o}(∇^{π_{1}^{∗}F_{M} ⊗π_{2}^{∗}F_{S^{1}}} )[M × S^{1}] = 0 .

We now turn to general flat bundles over M × S^{1}. Such a bundle F is given by a representation of π_{1}(M) × ℤ on ℂ^{r}, with ℤ = π_{1}(S^{1}) and r = rk F. Let A ∈ M_{r}(ℂ) denote the action of a fixed generator of ℤ. The action of π_{1}(M) respects the direct sum decomposition of ℂ^{r} into generalised eigenspaces, so we may assume that A ≡ λ id modulo strictly upper triangular matrices. Thus ℂ^{r} is filtered by spaces

0 = ker((A − λ)^{0}) ⊂ ⋯ ⊂ ker((A − λ)^{r}) = ℂ^{r} ,

and this filtration is preserved by the action of π_{1}(M). By [L], odd Chern classes are well-defined in the K-theory of flat vector bundles. This implies that ch^{o}(F) is the sum of the odd Chern classes of the flat bundles associated to the subquotients ker((A − λ)^{j})/ ker((A − λ)^{j−1}), which are of the form discussed in (5.30). Thus, we have established homotopy invariance in general.

If Φ^{k} is homotopic to the identity, the homomorphism property together with the homotopy invariance above implies that

k ch^{o}(∇^{T_{Φ}F} )[T_{Φ}M] = ch^{o}(∇^{T_{Φ^{k}} F} )[T_{Φ^{k}}M] = 0 .

Assume that g^{F} is a parallel metric, then Φ^{∗}g^{F} is also parallel, and we may interpolate by a family g_{t}^{F} of metrics that are parallel on M × {t} for each t. Then ch^{o}(∇^{T_{Φ}F} , g^{T_{Φ}F}) ∈ Ω^{1}(T_{Φ}M), in particular

ch^{o}(∇^{T_{Φ}F} )[T_{Φ}M] = 0 . □

We now want state the main application of Subsections 5.c and 5.d, which allows us to give many examples of manifolds M, B that allow infinitely many smooth maps M → B that are pairwise conjugate by homeomorphisms, but not by diffeomorphisms. Whenever the dimensions match and B is compact and orientable, we can construct infinitely many proper submersions M → B that are homeomorphic but not diffeomorphic as bundles. Let us write q_{j} = p_{jj_{0}} ∘ Θ_{jj_{0}}, where j_{0} > 0 and Θ_{jj_{0}} are given by Proposition 5.27.

5.31. Theorem. Let p: M → B be a proper submersion. Assume that B is compact, orientable, and of dimension 4k > 0. Let the bundles (q_{j}: M → B)_{j∈ℤ} with q_{0} = p be constructed as above. Assume that for i, j ∈ ℤ with i ≠ j, there exists diffeomorphisms Φ ∈ Diff (M) and Ψ ∈ Diff^{+}(B) such that the diagramme

M ⟶^{Φ} M
q_{i}↓  ↓^{q_{j}}
B ⟶^{Ψ} B

commutes. Then Ψ ∈ Diff^{+}(B; H) for H = H^{∗}(M/B; ℂ) → B, and

ch^{o}(∇^{T_{Ψ}H})[T_{Ψ}B] = 2^{2−k} (j − i) j_{0} b_{k} ζ′(−2k) .

In particular, this is only possible if H → B does not carry a parallel metric, and if Ψ does not represent a torsion element in the mapping class group.

Proof of Theorem 0.2. By Proposition 5.8, the maps q_{j} are pairwise homeomorphic as bundles, in the sense of Definition 5.7. In particular, they are also homeomorphic as maps from M to B. Proposition 5.29 and Theorem 5.31 have the following two easy consequences.

(1) If the bundle H = H^{∗}(M/B; ℂ) carries a parallel metric, then the maps q_{j} are not pairwise conjugate by diffeomorphisms, in particular, they are not pairwise isomorphic as smooth fibre bundles.

(2) If the bundle H does not admit a parallel metric, then the maps q_{j} are not pairwise isomorphic as smooth fibre bundles, because id_{B} clearly is a torsion element in the mapping class group.

