MORSE THEORY AND HIGHER TORSION INVARIANTS II

Sebastian Goette

Abstract. Let p: M → B be a family of compact manifolds equipped with a unitarily flat vector bundle F → M. We generalize Igusa’s higher Franz-Reidemeister torsion τ(M/B; F) to the case that the fibre-wise cohomology H^{∗}(M/B; F) → B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the difference of τ(M/B; F) and the higher analytic torsion T (M/B; F). We also generalise the examples given in [G].

This is the second 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 the first part [G], we evaluated the Bismut-Lott torsion form T for families that carry a fibre-wise Morse function, thus extending earlier work with Bismut ([BG]). Here, we relate τ and T for families with fibre-wise Morse functions in those situations where both invariants are defined.

We start by recalling some notation and results of [G]. Let p: M → B be a family of compact manifolds, let F → M be a flat vector bundle, and let h: M → ℝ be a fibre-wise Morse function. Then the fibre-wise critical points of h form a covering p̂: C → B. Let o(T^{u}X) → C denote the orientation bundle of the unstable vertical tangent bundle at C, and construct a vector bundle

V = p̂_{∗}(F |_{C} ⊗ o(T^{u}X)) .

Then V carries a flat connection ∇^{V} induced by ∇^{F}. In [G], we defined a family Thom-Smale complex by constructing a flat superconnection A′ on V. We also constructed a generalised “integration over the fibre”, i.e., an Ω^{∗}(B)-linear cochain map I: (Ω^{∗}(M; F), ∇^{F}) → (Ω^{∗}(B; V), A′). This cochain map naturally identifies the fibre-wise cohomology H = H^{∗}(M/B; F) → B, equipped with the flat Gauß-Manin connection ∇^{H}, with the bundle H^{∗}(V, a′_{0}) → B as follows. Both Ω^{∗}(M; F) and Ω^{∗}(B; V) can be filtered by horizontal degree. The cochain map I respects these filtrations and induces isomorphisms of the E_{p}-terms of the associated spectral sequences for p ≥ 1. The E_{1}-term of both spectral sequences is isomorphic to (Ω^{∗}(B; H), ∇^{H}).

A metric g^{F} on F induces a metric g^{V} on V, and the Morse function h induces an endomorphism h^{V} of V. Using these data, we defined an analytic torsion form

T (A′, g^{V} , h^{V}) ∈ Ω^{∗}(B) .

The metric g^{V} induces a metric g_{V}^{H} on H by finite-dimensional Hodge theory. Let ch^{o}(∇^{F} , g^{F}) denote the characteristic form of F → M defined in [BL], written in the normalisation of [BG]. Then

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

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.

Research at MSRI is supported in part by NSF-grant DMS-9701755

In particular, T (A′, g^{V} , h^{V}) generalises the analytic torsion forms on flat families of chain complexes defined in [BL].

In order to state the main theorem of [G], let T^{H}M ⊂ T M be a horizontal complement of the vertical tangent bundle T X → M of M, and let g^{T X} be a vertical metric on T X. Then there exists a unique natural connection g^{T X} on T X which generalises the Levi-Civita connection on a single manifold X. Let

T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) ∈ Ω^{∗}(B) .

be the analytic torsion form of Bismut and Lott ([BL]). The metrics g^{T X} and g^{F} induce an L_{2}-metric on the space Ω^{∗}(M/B; F) of fibre-wise differential forms. By Hodge theory, we get a metric g_{L_{2}}^{H} by identifying H → B with the bundle of fibre-wise harmonic forms. Let c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H}) be the Chern-Simons class associated to the characteristic class ch^{o}(∇^{H}). Let ψ(∇^{T X}, g^{T X}) denote the Matthai-Quillen current on the bundle T X → M. Finally, we define an additive characteristic class

0J = \frac{1}{2} ∑_{k≥1} ζ′(−2k) ch^{[4k]}

as in [BG], where ζ is the Riemann zeta function and ch^{[4k]} is the homogeneous part of the Chern class with coefficients in ℝ of degree 4k. The following theorem generalises the main theorem of [BG], where we assumed in addition that h admits a fibre-wise Smale gradient field.

0.1. Theorem ([G]). 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 .

As application, we used analytic torsion forms to detect infinite families of pairwise homeomorphic but not diffeomorphic fibre bundles, which were constructed using Hatcher’s example of a topologically trivial, but smoothly non-trivial bundle over 4k with fibre S^{2l−1}. The version we present here is slightly more general, since in the second statement, we can now work with an arbitrary unitarily flat F in place of the trivial bundle.

0.2. Theorem (cf. [G]). 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.

If M moreover carries a unitarily flat vector bundle F such that the fibre-wise cohomology bundle H^{∗}(M/B; F) → B admits a parallel metric with respect to the Gauß-Manin connection, then the p_{j} are pairwise not diffeomorphic as maps.

In the present paper, we restrict our attention to the case that both bundles F → M and H → B carry parallel metrics. In this case, we can use the higher analytic torsion T (T^{H}M, g^{T X}, ∇^{F} , g^{F}) to define a cohomology class T (M/B; F) ∈ H^{∗}(B; ℝ), see [G], (2.93). We compare this class to a generalisation τ(M/B; F) ∈ H^{∗}(B; ℝ) of Igusa’s higher Franz Reidemeister torsion in a normalisation adapted to [BG].

0.3. Theorem. 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) + ∫_{M/B} e(T X) ^{0}J(T X) rk F ∈ H^{∗}(B; ℝ) .

We also prove that τ(M/B; F) and T (M/B; F) induce the same universal class τ(M(ℂ), U) on the unitary Whitehead space Wh^{u}(M(ℂ), U), which contains a copy of Igusa’s acyclic Whitehead space Wh^{h}(M(ℂ), U).

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

The present paper contains the proof of Theorem 0.3 and the new version of Theorem 0.2, together with a generalisation of Igusa’s approach to the situation that the two flat bundles F → M and H → B admit a parallel metric. It is organised as follows.

We show in Section 6 that the construction of (V, A′, h^{V}) in Section 1 of [G] gives rise to a map ξ from the base space B to Igusa’s Whitehead space Wh(M(ℂ), GL(ℂ)) up to homotopy, which equals Igusa’s map ξ_{p,h,F}: B → Wh(M(ℂ), GL(ℂ)). Here, GL(ℂ) = lim_{→} GL_{n}(ℂ) = M(ℂ)^{×} denotes the stabilised general linear group. Conversely, each such map 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, Wh(M(ℂ), GL(ℂ)) is a universal space for flat vector bundles with the extra structure described above.

We give a different describtion of the finite dimensional torsion T (A′, g^{V} , h^{V}) in Section 7, where we show that T (A′, g^{V} , h^{V}) = T (∇^{V} , A′, g^{V}) depends only on the existence of local filtrations, not on h^{V}. If g^{V} is parallel and the fibre-wise cohomology H → B vanishes, then T (∇^{V} , A′, g^{V}) is the pull-back ξ_{p,h,F}^{∗} T (M(ℂ), U) of a universal higher torsion class T (M(ℂ), U) on the acyclic Whitehead space Wh^{h}(M(ℂ), U). We also decompose the higher torsion T (∇^{V} , A′, g^{V}) of a globally filtered family of complexes into the higher torsions of the steps of the associated spectral sequence, mimicking Ma’s result in [Ma].

We show in Section 8 that the Igusa’s universal torsion class τ(M(ℂ), U) on the Whitehead space Wh^{h}(M(ℂ), U) coincides up to normalisation with our universal class T (M(ℂ), U). By Igusa’s two-index theorem, it is enough to compare the two classes on the stabilised acyclic twoindex Whitehead space Wh^{h,{0,1},s}(M(ℂ), U), which is homotopy equivalent to Wh^{h}(M(ℂ), U). The construction of τ(M(ℂ), U) relies heavily on cyclic homology, whereas T (A′, g^{V} , h^{V}) is constructed using a double transgression of the Chern character. The comparison of both classes proceeds by showing that the components of Igusa’s class τ(M(ℂ), U) have the same transgression properties as the corresponding terms in T (M(ℂ), U), and can thus be identified in suitable abstract acyclic cochain complexes. In the fibre-wise acyclic case, Theorem 0.3 now follows from Igusa’s framing principle and Theorem 0.1.

We discuss the case that the two flat bundles F → M and H → B admit a parallel metrics. In Section 9, we construct another Whitehead space Wh^{u}(M(ℂ), U) which is the universal space of flat bundles V → B with the additional structures (A′, g^{V}) described above, together with a parallel metric g^{H} on the fibre-wise cohomology H → B. The space Wh^{u}(M(ℂ), U) is homotopy equivalent to a covering of Igusa’s original Wh(M(ℂ), U). The universal torsion class T (M(ℂ), U) can be extended naturally to Wh^{u}(M(ℂ), U). We also give an acyclic cone construction, by which the the torsion form T (M(ℂ), U) can be pulled back from a fibre-wise acyclic situation.

Moreover, the class T (M/B; F, h) of Definition 2.64 in [G] pulls back from a universal class on Wh^{u}(M(ℂ), U). Combining this with a natural lift of Igusa’s map ξ to the unitary Whitehead space Wh^{u}(M(ℂ), U), we obtain a new class τ(M/B; F) ∈ H^{∗}(B; ℝ) for all families M → B of smooth compact manifolds with boundary ∂M = M_{0} ∪ M_{1} which carry unitarily flat vector bundles F → M whose relative homology H^{∗}(M/B, M_{0}/B; F) is again unitarily flat.

Finally in Section 10, we show that Igusa’s framing principle of [I2] extends to the case that F → M and H → B carry parallel metrics. We then use Igusa’s framed function theorem of [I1] and the framing principle to finally establish Theorem 0.3. We prove a cut-and-paste formula for τ(M/B; F) and apply it to extend the examples considered in Section 5 of [G] and prove the generalised second statement of Theorem 0.2. Note that a more precise quantitative statement is given in Theorem 10.20. The proof of Theorem 10.20 uses Theorem 0.3 to switch from τ(M/B; F) to T (M/B; F) as needed.

We will generously cite [G]. Therefore, the numbering of the sections in this paper start with 6. Sections 1 to 5 always refer to [G], as well as all theorems, propositions, equations … whose first number is 1–5.

We are grateful to J.-M. Bismut, U. Bunke and K. Igusa for fruitful discussions and their interest in this project. We also thank the Mathematical Sciences and Research Institute at Berkeley and Igusa’s family for their hospitality.

6. The universal torsion class on the Whitehead space

In [I2], Igusa defines the Whitehead space as the homotopy fibre of a natural map from the stable homotopy of the classifying space BG of a subgroup G of the group of units R^{×} of a ring R into the algebraic K-theory space BGL(R)^{+} of R itself. In particular

Wh^{h}(R, G) ⟶ Q(BG_{+}) ⟶ \overline{ℤ} × BGL(R)^{+}

is a homotopy fibre sequence. Here BG_{+} is the universal classifying space of G with an extra base point, and Q denotes the stable homotopy functor Ω^{∞}S^{∞}. The space \overline{ℤ} × BGL(R)^{+} ⊂ K(R) gives the algebraic K-theory of R in degrees ≥ 1, where \overline{ℤ} denote the image of ℤ = K_{0}(ℤ) in K_{0}(R), BGL(R) is the classifying space for the general linear group GL(R) = lim_{⟶} GL(n, R)_{δ}, and ( ·)^{+} denotes Quillen’s plus construction. Note that all groups and rings here carry the discrete topology.

In Chapter 3 of [I2], Igusa constructs a model for the Whitehead space out of “systems of higher homotopies”, which are precisely the “combinatorial flat superconnections” that we encountered in Proposition 1.33. In Subsections 6.a and 6.b, we review Igusa’s construction of the simplicial Whitehead space as the geometric realisation of a simplicial category Wh^{simp}(R, G).

Let p: M → B be a proper submersion equipped with a fibre-wise Morse function h: M → ℝ and a flat vector bundle F → M. We show in Section 6.c that our construction of a combinatorial flat superconnection in Section 1.c gives a simplicial functor Ξ = Ξ_{p,h,F} from a certain simplicial category S associated to B to Wh^{simp}(R, G), which is homotopy equivalent to the corresponding functor constructed in [I2], Chapter 4.

If the ring R is an ℝ-subalgebra of the ring M_{n}(ℂ) of (n × n)-matrices over ℂ, and G ⊂ R^{×} is any subgroup, we develop two more models for the Whitehead space in Section 6.d, given by differential superconnections and by mixed combinatorial and differential superconnections. We show that all these models are homotopy equivalent, so that we may speak of “the” Whitehead space Wh(R, G). We want to point out that this equivalence of Whitehead categories has been investigated independently by K. Igusa in [I3], where he interprets both the map into the Whitehead space and the bundle of the fibre-wise cohomology as A_{∞}-functors.

In Section 7, we will show that the form T_{f}(A′, g^{V} , h^{V}) of Definition 2.46 is given by a universal cohomology class on a Whitehead space if the family of cochain complexes (V, a_{0}) → B is fibrewise acyclic. Since we may choose any of the models above for this Whitehead space, this justifies calling the form T_{f}(A′, g^{V} , h^{V}) a “combinatorial torsion form”, although it is defined in terms of a differential superconnection. In Section 9, this construction will be generalised to the case that H = H^{∗}(V, a_{0}) → B carries a parallel metric.

6.a. Simplicial sets and categories. We recall the definitions of simplicial sets and simplicial categories as well as some elementary facts and examples needed later on.

6.1. Definition. Let Δ denote the small category whose objects are the sets [k] = {0, …, k}, and whose morphisms are all mappings f: [j] → [k] with f(i + 1) ≥ f(i) for all 0 ≤ i < j. If C denotes any category, then a simplicial object in C is a contravariant functor from Δ to C, and a simplicial morphism between to simplicial objects in C is a natural transformation.

Simplicial sets and simplicial maps are simplicial objects and morphisms in the category of sets and set maps. Simplicial categories and simplicial functors are simplicial objects and morphisms in the category of small categories and covariant functors.

If S is a simplicial set, then S_{k} = S[k] is the set of elements of degree k, and Sf: S_{k} → S_{j} a (generalised) face operator for each morphism f: [j] → [k] in Δ. This terminology is compatible with Definition 1.11 in [G] of a smooth simplicial complex, where S_{k} was a set of smooth ksimplices σ: Δ_{k} → B. To comply with Definition 6.1, we must add degenerated simplices and degeneracy maps to the complex defined in Definition 1.11.

If S is a simplicial category, we still write S_{k} for the set of objects of degree k. The morphisms of S_{k} commute with the face operators S_{k} → S_{l}.

The geometric realisation is a functor from the category of simplicial sets to the category of topological spaces, cf. [Lo]. The geometric realisation of a small category is the realisation of its nerve, which is again a simplicial set. Similarly, the geometric realisation of a simplicial category is the realisation of its nerve, which is now a bisimplicial set.

6.2. Example. Let S be a smooth simplicial complex on B as in Definition 1.11. Then we define a simplicial category S:

(1) the set of objects of degree k is

S_{k} = { (σ, σ′) | σ′ ∈ S_{k} is a generalised face of σ ∈ S_{l} for some l } ;

(2) there is a unique morphism from (σ, σ′) to (τ, τ′) iff σ is a generalised face of τ and σ′ = τ′, and none otherwise;

(3) the generalised face operators f act by

(σ, σ′) ⟼ (σ, σ′_{f(0),...,f(j)}) .

If S is sufficiently fine, then the geometric realisation of S is homeomorphic to the set \overline{B} ⊂ B × B of (1.39). In particular, B ≅ |S| is then a deformation retract of |S|.

Note that we have implicitly worked with S and |S| in Sections 1.d and 1.e. We will describe the various complexes and superconnections constructed in Section 1 as simplicial functors from S to several simplicial categories that are models for Igusa’s Whitehead space.

6.b. The simplicial Whitehead category. We start by recalling Igusa’s simplicial Whitehead category Wh^{simp}(R, G) of [I2], Chapter 3. The typical object of this category is a system of coefficients a_{αβ}(i_{0}, …, i_{j}) as in Proposition 1.33. In this subsection, R is an associative ring with a unit, and G ⊂ R^{×} is any subgroup.

6.3. Definition. A graded poset (graded partially ordered set) is a finite ℤ-graded set P = ⋃_{n} P^{n} that is partially ordered by a relation ≺ which is independent of the ℤ-grading. For α ∈ P^{n}, we set ind(α) = n. If P, Q are graded posets, a graded closed map from P to Q is a map f: P → Q such that for all α, β ∈ P, ind(f(α)) = ind(α), and f(α) ≺_{Q} f(β) implies α ≺_{P} β.

In particular, a graded bijection P to Q is closed iff the partial relation ≺_{P} is finer than the pullback of ≺_{Q}.

6.4. Definition. Let P be a graded poset, and let V be a left R-module. Let V^{P} = ⨁_{α∈P} V_{α} be a direct sum of left R-modules with V_{α} ≅ V for all α ∈ P. A system of V -valued P -vectors v in degree k assigns to each sequence 0 ≤ i_{0} ≤ ⋯ ≤ i_{l} ≤ k an element v(i_{0}, …, i_{l}) ∈ V. A system of P -matrices a in degree k assigns to each sequence 0 ≤ i_{0} ≤ ⋯ ≤ i_{l} ≤ k a matrix a(i_{0}, …, i_{l}) with coefficients in R, with rows and columns indexed by P. We call a system of matrices (lower) triangular iff a_{αβ}(i_{0}, …, i_{l}) = 0 unless β ≺ α. Let V (k) and M_{P} (R; k) be the spaces of systems of P -vectors and of P -matrices in degree k, and let T_{P} (R; k) ⊂ M_{P} (R; k) be the subspace of triangular systems.

We define a simplicial coboundary operator δ on V (k) by

(1)  (δv)(i_{0}, …, i_{l}) = ∑_{j=0}^{l}(−1)^{j} v(i_{0}, …, î_{j}, …, i_{l}) ,

and similarly on M_{P} (R; k). We let M_{P} (R; k) act on V (k) by multiplication

(2)  (a ∗ v)_{α}(i_{0}, …, i_{l}) = ∑_{β∈P} ∑_{j=0}^{l}(−1)^{(l−j) (ind α−ind β)}a_{αβ}(i_{0}, …, i_{j}) v_{β}(i_{j}, …, i_{l}) ,

and similarly, we define a multliplication on M_{P} (R; k).

6.5. Remark. The sign factor indicates that we have a “horizontal” grading, given by the degree of the simplices we evaluate on, and a “vertical” grading, given by the grading of P. This is compatible with our orientation convention in Section 1.c. We leave it to the reader to check that “∗” is associative and that “δ” is an odd derivation with respect to the total grading:

(a ∗ b) ∗ v = a ∗ (b ∗ v) ,  δ(a ∗ v) = (δa) ∗ v + (−1)^{deg a} a ∗ (δv) ,
(a ∗ b) ∗ c = a ∗ (b ∗ c) ,  and  δ(a ∗ b) = (δa) ∗ b + (−1)^{deg a} a ∗ (δb) ,

where we set

((−1)^{deg a}a)_{αβ}(i_{0}, …, i_{j}) = (−1)^{j+ind α−ind β} a_{αβ}(i_{0}, …, i_{j}) .

6.6. Definition. If a is an element of M_{P} (R; l), we call δ + a a simplicial P -superconnection of degree l. We say that δ + a is flat if

(1)  (δ + a ∗)^{2} = 0

as an operator on M_{P} (R; l). We say that δ + a is of total degree 1 if

a_{αβ}(i_{0}, …, i_{j}) = 0  unless  ind α + j = ind β + 1 .

We say that δ + a is triangular if a is lower triangular with respect to ≺_{P}.

Let δ + a be a flat simplicial P -superconnection. If V = ⨁_{α∈P} V_{α} is a direct sum of left Rmodules and (C^{∗}, ∂) is a cochain complex, then a system of cochain maps I: (C, ∂) → (V (l), δ + a) is an element I ∈ (Hom_{ℤ}(C, V))^{P} (l), such that

(2)  (δ + a ∗) ∘ I = I ∘ ∂ .

We say that I is of degree k, if I_{α}(i_{0}, …, i_{j})|_{C^{n}} = 0 unless ind α + j = n + k.

6.7. Example. Let S be a smooth simplicial complex on B, and let a_{αβ}(σ) be defined as in Section 1.c. For each simplex σ ∈ S_{l}, let P be the set of leaves of p̂: C(|σ|) → |σ|, which is partially ordered by ≺_{σ}. If we set

(6.8)  a_{σ}(i_{0}, …, i_{k}) = \{ \begin{matrix}a(σ_{i_{0},...,i_{k}}) & if 0 ≤ i_{0} < ⋯ < i_{k} ≤ l, and \\ 0 & otherwise, \end{matrix}

then a ∈ T_{P} (ℤ; l), and δ + a is a flat triangular simplicial superconnection δ + a_{σ} of total degree 1 by Proposition 1.33 (1) and (3).

Assume that the bundle T^{u}X → C is oriented, and define I_{σ}: Ω^{∗}(M) → ℝ^{P} (l) by

(6.9)  I_{σ,α}(i_{0}, …, i_{j}) ω = \{ \begin{matrix}∫_{M_{α}(σ_{i_{0},...,i_{j}})} ω & if 0 ≤ i_{0} < ⋯ < i_{k} ≤ l, and \\ 0 & otherwise. \end{matrix}

Then I_{σ}: (Ω^{∗}(M), d^{M}) → (ℝ^{P} (l), δ + a) is a system of cochain maps of degree 0 by Stokes’ theorem and Proposition 1.33 (2).

Let G be a subgroup of the multiplicative group R^{×}. As a model for the Whitehead space mentioned at the beginning of this section, we construct a simplicial category Wh^{simp}(R, G). In fact, we first define the pre-Whitehead category Wh^{′simp}(R, G). The Whitehead category itself is obtained from Wh^{′simp}(R, G) by adding the so-called stabilisation morphisms, and the (pre-) Whitehead space is just the geometric realisation of the (pre-)Whitehead category. This category is defined precisely as in [I2], Chapter 3. However, we use a different, but equivalent sign convention, cf. Remark 1.34.

6.10. Definition. The simplicial pre-Whitehead category Wh^{′simp}(R, G) is defined as follows.

(1) The objects Wh_{k}^{′simp}(R, G) of degree k are pairs (P, a) where P is a graded poset, and δ + a is a flat triangular simplicial superconnection of total degree 1 with a ∈ T_{P} (R; k), such that a(i_{0}, …, i_{l}) = 0 unless i_{0} < ⋯ < i_{l}.

(2) For (P, a), (Q, b) ∈ Wh_{k}^{′simp}(R, G), the morphisms from (P, a) to (Q, b) are pairs (f, g) of a graded closed bijection f: P → Q and a map g: P → G, such that

b_{f(α),f(β)}(i_{0}, …, i_{l}) = g_{α} a_{αβ}(i_{0}, …, i_{l}) g_{β}^{−1}

for all α, β ∈ P and all 0 ≤ i_{0} < ⋯ < i_{l} ≤ k.

(3) face operators are given by re-indexing:

F_{j_{0}...j_{m}}(P, a) = (P, b) ∈ Wh_{m}^{′simp}(R; G)  with  b(i_{0}, …, i_{l}) = a(j_{i_{0}}, …, j_{i_{m}})

for all sequences 0 ≤ j_{0} ≤ ⋯ ≤ j_{m} ≤ k and 0 ≤ i_{0} ≤ ⋯ ≤ i_{l} ≤ m.

The simplicial pre-Whitehead space Wh^{′simp}(R, G) is the geometric realisation of the simplicial pre-Whitehead category.

In order to keep the category Wh^{′simp}(R, G) small, we agree that all our posets P are subsets of some large fixed set. Since the poset P is mainly a book-keeping device, this convention will not affect the homotopy type of any space that we construct.

6.11. Remark. Fix a complex representation ρ: R → M_{r}(ℂ). Then the simplicial pre-Whitehead space carries two natural ℤ-graded flat complex vector bundles.

(1) There is a natural ℤ-graded flat bundle V → Wh^{′simp}(R, G) with fibre (ℂ^{r})^{P} over each point of |(P, a)| for each k and each (P, a) ∈ Wh_{k}^{′simp}(R, G). The structure group of V is the group of G-monomial matrices.

(2) Because a(i)^{2} = 0, we also have cohomology groups H^{∗}(V^{P} , a(i)) for each 0 ≤ i ≤ k. Flatness of δ + a implies that id +a(i_{0}, i_{1}) induces a cochain isomorphism from (V^{P} , a(i_{1})) to (V^{P} , a(i_{0})), so we get isomorphisms

(id +a(i_{0}, i_{1}))_{∗}: H^{∗}(V^{P} , a(i_{1})) ⟶ H^{∗}(V^{P} , a(i_{0}))

for 0 ≤ i_{0} ≤ i_{1} ≤ l. For i_{0} < i_{1} < i_{2}, the maps a(i_{0}, i_{1}, i_{2}) are cochain homotopies between (id +a(i_{0}, i_{1})) ∘ (id +a(i_{1}, i_{2})) and (id +a(i_{0}, i_{2})), so all spaces H^{∗}(V^{P} , a(i)) can be identified in a compatible way. Morphisms (f, g): (P, a) → (Q, b) induce maps

(f, g)_{∗}(i): H^{∗}(V^{P} , a(i)) ⟶ H^{∗}(V^{Q}, b(i))

that are compatible with the isomorphisms (id +a(i_{0}, i_{1}))_{∗} above. This way, we obtain a natural ℤ-graded flat bundle

H = H^{∗}(V, a) ⟶ Wh^{′simp}(R, G)

of cohomology groups. Note that the structure group of H is a subgroup of GL(R), which is in general not related to G.

So far, each connected component of Wh^{′simp}(R, G) is built from graded posets P having the same number n_{k} of elements degree k for each k. Using Igusa’s stabilisation construction, we add more morphisms to the underlying categories Wh^{′simp}(R, G) between connected components of Wh^{′simp}(R, G). This gives us the (stabilised) Whitehead space Wh^{simp}(R, G). These extra morphisms are necessary to account for the changes of the family Thom-Smale complex that occur at birth-death singularities. We will check that all our constructions are compatible with stabilisation. In particular, we prove that the universal torsion class T (R, G) we are going to define in Section 7.b extends to Wh(R, G).

The only new ingredients in the final construction of Wh(R, G) are elementary expansions of simplicial superconnections. Let P^{[j]} denote the poset consisting of two elements α^{−} in degree j and α^{+} in degree j + 1, with α^{−} ≺ α^{+}. Let a^{[j]} ∈ M_{P^{[j]}}(R, k) denote the system of P^{[j]}-matrices given by

(6.12)  a^{[j]}(i_{0}, …, i_{l}) = \{ \begin{matrix}\binom{0 1}{0} & if l = 0, and \\ ( \begin{matrix}0 \\ & 0 \end{matrix} ) & if l ≥ 1. \end{matrix}

Then (P^{[j]}, a^{[j]}) is an object in Wh_{k}^{′simp}(R, G), which we call the elementary object of vertical degree j in Wh_{k}^{′simp}(R, G).

We can define a direct sum on the objects of Wh_{k}^{′simp}(R, G) by setting

(P, a) ⊕ (P′, a′) = (P ∪ P′, a ⊕ a′) .

Here, the partial ordering on P ∪ P′ is the union of the partial orderings on P and P′, so α ≺ β iff either α, β ∈ P and α ≺ β in P, or α, β ∈ P′ and α ≺ β in P′.

6.13. Definition. Let (P, a) be an object of Wh_{k}^{′simp}(R, G). Then the object (P, a) ⊕ (P^{[j]}, a^{[j]}) is called the elementary expansion of (P, a) in degree j.

We can now define the stabilised simplicial Whitehead category.

6.14. Definition. The (stabilised) simplicial Whitehead category Wh^{simp}(R, G) has the same objects and face operators as Wh^{′simp}(R, G). Its morphisms are iterated formal compositions of Wh^{′simp}(R, G)-morphisms and elementary expansions

(P, a) ⟶ (P, a) ⊕ (P^{[j]}, a^{[j]}) .

The simplicial Whitehead space Wh^{simp}(R, G) is the geometric realisation of the simplicial Whitehead category.

6.15. Remark. Note that there is a natural inclusion

Wh^{′simp}(R, G) ⟶ Wh^{simp}(R, G) ,

since Wh^{simp}(R, G) has more morphisms than Wh^{′simp}(R, G), and thus Wh^{simp}(R, G) has additional bi-simplices. We will call a flat bundle or a cohomology class on Wh^{′simp}(R, G) compatible with stabilisation if it is the pull-back of a corresponding object on Wh^{simp}(R, G).

The morphisms of the stabilised category still induce isomorphisms on the level of cohomology, because expansion pairs do not contribute to the cohomology. In particular, we still have a welldefined bundle H → Wh^{simp}(R, G) as in Remark 6.11 (2), which pulls back to H. However, the vector bundle V → Wh^{′simp}(R, G) extends to Wh^{simp}(R, G) only as an equivalence class of vector bundles up to elementary expansions. This means that we will not define higher torsion classes directly on Wh^{simp}(R, G), but we will instead look for classes on H → Wh^{′simp}(R, G) that are compatible with stabilisation.

We also recall the definition of the acyclic Whitehead space, which plays an important role in [I2].

6.16. Definition. The acyclic simplicial Whitehead category Wh^{simp,h}(R, G) is the full subcategory of Wh^{simp}(R, G) containing all objects (P, a) such that the complex (V^{P} , a(0)) is acyclic. The acyclic simplicial Whitehead space Wh^{simp,h}(R, G) = |N Wh^{simp,h}(R, G)| is the geometric realisation of this category.

We remark that Wh^{simp,h}(R, G) is the union of all connected components of Wh^{simp}(R, G) for which the bundle H vanishes.

6.c. A functor to the Whitehead category. Let M → B be a bundle of compact manifolds, and let F → M be a flat bundle with structure group G ⊂ R^{×}, such that ρ|_{G}: G → GL_{r}(ℂ) is faithful (injective). E.g., if F is a complex vector bundle, we may take R = M_{r}(ℂ) to be the ring of complex r × r-matrices, and G ⊂ GL_{r}R, where r is the complex rank of F. If F is unitarily flat, then G ⊂ U(r). In general, if F is associated to an r-dimensional complex representation of π_{1}(M) that factors over a group G, so

π_{1}(M) ⟶ G ⟶^{ρ′} GL_{r}(ℂ) ,

we may take R = ℤ(G) the group ring of G, and the representation ρ: ℤ(G) → M_{r}(ℂ) induced by ρ′.

Let h be a fibre-wise Morse function on M with fibre-wise critical set C, and let S be a smooth simplicial structure on B as in Definition 1.11, which is fine enough to satisfy (1.36). We want to interpret the construction in Sections 1.c and 1.d of [G] as the construction of a functor from the simplicial category S of Example 6.2 to Wh^{′simp}(R, G). Throughout this subsection, we fix a ring R and a complex representation ρ: R → M_{r}(ℂ). Then there are natural flat complex vector bundles V, H → Wh^{′simp}(R, G) as in Remark 6.11. Note that unless T^{u}X → C is oriented, we have to make sure that −1 ∈ ρ′(G) by enlarging G if necessary.

Supposing that M is connected, let us fix a basepoint o ∈ M. For each σ ∈ S, let P (σ) denote the set of connected components of C|_{M̊_{α}(σ)}, which we order by the partial relation ≺_{σ} of (1.14). If τ is a face of σ, then the relation ≺_{σ} is finer than ≺_{τ}, so the natural identification

f_{τ,σ}: P (σ) ⟶ P (τ)

is a closed bijection.

We take the CW structure on M with cells M_{α}(σ) for each simplex σ ∈ S and each fibre-wise critical level C_{α} of h over |σ| as in Section 1.c. For each α ∈ P (σ), we fix a path γ_{α}(σ) from o to some point in M̊_{α}(σ). Let us identify the space O_{F} (M_{α}(σ)) of parallel sections of F |_{M̊_{α}(σ)} with F_{o} by parallel translation along γ_{α}(σ). We also fix an orientation o_{α}(σ) of T^{u}X|_{C_{α}(σ̊)} and set

V (σ) = ⨁_{α} V_{α}(σ) .

Working with the simplicial set S of Example 6.2, for each σ ∈ S_{k} we have to construct an element a ∈ T_{P (σ)}(R; k) such that δ + a is a flat simplicial superconnection of total degree 1. Therefore, we fix orientations on the bundles T^{u}X|_{C_{α}(σ̊)} and regard the coboundary operator δ + a constructed in Proposition 1.33 for trivial coefficients, and in Corollary 1.45 for the general case. The coefficient a_{αβ}(σ_{i_{0}...i_{j}}) arises as a sum of contributions of flow lines of W going from C_{α}(σ_{i_{0}}) to C_{β}(σ_{i_{j}}). Together with γ_{α}(σ), γ_{β}(σ) and paths inside C_{α}(|σ|) and C_{β}(|σ|), these connecting lines give rise to a loop in M. Since parallel translation along this loop acts by an element of G, we may regard a_{αβ}(σ_{i_{0}...i_{j}}) as an element of ℤ(G), or of any ring R containing G. By Proposition 1.33 (1), (3) and by Corollary 1.45 (2), (3), these coefficients give rise to a flat triangular simplicial superconnection δ + a_{σ} of total degree 1.

Now to each pair (σ, σ′) ∈ S, where σ′ = σ_{i_{0}...i_{j}} is a face (or degeneracy) of σ, we associate the graded poset P (σ) and the restriction of the simplicial superconnection δ + a_{σ} to σ′ following Definition 6.10 (3). Set

(6.17)  Ξ_{p,h,F}^{simp} (σ, σ′) = (P (σ), a_{σ}|_{σ′}) ∈ Wh_{j}^{′simp}(R, G) .

Face operators in S map (σ, σ′) to (σ, σ′′), where σ′′ is a face of σ′. This implies that Ξ_{p,h,F}^{simp} commutes with generalised face operators.

6.18. Proposition. The assigment Ξ_{p,h,F}^{simp} constructed above is a simplicial functor from S to the simplicial pre-Whitehead category Wh^{′simp}(R, G). The flat bundles V → B and H → B are naturally isomorphic to the pull-backs by |Ξ_{p,h,F} | of the bundles V, H → Wh^{′simp}(R, G) of Remark 6.11. If S′ is another sufficiently fine smooth simplicial complex on B, S′ is the corresponding simplicial category, and Ξ_{p,h,F}^{simp′}: S′ → Wh(R; G) is another functor constructed as above, possibly with different choices than before, then the diagramme

\begin{aligned}& |S| \\ B & & Wh^{′simp}(R; G) \\ & |S′| \end{aligned}∼ ∼_{Ξ_{p,h,F}^{simp}}_{Ξ_{p,h,F}^{simp′}}

commutes up to homotopy, where the maps on the left are constructed using partitions of unity as in (1.63).

6.19. Remark. Conversely, given S and S as before, a functor Ξ: S → Wh^{′simp} induces

(1) a flat ℤ-graded complex vector bundle (V, ∇^{V}) → B of the form V = p̂_{∗}V̂, where V̂ → C is a vector bundle of rank r with structure group G on some finite cover p̂: C → B of B; and

(2) a flat simplicial superconnection δ + a of total degree 1 acting on the space of V -valued simplicial cochains, which is triangular over each simplex σ with respect to some partial ordering on the leaves of C|σ̊.

In other words, the coefficients a_{αβ}(σ) satisfy Proposition 1.33 (1) and (3). Note that property (2) in Proposition 1.33 is meaningless here because we cannot reconstruct the space M from V and a.

This means that Wh^{simp}(R, G) = | Wh^{simp}(R; G)| is a “universal space” for the flat triangular simplicial superconnections of total degree 1 described above. The geometric realisation

ξ_{p,h,F}^{simp} = |Ξ_{p,h,F}^{simp} |: B ⟶ Wh^{simp}(R, G)

is well-defined up to homotopy; it is the “classifying map” for the given flat triangular simplicial superconnection.

Proof of Proposition 6.18. Recall that there is an S-morphism from (σ, σ′) to (τ, τ′) iff τ is a face of σ and σ′ = τ′, and that this morphism is unique. We noted above that the natural identification f_{τ,σ}: P (σ) → P (τ) is a closed bijection of graded posets.

For α ∈ P (σ), the trivialisations of V_{α}(σ) and V_{f_{τ,σ}α}(τ) differ by an element g of G. If the orientations on C_{α}(σ) chosen above restricts to the one chosen on C_{f_{τ,σ}α}(σ), we set g_{τ,σ;α} = g, otherwise we set g_{τ,σ;α} = −g.

Note that by Proposition 1.33 (1), a_{σ;α,β}(σ′′) = 0 unless f_{τ,σ}β ≺_{τ} f_{τ,σ}α. Now it is easy to check that

a_{τ;f_{τ,σ}α,f_{τ,σ}β}(σ′′)g_{τ,σ;α} ∘ a_{σ;α,β}(σ′′) ∘ g_{τ,σ;β}^{−1}

for all generalised faces σ′′ of σ′. Thus, the pair

Ξ_{p,h,F}^{simp} ((σ, σ′) ⟶ (τ, σ′)) = (f_{τ,σ}, g_{τ,σ})

is a morphism in Wh^{′simp}(R; G). Functoriality of Ξ_{p,h,F}^{simp} is now trivial.

To show that ξ_{p,h,F}^{simp} is well-defined up to homotopy, we regard two smooth simplicial complexes on B × \{0\} and B × \{1\}. These can be combined to a smooth simplicial complex on \overline{B} = B × [0, 1] with corresponding simplicial category \overline{S}. We may now construct a functor

\overline{Ξ}_{p,h,F}^{simp} : \overline{S} ⟶ Wh^{′simp}(R, G)

that restricts to given functors on simplices in B × {0} and B × {1}. Our claim follows from the relativity of the constructions in Section 1, see e.g. the proof of Proposition 1.70. □

Next, we want to show that the functor Ξ_{p,h,F}^{simp} : S → Wh^{′simp}(R, G) constructed above is related to the one constructed in [I2], Section 4.3, by a natural transformation. This will then imply that the realisations are homotopic.

We therefore go back to Section 1.c, where we constructed a cell complex consisting of one cell M_{α}(σ) for each simplex σ of a simplicial decomposition S of B and each fibre-wise critical level α of h over σ̊ ⊂ B. We may assume that S is sufficiently fine for the constructions both in Section 1 and in Chapter 4 of [I2], e.g., we may take S to be the first barycentric subdivision of a simplicial complex satisfying (1.13) and (1.36). This implies that any sequence of simplices σ ⊃ τ_{0} ⊃ ⋯ ⊃ τ_{k} in [I2] corresponds to a single simplex in Section 1.

The next technical issue is the choice of partial ordering. In Section 1, we work with the partial order ≺_{σ} of (1.14), while Igusa uses a “standard” and a “minimal order. The standard order is defined by saying that

α <_{σ}^{s} β  iff  h_{α} < h_{β}  on |σ|.

The minimal order is generated by α <^{min} β if there exists a negative gradient flow line from C_{β}(σ) to C_{α}(σ) for a fixed vertical metric g^{T X}.

Note that both ≺_{σ} and <_{σ}^{min} are refinements of the corresponding orders on the faces of σ, while <_{σ}^{s} behaves the other way. Also, the standard order is always finer than ≺, which is in turn finer than the minimal order by Remark 1.15. If (P, a) is an object of Wh^{simp}(R, G), then there is a morphism from (P, a) to the object (P^{min}, a), where P^{min} is the same set as P, but equipped with the minimal ordering with respect to a. Let Ξ_{p,h,F}^{min} : S → Wh^{′simp}(R, G) be the corresponding functor, then there is a natural transformation from Ξ_{p,h,F}^{simp} to Ξ_{p,h,F}^{min} , so their geometric realisations are homotopic.

But the functor Ξ_{p,h,F}^{min} gives an admissible choice of elements Ξ^{min}(σ) ∈ M_{P^{min}(σ)}(R; k) for the construction leading to Igusa’s Theorem 4.3.8 (recall that σ is a simplex of the barycentric subdivision, so it corresponds to a sequence of simplices in [I2]). In particular, the realisation of the functor Ξ_{p,h,F}^{simp} : S → Wh^{′simp}(R, G) is homotopic to the pullback of the realisation of Igusa’s Ξ^{min} to |S| by Theorem 4.3.8 in [I2]. Let us summarize this.

6.20. Theorem. The constructions in Chapter 4 of [I2] and in Section 1 of this paper give rise to homotopic maps from |S| ∼ B to Wh^{′simp}(R, G).

6.d. The differential and the mixed model of the Whitehead space. We introduce two more models Wh^{diff}(R, G) and Wh^{mix}(R, G) of the Whitehead space (note that our notion of Wh^{diff}(R, G) differs from Igusa’s). Because we have to interpolate and differentiate the coefficients of mixed and differential superconnections, the ring R must now be an ℝ-algebra (typically, R = M_{r}(𝕜) for 𝕜 = ℝ, ℂ or ℍ). However, we may still work with any suitable group G ⊂ R^{×}. We still fix a complex representation ρ: R → M_{r}(ℂ).

Let Δ_{k} denote the standard affine l-simplex, and let f_{i_{0}...i_{l}}: Δ_{l} → Δ_{k} be the unique affine map sending the j-th vertex of Δ_{l} to the i_{j}-th vertex of Δ_{k}. Let T_{P} (R) ⊂ M_{P} (R) denote the algebra of P -triangular matrices, cf. Definition 6.4. For a superconnection A′ on the trivial bundle V^{P} ×Δ_{k} → Δ_{k}, we write

A′ = d + ∑_{j} a′_{j}

with a′_{j} ∈ Ω^{j}(Δ_{k}; End V^{P}) as usual. Recall that superconnections of total degree 1 were defined in Definition 2.21.

6.21. Definition. The differential pre-Whitehead category Wh^{′diff}(R, G) is defined as follows.

(1) The objects Wh_{k}^{′diff}(R, G) of degree k are pairs (P, a′) where P is a graded poset, and d+a′ is a flat superconnection of total degree 1 on the bundle Δ_{k} × (ℂ^{r})^{P} → Δ_{k} with a′ ∈ Ω^{∗}(Δ_{k}, T_{P} (R)).

(2) For (P, a′) and (Q, b′) ∈ Wh_{k}^{′diff}(R, G), the morphisms from (P, a′) to (Q, b′) are pairs (f, g) of a graded closed bijection f: P → Q and a map g: P → G, such that

b′_{f(α),f(β)} = g_{α} a′_{αβ} g_{β}^{−1} ∈ Ω^{∗}(Δ_{k}, R) for all α, β ∈ P.

(3) Generalised face operators F_{i_{0}...i_{l}} act by pulling d + a′ back by the corresponding affine map f_{i_{0}...i_{l}}: Δ_{l} → Δ_{k}, so

F_{i_{0}...i_{l}}(P, a′) = (P, f_{i_{0}...i_{l}}^{∗}a′) .

The differential pre-Whitehead space Wh^{diff}(R, G)is the geometric realisation of the differential pre-Whitehead category.

6.22. Remark. As in Remark 6.11, the complex representation ρ gives rise to a flat bundle V → Wh^{′diff}(R, G). Over each geometric simplex of Wh^{′diff}(R, G), we consider the space of smooth Vvalued forms. Define Ω^{∗}(Wh^{′diff}(R, G), V) to be the set of functions that assign a smooth V-valued form ω(σ) to each simplex σ of Wh^{′diff}(R, G), such that ω(σ′) = ω(σ)|_{σ′} for each face σ′ of σ. Then d + a′ induces a piece-wise smooth flat superconnection A′ on Ω^{∗}(Wh^{′diff}(R, G), V). Let H = H^{∗}(V, a′_{0}) → Wh^{′diff}(R, G) be the flat bundle of the fibre-wise cohomology of (Ω^{∗}(Wh^{′diff}; V), A′), then H also carries a natural flat connection, which is induced by A′.

Arguing as in Section 6.c, the differential pre-Whitehead space is universal for the kind of S-piece-wise smooth flat superconnections on vector bundles over simplicial complexes that we discussed at the beginning of Section 1.f. Similarly, we now define the mixed pre-Whitehead space, which is universal for the mixed superconnections constructed in Proposition 1.51. Let Δ_{k\j} denote the span of the vertices j, j + 1, …, k of Δ_{l}.

6.23. Definition. Let P be a graded poset. A mixed system of triangular P -matrix-valued forms on Δ_{k} is a pair (a, a′), where a ∈ T_{P} (R; k) is a triangular system of P -matrices, and a′ assigns a differential form a′(i_{0}, …, i_{l−1}) ∈ Ω^{∗}(Δ_{k\i_{l−1}}; T_{P} (R)) to each sequence 0 ≤ i_{0} ≤ ⋯ ≤ i_{l−1} ≤ k with 0 ≤ l, with

(1)  a′(i_{0}, …, i_{l−1})|_{Δ_{k}(i_{l})} = a(i_{0}, …, i_{l})

if i_{l−1} ≤ i_{l} ≤ k. Let M_{P}^{mix}(Δ_{k}; R) denote the space of all mixed systems of triangular P -matrix-valued forms with coefficients in R.

We define two operators δ and d on M_{P}^{mix}(Δ_{k}; R) by δ(a, a′) = (δa, δ′a′) and d(a, a′) = (0, d′a′) with

(2)  (δ′a′)(i_{0}, …, i_{l−1}) = ∑_{j=0}^{l−1}(−1)^{j} a′(i_{0}, …, î_{j}, …, i_{l−1}) + (−1)^{l} a(i_{0}, …, i_{l−1}) ,
(3)  (d′a′)(i_{0}, …, i_{l−1}) = (−1)^{l}d(a′(i_{0}, …, i_{l−1})) .

We also define a multiplication on M_{P}^{mix}(Δ_{k}; R) by (a, a′) ∗ (b, b′) = (a ∗ b, a′ ∗′ b′), with

(4) (a′ ∗′ b′)_{αγ}(i_{0}, …, i_{l−1})

= ∑_{j=0}^{l−1} ∑_{β∈P}(−1)^{(l−j)(ind α−ind β)} a_{αβ}(i_{0}, …, i_{j}) b′_{βγ}(i_{j}, …, i_{l−1})

+ (a′(i_{0}, …, i_{l−1}) b′(∅))_{αγ} .

In (2) and (4), a(i_{0}, …, i_{j}) is regarded as a constant T_{P} (R)-valued function on Δ_{k\i_{j}}.

6.24. Remark. We collect some properties of the definition above.

(1) Note that (2), (3) and (4) are compatible with (1). Note also that by (1), it would have been enough to specify a′ instead of (a, a′), this is why we may write δ′a′ instead of δ′(a, a′) and a′ ∗′ b′ instead of (a, a′) ∗′ (b, b′). We have chosen the notation (a, a′) mainly for compatibility with Section 1.e, see also Remark 1.52.

(2) We remark in particular that a′(∅) ∈ Ω^{∗}(Δ_{k}; End V^{P}), so that we can define a superconnection A′ = d + a′(∅) on the trivial bundle with fibre V^{P}. Let a′_{j} = a′(∅)^{[j]} ∈ Ω^{j}(Δ_{k}; End V^{P}) denote its component of degree j.

(3) The sign factors indicate that we have again two gradings: first, the simplicial degree, which is l for a′(i_{0}, …, i_{l−1}) and a(i_{0}, …, i_{l}); second, we have the sum of the exterior degree and the degree given by P on Ω^{∗}(Δ_{k\i_{l−1}}; T_{P} (R)). Thus, (a, a′) ∈ M_{P}^{mix}(Δ_{k}; R) is of total degree m iff

a′_{αβ}(i_{0}, …, i_{l−1}) ∈ Ω^{m−l−ind α+ind β}(Δ_{k\i_{l−1}}) .

Note that “d” and “δ” are super-commuting odd derivations with respect to the total degree, and that “∗” is associative.

6.25. Definition. The mixed pre-Whitehead category Wh^{′mix}(R, G) is defined as follows.

(1) The objects Wh_{k}^{′simp}(R, G) of degree k are triples (P, a, a′), where P is a graded poset, and (a, a′) ∈ M_{P}^{mix}(Δ_{k}; R) is of total degree 1 and satisfies

a(i_{0}, …, i_{l}) = a′(i_{0}, …, i_{l−1}) = 0

unless i_{0} < ⋯ < i_{l}, and

d(a, a′) + δ(a, a′) + (a, a′) ∗ (a, a′) = 0 .

(2) For (P, a) and (Q, a′) ∈ Wh_{k}^{′mix}(R, G), the morphisms from (P, a) to (Q, a′) are pairs (f, g) of a graded closed bijection f: P → Q and a map g: P → G, such that

a′_{f(α),f(β)}(i_{0}, …, i_{l−1}) = g_{α} a_{αβ}(i_{0}, …, i_{l−1}) g_{β}^{−1} ∈ Ω^{∗}(Δ_{k\i_{l−1}}, R)

for all α, β ∈ P and all 0 ≤ i_{0} ≤ ⋯ ≤ i_{l−1} ≤ k.

(3) For a sequence 0 ≤ j_{0} ≤ ⋯ ≤ j_{m} ≤ k, the face operator F_{j_{0}...j_{m}}: Wh_{k}^{′mix}(R, G) → Wh_{m}^{′mix}(R, G) is given by F_{j_{0}...j_{m}}(P, a, a′) = (P, b, b′), with

b(i_{0}, …, i_{l}) = a(j_{i_{0}}, …, j_{i_{l}})  and  b′(i_{0}, …, i_{l−1}) = f_{j_{0}...j_{m}}^{∗}(a(j_{i_{0}}, …, j_{i_{l−1}}))

for all 0 ≤ i_{0} ≤ ⋯ ≤ i_{l−1} ≤ m. The mixed Whitehead space Wh^{′mix}(R, G) is the geometric realisation of the mixed Whitehead category.

6.26. Remark. As in Remark 6.11 and Remark 6.22, the complex representation ρ gives rise to flat bundles V → Wh^{′mix}(R, G) and H → Wh^{′mix}(R, G). Here, V is now equipped with a mixed superconnection.

6.27. Example. The coefficients a′(σ, ·) constructed in Proposition 1.51 clearly give rise to an object (P_{σ}, a_{σ}, a′_{σ}) ∈ Wh_{k}^{mix}(R; G) if σ ∈ S_{k}, by

a_{σ}(i_{0}, …, i_{l}) = a(σ_{i_{0}...i_{l}})  and  a′_{σ}(i_{0}, …, i_{l−1}) = σ^{∗}a′(σ, σ_{i_{0}...i_{l−1}}) .