This proves Theorem 0.2. □

Proof of Theorem 5.31. We know from Proposition 5.8 and the construction in Subsections 5.c and 5.d that the fibre bundles q_{i}, q_{j}: M → B are homeomorphic to M → B by a homeomorphism that is supported on a contractible subset of M. To avoid confusion, we will write q_{i} and q_{j} instead of M/B whenever necessary. In particular, the fibrewise cohomologies H^{∗}(q_{i}; F) and H^{∗}(q_{j}; F) → B can be identified with the original H → B. By functoriality of the fibre-wise cohomology, the diffeomorphism Ψ lifts to Diff^{+}(B; H).

If g_{0}^{H} is any metric on H, the class

T (T^{H}M, g^{T X}, d, g^{triv}) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{0}^{H}) ∈ Ω^{even}(B)/dΩ^{odd}(B)

is independent of the choices of T^{H}M and g^{T X} by Corollary 2.17 and Theorem 2.48. Since we assume that dim B > 0, we obtain an invariant

T (M/B, ℂ; g_{0}^{H}) = ∫_{B}(T (T^{H}M, g^{T X}, d, g^{triv}) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{0}^{H})) ,

which depends on the choice of g_{0}^{H} by

(5.32)  T (M/B, ℂ; g_{1}^{H}) − T (M/B, ℂ; g_{0}^{H}) = − ∫_{B} c̃h^{o}(∇^{H}, g_{0}^{H}, g_{1}^{H}) .

By naturality of torsion, we find that

(5.33)  T (q_{j}, ℂ; g_{0}^{H}) = T (q_{i}, ℂ; Ψ^{∗}g_{0}^{H}) .

On the other hand, it follows from (5.22) that

(5.34)  T (q_{j}, ℂ, g_{0}^{H}) − T (q_{i}, ℂ, g_{0}^{H}) = 2^{2−k}(j − i) j_{0} b_{k} ζ′(−2k) u ,

where u is the positive generator of H^{4k}(B; ℤ). The formula in the theorem follows from (5.28) and (5.32)–(5.34). The remaining conclusions are a consequence of Proposition 5.29. □

5.f. Hatcher’s example and Lott’s secondary K-theory. We reformulate Corollary 5.23 in the language of Lott’s secondary K-theory \overline{K}_{R}^{0} of [L], cf. Section 3.c. We will see that the different maps p_{j}: M → B constructed in Subsections 5.d and 5.e give rise to different pushdown maps p_{j!}: \overline{K}_{R}^{0}(M) → K̂_{R}^{0}(B), at least if π_{1}(B) is finite. We also discuss this fact in the light of Conjecture 1 in [L], which relates the pushdown p_{!} to a Becker-Gottlieb transfer.

Thus, let p: M → B be a smooth fibre bundle with compact fibres, and assume that B is compact oriented with dim B = 4k and the dimension of the fibre is odd and sufficiently large. Let (q_{i}: M → B)_{i∈ℤ} with q_{0} = p denote the maps considered in Theorem 5.31. We take R to be one of ℤ, ℝ, ℂ for simplicity and let ρ: R → ℂ be the inclusion. Then by (3.25) and Theorem 3.26,

(5.35)  q_{i!}[M × R, g^{triv}, 0] = q_{0!}[M × R, g^{triv}, 0] + [0, 0, i u]

for some nonzero u ∈ H^{4k}(S^{4k}, ℝ). By Proposition 5.37 below, the elements of \overline{K}_{R}^{0}(B) given by (5.35) are pairwise different, at least if π_{1}(B) is finite.

In the following, we let B be a smooth manifold with finite fundamental group. Then we claim that there exists a well-defined map

ĉh^{o}: K̂^{0} ⟶ Ω^{even,≥2}(B)/dΩ^{odd}(B) .

To define ĉh^{o} on a generator (F, g^{F} , η), we choose a parallel metric g_{0}^{F} on F_{ℂ}, which is possible because B has finite fundamental group, and we set

(5.36)  ĉh^{o}(F, g^{F} , η) = c̃h^{o}(∇^{F} , g_{0}^{F} , g^{F} )^{[≥2]} − η^{[≥2]} .

This is independent of g_{0}^{F} because for two parallel metrics g_{0}^{F}, g_{1}^{F} on F, we have c̃h^{o}(∇^{F} , g_{0}^{F} , g_{1}^{F}) ∈ ℝ ⊂ Ω^{0}(B) modulo exact forms.