Mimicking the proof of Proposition 6.18, we can show that this defines a functor Ξ_{p,h,F}^{mix} : S → Wh^{′mix}(R; G).

To pass to the full Whitehead category, we define elementary extensions of differential and mixed superconnections. Let P^{[j]} and a^{[j]} be given as in (6.12). Similarly, we define a′ as in Definition 6.23 with only one non-zero component

a_{0}^{′[j]} = a^{′[j]}(∅) = ( _{0}^{0 1} ) .

We also use the symbol “⊕” as in Definition 6.13.

6.28. Definition. The elementary expansions of (P, a′) ∈ Wh_{k}^{′diff} and (P, a, a′) ∈ Wh_{k}^{′mix} are given by (P, a′) ⊕ (P^{[j]}, a^{′[j]}) and (P, a, a′) ⊕ (P^{[j]}, a^{[j]}, a^{′[j]}), respectively.

6.29. Definition. The (stabilised) differential Whitehead category Wh^{diff}(R, G) has the same objects and face operators as Wh^{′diff}(R, G). Its morphisms are iterated formal compositions of Wh^{′diff}(R, G)-morphisms and elementary expansions. The (stabilised) differential Whitehead space Wh^{diff}(R, G) is the geometric realisation of Wh^{diff}(R, G).

The (stabilised) mixed Whitehead category Wh^{mix}(R, G) and the (stabilised) mixed Whitehead space Wh^{mix}(R, G) are defined analogously.

As before, we define the full subcategories Wh^{diff,h}(R, G) ⊂ Wh^{diff}(R, G) and Wh^{mix,h}(R, G) ⊂ Wh^{mix}(R, G) consisting of all fibre-wise acyclic objects.

6.30. Remark. The category Wh^{mix}(R; G) encompasses both Wh^{simp}(R; G) and Wh^{diff}(R; G). In fact, there are forgetful functors

Φ_{1}: Wh^{mix}(R; G) ⟶ Wh^{simp}(R; G)  with  (P, a, a′) ⟼ (P, a) ,
and  Φ_{2}: Wh^{mix}(R; G) ⟶ Wh^{diff}(R; G)  with  (P, a, a′) ⟼ (P, a′(∅)) .

If (f, g) is a morphism in Wh^{mix}(R; G), then Φ_{1}(f, g) and Φ_{2}(f, g) are still given by (f, g) in the corresponding category.

It follows from the proof of Proposition 1.51 that Φ_{1} has a right-inverse on objects. We will now show that in fact, both functors have right-inverses as functors, which make them homotopy equivalences on the level of realisations. This implies in particular that during the passage from a simplicial superconnection on V → B to a smooth one in Section 1, up to homotopy, no information was lost. It also implies that the simplicial, mixed, and smooth superconnections of Section 1 are all “classified” by maps into the same pre-Whitehead space Wh′(R; G) that was used by Igusa to construct his torsion classes τ_{∗}(M/B; F).

6.31. Theorem. The forgetful functors

Φ_{1}: Wh^{mix}(R; G) ⟶ Wh^{simp}(R; G)  and  Φ_{2}: Wh^{mix}(R; G) ⟶ Wh^{diff}(R; G)

have homotopy inverses Ψ_{1} and Ψ_{2} on realisations.

6.32. Remark. For each complex representation ρ of R, we clearly have

V ≅ |Φ_{1}|^{∗} V ≅ |Φ_{2}|^{∗} V ⟶ Wh^{′mix}(R, G) ,
and  H ≅ |Φ_{1}|^{∗} H ≅ |Φ_{2}|^{∗} H ⟶ Wh^{mix}(R, G)

as flat bundles. Also by construction, the flat simplicial superconnection on V and the flat differential superconnection on V arise by forgetting parts of the mixed superconnection on V → Wh^{′mix}(R, G).

Proof of Theorem 6.31. We start with the functor Φ_{1}, and we work on the level of pre-Whitehead categories, taking care that all our constructions are compatible with stabilisation. The construction in the proof of Proposition 1.51 assigns to each object (P, a) of Wh^{′simp}(R; G) an object (P, a, a′) of Wh^{′mix}(R; G). However, this construction is not natural, so it is not clear that we obtain a functor from Wh^{′simp}(R; G) to Wh^{′mix}(R; G). Since morphisms are unaffected by generalised face operators and by the functor Φ_{1}, it is clear that a right-inverse Ψ_{1} of Φ_{1} must also map each Wh^{′simp}(R; G)-morphism (f, g) to the Wh^{′mix}(R; G)-morphism (f, g).

We proceed by induction over the simplicial degree. In degree 0, the functor Φ_{1} restricts to an identity of categories

Wh_{0}^{mix}(R; G) = Wh_{0}^{simp}(R; G) ,

so Ψ_{1,0} = Φ_{1,0}^{−1} is simply the inverse of this identity. We assume now that we have constructed functors Ψ_{1,k′}: Wh_{k′}^{simp}(R; G) → Wh_{k′}^{mix}(R; G) for all k′ < k, such that Φ_{1} ∘ Ψ_{1,k′} is the identity on Wh_{k′}^{simp}(R; G).

Given any object (P, a) ∈ Wh_{k}^{simp}(R; G), we want to construct Ψ_{1,k}(P, a) ∈ Wh^{mix}(R; G). If (P, a) is degenerated, then Ψ_{1,k}(P, a) is already determined by compatibility with generalised face operators. Otherwise, we first replace P by the graded poset P′ which is identical with P as a graded set, but carries the minimal partial ordering that supports a. In other words, we have α ≺′ β iff there exists 0 ≤ i_{0} < ⋯ < i_{l} ≤ k such that a_{αβ}(i_{0}, …, i_{l}) ≠ 0. Then it follows from Definition 6.10 (2) that any morphism (f, g): (P′, a) → (Q, b) is invertible. Also, assume that (Q, b) lies in the same connected component of the nerve N Wh_{k}^{′simp}(R; G), then there exists a morphism (Q, b) → (P′, a).

Assume that we can construct an Aut(P′, a)-invariant object (P′, a, a′) ∈ Wh_{k}^{simp}(R; G), such that

F_{j_{0}...j_{k′}} (P′, a, a′) = Ψ_{1,k′} F_{j_{0}...j_{k′}} (P′, a)

for all proper face operators F_{j_{0}...j_{k′}} with k′ < k. Then for any morphism (f, g): (Q, b) → (P′, a), we define

Ψ_{1,k}(Q, b) = (Q, b, b′)  with  b′_{αβ}(i_{0}, …, i_{l−1}) = g_{α}^{−1} ā′_{f^{−1}(α),f^{−1}(β)}(i_{0}, …, i_{l−1}) g_{β} .

By minimality of P′, the coefficients of b′ are Q-triangular. Also by minimality and because all morphisms are bijective, any two morphisms (f, g), (f′, g′): (Q, b) → (P′, a) are related by an automorphism of (P′, a). Since (P′, a, a′) is Aut(P′, a)-invariant, it does not matter which morphism (f, g): (Q, b) → (P′, a) we choose to define b′.

We first do all this only for objects (P′, a) that are not elementary expansions of other objects. Compatibility with stabilisation then forces us to extend Ψ_{1,k} in a prescribed way to all remaining objects in Wh_{k}^{simp}(R, G). It is now easy to check that our definition of Ψ_{1,k} is compatible with face operators, morphisms and elementary expansions.

In order to construct a′, we proceed as in the proof of Proposition 1.51, keeping in mind that a′ should be invariant under all automorphisms of a. Assume that 0 ≤ i_{0} < … i_{l−1} ≤ k with i_{l−1} > l − 1, then a′(i_{0}, …, i_{l−1}) is already completely determined by Ψ_{1,k′} F_{i_{0},...,i_{l−1},i_{l−1}+1,...,k}(P′, a). Since

Aut(P′, a) ⊂ Aut(F_{i_{0},...,i_{l−1},i_{l−1}+1,...,k}(P′, a)) ,

the coefficient a′(i_{0}, …, i_{l−1}) is Aut(P′, a)-invariant. Let us assume that we already inductively constructed all a′(0, …, l′ −1) with l < l′ ≤ k+1. Then the restriction of a′(0, …, l−1) to ∂Δ_{k\l−1} is determined by induction and by (1.55) (for l ≥ 1) or (1.58) (for l = 0), and is also Aut(P′, a)- invariant. Because Aut(P′, a) acts linearly on T_{P} (R), the fixpoint set is an ℝ-subalgebra of T_{P} (R). We can thus find an invariant continuation of a′(0, …, l − 1)|_{∂Δ_{k\l−1}} to Δ_{k\l−1}. This completes the construction of (P′, a, a′) ∈ Wh_{k}^{mix}(R; G).

This completes the construction of Ψ_{1}. It remains to show that Ψ_{1} ∘Φ_{1} is simplicially homotopic to the identity on Wh_{k}^{mix}(R; G). Such a homotopy can be constructed by the same procedure as above, see e.g. the proofs of Proposition 1.70 and Proposition 6.18. Details are left to the reader.

Now, consider the functor Φ_{2}: Wh^{mix}(R; G) → Wh^{simp}(R; G). The general arguments are the same as in the construction of Ψ_{1}, so it is enough to give an inductive Aut(P, a′)-invariant construction of (P, ā, ā′) = Ψ_{1}(P, a′), where (P, a′) ∈ Wh_{k}^{diff}(R, G) is neither degenerated nor an elementary expansion, and P carries the minimal ordering that supports a′.

Note that ā′ determines ā by Definition 6.23 (1), so we only construct ā′. As before, all coefficients ā′(i_{0}, …, i_{l−1}) with 0 ≤ i_{0} < ⋯ < i_{l−1} < k and i_{l−1} > l − 1 are inductively fixed by compatibility with face operators, and of course, they are Aut(P, a′)-invariant. We will construct the coefficients ā′(0, …, l − 1) by induction over l. For l = 0, we clearly have ā′(∅) = a′. For l > 0, we already know the restriction of ā′(0, …, l − 1) to the set

⋃_{j=l}^{k} f_{l−1,...,ĵ,...,k}(Δ_{k−l}) = ∂Δ_{k\l−1} \ Δ̊_{k\l} .

Note that this set is a deformation retract of Δ_{k\l−1}. By Definition 6.25 (1), we need an Aut(P, a′)- invariant solution ā′(0, …, l − 1) of the system of linear partial first-order differential equations

(6.33)  (−1)^{l} d(ā′_{αβ}(0, …, l − 1)) = −(∂′ā′ + a′ ∗′ a′)_{αβ}(0, …, l − 1) ,

such that ā′(0, …, l − 1) has degree 1 − l in Ω^{∗}(Δ_{k\l−1}; T_{P} (R)) and restricts to the given initial data on ∂Δ_{k\l−1} \Δ̊_{k\l}. By triangularity of ā′, this system can be solved inductively in the distance of α and β in the partial ordering of P. The integrability condition is inductively given by

0 = (d′(∂′ā′ + ā′ ∗′ ā′))_{αβ}(0, …, l − 1) ,

which follows from the equations

0 = (d′ā′ + δ′ā′ + ā′ ∗′ ā′)_{γδ}(i_{0}, …, i_{m})

for those indices γ, δ and those sequences 0 ≤ i_{0} < ⋯ < i_{m} ≤ k where all coefficients are already known, because

(d′(∂′ā′ + ā′ ∗′ ā′))_{αβ}(0, …, l − 1)

= (−∂′d′ā′ + (d′ā′) ∗′ ā′ − ā′ ∗′ d′ā′)_{αβ}(0, …, l − 1)

= (∂′(∂′ā′ + ā′ ∗′ ā′) − (∂′ā′ + ā′ ∗′ ā′) ∗′ ā′ + ā′ ∗′ (∂′ā′ + ā′ ∗′ ā′))_{αβ}(0, …, l − 1)

= 0 .

In order to get initial data for a Cauchy problem, we have to specify all coefficients of ā′(0, …, l− 1) along ∂Δ_{k\l−1} \ Δ̊_{k\l}, including those that involve directions normal to this set. It is clear that this is possible. Taking for ā′(0, …, l − 1) any Aut(P, a′)-invariant solution of (6.33), we have finished the inductive construction of (P, ā, ā′) = Ψ_{2}(P, a′).

This shows that Φ_{2} has a right-inverse Ψ_{2}. By the same argument as above, the simplicial functor Ψ_{2} ∘ Φ_{2} is simplicially homotopic to the identity on Wh^{mix}(R; G). □

6.34. Remark. In [I3], Igusa constructs a simplicial functor Wh^{diff}(R, G) → Wh^{simp}(R, G) using Chen’s iterated integrals. It seems that on realisations, Igusa’s functor is homotopic to Φ_{1} ∘ Ψ_{2}.

7. Properties of the finite-dimensional higher torsion

In this chapter, we collect some elementary properties of the Chern torsion classes T (A′, g^{V} , h^{V}) introduced in Section 2.

We start by giving a description of T (A′, g^{V} , h^{V}) = T (∇^{V} , A′, g^{V}) in Subsections 7.a and 7.b, which does not make use of the function h^{V}. This is important for two reasons. First, it allows us to define a torsion class T (R, G) on the acyclic Whitehead space Wh^{h}(R, G) in Definition 7.29, and as well on the unitary Whitehead space Wh^{u}(R, G) in Section 9. Second, the new representation of T (∇^{V} , A′g^{V}) will allow us to compare Igusa’s higher Franz-Reidemeister torsion and T on a suitable Whitehead space in Section 8.

We then analyse the behaviour of T (∇^{V} , A′, g^{V}) on globally filtered complexes. Applications include a gluing formula for the combinatorial torsion, an extension of the examples in Section 5, and a reduction of the unitary higher torsion to an acyclic situation in Section 9.

7.a. The Chern normalisation of the torsion forms. We give a formula for finite-dimensional the Chern torsion form of Section 2 as an integral of the Chern form of a superconnection over a two-dimensional domain. This is the first step in finding a formula for T (A′, g^{V} , h^{V}) that does not involve the endomorphism h^{V}.

Let f(x) = x e^{x^{2}}, and recall that by Definition 2.46 and Definition 2.91,

T (A′, g^{V} , h^{V} ) = ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} T_{f}(A′, g^{V} , h^{V} ) ds

= − ∫_{0}^{1}(\frac{4s(1 − s)}{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} ds

− ∫_{0}^{1}(\frac{4s(1 − s)}{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} ds .

(7.1)

Let \overline{A}′ = A′+ \frac{∂}{∂s} ds+ \frac{∂}{∂t} dt+ \frac{∂}{∂T} dT be the extension of the flat superconnection A′ = ∇^{V} +a_{0}+… to the pullback ^{1}\overline{B} = B × (0, 1) × ℝ_{>}^{2}. We define a metric

1g\overline{V} = g^{V} ((\frac{t}{4s(1 − s)})^{N^{V}} T^{h^{V}} · , · ) ,

which coincides with the metric of (2.39) at s = \frac{1}{2}. Let ^{1}\overline{A}′′ denote the adjoint of \overline{A}′ with respect to ^{1}g\overline{V}, then

\begin{aligned}^{1}\overline{A}′′ & = ∇^{V,∗} + ∑_{j}(\frac{t}{4s(1 − s)})^{1−j} T^{−h^{V}} a′′_{j} T^{h^{V}} \\ & + log T dh^{V} + (\frac{∂}{∂s} + \frac{N^{V}}{1 − s} − \frac{N^{V}}{s} ) ds + (\frac{∂}{∂t} + \frac{N^{V}}{t} ) dt+(\frac{∂}{∂T} + \frac{h^{V}}{T} ) dT . \end{aligned}

Conjugating everything by (\frac{t}{4(1−s)^{2}} )^{\frac{N^{V}}{2}} T^{\frac{h^{V}}{2}} gives two flat superconnections

\begin{aligned}Ã′^{1} & = ∇^{V} + ∑_{j}(\frac{t}{4(1 − s)^{2}} )^{\frac{1−j}{2}} T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} − \frac{log T}{2} dh^{V} \\ & + (\frac{∂}{∂s} − \frac{N^{V}}{1 − s}) ds + (\frac{∂}{∂t} − \frac{N^{V}}{2t} ) dt+(\frac{∂}{∂T} − \frac{h^{V}}{2T} ) dT , \end{aligned} (7.2)

\begin{aligned}Ã′′^{1} & = ∇^{V,∗} + ∑_{j}(\frac{t}{4s^{2}} )^{\frac{1−j}{2}} T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} + \frac{log T}{2} dh^{V} \\ & + (\frac{∂}{∂s} − \frac{N^{V}}{s} ) ds + (\frac{∂}{∂t} + \frac{N^{V}}{2t} ) dt+(\frac{∂}{∂T} + \frac{h^{V}}{2T} ) dT . \end{aligned} (7.3)

As in [BG], we consider the superconnection

\begin{aligned}Ã^{1} & = (1 − s) ^{1}Ã′ + s ^{1}Ã′′ \\ & = ∇^{\overline{V} ,u} + ∑_{j}(\frac{t}{4})^{\frac{1−j}{2}} ((1 − s)^{j} T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} + s^{j} T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} ) \\ & + (2s − 1) (\frac{ω^{V}}{2} + \frac{log T}{2} dh^{V} + \frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ) − 2N^{V} ds . \end{aligned} (7.4)

Note that ^{1}Ã contains no negative powers of 1 − s or s, so we can extend it to B × [0, 1] × ℝ_{>}^{2}. Because both ^{1}Ã′ and ^{1}Ã′′ are flat, we find that

(7.5)  ^{1}Ã^{2} = −s(1 − s) (^{1}Ã′′ − ^{1}Ã′)^{2} − (^{1}Ã′′ − ^{1}Ã′) ds .

7.6. Remark. Formula(7.5) is well suited for describing ^{1}Ã^{2} in B × (0, 1) × [1, ∞) × ℝ. For the restriction to s = 0, we obtain instead

\begin{aligned}^{1}Ã|_{s=0} & = ^{1}Ã′|_{s=0} + \frac{\sqrt{t}}{2} T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}} \\ and ^{1}Ã^{2}|_{s=0} & = [^{1}Ã′|_{s=0} , \frac{\sqrt{t}}{2} T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}} ] . \end{aligned} (7.7)

Note that ^{1}Ã^{2}|_{s=0} is in triangular shape with respect to the ℤ-grading of V, with the diagonal term given by \frac{t}{4} [T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} , T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}} ]. This has two consequences.

(1) The kernel of (^{1}Ã^{2})^{[0]} does not jump at s = 0, and

(2) The restriction of str(^{1}Ã^{2})^{[0]} to s = 0 is of degree 0. Similar observations are of course true at s = 1.

7.8. Proposition. Up to exact forms on B, the Chern analytic torsion form is given by

T (A′, g^{V} , h^{V} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(s,t)∈[0,1]×[1,∞)}(\frac{1}{2} str_{V} (e^{−^{1}Ã^{2}}) − χ′(H) \frac{ds dt}{2t}

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

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,T)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{1}Ã^{2}}) − str_{V} (h^{V} ) \frac{ds dT}{2T}

− (χ′(V) − χ′(H)) ((f′(\sqrt{−T} /2) − 1) \frac{ds dT}{2T} ) .

Proof. Comparing Definition 2.46 and Definition 2.91 with the formula above, we easily see that the proposition holds in degree 0. In particular, we do not need to investigate any correction terms.

Let ^{1}Ã′ and ^{1}Ã′′ be defined by (7.2), (7.3), and set

1X̃ = \frac{1}{2} (^{1}Ã′′ − ^{1}Ã′) = ^{1}X̃_{s,t,T} + (\frac{N^{V}}{2(1 − s)} − \frac{N^{V}}{2s} ) ds + \frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ,

then

(7.9)  ^{1}X̃^{2} = ^{1}X̃_{s,t,T}^{2} + [(\frac{N^{V}}{2(1 − s)} − \frac{N^{V}}{2s} ) ds + \frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT , ^{1}X̃_{s,t,T} ] .

We calculate the following contribution to the integrand in formula (7.1) for T (A′, g^{V} , h^{V}):

(7.10) ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} str((\frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ) f′(X̃_{t})) ds

= ∫_{0}^{1} str((\frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ) f′(\sqrt{s(1 − s)} (Ã′′_{\frac{t}{4s(1−s)} ,T} − Ã′_{\frac{t}{4s(1−s)} ,T} ))) ds

= ∫_{0}^{1} str((\frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ) f′(\sqrt{s(1 − s)} (Ã′′_{\frac{t}{4s^{2}} ,T} − Ã′_{\frac{t}{4(1−s)^{2}} ,T} ))) ds

= ∫_{0}^{1} str((\frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ) f′(\sqrt{4s(1 − s)} ^{1}X̃_{s,t,T} )) ds .

For the second equation, we have conjugated by (\frac{s}{1−s})^{\frac{N^{V}}{2}} within the trace.

On the other hand, by (7.5) and (7.9), and because the supertrace of an even positive power of an odd operators vanishes, we have

\frac{1}{2} str(e^{−^{1}Ã^{2}}) = \frac{1}{2} str(e^{4s(1−s) ^{1}X̃^{2}+2 ^{1}X̃ ds})

= χ(V) + \frac{1}{\sqrt{4s(1 − s)}} str(f(\sqrt{4s(1 − s)} ^{1}X̃)) ds

= χ(V) + \frac{1}{\sqrt{4s(1 − s)}} str(f(\sqrt{4s(1 − s)} ^{1}X̃_{s,t,T} )) ds

+ str((\frac{N^{V}}{2t} dt + \frac{h^{V}}{2T} dT ) f′(\sqrt{4s(1 − s)} ^{1}X̃_{s,t,T} )) ds + …,

(7.11)

where the remaining term involves ds, dt and dT. The previous equation and (7.10) together with Definition 2.46 and Definition 2.91 prove the proposition. □

7.b. A new formula for the Chern torsion classes. We now want to get rid of the endomorphism h^{V} in our formula for T (A′, g^{V} , h^{V}). This implies that we have to replace the last integral in Proposition 7.8 by something different. This is done in two steps, Proposition 7.18 and Theorem 7.23.

First, let

2\overline{V} ⟶ ^{2}\overline{B} = B × (0, 1) × ℝ_{>} × [0, 1]

be the pullback of V → B, and let (s, T, u) be the coordinates on (0, 1) × ℝ_{>} × [0, 1]. We extend the superconnections Ã′|_{t=1/4} and Ã′′|_{t=1/4} of (2.40) and ^{1}Ã|_{t=1} to ^{2}\overline{B}, and set

(7.12)  Ã = (1 − s) Ã′ + s Ã′′ .

We then consider the superconnection

\begin{aligned}Ã^{2} & = u ^{1}Ã|_{t=1} + (1 − u) Ã|_{t=1/4} \\ & = ∇^{\overline{V} ,u} + ∑_{j} 2^{j−1} (((1 − s)^{j}u + (1 − s)(1 − u)) T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} \\ & + (s^{j}u + s(1 − u)) T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} ) \\ & + (2s − 1) (\frac{ω^{V}}{2} + \frac{log T}{2} dh^{V} + \frac{h^{V}}{2T} dT ) − 2u N^{V} ds . \end{aligned} (7.13)

As T → 0, this superconnection becomes singular. Note that by Definition 2.35, there exists ε > 0 with

(7.14)  T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} = O(T^{\frac{ε}{2}} )  and  T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} = O(T^{\frac{ε}{2}} ) .

The remaining terms in ^{2}Ã define a new superconnection

2Ã_{0} = ∇^{\overline{V} ,u} + (2s − 1) ^{2}X̃_{0} − 2u N^{V} ds ,

where

2X̃_{0} = \frac{ω^{V}}{2} + \frac{log T}{2} dh^{V} + \frac{h^{V}}{2T} dT .

Note that ^{2}X̃_{0}^{2} = ω^{V} /4. Because N^{V} commutes with ^{2}Ã_{0}, the curvature of ^{2}Ã_{0} is given by

2Ã_{0}^{2} = −s(1 − s) (ω^{V} )^{2} − 2 ^{2}X̃_{0} ds − 2 N^{V} du ds

Let us now compute the (unnormalised) Chern character form of ^{2}Ã_{0}. We find that

str(e^{−^{2}Ã_{0}^{2}}) = str(e^{s(1−s) (ω^{V} )^{2}}) − 2 str(N^{V} e^{s(1−s) (ω^{V} )^{2}}) du ds

+ \frac{1}{\sqrt{s(1 − s)}} str(f(\sqrt{4s(1 − s)} ^{2}X̃_{0})) ds

= χ(V) − 2χ′(V) du ds + d str(h^{V} log T ) ds

+ str(ω^{V} e^{s(1−s) (ω^{V} )^{2}}) ds .

(7.15)

Here we have used that positive even powers of odd operators vanish, that [∇^{V,u}, ω^{V} ] = 0, and that ω^{V}, h^{V} and N^{V} supercommute pairwise.

As in (2.45), it follows that there exists an ε′ > 0 with

(7.16) str(e^{−^{2}Ã^{2}}) − d str(h^{V} log T ) ds

= χ(V) − 2χ′(V) du ds + str(ω^{V} e^{s(1−s) (ω^{V} )^{2}}) ds + O(T^{ε′}) + O(T^{ε′}) \frac{dT}{T} ,

where the terms O(T^{ε′}) do not contain the exterior variable dT.

For T_{0} > 0, consider the polyhedron ^{2}D = [0, 1] × [T_{0}, 1] × [0, 1].

T  u s  1 1  1 T_{0}

By Stokes’ theorem, the form

−(2πi)^{− \frac{N^{B}}{2}} ∫_{(B×∂^{2}D)/B} \frac{1}{2} (str(e^{−^{2}Ã^{2}}) − str(h^{V} ) \frac{dT ds}{T} + 2χ′(V) du ds)

is exact. We study the behaviour of the integrals over the six faces as T_{0} → 0. This will allow us to replace one contribution to the Chern torsion class by two contributions which are easier to handle.

α) The faces s = 0 and s = 1. Let us consider the restriction to s = 0, where by (7.7), (7.13) and (7.12),

2Ã|_{s=0} = Ã′|_{t=1/4} + \frac{u}{2} T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}}
and  ^{2}Ã^{2}|_{s=0} = \frac{u}{2} [Ã′|_{t=1/4} , T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}} ] − \frac{1}{2} T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}} du

because (Ã′)^{2} = (a′′_{0})^{2} = 0. Note that ^{2}Ã^{2}|_{s=0} is in triangular shape with respect to N^{V}, with diagonal term

\frac{u}{4} [T^{\frac{h^{V}}{2}} a′_{0} T^{− \frac{h^{V}}{2}} , T^{− \frac{h^{V}}{2}} a′′_{0} T^{\frac{h^{V}}{2}} ] .

This implies that the supertrace of e^{−^{2}Ã^{2}} contains neither dT nor du. A similar statement holds at s = 1. In particular

(7.17)  0 = ∫_{(s,T)∈[T_{0},1]×[0,1]} \frac{1}{2} str(e^{−^{2}Ã^{2}|_{s=0}}) = ∫_{(s,T)∈[T_{0},1]×[0,1]} \frac{1}{2} str(e^{−^{2}Ã^{2}|_{s=1}}) .

β) The face u = 0. With (2.40), (2.41) and (7.5), we have

2Ã|_{u=0} = (1 − s) Ã′|_{t=1/4} + s Ã′′|_{t=1/4}
and  ^{2}Ã^{2}|_{u=0} = −4s(1 − s) X̃|_{t=1/4}^{2} − 2 X̃|_{t=1/4} ds .

Again by (7.16), we know that the following integral exists:

∫_{(s,T)∈[0,1]^{2}}(\frac{1}{2} str(e^{−^{2}Ã^{2}|_{u=0}}) − str(h^{V} ) \frac{dT ds}{2T}) .

γ) The face u = 1. By (7.4) and (7.13),

2Ã|_{u=1} = ^{1}Ã|_{t=1} .

In particular, (7.10) and (7.11) imply that

∫_{(s,T)∈[0,1]×[T_{0},1]} \frac{1}{2} str(e^{−^{2}Ã^{2}|_{u=1}}) = − ∫_{T_{0}}^{T} ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} str(h^{V} f′(X̃_{t})) \frac{dT}{2T} .

By (7.16),, we know that the following integral exists:

∫_{(s,T)∈[0,1]×[T_{0},1]}(\frac{1}{2} str(e^{−^{2}Ã^{2}|_{u=1}}) − str(h^{V} ) \frac{dT}{2T} ds) .

δ) The face T = 1. The integral

∫_{(s,u)∈[0,1]^{2}}(\frac{1}{2} str(e^{−^{2}Ã^{2}|_{T =1}}) + χ′(V) du ds)

exists and can be nonzero. It will become a contribution to T (A′, g^{V} , h^{V}) in the next intermediate step.

ε) The face T = T_{0}. From (7.16), we conclude that

lim_{T_{0}→0} ∫_{(s,u)∈[0,1]^{2}}(\frac{1}{2} str(e^{−^{2}Ã^{2}|_{T =T_{0}}} ) + χ′(V) du ds) = 0 .

Stokes’ theorem and α–ε) imply the following intermediate result.

7.18. Proposition. Up to exact forms on B, the Chern analytic torsion T (A′, g^{V} , h^{V}) can be expressed as

T (A′, g^{V} , h^{V} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(s,t)∈[0,1]×[1,∞)}(\frac{1}{2} str_{V} (e^{−^{1}Ã^{2}}) − χ′(H) \frac{dt}{2t} ds

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

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,u)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{T =1}}) + χ′(V) du ds)

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,T)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{u=0}}) − str_{V} (h^{V} ) \frac{dT ds}{2T}

− (χ′(V) − χ′(H)) (f′(\sqrt{−t}/2) − 1) \frac{dT ds}{2T}) .

We are now in a position to finally eliminate h^{V} from our representation of T (A′, g^{V} , h^{V}). For the final step, we let

3\overline{V} ⟶ ^{3}\overline{B} = B × [0, 1] × ℝ_{>} × [0, 1]

be the pullback of V → B, and let (s, T, v) be the coordinates on [0, 1] × ℝ_{>} × [0, 1]. We define a superconnection

\begin{aligned}Ã^{3} & = v ((1 − s) Ã′|_{t=1/4} + s Ã′′|_{t=1/4}) + (1 − v) ((1 − s) ∇\overline{V} + s ∇^{\overline{V} ,∗}) \\ & = ∇^{\overline{V} ,u} + v ∑_{j} 2^{j−1} ((1 − s) T^{\frac{h^{V}}{2}} a′_{j} T^{− \frac{h^{V}}{2}} + s T^{− \frac{h^{V}}{2}} a′′_{j} T^{\frac{h^{V}}{2}} ) \\ & + (2s − 1) (\frac{ω^{V}}{2} + v \frac{log T}{2} dh^{V} + v \frac{h^{V}}{2T} dT ) . \end{aligned} (7.19)

As above, we use those terms that do not vanish as T → 0 to define a new superconnection

3Ã_{0} = ∇^{\overline{V} ,u} + (2s − 1) (\frac{ω^{V}}{2} + v \frac{log T}{2} dh^{V} + v \frac{h^{V}}{2T} dT ) .

Similar as above, we calculate

3Ã_{0}^{2} = −s(1 − s)(ω^{V} )^{2} − ω^{V} ds − \frac{1}{2} d(h^{V} log T ) d((2s − 1)v) .

As above, this implies that there exists an ε′ > 0 such that

(7.20) str(e^{−^{3}Ã^{2}}) − \frac{1}{2} d str(h^{V} log T ) d((2s − 1)v)

= χ(V) + str(ω^{V} e^{s(1−s) (ω^{V} )^{2}}) ds + O(T^{ε′}) + O(T^{ε′}) \frac{dT}{T} ,

Let ^{3}D = [0, 1]^{3}, with coordinates s, T and v. By Stokes’ theorem, the form

−(2πi)^{− \frac{N^{B}}{2}} ∫_{(B×∂^{3}D)/B} \frac{1}{2} (str(e^{−^{3}Ã^{2}}) − str(h^{V} ) \frac{dT}{2T} d((2s − 1)v))

is exact. We study the behaviour of the integrals over the six faces of ^{3}D as T_{0} → 0. This will allow us to replace one contribution to the Chern torsion class by another one that is easier to handle.

α) The faces s = 0 and s = 1. Let us consider the restriction to s = 0, where by (7.19),

3Ã|_{s=0} = v Ã′|_{t=1/4} + (1 − v) ∇\overline{V}
and  ^{3}Ã^{2}|_{s=0} = −4v(1 − v) (Ã′|_{t=1/4} − ∇\overline{V} )^{2} − 2 (Ã′|_{t=1/4} − ∇\overline{V} ) dv .

Note that (Ã′|_{t=1/4} − ∇\overline{V} ) is in triangular shape with respect to the local partial ordering induced by h^{V}, and the diagonal term is just \frac{1}{2} d(h^{V} log T). A similar statement holds at s = 1. In particular

0 = ∫_{(T,v)∈[T_{0},1]×[0,1]} \frac{1}{2} (str(e^{−^{3}Ã^{2}|_{s=0}}) + str(h^{V} ) \frac{dT}{2T} dv))

= ∫_{(T,v)∈[T_{0},1]×[0,1]} \frac{1}{2} (str(e^{−^{3}Ã^{2}|_{s=1}}) − str(h^{V} ) \frac{dT}{2T} dv)) .

MORSE THEORY AND HIGHER TORSION INVARIANTS II 25

β) The face v = 0. With (2.40), (2.41) and (7.5), we have

(7.21)  ^{3}Ã|_{v=0} = (1 − s) ∇\overline{V} + s ∇^{\overline{V} ,∗}  and  ^{3}Ã^{2}|_{v=0} = −4s(1 − s) (ω^{V} )^{2} − 2 ω^{V} ds .

Clearly, ^{3}Ã^{2}|_{v=0} does not contain the exterior variable dT, so

∫_{(s,T)∈[0,1]^{2}} \frac{1}{2} str(e^{−^{3}Ã^{2}|_{v=0}}) = 0 .

γ) The face v = 1. By (7.13) and (7.19),

3Ã|_{v=1} = ^{2}Ã|_{u=1} .

By (7.16), we know that the following integrals exist and are equal:

∫_{(s,T)∈[0,1]^{2}} \frac{1}{2} (str(e^{−^{2}Ã^{2}|_{u=0}}) − 2 str(h^{V} ) \frac{dT}{2T} ds)

= ∫_{(s,T)∈[0,1]^{2}} \frac{1}{2} (str(e^{−^{3}Ã^{2}|_{v=1}}) − 2 str(h^{V} ) \frac{dT}{2T} ds) .

δ) The face T = 1. The integral

∫_{(s,v)∈[0,1]^{2}} \frac{1}{2} str(e^{−^{3}Ã^{2}|_{T =1}})

exists and can be nonzero. It will become a contribution to T (A′, g^{V} , h^{V}).

ε) The face T = T_{0}. From (7.20), we conclude that

lim_{T_{0}→0} ∫_{(s,v)∈[0,1]^{2}} \frac{1}{2} str(e^{−^{3}Ã^{2}|_{T =T_{0}}} ) = 0 .

We may thus define a Chern torsion form without using h^{V}. 7.22. Definition. The finite dimensional Chern torsion class is modulo exact forms defined as

T (∇^{V} , A′, g^{V} ) = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(s,t)∈[0,1]×[1,∞)}(\frac{1}{2} str_{V} (e^{−^{1}Ã^{2}}) − χ′(H) \frac{dt ds}{2t}

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

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,u)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{T =1}}) − χ′(V) du ds)

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,v)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{T =1}})

− (χ′(V) − χ′(H)) (f′(\sqrt{−t}/2) − 1) \frac{dv ds}{2v}) .

Then the considerations above immediately prove the following result.

7.23. Theorem. The two definitions of a finite-dimensional Chern torsion form agree, so up to exact forms on B,

T (A′, g^{V} , h^{V} ) = T (∇^{V} , A′, g^{V} ) .

If A′ = ∇^{V} + a′_{0} with a′_{0} ∈ Γ(End^{1} V), we let T^{BL}(A′, g^{V}) denote the Bismut-Lott torsion form of [BL] in the Chern normalisation of [BG], cf. Section 2.i. Together with Remark 2.47, Theorem 7.23 has the following consequence.

7.24. Corollary. Let A′ = ∇^{V} + a′_{0} with a′_{0} ∈ Γ(End^{1} V), then

T (∇^{V} , A′, g^{V} ) = T^{BL}(A′, g^{V}) .

7.25. Theorem. Let (V, ∇^{V}) → B be a flat ℤ-graded vector bundle with a flat superconnection A′ such that A′ − ∇^{V} ∈ Ω^{∗}(B; End V) takes values in a nilpotent subalgebra of End V, which may vary over B. Then for any metric g^{V} on V, we have

dT (∇^{V} , A′, g^{V} ) = ch^{o}(∇^{V} , g^{V} ) − ch^{o}(∇^{H}, g^{H}) .

Proof. By Definition 7.22 and the preceeding arguments, we find that

(7.26) dT (∇^{V} , A′, g^{V} ) = \frac{1}{2} (2πi)^{\frac{1−N^{B}}{2}} (− lim_{t→∞} ∫_{s∈[0,1]} str_{V} (e^{−^{1}Ã^{2}|_{T =1}})

+ ∫_{s∈[0,1]} str_{V} (e^{−^{3}Ã^{2}|_{{v=0 \atop T =1}}} ) + ∫_{t∈[1,∞)} str_{V} (e^{−^{1}Ã^{2}|_{T =1}})|_{s=0}^{1}

+ ∫_{u∈[0,1]} str_{V} (e^{−^{2}Ã^{2}|_{T =1}})|_{s=0}^{1} + ∫_{v∈[0,1]} str_{V} (e^{−^{3}Ã^{2}|_{T =1}})|_{s=0}^{1}) .

Arguing as in (7.10) and (7.11), we compute

(2πi)^{\frac{1−N^{B}}{2}} lim_{t→∞} ∫_{s∈[0,1]} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{T =1}})

= (2πi)^{\frac{1−N^{B}}{2}} lim_{t→∞} ∫_{s∈[0,1]} \frac{1}{\sqrt{4s(1 − s)}} str(f(\sqrt{4s(1 − s)} ^{1}X̃_{s,t,1})) ds

= (2πi)^{\frac{1−N^{B}}{2}} lim_{t→∞} ∫_{s∈[0,1]} \frac{1}{\sqrt{4s(1 − s)}} str(f(\sqrt{4s(1 − s)} X̃_{\frac{t}{4s(1−s)}} )) ds

= lim_{t→∞} ∫_{s∈[0,1]}(\frac{4s(1 − s)}{2πi})^{\frac{N^{B}−1}{2}} str(f(X̃_{t})) ds = ch^{o}(∇^{H}, g_{V}^{H})

as in the proof of Theorem 2.48. On the other hand, we have

(2πi)^{\frac{1−N^{B}}{2}} ∫_{s∈[0,1]} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{{v=0 \atop T =1}}} )

= (2πi)^{\frac{1−N^{B}}{2}} lim_{v→0} ∫_{s∈[0,1]} \frac{1}{\sqrt{4s(1 − s)}} str_{V} (f(\sqrt{s(1 − s)} ω^{V} )) ds = ch^{o}(∇^{V} , g^{V} ) .

We still have to show that the remaining terms on the right hand side of (7.26) vanish. By Remark 7.6, the restriction of str_{V} (e^{−^{1}Ã^{2}}) to s = 0 and s = 1 is of degree 0 and hence does not contain dt, so

∫_{t∈[1,∞)} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{T =1}})|_{s=0}^{1} = 0 .

By (7.15), the integrand str_{V} (e^{−^{2}Ã^{2}}) involves no term that contains du but not ds, so

∫_{u∈[0,1]} \frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{T =1}})|_{s=0}^{1} = 0 .

Finally, we have

∫_{v∈[0,1]} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{{s=0 \atop T =1}}} )

= ∫_{v∈[0,1]} \frac{1}{\sqrt{4v(1 − v)}} str_{V} (f(\sqrt{4v(1 − v)} (Ã′|_{t=1/4} − ∇^{V} ))) dv = 0 ,

because Ã′ − ∇^{V} takes values in a nilpotent subalgebra of End V. Our theorem follows from the equations above. □

Theorem 7.25 shows that T (∇^{V} , A′, g^{V}) behaves as expected even without the existence of a Morse endomorphism h^{V}. Proceeding as in the proof of Corollary 2.63, we can therefore draw the following conclusion.

7.27. Corollary. Let (V, ∇^{V}) → B be a flat ℤ-graded vector bundle with a flat superconnection A′ such that A′ − ∇^{V} ∈ Ω^{∗}(B; End V) takes values in a nilpotent subalgebra of End V, which may vary over B. Assume that both V and H = H^{∗}(V, a′_{0}) carry parallel metrics, then the form

T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})

is even, closed and real, and its cohomology class is independent of the choice of parallel metrics in degree ≥ 2.

7.28. Remark. Let us state some properties of Definition 7.22 above.

(1) The formula for T (∇^{V} , A′, g^{V}) makes essential use of the Chern normalisation. Integrating over two-dimensional domains instead of one-dimensional contours as in Definition 2.46 allows a much larger choice of superconnections to use in the definition. In fact, the integrals in Definition 7.22 do not arise from applying the Chern normalisation operator ∫_{0}^{1}(s(1 − s))^{N^{B}} ds to a one-dimensional contour integral.

(2) We have to include the flat reference connection ∇^{V} in the notation T (∇^{V} , A′, g^{V}) because ∇^{V} is used for the construction of the superconnection ^{3}Ã in (7.19) above.

(3) We had to assume that A′−∇^{V} is an inhomogeneous differential form with values in a nilpotent subalgebra of End V, which may vary over B. In fact, we could define T (∇^{V} , A′, g^{V}) without this assumption. However, we would then be unable to prove that

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

By the way, we do not have to demand that the nilpotent subalgebra mentioned above can locally be chosen parallel. This is because by flatness of A′,

[∇^{V} , a′_{k}] = −a′_{0}a′_{k+1} − ⋯ − a′_{k+1}a′_{0}

is again an element of the subalgebra.

Let from now on T (∇^{V} , A′, g^{V}) denote the form constructed in Theorem 7.23, which can be used to define a universal torsion class on the acyclic Whitehead space. We assume that G is a subgroup of a unitary group, so that the bundle V → Wh^{h}(R, G) carries a canonical parallel metric g^{V}.

7.29. Definition. Define T (R, G) ∈ H^{even}(Wh^{h}(R, G), ℝ) to be the class represented by the cocycle which assigns to every simplex Δ in Wh^{diff}(R, G) the integral

∫_{Δ} T (∇^{V}, A′, g^{V}) .

Then T (R, G)^{[0]} equals the fibre-wise Franz-Reidemeister torsion of the acyclic complex (V, a_{0}). Another possibility would have been to find an adapted endomorphism h^{V} over |Wh^{diff}(R, G)|. Here the difficulty is that one has to make sure that Definition 2.35 (3) is satisfied. This is possible if h^{V} is chosen suitably and the tautological superconnection A′ on V → |Wh^{diff}(R, G)| is pulled back to the interior of each simplex using cut-off functions as in Definition 1.65. Of course, by Theorem 7.23, the class obtained this way would be cohomologous to the class T (R, G) defined in Definition 7.29 above.

Let (V, A′, h^{V}) be a family Thom-Smale complex over B such that ∇^{V} is unitarily flat and fibre-wise acyclic. By Proposition 6.18 and Remark 6.19, there exists a “classifying functor” Ξ = Ξ_{p,h,F}^{diff} : S → Wh^{diff,h}(R, G), inducing a classifying map ξ: B → Wh^{h}(R, G) up to homotopy, such that

(V, ∇^{V} , A′, g^{V}) ≅ ξ^{∗}(V, ∇^{V}, A′, g^{V}) .

7.30. Corollary. Let S be a simplicial complex on B and let Ξ: S → Wh^{diff,h}(R, G) be a simplicial map. Let (V, ∇^{V} , A′, g^{V}) = Ξ^{∗}(V, ∇^{V}, A′, g^{V}), then

(1)  T (∇^{V} , A′, g^{V} ) = |Ξ|^{∗} T (R, G) ∈ H^{even}(|S| , ℝ) .

In particular, if p: M → B is a proper submersion with a fibrewise Morse-function h: M → ℝ, if (F, ∇^{F} , g^{F}) → M is a unitarily flat and fibrewise acyclic vector bundle, and if ξ: B → Wh(R, G) is constructed as Chapter 4 of [I2], cf. Remark 6.19, then

(2)  T (M/B, F, h) = ξ^{∗}T (R, G)^{[≥2]} .

Proof. Claim (1) is an immediate consequence of Theorem 7.23 and Definition 7.29. By Theorem 6.20, Igusa’s functor Ξ is homotopic to the functor constructed in Section 1. Thus, (2) follows from Definition 2.64 and (1). □

7.c. Higher torsion and filtrations. If there is a filtration of the vector bundle V that is compatible with ∇^{V} and A′, and preserves the local splitting of V of Definition 2.35, then we show that the higher torsion T (∇^{V} , A′, g^{V}) can be computed using higher torsions associated to a spectral sequence. Such filtered complexes arise in the following situations.

(1) In the case that the fibre X of p: N → B is itself fibred by p = p_{2} ∘ p_{1}, with p_{1}: N → M and p_{2}: M → B, the filtration of Ω^{∗}(M/B; F) by relative horizontal degree gives a family of Leray spectral sequences over B. The analytic torsion of such a multifibration was treated by Ma in [Ma]. Here, we recover Ma’s result in the case that both fibrations admit fibre-wise Morse functions, if the fibre-wise Morse function on M with respect to h_{2} admits a Smale gradient field.

(2) If the bundle p: M → B admits a fibre-wise function such that 0 is a regular value on each fibre of p, one has a decomposition M = M^{+} ∪ M^{−} with M^{−} = h^{−1}(−∞, 0] and M^{+} = h^{−1}[0, ∞). Under certain assumptions on H, Igusa proves in [I2], Chapter 5 that the higher Franz-Reidemeister torsion of M → B equals the sum of the torsions of M^{+} → B and M^{−} → B already on the level of maps from B to Wh(R, G). If h can be chosen to be Morse on every fibre of p, we obtain a gluing formula for T_{ch}(M/B; F, h). A gluing formula for the infinitesimal equivariant analytic torsion in a similar situation was proved by Bunke in [Bu].

(3) Suppose that F → M and H = H^{∗}(M/B; F) admit parallel metrics, then we get a filtered complex on V̂ = V ⊕ H → B which is fibre-wise acyclic. Here, the spectral sequence degenerates, and T_{ch}(M/B; F, h) = T_{ch}(∇V̂ , Â′, gV̂ ). In other words, the torsion class on the acyclic Whitehead space determines the torsion class on the unitary Whitehead spaces that will be defined in Section 9.

We will prove a finite dimensional analogue of Ma’s results in [Ma], Section 3.2, where the fibre-wise Leray spectral sequence of a family of submersions is considered.

Suppose that the vector bundle (V, ∇^{V} , A′) admits a metric g^{V} and a decreasing graded filtration

(7.31)  F^{0}V^{j} = V ⊃ ⋯ ⊃ F^{k_{0}}V^{j} = 0

that is compatible with ∇^{V} and A′ = ∇^{V} +a′, and put F^{p}V = V for k ≤ 0 and F^{p}V = 0 for p ≥ k_{0}. Then let (E_{k}, d′_{k}) → B be the spectral sequence associated to the filtered complex (V, a′_{0}). There is an associated filtration

(7.32) F^{0}H = H ⊃ ⋯ ⊃ F^{k_{0}}H = 0  with  F^{p}H = (ker(a′_{0}) ∩ F^{p}V ) / (im(a′_{0}) ∩ F^{p}V ) .

The following result has been proved by Ma in [Ma] in an infinite-dimensional setting.

7.33. Proposition. For k = 0, let the operator d′_{0} on E_{0}^{p} = F^{p}V/F^{p+1}V be induced by a′_{0}. There are a flat connection ∇^{E_{0}} and a flat superconnection D′_{0} of total degree 1 on E_{0}, induced by ∇^{V} and A′, such that D′_{0} starts with d′_{0} in degree 0, and D′_{0} − ∇^{E_{0}} is a differential form with values in some nilpotent subalgebra of End E_{0} which may vary over B.

For k ≥ 0, we have a vector bundle E_{k+1} = (H^{∗}(E_{k}, d′_{k})) → B. If ∇^{E_{k+1}} denotes the Gauß-Manin connection induced by D′_{k} then ∇^{E_{k+1}} is flat, and the fibre-wise differential d′_{k} is parallel with respect to ∇^{E_{k+1}}. In particular, the superconnection D′_{k+1} defined by D′_{k+1} = ∇^{E_{k+1}} + d′_{k+1} is flat and of total degree 1.

The spectral sequence E_{k} converges to E_{∞} = E_{k_{0}} with E_{∞}^{p,q} = F^{p}H^{q}/F^{p+1}H^{q} and ∇^{E_{∞}} is the connection induced by ∇^{H}. □

Let g^{V} be a metric on V and identify E_{0}^{p} with (F^{p+1}V)^{⊥} ∩ F_{p}V. This allows us to identify

(7.34)  V ≅ E_{0} = ⨁_{p} E_{0}^{p} ,

and g^{V} induces a metric g^{E_{0}} on E_{0}. We inductively define metrics on E_{k} as follows. Assume that g^{E_{k}} is already defined, and let d_{k}^{′∗} denote the adjoint of d′_{k} with respect to g^{E_{k}}. Then we can identify E_{k+1} = H(E_{k}, d′_{k}) with the (d′_{k}+d_{k}^{′∗})-harmonic elements of E_{k}. This induces a metric g^{E_{k+1}} on E_{k+1}.

By assumption, the spectral sequence converges to E_{k_{0}+1} = E_{∞} → B, and we obtain a metric g^{E_{∞}} = g^{Ek_{0}+1}. A metric g^{H} on H allows us to identify E_{∞}^{p+q} with F^{p}H^{p+q} ∩ (F^{p+1}H^{p+q})^{⊥} similar as above.