It is easy to check that (5.36) complies with (3.27) due to the properties of c̃h^{o}. We check that (5.36) is compatible with the K̂_{R}^{0}-relations (3.22). Regard a short exact sequence as in (3.20). By (3.27), we may choose parallel metrics g^{F_{i}} on the bundles F_{i,ℂ}. Then the associated Bismut-Lott torsion of (3.21) vanishes in degree ≥ 2, so (5.36) is compatible with (3.22) and the map ĉh^{o} is well-defined.

5.37. Proposition. Assume that π_{1}(B) is finite, then

\overline{K}_{R}^{0}(B) ≅ K_{R}^{0}(B) ⊕ H^{even,≥2}(B, ℝ) ,

and the combined map

H^{even,≥2}(B, ℝ) ⟶ \overline{K}_{R}^{0}(B) ⟶^{ĉh^{o}} H^{even,≥2}(B, ℝ)

is multiplication by −1.

Proof. Because B has finite fundamental group, the sequence (3.23) becomes

H^{even}(B, ℝ) ⟶ \overline{K}_{R}^{0}(B) ⟶ K_{R}^{0}(B) ⟶ 0 .

Since any element of \overline{K}_{R}^{0}(B) can be represented as a sum of K̂_{R}^{0}-generators (F, g^{F} , η) with parallel g^{F} and η ∈ H^{even}(B, ℝ), the proposition follows from the fact that ĉh^{o} is well-defined. □

5.38. Remark. Lott conjectured in [L] that p_{!} is related to a Becker-Gottlieb transfer. More precisely, let F_{R}^{∗} denote the homotopy fibre of the map

K^{∗}(R) ⟶^{ρ_{∗}} K^{∗}(ℂ) ⟶ ∏_{j} K(ℝ, 2j + 1) ,

where K^{∗} denotes the spectrum of algebraic K-theory, and K(ℝ, 2j + 1) denotes the Eilenberg-McLane spectrum of odd real homology. Here, the left arrow is induced by the representation ρ: R → End(C^{n}), and the right arrow is the Borel regulator map. Then F_{R}^{∗} is again a generalised cohomology theory, and the Becker-Gottlieb transfer gives a homomorphism

(5.39)  τ_{BG}^{∗}: F_{R}^{0}(M) ⟶ F_{R}^{0}(B) .

Lott conjectures the existence of a natural map \overline{K}_{R}^{0} → F_{R}^{0} that fits into commutative diagrammes

(5.40)

H^{even}(M; ℝ) ⟶ \overline{K}_{R}^{0}(M) ⟶ K_{R}^{0}(M)
↓  ↓  ↓
H^{even}(M; ℝ) ⟶ F_{R}^{0}(M) ⟶ K_{alg,R}^{0}(M) ,

cf. (3.23), and

(5.41)  \begin{matrix}\overline{K}_{R}^{0}(M) & ⟶ & F_{R}^{0}(M) \\ ^{p_{!}}↓ & & ↓^{τ_{BG}^{∗}} \\ \overline{K}_{R}^{0}(B) & ⟶ & F_{R}^{0}(B) & . \end{matrix}

The Becker-Gottlieb transfer itself is a stable map τ_{BG}: B^{+} → M^{+}, which can be defined without referring to the smooth structure of q: M → B, see [BeG]. This implies that τ_{BG}^{∗} in (5.39) is the same map for all q_{i}, i ∈ ℤ, whereas the maps q_{i!}: \overline{K}_{R}^{0}(M) → \overline{K}_{R}^{0}(B) are pairwise different by (5.35) and Proposition 5.37 if π_{1}(B) is finite. We conclude that the \overline{K}_{R}^{0}-pushdown contains more information than the Becker-Gottlieb transfer τ_{BG}^{∗} in F_{R}^{0}. Moreover, if a natural map \overline{K}_{R}^{0} → F_{R}^{0} satisfying (5.40) and (5.41) above exists, then the element [0, 0, u] ∈ \overline{K}_{R}^{0}(B) of (5.35) lies the kernel of \overline{K}_{R}^{0}(B) → F_{R}^{0}(B) for all closed oriented B of dimension 4k.

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NWF 1 Mathematik, Universität Regensburg, 93040 Regensburg, Germany

   E-mail address: sebastian.goette@mathematik.uni-regensburg.de