7.35. Definition. Let (V, ∇^{V}) be a flat vector bundle with a ∇^{V} -parallel filtration F^{∗}V. A metric g^{V} is called adapted to F^{∗}V if the subbundles (F^{p+1}V)^{⊥} ∩ F_{p}V are also ∇^{V} -parallel.

7.36. Remark. We collect some examples and properties of adapted metrics.

(1) Any ∇^{V} -parallel metric is adapted to F^{∗}V.

(2) Suppose that (V, A′, h^{V}) is a family Thom-Smale complex. If each subbundle F^{p} can locally be written as the direct sum of some of the bundles V_{α}^{j} of Definition 2.35, then any metric g^{V} adapted to (V, A′, h^{V}) in the sense of Definition 2.37 is in particular adapted to F^{∗}V. This will be the case in all the applications we have in mind.

(3) Suppose g^{V} is adapted to F^{∗}V. Then the adjoint ∇^{V,∗} of ∇^{V} with respect to g^{V} also respects the filtration F^{∗}V and the decomposition (7.34), and we have

ch^{o}(∇^{V} , g^{V} ) = ∑_{p,q}(−1)^{p+q} ch^{o}(∇^{E_{0}^{p}}, g^{E_{0}^{p}}) = ch^{o}(∇^{E_{0}}, g^{E_{0}}) .

Similarly, if g_{0}^{V} and g_{1}^{V} are two adapted metrics that induce the same splitting (7.34), then g_{t}^{V} = (1 − t) g_{0}^{V} + t g_{1}^{V} is adapted for all t, so modulo exact forms,

c̃h^{o}(∇^{V} , g_{0}^{V} , g_{1}^{V} ) = c̃h^{o}(∇^{E_{0}}, g_{0}^{E_{0}}, g_{1}^{E_{0}}) .

(4) The bundle H → B is filtered by (7.32), but even if g^{V} is adapted to F^{∗}V, it is not clear that there is a metric g^{H} adapted to F^{∗}H. However, if we assume that g^{H} is adapted, we can use g^{H} to identify

H^{p,q} := F^{p}H^{p+q} ∩ (F^{p+1}H^{p+q})^{⊥} ≅ F^{p}H^{p+q} / F^{p+1}H^{p+q} = E_{∞}^{p,q}

as flat vector bundles with respect to ∇^{H} and ∇^{E_{∞}}, and to define

c̃h^{o}(∇^{H}, g^{H}, g^{E_{∞}}) = ∑_{p,q}(−1)^{p+q} c̃h^{o}(∇^{E_{∞}^{p,q}}, g^{H^{p,q}}, g^{E_{∞}^{p,q}}) .

We can now state the main result of this subsection, which we prove below.

7.37. Theorem. Suppose that g^{V} is adapted to the filtration F^{∗}V, and that g^{H} is adapted to the induced filtration on H. Then the torsion of V can be computed as

T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})

= T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) + ∑_{k=1}^{k_{0}} T^{BL}(D′_{k}, g^{E_{k}}) + c̃h^{o}(∇^{E_{∞}}, g^{H}, g^{E_{∞}}) .

7.38. Remark. Let N ⟶^{p_{1}} M ⟶^{p_{2}} B be a family of proper submersions as initem (1) at the beginning of this subsection. Assume that N carries a fibre-wise Morse function h_{1} with respect to π_{1}, and that M carries a fibre-wise Morse function h_{2}. Over each compact subset of B, h = ε, h_{1} + h_{2} ∘ π_{1} becomes a fibre-wise Morse function with respect to π = π_{2} ∘ π_{1} for ε small enough. The family-Thom-Smale complex of h carries a natural filtration by horizontal degree with respect to π_{1} if h_{2} admits a fibre-wise Morse-Smale gradient field. In this case, the filtration gives rise to the Leray spectral sequence of N_{b} → M_{b} for each b ∈ B.

In this situation, Theorem 7.37 implies that Theorem 0.1 is compatible with Ma’s formula for the analytic torsion of iterated fibre bundles in [Ma]. We leave the details to the interested reader. The necessity for the existence of a Morse-Smale gradient field for h_{2} is reflected by the fact that already the E_{1}-term in the Leray spectral of the filtered complex V associated with h carries a superconnection of the form ∇^{E_{1}} + d′_{1}.

Note that if h_{2} does not admit a fibre-wise Morse-Smale gradient field, we still get local filtrations over subsets of B by the values of h_{2}. However, the type of the sectral sequence changes as b ∈ B varies, and we are situation similar to the one describe in Section 1.

7.d. A proof of Theorem 7.37. If g^{V} is adapted, we may define a ∇^{V} -parallel number operator N^{F} on V, which acts on F^{p} ∩ (F^{p+1})^{⊥} by p. We define a new metric g_{r}^{V} on V by

(7.39)  g_{r}^{V} = g^{V} r^{N^{F}} = ⨁_{p} r^{p} g^{V} |_{E_{0}^{p}} .

Let g_{r}^{E_{k}} denote the metrics on the bundles E_{k} → B induced by g_{r}^{V} as above. Because we have identified E_{k+1}^{p,q} with a subbundle of E_{k}^{p,q} in order to define g^{E_{k+1}}, it follows by induction that

g_{r}^{E_{k}^{p,q}} = g^{E_{k}^{p,q}} r^{N^{F}} = r^{p} g^{E_{k}^{p,q}}

for all k ≥ 0, including k = ∞.

Clearly, the statement of our theorem is compatible with variation of g^{V} by metrics that preserve the splitting (7.34) of V. We may therefore prove the theorem by replacing g^{V} with g_{r}^{V} for some small r > 0, and considering the limit r → 0. Note that modulo exact forms,

c̃h^{o}(∇^{V} , g^{V} , g_{r}^{V} ) = \frac{1}{2} ∫_{1}^{r} str_{V} (N^{F} ) \frac{dr}{r} = \frac{log r}{2} str_{V} (N^{F} )

because N^{F} is parallel with respect to ∇^{V} and ∇^{V,∗}. We still denote the induced number operators on E_{k} by N^{F}. A careful count shows that

str_{E_{k}}(N^{F} ) = str_{E_{k+1}}(N^{F} ) + k (χ′(E_{k+1}) − χ′(E_{k})) ,

so

(7.40)  c̃h^{o}(∇^{V} , g^{V} , g_{r}^{V} ) = \frac{log r}{2} (str_{E_{∞}}(N^{F}) + ∑_{k=1}^{∞}(χ′(E_{k}) − χ′(E_{∞}))) .

If A′′ denotes the adjoint of A′ with respect to g^{V}, then the adjoint of A′ with respect to g_{r}^{V} is given by

A′′_{r} = r^{−N^{F}} A′′ r^{N^{F}} .

Conjugating both A′ and A′′_{r} with r^{\frac{N^{F}}{2}} gives new superconnections

Ã′_{r} = r^{\frac{N^{F}}{2}} A′ r^{− \frac{N^{F}}{2}}  and  Ã′′_{r} = r^{− \frac{N^{F}}{2}} A′′ r^{\frac{N^{F}}{2}} .

Let

Ã_{r} = \frac{1}{2} (Ã′_{r} + Ã′′_{r})  and  X̃_{r} = \frac{1}{2} (Ã′′_{r} − Ã′_{r}) .

We note that under the identification of (7.34),

(7.41)  lim_{r→0} Ã′_{r} = D′_{0}  and  lim_{r→0} Ã′′_{r} = D′′_{0} ,

where D′′_{0} is the adjoint of D′_{0} with respect to g^{E_{0}}, and thus also with respect to all g_{r}^{E_{0}} for all r > 0. If we define superconnections ^{1}Ã_{r}, ^{2}Ã_{r} and ^{3}Ã_{r} on V and ^{1}D̃_{0}, ^{2}D̃_{0} and ^{3}D̃_{0} on E_{0} as in (7.4), (7.13), (7.19) at T = 1, then similarly,

(7.42)  lim_{r→0} ^{j}Ã_{r} = ^{j}D̃_{0}

for j = 1, 2, 3.

Equation (7.42) immediately clarifies the following behaviour of two of the three integrals in Definition 7.22. We have

(7.43)  lim_{r→0} ∫_{(s,u)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{2}Ã_{r}^{2}}) = ∫_{(s,u)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{2}D̃_{0}^{2}}) ,

and

lim_{r→0} ∫_{(s,v)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{3}Ã^{2}}) − (χ′(V) − χ′(H)) (f′(\sqrt{−v}/2) − 1) \frac{dv ds}{2v})

= ∫_{(s,v)∈[0,1]^{2}}(\frac{1}{2} str_{V} (e^{−^{3}D̃_{0}^{2}}) − (χ′(E_{0}) − χ′(E_{1})) (f′(\sqrt{−v}/2) − 1) \frac{dv ds}{2v})

− (χ′(E_{1}) − χ′(E_{∞})) ∫_{0}^{1}(f′(\sqrt{−v}/2) − 1) \frac{dv}{2v} .

(7.44)

In particular, in the limit we recover two of the three terms in the definition of T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) up to a simple correction term.

If we write

α_{r,t} = (str_{V} (N^{V} f′(X̃_{r,t})) − χ′(H)) \frac{dt}{2t} ,

then by its construction in Section 7.a, the remaining integral takes the form

(7.45) ∫_{(s,t)∈[0,1]×[1,∞)}(\frac{1}{2} str_{V} (e^{−^{1}Ã_{r}^{2}}) − χ′(H) \frac{ds dt}{2t} − (χ′(V) − χ′(H)) f′(\sqrt{−t}/2) \frac{ds dt}{2t})

= ∫_{0}^{1}(4s(1 − s))^{\frac{N^{B}}{2}} ds ∫_{1}^{∞} α_{r,t} − (χ′(E_{0}) − χ′(E_{∞})) ∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t} .

We will split the integral over α_{r,t} into finitely many subintervals, which we will rescale differently.

Similar as in [MM], [D] or [Ma], we have to control the “small eigenvalues” of ã_{0,r} = \frac{1}{2} (ã′_{0,r}+ã′′_{0,r}) as r → 0. Let us write

a′_{j} = ∑_{k} a_{j}^{′(k)}  with  a_{j}^{′(k)}|_{E_{0}^{p,q}}: E_{0}^{p,q} ⟶ E_{0}^{p+k,q+1−k} ,
a′′_{j} = ∑_{k} a_{j}^{′′(k)}  with  a_{j}^{′′(k)}|_{E_{0}^{p,q}}: E_{0}^{p,q} ⟶ E_{0}^{p−k,q+k−1} ,
and  x_{j} = ∑_{k} x_{j}^{(k)}  with  x_{j}^{(k)} = (a_{j}^{′′(k)} − a_{j}^{′(k)}) / 2

for all p, q. Then the components ã_{j,r} and x̃_{j,r} of Ã_{r} − ∇^{E_{k},u} and X̃_{r} in Ω^{j}(B; End V) are given by

(7.46)  ã_{j,r} = ∑_{k} r^{\frac{k}{2}} a_{j}^{(k)}  and  x̃_{j,r} = ∑_{k} r^{\frac{k}{2}} x_{j}^{(k)} .

We define a superconnection D_{k} = \frac{1}{2} (D′_{k} + D′′_{k}) on E_{k}, and let d_{k} = \frac{1}{2} (d′_{k} + d′′_{k}) denote its component of degree 0. Similarly, we construct fibrewise operators

Y_{k} = \frac{1}{2} (D′′_{k} − D′_{k}) = y_{k} + \frac{ω_{k}}{2} ∈ End E_{k}

for k ≥ 1 with y_{k} = \frac{1}{2} (d′′_{k} − d′_{k}) and ω_{k} = ∇^{E_{k},∗} − ∇^{E_{k}}. We put

Y_{k,t} = \sqrt{t} y_{k} + \frac{ω^{E_{k}}}{2} .

7.47. Definition. Set

U_{ε,k} = { z ∈ ℂ | d(z, Spec(y_{k})) > ε } .

Let Π_{k} ∈ End E_{0} denote the orthogonal projection onto E_{k}, regarded as a subset of V ≅ E_{0}. The evaluation of (7.45) is based on a quantitative version of a result of Bismut-Berthomieu and Ma in our finite-dimensional setting. We will prove the following theorem in Section 7.e.

7.48. Theorem ([BB], Theorem 6.5; [Ma], Theorem 2.2). There exist r_{0} > 0, c > 0 and C > 0 such that for all r ∈ (0, r_{0}), k ≥ 1 and all λ ∈ U_{ε,k} with max(|λ| , ε^{−1}) < c r^{− \frac{1}{2}} , one has

∥(λ − r^{− \frac{k}{2}} x̃_{0,r})^{−1} − Π_{k} (λ − y_{k})^{−1} Π_{k}∥ ≤ C \sqrt{r} ε^{−1} (1 + max(|λ| , ε^{−1})) ,

and for r ∈ (0, r_{0}), k = 0 and λ ∈ U_{ε,0} with ε^{−1} < c r^{− \frac{1}{2}} ,

∥(λ − x̃_{0,r})^{−1} − (λ − x̃_{0,0})^{−1}∥ ≤ C \sqrt{r} ε^{−1} (1 + ε^{−1}) .

Let us fix constants 0 < c_{1} < c_{2} < ∞ such that for all k,

Spec(Y_{k}) \ {0} ⊂ i[c_{1}, c_{2}] ∪ −i[c_{1}, c_{2}] .

Let Γ_{r,t} = Γ_{r,t}^{1} ∪ Γ_{r,t}^{2} ∪ Γ_{r,t}^{3} ⊂ ℂ be the contour depicted in Figure 7.49.

− \frac{ε}{\sqrt{t}} 0 \frac{ε}{\sqrt{t}} ic_{1} ic_{2}^{\frac{ic1}{2\sqrt{r}}} −ic_{1} −ic_{2} − \frac{ic1}{2\sqrt{r}}  Γ_{r,t}^{3} Γ_{r,t}^{2} Γ_{r,t}^{1} Γ_{r,t}^{2} Γ_{r,t}^{3}

Figure 7.49. The contour Γ_{r,t} = Γ_{r,t}^{1} ∪ Γ_{r,t}^{2} ∪ Γ_{r,t}^{3} ⊂ ℂ.

We split the integral (7.45) as

∫_{1}^{∞} α_{r,t} = ∫_{1}^{r^{− \frac{1}{2}}} α_{r,t} + ∑_{k=1}^{k_{0}−1} ∫_{r^{−k+ \frac{1}{2}}}^{r^{−k− \frac{1}{2}}} α_{r,t} + ∫_{r^{−k_{0}+ \frac{1}{2}}}^{∞} α_{r,t}

= ∫_{1}^{r^{− \frac{1}{2}}} α_{r,t} + ∑_{k=1}^{k_{0}−1} ∫_{r^{\frac{1}{2}}}^{r^{− \frac{1}{2}}} α_{r,r^{−k}t} + ∫_{r^{\frac{1}{2}}}^{∞} α_{r,r^{−k_{0}} t} .

For k ≥ 0, r ∈ (0, r_{0}) with r_{0} sufficiently small and for t ∈ (t^{\frac{r}{2}} , t^{− \frac{r}{2}} ), we consider

α′_{k,r,t} := (str_{V} (N^{V} f′(X̃_{r,r^{−k}t})) − χ′(E_{k+1})) \frac{dt}{2t}

= (str_{V} (N^{V} f′(t^{\frac{1}{2}} r^{− \frac{k}{2}} x̃_{0,r} + x̃_{1,r} + t^{− \frac{1}{2}} r^{\frac{k}{2}} x̃_{2,r} + …)) − χ′(E_{k+1})) \frac{dt}{2t} ,

such that

(7.50)  ∫_{r^{\frac{1}{2}}}^{r^{− \frac{1}{2}}} α_{r,r^{−k}t} = ∫_{r^{\frac{1}{2}}}^{r^{− \frac{1}{2}}} α′_{k,r,t} − (χ′(E_{k+1}) − χ′(E_{∞})) \frac{log r}{2} .

We will treat the integrals over t ∈ [r^{\frac{1}{2}} , 1] and t ∈[1, r^{− \frac{1}{2}} ] separately. For t ∈[1, r^{− \frac{1}{2}} ] and k ≥ 0, we write

f′(X̃_{r,r^{−k}t}) = \frac{1}{2πi} ∫_{Γ_{r,t}^{1}∪Γ_{r,t}^{2}∪Γ_{r,t}^{3}} \frac{f′(\sqrt{t} λ)}{λ − r^{− \frac{k}{2}} x̃_{0,r}}

∑_{j=0}^{∞}(t^{− \frac{1}{2}} (x̃_{1,r} + O(r^{\frac{k}{2}} t^{− \frac{1}{2}} )) \frac{1}{λ − r^{− \frac{k}{2}} x̃_{0,r}} )^{j} dλ .

Let us note that because Γ_{r,t} depends explicitly on t, the real part of \sqrt{t} λ remains bounded by ε, so that f(\sqrt{t} λ) remains uniformly bounded, too.

For λ ∈ Γ_{r,t}^{1} and k ≥ 1, Theorem 7.48 implies that

∥\frac{f′(\sqrt{t} λ)}{λ − r^{− \frac{k}{2}} x̃_{0,r}} ∑_{j=0}^{∞}(t^{− \frac{1}{2}} (x̃_{1,r} + O(r^{\frac{k}{2}} t^{− \frac{1}{2}} )) \frac{1}{λ − r^{− \frac{k}{2}} x̃_{0,r}} )^{j} − \frac{f′(\sqrt{t} λ)}{λ − t^{− \frac{1}{2}} Y_{k,t}} ∥ ≤ C \sqrt{r} t .

Integrating over the contour Γ_{r,t}^{1} of length O(t^{− \frac{1}{2}} ) gives

∥∫_{Γ_{r,t}^{1}}(\frac{f′(\sqrt{t} λ) dλ}{λ − t^{− \frac{1}{2}} X̃_{r,r^{−k}t}} − \frac{f′(\sqrt{t} λ) dλ}{λ − t^{− \frac{1}{2}} Y_{k,t}} )∥ ≤ C \sqrt{rt} .

For t < r^{− \frac{1}{2}} , we have \sqrt{r} < t^{−1}, so the integral over t ∈ [1, r^{− \frac{1}{2}} ] of the difference above converges. By [BL], we already know that

∫_{1}^{∞}(∫_{Γ_{r,t}^{1}} str_{V} (N^{V} \frac{f′(\sqrt{t} λ) dλ}{λ − t^{− \frac{1}{2}} Y_{k,t}} ) − χ′(E_{k+1})) \frac{dt}{2t}

converges, so we have uniform convergence for the contribution from Γ_{r,t}^{1} to (7.50) for t ∈ [1, r^{− \frac{1}{2}} ].

The reasoning for the integral over Γ_{r,t}^{2} is similar, but we use that |Im λ| > c > 0, so f(\sqrt{t} λ) decays rapidly as t → ∞ uniformly on Γ_{r,t}^{2}. Finally the integral over Γ_{r,t}^{3} vanishes as r → 0 because |Im λ| > r^{− \frac{1}{2}} c. This shows that

lim_{r→0} ∫_{1}^{r^{− \frac{1}{2}}} α′_{k,r,t} = ∫_{1}^{∞}(str(N^{V} f′(Y_{k,t})) − χ′(E_{k+1})) \frac{dt}{2t} .

For k = 0, we similarly recover the remaining contribution to T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) up to divergencies and correction terms, because

lim_{r→0}(∫_{1}^{r^{− \frac{1}{2}}} α_{r,t} + \frac{log r}{4} (χ′(E_{1}) − χ′(E_{∞})))

= ∫_{1}^{∞}(str_{V} (N^{E_{0}} f′(X̃_{0,t})) − χ′(E_{1}) − (χ′(E_{0}) − χ′(E_{1})) f′(\sqrt{−t}/2)) \frac{dt}{2t}

+ (χ′(E_{0}) − χ′(E_{1})) ∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t} .

(7.51)

We use the contour Γ_{r,1} for all t ∈ [r^{\frac{1}{2}} , 1] and k ≥ 1. Using Theorem 7.48 and the results of [BL], we can again prove uniform convergence, and we find that

lim_{r→0} ∫_{r^{\frac{1}{2}}}^{1} (str(N^{V} f′(X̃_{r,r^{−k}t})) − χ′(E_{k}) f′(\sqrt{−t}/2)) \frac{dt}{2t}

= ∫_{0}^{1}(str(N^{V} f′(Y_{k,t})) − χ′(E_{k}) f′(\sqrt{−t}/2)) \frac{dt}{2t} .

Thus for k ≥ 1,

lim_{r→0}(∫_{r^{\frac{1}{2}}}^{r^{− \frac{1}{2}}} α_{r,r^{−k}t} + \frac{log r}{4} (χ′(E_{k}) − χ′(E_{k+1})) + \frac{log r}{2} (χ′(E_{k+1}) − χ′(E_{∞})))

= ∫_{0}^{∞}(str_{V} (N^{E_{k}} f′(Y_{k,t})) − χ′(E_{k+1}) − (χ′(E_{k}) − χ′(E_{k+1})) f′(\sqrt{−t}/2)) \frac{dt}{2t}

+ (χ′(E_{k}) − χ′(E_{k+1})) (∫_{1}^{∞} f′(\sqrt{−t}/2) \frac{dt}{2t} + ∫_{0}^{1}(f′(\sqrt{−t}/2) − 1) \frac{dt}{2t}) .

(7.52)

Modulo divergencies and correction terms in degree 0, we find T^{BL}(D′_{k}, g^{E_{k}}) in the limit. Finally, we regard the last contribution

∫_{√\overline{r}}^{∞} α_{r,r^{−k_{0}} t} .

Because E_{k_{0}} = E_{∞}, we know that

Spec(r^{− \frac{k_{0}}{2}} x̃_{0,r}) ⊂ \{0\} ∪ i(−∞, −r^{− \frac{1}{2}} \frac{c_{1}}{2} ) ∪ i(r^{− \frac{1}{2}} \frac{c_{1}}{2} , ∞) .

For λ ∈ Γ_{r,t}^{1}, we therefore have

∥(λ − r^{− \frac{k_{0}}{2}} x̃_{0,r})^{−1} − Π_{ker x̃_{0,r}} λ^{−1} Π_{ker x̃_{0,r}} ∥ ≤ C \sqrt{r}

and

∥(λ − r^{− \frac{k_{0}}{2}} X̃_{r,t})^{−1} − Π_{ker x̃_{0,r}} λ^{−1} Π_{ker x̃_{0,r}} ∥ ≤ C \sqrt{r/t} .

Since only integrals of f′(\sqrt{t} λ) (λ − r^{− \frac{k_{0}}{2}} X̃_{r,t})^{−1} over Γ_{r,t}^{1} and Γ_{r,t}^{3} contribute to f′(X̃_{r,r^{− \frac{k_{0}}{2}} t}), we

conclude that

(7.53)  lim_{r→0} ∫_{r^{\frac{1}{2}}}^{∞} α_{r,r^{−k_{0}} t} = 0 .

Combining (7.43), (7.44), (7.51), (7.52) and (7.53), we can compute the asymptotic behaviour of T (∇^{V} , A′, g_{r}^{V}) by

(7.54) lim_{r→0}(T (∇^{V} , A′, g_{r}^{V} ) − \frac{log r}{2} ∑_{k=1}^{k_{0}−1}(χ(E_{k}) − χ(E_{∞})))

= T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) + ∑_{k=1}^{k_{0}−1} T^{BL}(D′_{k}, g^{E_{k}}) .

Moreover, we can extend both summations over k to infinity.

It remains to study the variation of c̃h^{o}(∇^{H}, g^{H}, g_{V,r}^{H}) as r → 0. Identifying H and E_{∞} using g^{H}, we find

(7.55) c̃h^{o}(∇^{H}, g^{H}, g_{V,r}^{H}) = c̃h^{o}(∇^{H}, g^{H}, g^{E_{∞}}) + c̃h^{o}(∇^{E_{∞}}, g^{E_{∞}}, g_{r}^{E_{∞}}) + c̃h^{o}(∇^{H}, g_{r}^{E_{∞}}, g_{V,r}^{H}) .

Clearly

(7.56)  c̃h^{o}(∇^{E_{∞}}, g^{E_{∞}}, g_{r}^{E_{∞}}) = \frac{log r}{2} str_{E_{∞}}(N^{F}) .

If we put H_{r} = ker a_{0,r} and H̃ = ker ã_{0,r} and let ∇^{H_{r}} and ∇H̃_{r} denote the connections induced by A′ and Ã′_{r}, then

c̃h^{o}(∇^{H}, g_{r}^{E_{∞}}, g_{V,r}^{H}) = c̃h^{o}(∇^{H_{r}}, (r^{\frac{N^{F}}{2}} Π_{k_{0}} r^{− \frac{N^{F}}{2}} |_{H_{r}})^{∗}g_{r}^{V} , g_{r}^{V} |_{H_{r}})
= c̃h^{o}(∇H̃_{r}, (Π_{k_{0}}|H̃_{r})^{∗}g^{V} , g^{V} |H̃_{r})

because E_{k_{0}} = E_{∞}. On the other hand, let ΠH̃_{r} denote the g^{V} -orthogonal projection onto H̃_{r}. Then

lim_{r→0} ΠH̃_{r} = \frac{1}{2πi} lim_{r→0} ∫_{Γ_{r,t}^{1}} \frac{dλ}{λ − r^{− \frac{k_{0}}{2}} ã_{0,r}} = Π_{k_{0}}

in all C^{k}-norms, so

lim_{r→0} c̃h^{o}(∇^{H}, g_{r}^{E_{∞}}, g_{V,r}^{H}) = lim_{r→0} c̃h^{o}(∇H̃_{r}, (Π_{k_{0}}|H̃_{r})^{∗}g^{V} , g^{V} |H̃_{r}) = 0 .

By (7.56) and (7.55),

(7.57)  lim_{r→0}(c̃h^{o}(∇^{H}, g^{H}, g_{V,r}^{H}) − \frac{log r}{2} str_{E_{∞}}(N^{F})) = c̃h^{o}(∇^{H}, g^{H}, g^{E_{∞}}) .

Now it follows from (7.40), (7.54) and (7.57) that

T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})

= lim_{r→0}(T (∇^{V} , A′, g_{r}^{V} ) + c̃h^{o}(∇^{V} , g_{r}^{V} , g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V,r}^{H}))

= T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) + ∑_{k=1}^{∞} T^{BL}(D′_{k}, g^{E_{k}}) + c̃h^{o}(∇^{H}, g^{H}, g^{E_{∞}}) .

This completes the proof of Theorem 7.37. □

7.e. A proof of the Berthomieu-Bismut convergence results. This chapter should be considered an appendix containing the proofs of some results that are analogous to results proved in [BB]. In particular, we prove Theorem 7.48 for k ≥ 1, however regarding the family of self-adjoint operators ã_{0,r} instead of x̃_{0,r}. The case k = 0 is an easy exercise that we leave to the reader.

7.58. Proposition ([BB], Theorem 6.1). For each r ≥ 1 and each k ≥ 1, there are inclusions E_{k,r}^{p,q} ⊂ E_{0}^{p,q} such that E_{k,r}^{p,q} = ker(d′_{k−1,r} + d′′_{k−1,r}) ⊂ E_{k−1,r}^{p,q}, and E_{k,r}^{p,q} is given by

(1) E_{k,r}^{p,q} = { v_{0} ∈ E_{0}^{p,q} | there exist v_{1}, …, v_{k−1} such that

∑_{i=0}^{j} ã_{0,r}^{(i)}v_{j−i} = 0 for j = 0, …, k − 1 } .

Let Π_{k,r}: E_{0} → E_{k,r} denote the orthogonal projection with respect to g. The operator d_{k,r} = \frac{1}{2} (d′_{k,r} + d′′_{k,r}) acts on E_{k,r} by

(2)  d_{k,r}v_{0} = Π_{k,r} ∑_{i=1}^{k} ã_{0,r}^{(i)}v_{j−i} .

There exists a linear map φ_{k}: E_{k,T} → V^{k−1} such that for v_{0} ∈ E_{k,r} and (v_{1}, …, v_{k−1}) = φ_{k}(v_{0}), one has

(3)  ∑_{i=0}^{j} ã_{0,r}^{′(i)} r^{\frac{j−i}{2}} v_{j−i} = 0  for j = 0, …, k − 1 .

Proof. The proof is a simple adaptation of the proof given in [BB]. □

It follows from this theorem that for all k both the subspace E_{k,r} ⊂ E_{0} and the operator

r^{− \frac{k}{2}} d_{k,r} ∈ End(E_{k,r})

are independent of r. Later on, we put

E_{k} = E_{k,r} ,  Π_{k} = Π_{k,r}  and  d_{k} = r^{− \frac{k}{2}} d_{k,r} ∈ End(E_{k}) .

If φ_{k}(v_{0}) = (v_{1}, …, v_{k−1}), we define φ_{k,r} by putting

φ_{k,r}(v_{0}) = (r^{\frac{1}{2}} v_{1}, …, r^{\frac{k−1}{2}} v_{k−1}) .

Next, we study the resolvent of ã_{0,r}. We begin with an approximation. Because (d′_{0,r})^{2} = (d′′_{0,r})^{2} = 0, the operators ρ d′_{0,r} + ρ̄d′′_{0,r} have the same square for all ρ ∈ S^{1} ⊂ ℂ and thus the same spectrum. For this reason,

iU_{ε,k} = { z ∈ ℂ | d(z, Spec(d_{k})) > ε } .

7.59. Proposition ([BB], Theorem 6.2). For each k ≥ 0 and each λ ∈ iU_{ε,k}, there exists a linear map

ψ_{k,r,λ}: V ⟶ V^{k+1}  with  v ⟼ (v_{0,r}, …, v_{k,r})

such that

(1)  v_{0,r} = (λ − d_{k})^{−1}Π_{k}v ∈ E_{k}

and

(2)  ∑_{i=0}^{j} ã_{0,r}^{(j−i)}v_{i,r} = 0  for j = 0, …, k − 1, and

(3)  r^{− \frac{k}{2}} ∑_{i=0}^{k} ã_{0,r}^{(k−i)}v_{i,r} = λv_{0,r} − v .

Moreover there exists a constant C > 0 such that

(4)  |v_{i,r}| ≤ C r^{\frac{i}{2}} ε^{−1} |v|  for i = 1, …, k .

Proof. As in the proof of Proposition 7.58, we first solve the equations (2) and

(2’)  Π_{k−l,r}(λv_{0,r} − ∑_{i=0}^{k} ã_{0,r}^{(k−i)}v_{i,r}) = Π_{k−l,r}v

with v_{k,r} = 0 by induction over l = 0, …, k − 1, and finally give a solution for l = k, where (3’) is equivalent to (3).

For l = 0, we have to find v_{0,r} ∈ E_{k} with

(λ − r^{− \frac{k}{2}} d_{k,r}) v_{0,r} = (λ − d_{k}) v_{0,r} = Π_{k,r}v ,

which gives (1). We put (v_{1,r}, …, v_{k−1,r}) = φ_{k,r}(v_{0}). Then by Proposition 7.58, equations (2) and (3’) hold for l = 0, and there exists a constant C such that

|v_{0,r}| ≤ C (ε (1 + |λ|))^{−1} |v|  and  |v_{i,r}| ≤ C r^{\frac{i}{2}} ε^{−1} |v|  for i = 1, …, k − 1 .

In the following steps, we have to choose w_{0} ∈ E_{k−l} such that

d_{k−l}w_{0} = r^{\frac{l}{2}} Π_{k−l,r}(λv_{0,r} − v − r^{− \frac{k}{2}} ∑_{i=0}^{k−1} ã_{0,r}^{(k−i)}v_{i,r}) ∈ E_{k−l+1}^{′⊥} ⊂ E_{k−l} .

The right hand side is still of order C r^{\frac{l}{2}} ε^{−1} |v|. We have |d_{k−l}^{−1}|_{E_{k−l+1}^{′⊥}} | ≤ C′, so that w_{0} is of

order C C′ r^{\frac{l}{2}} ε^{−1} |v|. For definiteness, we may demand that s_{l} + w_{0} ∈ E_{k−l+1}^{′⊥} without changing the estimate above. We take (w_{1}, …, w_{l−1}) = φ_{k−l,r}(w_{0}) and replace (v_{0,r}, …, v_{k−1,r}) by

(v_{0,r}, …, v_{l−1,r}, v_{l,r} + w_{0}, …, v_{k−1,r} + w_{l−1}) .

For this new tupel, (2) and (3’) hold for l and (4) holds with a new constant C.

Finally for k = l, it remains to determine v_{k,r} ∈ E_{1}^{′⊥} with

ã_{0}^{(0)} v_{k,r} = r^{\frac{k}{2}} (λv_{0,r} − v) − ∑_{i=0}^{k−1} ã_{0,r}^{(k−i)}v_{i,r} ∈ E_{1}^{′⊥} .

This implies that (2) and (3) hold, and (4) holds with a new constant C, so we are done. □

If we put

ψ̄_{k,r,λ} = v_{0,r} + ⋯ + v_{k,r} ,

then Proposition 7.59 immediately implies

(7.60)  (λ − r^{− \frac{k}{2}} ã_{0,r}) ψ̄_{k,r,λ}(v) = v + λ ∑_{i=1}^{k} v_{i,k} − r^{− \frac{k}{2}} ∑_{j=k+1}^{∞} ∑_{i=0}^{k} ã_{0,r}^{(j−i)} v_{i,r} .

Note that the first sum is finite because our filtration F^{p}V has finite length. As in [BB], this gives the main result on the asymptotics of the operator ã_{0,r}.

7.61. Theorem ([BB], Theorem 6.5). There exist r_{0} > 0, c > 0 and C > 0 such that for all k ≥ 1, all r ∈ (0, r_{0}) and all λ ∈ iU_{ε,k} with max(|λ| , ε^{−1}) < c r^{− \frac{1}{2}} , one has

∥(λ − r^{− \frac{k}{2}} ã_{0,r})^{−1} − Π_{k} (λ − d_{k})^{−1} Π_{k}∥ ≤ C \sqrt{r} ε^{−1} (1 + max(|λ| , ε^{−1})) . □

For k = 0, we may replace max(|λ| , ε^{−1}) simply by ε^{−1} in the theorem. Theorem 7.48 follows from Theorem 7.61 by replacing ã_{0,r} by x̃_{0,r} etc.

Proof. Write (7.60) as

(λ − r^{− \frac{k}{2}} ã_{0,r}) ψ̄_{k,r,λ}(v) = v + R_{k,r,λ}(v) ,

then Proposition 7.59 implies that the operator norms of ψ_{k,r,λ} and R_{k,r,λ}(v) satisfy the estimates

∥ψ_{k,r,λ} − Π_{k} (λ − d_{k})^{−1} Π_{k}∥ = O(\sqrt{r}/ε)  and  ∥R_{k,r,λ}∥ = O(\sqrt{r} max(|λ| , ε^{−1})) .

The “max” comes from the fact that the ε^{−1} in Proposition 7.59 becomes relevant only if |λ| is small (for large |λ|, we may may choose a large value of ε). Let us fix c > 0 and r_{0} > 0 suitably small and let us assume that r < r_{0} and max(|λ| , ε^{−1}) ≤ c r^{− \frac{1}{2}} . Then (7.60) gives us a series converging to the resolvent of r^{\frac{k}{2}} ã_{0,r}, and we estimate

\begin{aligned}(λ − r^{− \frac{k}{2}} ã_{0,r})^{−1} & = ∑_{k=0}^{∞} ψ_{k,r,λ} (−R_{k,r,λ})^{k} \\ & = (Π_{k} (λ − d_{k})^{−1} Π_{k} + O(\sqrt{r}/ε)) (1 + O(\sqrt{r} max(|λ| , ε^{−1}))) . □ \end{aligned}

As in [BB], this implies that for r sufficiently small, the operator ã_{0} has eigenvalues of order r^{\frac{k}{2}} that correspond precisely to the eigenvalues of d_{k}. The condition on |λ| guarantees that λ stays away from the eigenvalues of r^{− \frac{k}{2}} ã_{0,r} corresponding to those of r^{\frac{k′−k}{2}} d_{k′} with k′ < k when taking the resolvent. Similarly, the restriction to ε^{−1} guarantees that λ avoids the eigenvalues of r^{− \frac{k}{2}} ã_{0,r} corresponding to those of r^{\frac{k′−k}{2}} d_{k′} with k′ > k.

8. Igusa’s higher Franz-Reidemeister torsion

In this chapter, we state and prove Theorem 8.1, by which Igusa’s higher Franz-Reidemeister torsion coincides with the higher torsion constructed in Definition 7.29.

Using Igusa’s two-index theorem, the proof of Theorem 8.1 is reduced to the case of a family Thom-Smale complex V = V^{0} ⊕ V^{1} living in only two degrees. We also give a sketch of the proof in Section 8.a.

We construct a family of flat superconnections from a given flat superconnection on a bundle V = V^{0} ⊕ V^{1} in Section 8.b. This family is used to decompose T (∇^{V} , A′, g^{V}) into four terms in Section 8.c.

One of these terms is a Bismut-Lott torsion class, and we show in Section 8.d that it is equal to Igusa’s class F in [I2]. The remaining three terms are polynomials in the components of A′, and their sum forms an L-class à la [BG], cf. Section 2.e. We identify these terms with Igusa’s classes G, K and \overline{K}^{∗} in Subsections 8.e and f.

8.a. Igusa’s two-index theorem and a proof of Theorem 8.1. We formulate the main result of this section and reduce its proof to the special case of a family Thom-Smale complex living in only two degrees.

Let τ = τ(M_{r}(ℂ), U(r)) and T = T (M_{r}(ℂ), U(r)) ∈ H^{even}(Wh^{h}(M_{r}(ℂ), U(r))) denote Igusa’s universal higher Franz-Reidemeister torsion, and the class constructed in Definition 7.29 above.

8.1. Theorem. The classes τ and T in H^{even}(Wh^{h}(M_{r}(ℂ), U(r))) coincide.

A proof of this theorem is given at the end of this subsection.

8.2. Definition. Let Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) be the full simplicial subcategory of the differential Whitehead category Wh^{diff}(M_{r}(ℂ), U(r)) consisting in degree k of all objects (P, a′) such that P = P^{0} ∪ P^{1} and such that a′_{0}: (ℂ^{r})^{P^{0}} → (ℂ^{r})^{P^{1}} is invertible at one (and thus at all) points of Δ_{k}. Let the acyclic differential two-index Whitehead space Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) be the geometric realisation of Wh^{diff}(M_{r}(ℂ), U(r)).

8.3. Remark. We can also construct simplicial and mixed models Wh^{simp,h,{0,1}}(M_{r}(ℂ), U(r)) and Wh^{mix,h,{0,1}}(M_{r}(ℂ), U(r)) of the acyclic two-index Whitehead space as in Section 6. However, the proof of Theorem 6.31 shows that these spaces are homotopy equivalent.

8.4. Fact (Two-Index Theorem). The acyclic two-index Whitehead space Wh^{h,{0,1}}(M_{r}(ℂ), U(r)) is a deformation retract of the full acyclic Whitehead space Wh^{h}(M_{r}(ℂ), U(r)).

This implies that it is sufficient to prove Theorem 8.1 for the restrictions of τ and T to the acyclic two-index Whitehead space Wh^{h,{0,1}}(M_{r}(ℂ), U(r)).

Proof. of Theorem 8.1. By the two-index theorem, it is enough to check the equality of τ and T on Wh^{h,{0,1}}(M_{r}(ℂ), U(r)). All constructions in this and the following subsections will be natural with respect to pull-back. Thus, me may and we will replace Wh^{h,{0,1}}(M_{r}(ℂ), U(r)) for notational convenience by an arbitrary manifold B equipped with a family Thom Smale complex of the special form

(8.5)  (V, ∇^{V} , A′) = (V^{0} ⊕ V^{1}, ∇^{V^{0}} ⊕ ∇^{V^{1}}, ∇^{V^{0}} ⊕ ∇^{V^{1}} + a_{0} + a_{1}^{0} + a_{1}^{1} + a_{2}) ,

where a_{0}, a_{1}^{0}, a_{1}^{1} and a_{2} are the components of A′ −∇^{V}. We assume that V carries an adapted metric g^{V} as in Definition 2.37 that is parallel. We will forget about the family of endomorphisms h^{V}; we will however remember that the coefficients of A′ − ∇^{V} take values in a nilpotent subalgebra of End V in a neighbourhood of each point of B, cf. Remark 7.28 (3) above.

There is a natural construction of Igusa’s higher Franz-Reidemeister torsion τ = τ(∇^{V} , A′, g^{V}) in this situation, which we will introduce in Subsections 8.d–f. In fact, it follows from Proposition 8.18 and Lemmata 8.38, 8.48 and 8.72 that

T (∇^{V} , A′, g^{V} )^{[≥2]} = (F + G + K − \overline{K}^{∗})^{[≥2]} + d(T̃ + G̃ + K̃ − \overline{K̃}^{∗})
= τ(∇^{V} , A′, g^{V})^{[≥2]} + d(T̃ + G̃ + K̃ − \overline{K̃}^{∗})

for certain naturally defined forms T̃, G̃ and K̃ depending only on ∇^{V}, A′, and g^{V}. This implies that the cochains obtained by integrating τ(∇^{V}, A′, g^{V})^{[≥2]} and T (∇^{V}, A′, g^{V})^{[≥2]} over the simplices of |Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r))| differ by an exact cochain. On the other hand, we know by [I2] and Theorem 2.48 that both τ(∇^{V}, A′, g^{V})^{[0]} and T (∇^{V}, A′, g^{V})^{[0]} coincide with the classical fibre-wise Franz-Reidemeister torsion of the complex (V, a_{0}). Because both τ and T are even classes, this proves Theorem 8.1. □

8.b. A family of flat superconnections. We now restrict our attention to family Thom-Smale complexes of the form (8.5).

Let us start by classifying all flat superconnections of total degree one with a prescribed term in degree zero on a vector bundle V = V^{0} ⊕ V^{1}. If ∇^{V^{0}} and ∇^{V^{1}} are connections on V^{0} and V^{1} and a_{0}: V^{0} → V^{1} is an isomorphism, we define new connections

(8.6)  a_{0}^{#}∇^{V^{1}} = −a_{0}^{−1} ∇^{V^{1}} a_{0}  and  a_{0#}∇^{V^{0}} = −a_{0} ∇^{V^{0}} a_{0}^{−1} .

By our sign convention, these are indeed connections, with

(a_{0}^{#}∇^{V^{1}})_{v} = a_{0}^{−1} ∇_{v}^{V^{1}} a_{0}  and  (a_{0#}∇^{V^{0}})_{v} = a_{0} ∇_{v}^{V^{0}} a_{0}^{−1}

for all v ∈ T B.

8.7. Lemma. Let V = V^{0} ⊕ V^{1} → B be a ℤ-graded vector bundle, let a_{0}: V^{0} → V^{1} be a vector bundle isomorphism, and let ∇^{1} be a connection on V^{1}. Then

A′ = a_{0} + a_{0}^{#}∇^{1} ⊕ ∇^{1} − a_{0}^{−1} (∇^{1})^{2}

defines a flat superconnection of total degree 1 on V. Moreover, all flat superconnections of total degree 1 on V with a given invertible a_{0} arise this way.

Proof. Let A′ = a_{0} + ∇^{0} ⊕ ∇^{1} + a_{2} be a superconnection of total degree 1. Flatness of A′ is equivalent to the following three conditions:

(1)  a_{0} ∇^{0} + ∇^{1} a_{0} = 0 ,
(2)  a_{2} a_{0} + (∇^{0})^{2} = 0  and  a_{0} a_{2} + (∇^{1})^{2} = 0 ,
(3)  and  ∇^{0} a_{2} + a_{2} ∇^{1} = 0 .

We assume that a_{0} is invertible. Then equation (1) is equivalent to

(1′)  ∇^{0} = a_{0}^{#}∇^{1} .

Given (1) or (1′), we have a_{0} (∇^{0})^{2} = (∇^{1})^{2} a_{0}, so equation (2) is equivalent to

(2′)  a_{2} = −a_{0}^{−1} (∇^{1})^{2} .

Finally if (1) or (1′) and (2) or (2′) hold, then (3) follows from the Bianchi identity for ∇^{1}, because

∇^{0} a_{2} + a_{2} ∇^{1} = −a_{0}^{−1} [∇^{1}, (∇^{1})^{2}] = 0 . □

In order to compare Igusa’s torsion class in the Chern normalisation with T (∇^{V} , A′, g^{V}), we consider a family of flat superconnections linking A′ to a “trivial” flat superconnection. Consider the bundle

\overline{V} = V × [0, 1] ⟶ \overline{B} = B × [0, 1] ,

and extend the connection ∇^{V} = ∇^{V^{0}} ⊕ ∇^{V^{1}} and the superconnection A′ trivially to \overline{V}. Let r be the coordinate of [0, 1], and suppose that the original superconnection takes the form

A′ = ∇^{V} + a_{0} + a_{1}^{0} + a_{1}^{1} + a_{2}

with an invertible a_{0}. We define a new superconnection on \overline{V} by

(8.8)  \overline{A}′ = a_{0} + a_{0}^{#}(∇^{\overline{V}^{1}} + r a_{1}^{1}) ⊕ (∇^{\overline{V}^{1}} + r a_{1}^{1}) − a_{0}^{−1}(∇^{\overline{V}^{1}} + r a_{1}^{1})^{2} .

8.9. Lemma. The superconnection \overline{A}′ is flat, and its restriction to r = 1 equals A′.

Proof. The first statement is a consequence of Lemma 8.7. The second statement also follows easily from Lemma 8.7, because in both cases it suffices to compare a_{0} and the connection acting on V^{1}. □

8.10. Remark. Note that the coefficients of \overline{A}′|_{r=1} − ∇\overline{V} = A′ − ∇^{V} take values in a nilpotent subalgebra of End V by assumption. Similarly, we have

\overline{A}′|_{r=0} − a_{0}^{#}∇^{V^{1}} ⊕ ∇^{V^{1}} = a_{0} ∈ Hom(V^{0}, V^{1}) ⊂ End V ,

and Hom(V^{0}, V^{1}) is of course a nilpotent subalgebra. Thus it makes sense to talk about the higher torsions

(8.11)  T (a_{0}^{#}∇^{V^{1}} ⊕ ∇^{V^{1}}, \overline{A}′|_{r=0}, g^{V} ) = T^{BL}(\overline{A}′|_{r=0}, g^{V} )  and  T (∇^{V} , \overline{A}′|_{r=1}, g^{V} ) .

Note however that in general there will be no meaningful torsion of A′|_{r} for some r ∈ (0, 1). Even worse, in general there will be no flat connection on \overline{V} which restricts to a_{0}^{#}∇^{V^{1}} ⊕ ∇^{V^{1}} at r = 0 and to ∇^{V} at r = 1.

Let \overline{A}′′ be the adjoint of \overline{A}′ with respect to g^{V}. Since we assume that g^{V} is unitarily flat with respect to ∇^{V}, we have ∇^{V^{1}} = ∇^{V^{1},∗}. We find that

A′′ = a_{0}^{∗} + a_{0#}^{∗}(∇^{V^{1}} + ra_{1}^{1∗}) ⊕ (∇^{V^{1}} + ra_{1}^{1∗}) − (∇^{V^{1}} + ra_{1}^{1∗})^{2} (a_{0}^{∗})^{−1} ,

where we agree that a_{1}^{1∗}(v) = −(a_{1}^{1}(v)) for v ∈ T B.

We extend the restrictions to the hyperplane T = 1 of the superconnections ^{1}Ã, ^{2}Ã and ^{3}Ã of (7.4), (7.13) und (7.19) to the bundles

\overline{V}^{1} ⟶ ^{1}\overline{B} = B × [0, 1]^{2} × [1, ∞)  with coordinates (r, s, t) ∈ [0, 1]^{2} × [1, ∞) ,
\overline{V}^{2} ⟶ ^{2}\overline{B} = B × [0, 1]^{3}  with coordinates (r, s, u) ∈ [0, 1]^{3} ,
and  ^{3}\overline{V} ⟶ ^{3}\overline{B} = B × [0, 1]^{3}  with coordinates (r, s, v) ∈ [0, 1]^{3} .

Therefore, set

\begin{aligned}Ã^{1} & = \frac{\sqrt{t}}{2} (a_{0} + a_{0}^{∗}) + (1 − s) a_{0}^{#}(∇^{\overline{V}^{1}} + ra_{1}^{1}) ⊕ (∇^{\overline{V}^{1}} + ra_{1}^{1}) \\ & + s a_{0#}^{∗}(∇^{\overline{V}^{1}} + ra_{1}^{1∗}) ⊕ (∇^{\overline{V}^{1}} + ra_{1}^{1∗}) \\ & − \frac{2}{\sqrt{t}} ((1 − s)^{2} a_{0}^{−1} (∇^{\overline{V}^{1}} + ra_{1}^{1})^{2} + s (∇^{\overline{V}^{1}} + ra_{1}^{1∗})^{2} (a_{0}^{∗})^{−1}) \\ & − 2N^{V} ds + (2s − 1) \frac{N^{V}}{2t} dt , \end{aligned} (8.12)

\begin{aligned}Ã^{2} & = \frac{1}{2} ((1 − s + us) a_{0} + (u + s − us) a_{0}^{∗}) \\ & + (1 − s) a_{0}^{#}(∇^{\overline{V}^{1}} + ra_{1}^{1}) ⊕ (∇^{\overline{V}^{1}} + ra_{1}^{1}) \\ & + s a_{0#}^{∗}(∇^{\overline{V}^{1}} + ra_{1}^{1∗}) ⊕ (∇^{\overline{V}^{1}} + ra_{1}^{1∗}) \\ & − 2 (1 − us) (1 − s) a_{0}^{−1} (∇^{\overline{V}^{1}} + ra_{1}^{1})^{2} \\ & − 2 (1 − u + us) s (∇^{\overline{V}^{1}} + ra_{1}^{1∗})^{2} (a_{0}^{∗})^{−1} − 2u N^{V} ds , \end{aligned} (8.13)

\begin{aligned}Ã^{3} & = \frac{v}{2} ((1 − s) a_{0} + s a_{0}^{∗}) \\ & + (1 − s) ((1 − r) a_{0}^{#}∇^{\overline{V}^{1}} + r (∇^{\overline{V}^{0}} + v a_{1}^{0})) ⊕ (∇^{\overline{V}^{1}} + rva_{1}^{1}) \\ & + s ((1 − r) a_{0#}^{∗}∇^{\overline{V}^{1}}) + r (∇^{\overline{V}^{0}} + v a_{1}^{0∗})) ⊕ (∇^{\overline{V}^{1}} + rva_{1}^{1∗}) \\ & − 2 (1 − s)v a_{0}^{−1} (∇^{\overline{V}^{1}} + ra_{1}^{1})^{2} − 2 sv (∇^{\overline{V}^{1}} + ra_{1}^{1∗})^{2} (a_{0}^{∗})^{−1} . \end{aligned} (8.14)

We state some elementary properties of these superconnections.

8.15. Proposition. The superconnections ^{1}Ã, ^{2}Ã and ^{3}Ã above satisfy the following properties. (1) We have ^{1}Ã|_{t=1} = ^{2}Ã|_{u=1} and ^{2}Ã|_{u=0} = ^{3}Ã|_{v=1}.

(2) The zero degree term of ^{1}Ã^{2} with respect to N^{B} is given by \frac{t}{4} [a_{0}, a_{0}^{∗}].

(3) The zero degree terms of ^{3}Ã|_{s=0}, ^{3}Ã|_{s=1} and ^{3}Ã|_{v=0} are nilpotent.

(4) The curvatures of ^{1}Ã|_{s=0}, ^{1}Ã|_{s=1} and ^{2}Ã|_{s=0}, ^{2}Ã|_{s=1} do not contain the exterior variable dr.

Proof. Claims (1), (2) and (3) are immediate from the definitions and Lemma 8.7.

To prove (4), note that at s = 0, we have

1Ã^{2}|_{s=0} = [(\frac{t}{4})^{\frac{N^{V}}{2}} A′ (\frac{t}{4})^{− \frac{N^{V}}{2}} , \frac{\sqrt{t}}{2} a_{0}^{∗}] ,
and  ^{2}Ã^{2}|_{s=0} = [2^{−N^{V}} A′ 2^{N^{V}} , \frac{u}{2} a_{0}^{∗}] .

The only term that could involve dr is in both cases a multiple of

[a_{0}^{−1} a_{1}^{1} dr , a_{0}^{∗}] .

But the only nozero components of a_{0}^{−1} a_{1}^{1} dr and a_{0}^{∗} map V^{1} to V^{0}, so the supercommutator vanishes. Clearly, a similar statement holds at s = 1. □

8.c. A comparison of two finite-dimensional higher torsions. In this subsection, we describe how we can use the above superconnections to construct and compare the two higher torsions of (8.11) in Proposition 8.18. This allows us to compare the definitions of Section 2 with those of Bismut and Lott in [BL] in Remark 8.19 (2). It also gives an expression for T (∇^{V} , A′, g^{V}) as a sum of four terms. Our strategy in the proof of Theorem 8.1 will be a comparison of each of these terms with one term in Igusa’s definition of τ, which will take the remainder of this section.

We start by constructing some elements in Ω^{∗}(B; End V^{0}) from the coefficient of the original superconnections A′ and its dual. These elements are used by Igusa to construct certain correction terms. If we write A′ = a_{0} + (∇^{V^{0}} + a_{1}^{0}) ⊕ (∇^{V^{1}} + a_{1}^{1}) + a_{2} as before, we set

(8.16)  x = a_{0}^{−1} [∇^{V} , a_{0}] ,  y = −a_{1}^{0} ,  z = −a_{2}a_{0} = (∇^{V^{0}} + a_{1}^{0})^{2} ,
and  x^{∗} = [∇^{V} , a_{0}^{∗}] (a_{0}^{∗})^{−1} ,  y^{∗} = −a_{1}^{0∗} ,  z^{∗} = −a_{0}^{∗}a_{2}^{∗} = (∇^{V^{1}} + a_{1}^{0^{∗}})^{2} .

Here we have used that ∇^{V} was assumed to be parallel with respect to g^{V}. Note that x, y, z above correspond up to sign to the End V^{0}-valued forms X, V, W in [I2], Section 2.4, whereas x^{∗} above corresponds to Y in [I2]. Igusa’s U can be expressed in terms of X, V, W and a_{0}.

Using flatness of A′ and the Bianchi-Identity, we can also compute the derivatives of the forms x, y, z. With our sign convention, we find

\begin{aligned}[∇^{V^{0}}, x] & = a_{0}^{−1} [∇^{V} a_{0}] a_{0}^{−1} [∇^{V} , a_{0}] = x^{2} , \\ [∇^{V^{0}}, y] & = −(∇^{V^{0}} − y)^{2} + y^{2} = y^{2} − z , \\ and [∇^{V^{0}}, z] & = [∇^{V^{0}} − y, (∇^{V^{0}} − y)^{2}] + [y, (∇^{V^{0}} − y)^{2}] = [y, z] . \end{aligned} (8.17)

The derivatives of the adjoint forms x^{∗}, y^{∗}, z^{∗} are given by similar formulas.

With these preliminaries, we can give yet another expression of the higher Chern analytic torsion of Definition 2.91 and Theorem 7.23. We restrict our attention to classes of forms on B in degree ≥ 2, because we want to ignore all the correction terms in Theorem 7.23.

8.18. Proposition. The higher Bismut-Lott torsion of \overline{A}′|_{r=0} = a_{0} +a_{0}^{#}∇^{V^{1}} ⊕∇^{V^{1}} is up to exact forms given by

T^{BL}(\overline{A}′|_{r=0}, g^{V} )^{[≥2]} = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(s,t)∈[0,1]×[1,∞)} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{r=0}})^{[≥2]}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,u)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{r=0}})^{[≥2]}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,v)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{r=0}})^{[≥2]} .

(1)

The higher torsion of \overline{A}′|_{r=1} = A′ is up to exact forms given by

T (∇^{V} , A′, g^{V} )^{[≥2]} = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(s,t)∈[0,1]×[1,∞)} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{r=1}})^{[≥2]}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,u)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{r=1}})^{[≥2]}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,v)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{r=1}})^{[≥2]} .

(2)

There is a naturally defined form T̃ such that

T^{BL}(\overline{A}′|_{r=0}, g^{V} )^{[≥2]} − T (∇^{V} , A′, g^{V} )^{[≥2]} = (G′ + K′ − \overline{K′}^{∗})^{[≥2]} + dT̃ ,
with  G′ = (2πi)^{− \frac{N^{B}}{2}} ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{v=0}}) ,
and  K′ = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{s=0}}) ,

(3)

where \overline{K′}^{∗} is constructed as in (2.5).

The forms G′, K′ and \overline{K′}^{∗} ∈ Ω^{∗}(B) are even. The form G′_{2k} is the supertrace of a polynomial in x and x^{∗} for each k, and

\begin{aligned}dG′ = ch^{o}(a_{0}^{#}∇^{V^{0}}, g^{V^{0}}) & − (2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr_{V^{0}}(x e^{r(1−r) x^{2}}) dr \\ & + (2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr_{V^{0}}(x^{∗} e^{r(1−r) (x^{∗})^{2}}) dr . \end{aligned} (4)

The form K′_{2k} is the supertrace of a polynomial in x, y, z, and w for each k, and

(5)  dK′ = (2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr_{V^{0}}(x e^{r(1−r) x^{2}}) dr .

The form \overline{K′}_{2k}^{∗} is the supertrace of a polynomial in x^{∗}, y^{∗}, z^{∗}, and w^{∗} for each k, and

(6)  d\overline{K′}^{∗} = (2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr_{V^{0}}(x^{∗} e^{r(1−r) (x^{∗})^{2}}) dr .

Note that x, y, z do not a priori commute pairwise. Thus, in each monomial, the order of variables is important.

8.19. Remark.

(1) This proposition is the first step in a comparison of Igusa’s torsion class τ with our T. The finite-dimensional Bismut-Lott torsion class T^{BL}(\overline{A}′|_{r=0}, g^{V}) corresponds to Igusa’s class F, see Lemma 8.38 below. The class G′ corresponds to the class G in [I2], Chapter 1, which is called K in [I2], Chapter 2, see Lemma 8.48 below. The class K′ corresponds to L in [I2], see Lemma 8.72 below. We have chosen to use the letter K instead, so that we do not confuse the class L of [BG] and Definition 2.19 above with Igusa’s L.

(2) We want to show that the classes G, K, and \overline{K}^{∗} together can be interpreted as an L-class in the sense of [BG], Section 2. Recall that we defined another superconnection ^{3}Ã in (7.19), this fits with our present ^{3}Ã along the common domain of definition B × {1}^{2} × [0, 1]^{2}, where r = T = 1 and the coordinates of [0, 1]^{2} are s and v. Let us define a family of flat superconnections (\overline{A}′_{ℓ})_{ℓ∈[0,1]} using (7.19) and (8.14) as

\overline{A}′_{ℓ} = \{ \begin{matrix}^{3}Ã|_{(r,s,v)=(0,0,3ℓ)} & 0 ≤ ℓ ≤ \frac{1}{3} , \\ ^{3}Ã|_{(r,s,v)=(3ℓ−1,0,1)} & \frac{1}{3} ≤ ℓ ≤ \frac{2}{3} , and \\ ^{3}Ã|_{(s,T,v)=(0,3−3ℓ,1)} & \frac{2}{3} ≤ ℓ ≤ 1 . \end{matrix}

Since the coefficient a_{0} degenerates at ℓ = 0 and at ℓ = 1, the integral defining the L-class as in Definition 2.19 and Definition 2.91 diverges in degree 0. Still, we can apply Stokes’ theorem to B × D_{3} and proceed as in the proof of Theorem 7.23 to find that

(G′ + K′ − \overline{K′}^{∗})^{[≥2]}

≡ \frac{1}{2} (2πi)^{− \frac{N^{B}}{2}} (− ∫_{(s,v)∈[0,1]^{2}} str_{V} (e^{−^{3}Ã^{2}|_{r=0}}) + ∫_{(r,s)∈[0,1]^{2}} str_{V} (e^{−^{3}Ã^{2}|_{v=1}})

+ ∫_{(s,v)∈[0,1]^{2}} str_{V} (e^{−^{3}Ã^{2}|_{r=1}}))^{[≥2]}

≡ \frac{1}{2} (2πi)^{− \frac{N^{B}}{2}} (− ∫_{(s,v)∈[0,1]^{2}} str_{V} (e^{−^{3}Ã^{2}|_{r=0}}) + ∫_{(r,s)∈[0,1]^{2}} str_{V} (e^{−^{3}Ã^{2}|_{v=1}})

+ ∫_{(s,T)∈[0,1]^{2}} str_{V} (e^{−^{3}Ã^{2}|_{v=1}}))^{[≥2]}

= −L_{ch}(\overline{A}′_{ℓ}, g^{V} )^{[≥2]}

modulo exact forms. This allows us to compare the finite-dimensional Bismut-Lott torsion classes with our torsion classes of Definition 2.46 and Remark 7.28 by

(8.20)  T (∇^{V} , A′, g^{V} ) = T^{BL}(a_{0} + a_{0}^{#}∇^{V^{1}} ⊕ ∇^{V^{1}}, g^{V} ) + L_{ch}(\overline{A}′_{ℓ}, g^{V} )^{[≥2]} .

Proof of Proposition 8.18. We recall that by Corollary 7.24 and Remark 8.10, we have

T^{BL}(\overline{A}′|_{r=0}, g^{V} ) = T (a_{0}^{#}∇^{V^{1}} ⊕ ∇^{V^{1}}, \overline{A}′|_{r=0}, g^{V} ) .

Now, our claims (1) and (2) are simple consequences of Theorem 7.23 and the definitions of ^{1}Ã, ^{2}Ã and ^{3}Ã in (8.12), (8.13) and (8.14).

To prove assertion (3), we apply Stokes theorem. We use (1) and (2) to find that

T (∇^{V} , A′, g^{V} )^{[≥2]} − T^{BL}(\overline{A}′|_{r=0}, g^{V} )^{[≥2]}

= −(2πi)^{− \frac{N^{B}}{2}} lim_{T →∞} ∫_{(s,t)∈[0,1]×[1,T]} \frac{1}{2} (str_{V} (e^{−^{1}Ã^{2}|_{r=1}}) − str_{V} (e^{−^{1}Ã^{2}|_{r=0}}))

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,u)∈[0,1]^{2}} \frac{1}{2} (str_{V} (e^{−^{2}Ã^{2}|_{r=1}}) − str_{V} (e^{−^{2}Ã^{2}|_{r=0}}))

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(s,v)∈[0,1]^{2}} \frac{1}{2} (str_{V} (e^{−^{3}Ã^{2}|_{r=1}}) − str_{V} (e^{−^{3}Ã^{2}|_{r=0}})) .

(8.21)

MORSE THEORY AND HIGHER TORSION INVARIANTS II 47

We now investigate the integrals over the remaining faces of B×∂([0, 1]^{2}×[1, T]) and B×∂([0, 1]^{3}), respectively. By Proposition 8.15 (1), we have

0 = ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} (str_{V} (e^{−^{1}Ã^{2}|_{t=1}}) − str_{V} (e^{−^{2}Ã^{2}|_{u=1}}))

= ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} (str_{V} (e^{−^{2}Ã^{2}|_{u=0}}) − str_{V} (e^{−^{3}Ã^{2}|_{v=1}})) .

(8.22)

By Proposition 8.15 (2),

(8.23)  0 = lim_{T →∞} ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{t=T}} )

because [a_{0}, a_{0}^{∗}] ∈ End V is invertible. By Proposition 8.15 (4), we find

0 = ∫_{(r,t)∈[0,1]×[1,T]} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{s=0}}) = ∫_{(r,t)∈[0,1]×[1,T]} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}|_{s=1}})

= ∫_{(r,u)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{s=0}}) = ∫_{(r,u)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{2}Ã^{2}|_{s=1}}) .

(8.24)

By Stokes theorem and (8.21)–(8.24),

T^{BL}(\overline{A}′|_{r=0}, g^{V} )^{[≥2]} − T (∇^{V} , A′, g^{V} )^{[≥2]}

= (2πi)^{− \frac{N^{B}}{2}} ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{v=0}})^{[≥2]}

− (2πi)^{− \frac{N^{B}}{2}} ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{s=0}})^{[≥2]}

+ (2πi)^{− \frac{N^{B}}{2}} ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{s=1}})^{[≥2]} .

(3)

Because

\overline{K′}^{∗} = −(2πi)^{− \frac{N^{B}}{2}} ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}|_{s=1}}) ,

this proves (3), with

T̃ = (2πi)^{− \frac{N^{B}+1}{2}} ∫_{(r,s,t)∈[0,1]^{2}×[1,∞)} \frac{1}{2} str_{V} (e^{−^{1}Ã^{2}})

+ (2πi)^{− \frac{N^{B}+1}{2}} ∫_{(r,s,u)∈[0,1]^{3}} \frac{1}{2} str_{V} (e^{−^{2}Ã^{2}})

+ (2πi)^{− \frac{N^{B}+1}{2}} ∫_{(r,s,v)∈[0,1]^{3}} \frac{1}{2} str_{V} (e^{−^{3}Ã^{2}}) .

We want to express G′ in terms of x and x^{∗}. The connection ^{3}Ã|_{v=0} acts on \overline{V}^{1} as ∇^{\overline{V}^{1}}, and on \overline{V}^{0} as

\begin{aligned}(^{3}Ã|_{v=0})|\overline{V}^{0} & = (1 − r) ((1 − s) a_{0}^{#}∇^{\overline{V}^{1}} + s a_{0#}^{∗}∇^{\overline{V}^{1}}) + r ∇^{V^{0}} \\ & = ∇^{\overline{V}^{0}} − (1 − r) ((1 − s) x + sx^{∗}) . \end{aligned} (8.25)

Together with (8.17), we find that (^{3}Ã|_{v=0})^{2}|\overline{V}^{0} is an element of the subalgebra of Ω^{∗}(B; End V^{0}) generated by x, x^{∗} and r, s, dr, ds. Because the term ^{3}Ã^{2}|_{v=0} is of total degree 2, the form G′_{k} is the trace of a polynomial in x, x^{∗} for each k.

By Stokes’ theorem and because ^{3}Ã|_{v=0}|\overline{V}^{1} = ∇^{\overline{V}^{1}} is flat,

\begin{aligned}dG′ & = (2πi)^{\frac{1−N^{B}}{2}} ∫_{s=0}^{1} \frac{1}{2} tr_{V^{0}}(e^{− ^{3}Ã^{2}|_{{r=0 \atop v=0}}} − e^{− ^{3}Ã^{2}|_{{r=1 \atop v=0}}} ) \\ & − (2πi)^{\frac{1−N^{B}}{2}} ∫_{r=0}^{1} \frac{1}{2} tr_{V^{0}}(e^{− ^{3}Ã^{2}|_{{s=0 \atop v=0}}} − e^{− ^{3}Ã^{2}|_{{s=1 \atop v=0}}} ) . \end{aligned} (8.26)

We analyse the right hand side term by term.

At r = 0, we have the connection

3Ã|_{{r=0 \atop v=0}} |\overline{V}^{0} = (1 − s) a_{0}^{#}∇^{\overline{V}^{1}} + s a_{0#}^{∗}∇^{\overline{V}^{1}} .

Note that a_{0#}^{∗}∇^{\overline{V}^{1}} is the adjoint connection of a_{0}^{#}∇^{\overline{V}^{1}} wit respect to g^{V^{0}} by (8.6). By Definition 2.89, this implies that

(8.27)  (2πi)^{\frac{1−N^{B}}{2}} ∫_{s=0}^{1} \frac{1}{2} str_{V} (e^{− ^{3}Ã^{2}|_{{r=0 \atop v=0}}} ) = ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V^{0}}) .

At r = 1, the connection ^{3}Ã|_{{r=1 \atop v=0}} = ∇\overline{V} is flat, and we have

(8.28)  ∫_{s=0}^{1} \frac{1}{2} tr_{V^{0}}(e^{−^{3}Ã^{2}|_{{r=1 \atop v=0}}} ) = 0 .

At s = 0, we find the connection ^{3}Ã|_{{s=0 \atop v=0}} |\overline{V}^{0} = ∇^{\overline{V}^{0}} − (1 − r) x with curvature given by

3Ã^{2}|_{{s=0 \atop v=0}} = −r(1 − r) x^{2} − x dr

by (7.5). Thus, we have

(8.29)  −(2πi)^{\frac{1−N^{B}}{2}} ∫_{r=0}^{1} \frac{1}{2} str_{V} (e^{− ^{3}Ã^{2}|_{{s=0 \atop v=0}}} ) = −(2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr_{V^{0}}(x e^{r(1−r) x^{2}}) dr .

Passing to adjoints with respect to g^{V^{1}} proves the analogous equation

(8.30)  (2πi)^{\frac{1−N^{B}}{2}} ∫_{r=0}^{1} \frac{1}{2} str_{V} (e^{− ^{3}Ã^{2}|_{{s=1 \atop v=0}}} ) = (2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr_{V^{0}}(x^{∗} e^{r(1−r) (x^{∗})^{2}}) dr .

Now, statement (4) follows from (8.26)–(8.30).

The remaining statements are proven analogously. Since we wish to describe the superconnection ^{3}Ã|_{s=0} and its curvature by elements of Ω^{∗}(B; End V^{0}), we use a_{0} to identify V^{1} with V^{0}, and continue to work with two copies of V^{0}. We may write ^{3}Ã|_{s=0} − ∇^{V^{0}} ⊕ a_{0#}∇^{V^{0}} as a 2 × 2-matrix with coefficients in Ω^{∗}(B; V^{1}) by

(8.31)  ^{3}Ã|_{s=0} = ( \begin{matrix}∇^{\overline{V}^{0}} − (1 − r) x − rv y & −2v (∇^{\overline{V}^{0}} − (1 − r) x − r y)^{2} \\ \frac{v}{2} & −∇^{\overline{V}^{0}} + (1 − rv) x + rv y \end{matrix} ) .

Recall that due to our sign convention, a_{0}^{−1} ∇^{V^{1}} a_{0} is the negative of a connection; this fact is responsible for the strange signs in the lower right corner. Together with (8.17), this shows that the curvature of ^{3}Ã|_{s=0} is a 2 × 2-matrix with coefficients in the subalgebra of Ω^{∗}(B; End V^{0}) generated by x, y, z and r, v, dr, dv. Because the term ^{3}Ã^{2}|_{v=0} of degree 0 with respect to N^{B} is nilpotent,

(∫_{(r,v)∈[0,1]^{2}} str(e^{−^{3}Ã^{2}|_{s=0}}))^{[k]}

is the supertrace of a polynomial in x, y, z for each k.

We prove (5) using Stokes’ theorem. We have

\begin{aligned}dK′ & = (2πi)^{\frac{1−N^{B}}{2}} ∫_{r=0}^{1} \frac{1}{2} str_{V} (e^{− ^{3}Ã^{2}|_{{s=0 \atop v=0}}} − e^{− ^{3}Ã^{2}|_{{s=0 \atop v=1}}} ) \\ & − (2πi)^{\frac{1−N^{B}}{2}} ∫_{s=0}^{1} \frac{1}{2} str_{V} (e^{− ^{3}Ã^{2}|_{{r=0 \atop s=0}}} − e^{− ^{3}Ã^{2}|_{{r=1 \atop s=0}}} ) . \end{aligned} (8.32)

The contribution from the line with s = 0 and v = 0 has been given in (8.29). We will show that the three remaining faces of ∂([0, 1]^{2}) do not contribute to dK′.

At v = 1, we have the flat superconnection

\begin{aligned}^{3}Ã|_{{s=0 \atop v=1}} & = \frac{1}{2} a_{0} + a_{0}^{#} (∇^{\overline{V}^{1}} + r a_{1}^{1}) ⊕ (∇^{\overline{V}^{1}} + r a_{1}^{1}) − 2a_{0}^{−1} (∇^{\overline{V}^{1}} + r a_{1}^{1}) \\ & = 2^{− \frac{N^{V}}{2}} \overline{A}′ 2^{\frac{N^{V}}{2}} , \end{aligned}

by (8.14), so this face does not contribute to dK′.

At r = 0 and r = 1, we get

\begin{aligned}^{3}Ã|_{{r=0 \atop s=0}} & = \frac{v}{2} a_{0} + a_{0}^{#} ∇^{\overline{V}^{1}} ⊕ ∇^{\overline{V}^{1}} , \\ and ^{3}Ã|_{{r=1 \atop s=0}} & = \frac{v}{2} a_{0} + (∇^{\overline{V}^{0}} + v a_{1}^{0}) ⊕ (∇^{\overline{V}^{1}} + v a_{1}^{1}) + 2v a_{2} . \end{aligned}

The curvatures ^{3}Ã^{2}|_{{r=0 \atop s=0}} and ^{3}Ã^{2}|_{{r=1 \atop s=0}} are constructed only from a_{0}, a_{1}, and a_{2}, which belong to a nilpotent subalgebra of End V. Thus, these two faces do not contribute to dK′ either, and the only contribution to dK′ stems from the face v = 0, and was calculated (up to sign) in (8.30).

Because K′ and \overline{K′}^{∗} are related by (2.5), the remaining claims about \overline{K′}^{∗} follow immediately. □

8.d. Igusa’s classes I and F. We prove that Igusa’s class I is a variant of the odd Chern class ch^{o}, and that by transgression, the class F equals the Bismut-Lott torsion class T^{BL}(A′, g^{V}). This is the first step in the programme outlined in Remark 8.19 (1).

We start with the classes c_{2k+1} and I_{2k+1} defined in [I2], Section 1.2, but using the Chern normalisation of [BG]. Let V = V^{0} ⊕ V^{1} → B be a ℤ-graded complex vector bundle with rk V^{0} = rk V^{1} = n, with a flat connection ∇^{V} = ∇^{V^{0}} ⊕ ∇^{V^{1}} and a parallel metric g^{V} = g^{V^{0}} ⊕ g^{V^{1}}. Let a_{0} ∈ Γ(B, Iso(V^{0}, V^{1}) ⊂ Ω^{0}(B; End^{1} V) be a family of isomorphisms.

Igusa’s form I_{2k+1} ∈ Ω^{2k+1}(B) is the real valued differential form given by

(8.33)  I_{2k+1} = \frac{k!}{2 (2πi)^{k} (2k + 1)!} tr(((a_{0}a_{0}^{∗})^{−1} [∇^{1}, a_{0}a_{0}^{∗}])^{2k+1}) .

We put I = ∑_{k=0}^{∞} I_{2k+1} ∈ Ω^{odd}(B).

The form I is closed because its exterior differential is the supertrace of a non-zero even power of an odd operator. Note that in order to have the same normalisation as in [BG], we have changed the coefficients with respect to [I2]. Integrating I and ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V_{0}}) over I gives simplicial cochains on Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) that we still denote by I and ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V_{0}}).

8.34. Lemma. We have

I = ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V^{0}}) .

In particular, the simplicial cocycles I and ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V_{0}}) agree on Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)).

Proof. Because ∇^{V^{1}} is selfadjoint with respect to g^{V^{1}}, the adjoint connection of a_{0}^{#}∇^{V^{1}} with respect to g^{V^{0}} is given by a_{0#}^{∗}∇^{V^{1}}, cf. (8.6). This implies that

(a_{0}^{#}∇^{V^{1}})^{∗} − a_{0}^{#}∇^{V^{1}} = a_{0}^{∗} (a_{0}a_{0}^{∗})^{−1} [∇^{V^{1}}, a_{0}a_{0}^{∗}] (a_{0}^{∗})^{−1} .

Our claims follow because by (8.27), because

I = ∫_{0}^{1} tr_{V^{1}}(\frac{(a_{0}a_{0}^{∗})^{−1} [∇^{V^{1}}, a_{0}a_{0}^{∗}]}{2} e^{s(1−s) \frac{((a_{0}a_{0}^{∗})^{−1} [∇^{V ^{1}} ,a_{0}a_{0}^{∗}])^{2}}{2πi}} ) ds

= ∫_{0}^{1} tr_{V^{0}}(\frac{(a_{0}^{#}∇^{V^{1}})^{∗} − a_{0}^{#}∇^{V^{1}}}{2} e^{s(1−s) \frac{((a_{0}^{#}∇^{V ^{1}} )^{∗}−a_{0}^{#}∇^{V ^{1}} )^{2}}{2πi}} ) ds = ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V^{0}}) . □

Using I_{2k+1}, Igusa defines another class F_{2k} by transgression in [I2], (1.4). Let \overline{B} = B × [0, 1], pull V and ∇^{1} back to \overline{B}, and with the coordinate u of [0, 1], put

\overline{I}_{2k+1} = \frac{k!}{2 (2πi)^{k} (2k + 1)!} tr(((a_{0}a_{0}^{∗})^{−u} [ \overline{∇}^{1}, (a_{0}a_{0}^{∗})^{u}])^{2k+1}) .

Igusa’s form F_{2k} ∈ Ω^{even}(B) is defined as

(8.35)  F_{2k} = − ∫_{\overline{B}/B} \overline{I}_{2k+1} .

We put F = ∑_{k=0}^{∞} F_{2k} ∈ Ω^{even}(B).

8.36. Remark. We note that \overline{I}_{2k+1} is closed and \overline{I}_{2k+1}|_{u=0} vanishes identically. It thus follows from Stokes’ theorem that

d F_{2k} = \overline{I}_{2k+1}|_{u=1} − \overline{I}_{2k+1}|_{u=0} = I_{2k+1} .

We define a superconnection of total degree one on V → B by

(8.37)  A′_{0} = \overline{A}′|_{r=0} = a_{0} + a_{0}^{#}∇^{V^{1}} ⊕ ∇^{V^{1}} ,

cf. Remark 8.10. Let T^{BL}(A′_{0}, g^{V}) denote the Bismut-Lott torsion form of (8.11). As above, we denote the corresponding forms and simplicial cochains on Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) by F and T^{BL}(A′_{0}, g^{V}).

8.38. Lemma. There is a natural form F̃ ∈ Ω^{odd}(B) such that

F = T^{BL}(A′_{0}, g^{V} ) + dF̃ .

In particular, the simplicial cochains F and T^{BL}(A′_{0}, g^{V}) on Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) differ by an exact cochain.

Proof. On the pullback of V to B × [0, 1], consider the metric

\overline{g}^{V} = (s g^{V^{0}} + (1 − s) a_{0}^{#}g^{V^{1}}) ⊕ g^{V^{1}} = (s g^{V^{0}} + (1 − s) g^{V^{1}}(a_{0} · , a_{0} ·)) ⊕ g^{V^{1}} .

Note that we have not used parallelity of g^{V^{0}} in the definitions of I and F. Thus, let \overline{I}, \overline{F} and T (\overline{A}′_{0}, \overline{g}^{V^{0}} ⊕ \overline{g}^{V^{1}}) denote the corresponding forms on B × [0, 1]. Then clearly

\overline{F} |_{B×\{0\}} = T^{BL}(\overline{A}′_{0}, \overline{g}^{V} )|_{B×\{0\}} = 0 .

We put

F̃ = − ∫_{s∈[0,1]}(\overline{F} − T^{BL}(\overline{A}′_{0}, ḡ^{V} )) .

Then by Lemma 8.34, (8.35), Remark 8.36 and Stokes’ Theorem, we have

F − T^{BL}(A′_{0}, g^{V} ) = (\overline{F} − T^{BL}(A′_{0}, g^{V} ))|_{s=1}
= ∫_{s∈[0,1]} d(\overline{F} − T^{BL}(\overline{A}′_{0}, ḡ^{V} )) − d ∫_{s∈[0,1]}(\overline{F} − T^{BL}(\overline{A}′_{0}, ḡ^{V} ))
= ∫_{s∈[0,1]}(\overline{I} − ch^{o}(a_{0}^{#}∇^{1}, ḡ^{V} )) + dF̃ = dF̃ . □

8.e. Igusa’s polynomial P and the class G. We interpret the classes G_{2k} of Chapter 1 (respectively K_{2k} in Chapter 2) of [I2] in our smooth setting and show that they equal the classes G′_{2k} of Proposition 8.18. This follows from Proposition 8.18 (4) and the acyclicity of a certain cyclic cochain complex already studied in [I2].

Recall the definition of x, x^{∗} ∈ Ω^{1}(B; End V^{0}) in (8.16). These forms correspond to −x and y in [I2], Lemma 1.3.1. It follows from the proof of Lemma 8.34 that

(8.39)  I = ch^{o}(a_{0}^{#}∇^{V^{1}}, g^{V^{0}}) = −(2πi)^{\frac{1−N^{B}}{2}} ∫_{0}^{1} \frac{1}{2} tr((x^{∗} − x) e^{s(1−s) (x^{∗}−x)^{2}}) ds .

The construction of G_{2k} relies on the existence of a certain homogeneous polynomial P_{2k} of degree 2k with

(8.40)  d tr(P_{2k}(x, x^{∗})) = \frac{k!}{2 (2k + 1)!} (tr((x^{∗})^{2k+1}) − tr(x^{2k+1}) − tr((x^{∗} − x)^{2k+1})) .

Note that x and x^{∗} do not commute, so the order of the xes and x^{∗}s in each monomial is important. There will in general be several monomials with the same number of xes and x^{∗}s involved.

We recall the definition of the polynomials P_{2k} in [I2], Section 1.3.1. Let A = ℂ × ℂ be the direct sum of two copies of the algebra ℂ, generated by u = (1, 0) and v = (0, 1). As in [I2], we consider the ℤ-graded cyclic complex

(8.41)  C^{∗} = ∏_{j=0}^{∞} A^{⊗(j+1)} / ((−1)^{j}z_{j} − id) ,

where z_{j}(a_{0} ⊗ ⋯ ⊗ a_{j}) = (a_{1} ⊗ ⋯ ⊗ a_{j} ⊗ a_{0}).

Let b: C^{∗} → C^{∗−1} denote the cyclic boundary operator,

(8.42)  b(a_{0} ⋯ a_{j}) = ∑_{i=0}^{j−1}(−1)^{i}a_{0} ⋯ (a_{i}a_{i+1}) ⋯ a_{j} + (−1)^{j} (a_{j}a_{0}) a_{1} ⋯ a_{j−1} .

We define a coboundary operator δ by δu = u · u, δv = v · v, and

δ(a_{0} ⋯ a_{j}) = ∑_{i=0}^{j}(−1)^{i}a_{0} ⋯ a_{i−1} · δa_{i} · a_{i+1} ⋯ a_{j} .

Clearly δu^{· k} = u^{· (k+1)} if k is odd and δu^{· k} = 0 if k is even, and similarly for v. Let the “weight” w ∈ End C^{∗} be the linear endomorphism that multiplies a monomial a_{0} ⋯ a_{j} with a_{i} ∈ {u, v} for all i by the number of indices i modulo j + 1, such that a_{i} ≠ a_{i+1}. By [I2], we have

(8.43)  b ∘ δ + δ ∘ b = w .

Note that w commutes with δ and b. In particular, the subcomplex generated by all mixed monomials (i.e., by all monomials of non-zero weight) is acyclic both with respect to b and with respect to δ.

Because δ(v − u) = [v + u, v − u], we have

δ((v − u)^{2k+1}) = ∑_{i=0}^{2k}(−1)^{i}(v − u)^{i} · [v + u, v − u] · (v − u)^{2k−i} = 0 .

Similarly,

δ(u^{2k+1}) = δ(v^{2k+1}) = 0 .

Igusa defines

(8.44)  P_{2k}(u, v) = −\frac{k!}{2 (2k + 1)!} \frac{b}{w} ((v − u)^{2k+1} + u^{2k+1} − v^{2k+1})

in [I2]. Because (v − u)^{2k+1} + u^{2k+1} − v^{2k+1} contains no mixed monomials, the polynomial P satisfies

(8.45)  ∂P = −\frac{k!}{2 (2k + 1)!} ((v − u)^{2k+1} + u^{2k+1} − v^{2k+1}) .

by (8.43) and (8.46). We put P = ∑_{k=1}^{∞} P_{2k}.

There is a ℂ-linear map φ: A → Ω^{1}(B, End F) with φ(u) = x and φ(v) = x^{∗}. Define

Φ: C^{∗} ⟶ Ω^{∗}(B)  with  Φ(a_{0} ⋯ a_{j}) = (2πi)^{− \frac{N^{B}}{2}} tr(φ(a_{0}) ⋯ φ(a_{j})) .

The sign factor (−1)^{j} in (8.41) is compatible with this construction because x and y are of odd degree. By (8.41), we clearly have

(8.46)  d ∘ Φ = \sqrt{2πi} Φ ∘ δ .

Igusa’s form G is defined as

(8.47)  G = Φ(P) .

Let G and G′ also denote the cochains obtained by integrating G and G′ over each simplex in Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)).

8.48. Lemma. The forms G and G′ agree up to the differential of a naturally defined form G̃. In particular, the cochains G and G′ on Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) agree up to an exact cochain.

Proof. We are going to prove the existence of polynomials P′_{2k} ∈ C^{2k} in u and v with the following properties.

(1) G′_{2k} = Φ(P′_{2k}).

(2) ∂P′_{2k} = \frac{k!}{2 (2k+1)!}(v^{2k+1} − v^{2k+1} − (v − u)^{2k+1}), and

(3) P′_{2k} contains only mixed monomials. In this case, we concluce that ∂(P − P′) = 0 by (8.45) and (2). Moreover, by (8.44) and (3), the polynomial P_{2k} −P′_{2k} contains no mixed monomials. Putting P̃ = \frac{b}{w} (P −P′), we find ∂P̃ = P −P′ by (8.43). Using (8.46), (8.47) and (1), our claim follows with G̃ = (2πi)^{− \frac{1}{2}} Φ(P̃), because

dG̃ = Φ(∂P̃) = Φ(P − P′) = G − G′ .

We work in the algebra Ω^{∗}([0, 1]^{2}, C^{∗}). Let us set

(8.49)  P′ = ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} e^{−(δ+d−(1−r) ((1−s) u+s v))^{2}} ∈ C^{∗} ,

where d denotes the exterior differential on [0, 1]^{2}. By Proposition 8.18 and (8.25), we get (1).

In the proof of Proposition 8.18 (4), we have only used (8.17), and the cyclic properties of the trace. This implies that all calculations remain valid in Ω^{∗}([0, 1]^{2}; C^{∗}), in particular, (2) follows from

(8.50) \begin{aligned}δP′ & = ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} [δ + d − (1 − r) ((1 − s) u + s v) , e^{−(δ+d−(1−r) ((1−s) u+s v))^{2}}] \\ & − ∫_{(r,s)∈[0,1]^{2}} \frac{1}{2} [d , e^{−(δ+d−(1−r) ((1−s) u+s v))^{2}}] \\ & = − ∫_{0}^{1} \frac{v − u}{2} e^{s(1−s) (v−u)^{2}} ds + ∫_{0}^{1} \frac{v}{2} e^{r(1−r) v^{2}} dr − ∫_{0}^{1} \frac{u}{2} e^{r(1−r) u^{2}} dr \\ & = ∑ \frac{k!}{2 (2k + 1)!} (v^{2k+1} − u^{2k+1} − (v − u)^{2k+1}) . \end{aligned}

Because positive even powers of u and v vanish in C^{∗}, the polynomial P′ contains only mixed monomials. This finishes the proof. □

8.f. Igusa’s polynomial Q and the class K. We translate some definitions of [I2], Chapter 2, to the language of flat superconnections. The reader is warned at this point that for compatibility with [BL], we work with bundles of cochain complexes. This means that the bundles C_{0}, C_{1} of [I2] are now named V^{1} and V^{0}. This is compatible with our definitions in Subsections d and e.

Let V^{0}, V^{1} → B be flat bundles with connection ∇, let f: V^{0} → V^{1} be a vector bundle isomorphism. Igusa assumes that for nearby points p, q ∈ B, there are smooth families E^{0}, E^{1} of isomorphisms E^{i}(p, q): V^{i}(q) → V^{i}(p) with E^{i}(q, p) = (E^{i}(p, q))^{−1} such that

(8.51)  \begin{matrix}V^{1}(p) & ⟵^{E^{1}(p,q)} & V^{1}(q) \\ ^{f(p)}↑ & ^{f(q)}↑ \\ V^{0}(p) & ⟵^{E^{0}(p,q)} & V^{0}(q) \end{matrix}

commutes.

Igusa also assumes that there is a family S of linear maps S(p, q, r): V^{1}(r) → V^{0}(p) for nearby points p, q, r ∈ B such that

\begin{aligned}S(p, q, r) f(r) & = E^{0}(p, q) E^{0}(q, r) − E^{0}(p, r) , and \\ f(p) S(p, q, r) & = E^{1}(p, q) E^{1}(q, r) − E^{1}(p, r) . \end{aligned} (8.52)

Moreover, he requires the following behaviour under permutations of p, q, r:

(8.53)  S(q, p, r) = −E^{0}(q, p) S(p, q, r) ,  and  S(p, r, q) = −S(p, q, r) E^{1}(r, q) .

Igusa demands that the coefficients of f, E^{0} − id, E^{1} − id and S are given by upper triangular matrices with respect to some local bases of V. This corresponds to property (3) in Definition 2.35.

We use the data f, E and S to construct a flat superconnection of total degree 1. We denote by ∇^{V^{i},q}, ∇^{V^{i},r} the derivation with respect to the variables q and r ∈ B. Then we define a_{0} ∈ Hom(V^{0}, V^{1}), a_{1}^{i} ∈ Ω^{1}(B; End(V^{i})) and a_{2} ∈ Ω^{2}(B; Hom(V^{1}, V^{0})) by

a_{0}(p) = f(p) ,
a_{1}^{i}(p)(X) = ∇_{X}^{V^{i},q}|_{q=p}E^{i}(p, q) ,
and  a_{2}(p)(X, Y) = −(∇_{X}^{V,q} ∇_{Y}^{V,r} − ∇_{Y}^{V,q} ∇_{X}^{V,r} − ∇_{[X,Y]}^{V,r})|_{r=q=p}S(p, q, r) ,

and define a superconnection of total degree 1 on V = V^{0} ⊕ V^{1} by

(8.54)  A′ = ∇ + a_{0} + a_{1} + a_{2} .

Then the coefficients of A′ are all strictly upper triangular.

8.55. Proposition. The superconnection A′ is flat.

Proof. Note that A′ is of total degree 1, so (A′)^{2} could have contributions in Ω^{k}(B, End V^{∗}), where k = 1, 2, 3.

Because E(q, p) = E(p, q)^{−1}, we have E(p, p) = id. We derive (8.51) with respect to q at q = p and obtain

(8.56)  0 = ∇_{X}^{V,q}|_{q=p}(E(p, q) f(q) − f(p) E(p, q)) = [∇ + a_{1}, a_{0}](p) ,

so (A′)^{2} has no component in degree 1.

By (8.53), we have S(p, p, q) = S(p, q, q) = 0. We derive (8.52) twice, and find

(8.57) 0 = (∇_{X}^{V,q} ∇_{Y}^{V,r} − ∇_{Y}^{V,q} ∇_{X}^{V,r} − ∇_{[X,Y]}^{V,r})|_{r=q=p}

(E(p, q) E(q, r) − E(p, r) − f(p) S(p, q, r) − S(p, q, r) f(r))
= ((∇^{V} + a_{1})^{2} + [a_{0}, a_{2}])(p)(X, Y) ,

so the homogeneous component of (A′)^{2} of degree 2 vanishes, too.

As in the proof of Lemma 8.7, the Bianchi identity for ∇ + a_{1} and equations (8.56) and (8.57) imply that (A′)^{2} has no component in degree 3 either. This proves the Proposition. □

8.58. Remark. In other words, there is a functor from Igusa’s category W^{diff}(ℂ^{m}, n) introduced in [I2], Chapter 2.2, to our category Wh^{diff,h,{0,1}}(R, G). Igusa proved that W^{diff}(ℂ^{m}, n) is homotopy equivalent to the category Wh^{simp,h,{0,1}}(R, G). It is easy to check that these functors are compatible with the homotopy equivalences constructed in Theorem 6.31.

Moreover, the final correction term K (called L in [I2]) is constructed using the variables x, y, z of (8.16) that are defined using the coefficients of the superconnection A′. In Igusa’s notation in [I2], Section 2.4.1, we have X = x, V = −y, and W = z. The form U ∈ Ω^{1}(B, End V^{1}) can be represented by U = a_{0} (y − x) a_{0}^{−1}, however, a careful analysis of Igusa’s construction shows that U is not really used in the definition of K. It is now clear from the above that Igusa’s torsion class on W^{diff}(ℂ^{m}, n) pulls back from Wh^{diff,h,{0,1}}(R, G).

The construction of K relies on a sequence of homogeneous polynomials Q_{2k} in x, y, z of degree 2k, where deg x = deg y = 1 and deg w = 2. These polynomials should satisfy

d tr(Q_{2k}(x, y, z) = \frac{k!}{2 (2k + 1)!} tr(x^{2k+1}) .

Again, the order of the variables in each monomial is important.

Let us recall some details of the construction of Q in [I2]. We consider the subalgebra A ⊂ M_{3}(ℂ) generated by the elements

o = ( \begin{matrix}1 \\ & 0 \\ & & 0 \end{matrix} ) ,  p = ( \begin{matrix}0 \\ & 0 \\ & & 1 \end{matrix} ) ,  and  q = ( \begin{matrix}0 \\ & 0 & 1 \\ & & 0 \end{matrix} ) .

This is a subalgebra of the algebra used in [I2], Section 2.4.1, where o, p, and q correspond to the generators x, v, w in [I2]. The corresponding algebra in [I2] has another generator u, however, u is never used in [I2], so we do not introduce a name for it here. We assign degrees |o| = |p| = 1 and |q| = 2. As in [I2], Section 2.4.1, we consider a new cyclic complex

\begin{aligned}C^{∗} & = ∏_{j=0}^{∞} A^{⊗(j+1)}/(z_{j} − id) , \\ where z_{j}(a_{0} ⊗ ⋯ ⊗ a_{j}) & = (−1)^{|a_{0}| (|a_{1}|+···+|a_{j}|)} (a_{1} ⊗ ⋯ ⊗ a_{j} ⊗ a_{0}) . \end{aligned} (8.59)

We define a coboundary operator δ on generators by

(8.60)  δo = o · o ,  δp = p · p − q ,  and  δq = p · q − q · p ,

and in general by

δ(a_{0} ⋯ a_{j}) = ∑_{i=0}^{j}(−1)^{i} a_{0} ⋯ a_{i−1} · δa_{i} · a_{i+1} ⋯ a_{j} .

Then δ increments the total degree by 1. Let b be the cyclic boundary operator given as in (8.42), and note that b decrements the total degree by 1 because q · q = 0. Let w ∈ End C^{∗} be the colour weight that multiplies each monomial a_{0} ⋯ a_{j} by twice the number of indices i modulo j + 1 such that a_{i} = o and a_{i+1} ≠ o. It is proved in [I2], Lemma 2.4.2, that

(8.61)  b ∘ δ + δ ∘ b = w .

Note that w commutes with b and δ, and that the subcomplex spanned by all mixed monomials a_{0} ⋯ a_{j} is acyclic. Here, a monomial is called mixed iff a_{i} = o for at least one index i, and a_{k} ≠ o for at least one index k, i.e., iff w(a_{0} ⋯ a_{j}) ≠ 0.

We now regard the subcomplex (C^{′∗}, δ) ⊂ (C^{∗}, δ) spanned by p and q. It follows that if a monomial has colour weight w = 0, then it is either of the form x^{2k+1}, or it belongs to C^{′∗}. By (8.60), the subcomplex C^{′∗} admits a filtration

(8.62)  C^{′∗} = F^{′0} ⊃ F^{′1} ⊃ …,

where F^{′k} is spanned by all monomials in p and q that contain q at least k times. As in [I2], we may write δ|_{C^{′∗}} = δ′ + δ′′, where

δ′: F^{′k} ⟶ F^{′k} ,  with  p ⟼ p^{2} ,  q ⟼ pq − qp ,
and  δ′′: F^{′k} ⟶ F^{′k+1} ,  with  p ⟼ −q ,  q ⟼ 0 .

We define a new cyclic boundary operator b′ on C^{′∗} of total degree −1 by

\begin{aligned}b′(a_{0} ⋯ a_{j}) & = ∑_{i=0}^{j−1}(−1)^{|a_{0}|+···+|a_{i−1}|} a_{0} ⋯ (a_{i} ∗ a_{i+1}) ⋯ a_{j} \\ & + (−1)^{|a_{j}| (|a_{0}|+···+|a_{j−1}|)} (a_{j} ∗ a_{0}) · a_{1} ⋯ a_{j−1} , \end{aligned}

which is on generators given by

p ∗ p = p ,  p ∗ q = q ∗ p = q ,  and  q ∗ q = 0 .

Let w′ be the weight operator that multiplies each monomial a_{0} ⋯ a_{j} by the number of indices i modulo j + 1 such that a_{i} = p, a_{i+1} = q. Then it it proved in [I2], Lemma 2.4.4, that

(8.63)  b′ ∘ δ′ + δ′ ∘ b′ = −w′ .

Because the only monomial of weight w′ = 0 of total degree 2k + 1 is v^{2k+1}, a simple spectral sequence argument gives

(8.64)  H^{odd}(F^{′k}, δ) = 0  for all k ≥ 1.

Note that C^{∗} admits an involution ι given by

(8.65)  o ⟷^{ι} −o ,  p ⟷^{ι} p − o ,  and  q ⟷^{ι} q .

To show that ι is a cochain map, note that

δ ⟷^{ι} δ − [o, ·]

on generators, however

∑_{i}(−1)^{|a_{0}|+···+|a_{i−1}|} a_{0} ⋯ a_{i−1} · (δa_{i} − o · a_{i} − (−1)^{|a_{i}|}a_{i} · o) · a_{i+1} ⋯ a_{j}

= ∑_{i}(−1)^{|a_{0}|+···+|a_{i−1}|} a_{0} ⋯ a_{i−1} · δa_{i} · a_{i+1} ⋯ a_{j} ,

so the action of δ on C^{∗} commutes with ι. However, the multiplication on A and thus the operators b and w are changed. In fact, the involution ι corresponds to passing from the superconnection A′ of degree 1 to a superconnection

a_{0}^{−1} + (∇^{V^{0}} + a_{1}^{0}) ⊕ (∇^{V^{1}} + a_{1}^{1})a_{0} a_{2} a_{0}

of degree −1.

Finally, we define a linear map φ: A → Ω^{∗}(B; End V^{0}) on generators by φ(o) = x, φ(p) = y, and φ(q). Define

(8.66) Φ: (C^{∗}, δ) ⟶ (Ω^{∗}(B), d)  by  Φ(a_{0} ⋯ a_{j}) = (2πi)^{− \frac{N^{B}}{2}} tr(φ(a_{0}) ⋯ φ(a_{1})) .

Then by (8.17) and (8.60), we have

(8.67)  d ∘ Φ = \sqrt{2πi} Φ ∘ δ .

The map Φ has a large kernel. We note that φ(p) = a_{1}^{0} and φ(q) = a_{2}a_{0} are strictly triangular with respect to the eigenspaces of h^{V^{0}}. We also note that

φ(o − p) = (∇^{V^{0}} + a_{1}^{0}) + a_{0}^{−1} ∇^{V^{1}} a_{0} = −a_{0}^{−1} a_{1}^{1} a_{0}

and φ(q) = a_{0}^{−1} (a_{0}a_{2}) a_{0} are strictly triangular with respect to the eigenspaces of a_{0}^{−1}h^{V^{1}}a_{0}. In other words,

(8.68)  C^{′∗} ⊂ ker(Φ)  and  ι(C^{′∗}) ⊂ ker(Φ) .

As noted after Lemma 2.4.5 in [I2], Igusa’s polynomials Q_{2k} in our normalisation satisfy

(8.69)  δQ_{2k} = \frac{k!}{2 (2k + 1)!} o^{2k+1} + R_{2k+1} − ιR_{2k+1}

for some R_{2k+1} ∈ C^{′∗}, where

(8.70)  R_{2k+1} ≡ \frac{k!}{2(2k + 1)!} p^{2k+1}  modulo F^{′1} .

Igusa’s form K_{2k} is defined as

(8.71)  K_{2k} = Φ(Q_{2k}) ∈ Ω^{2k}(Wh^{h,{0,1}}(M_{r}(ℂ), U(r))) .

We put K = ∑_{k} K_{2k}. Let K and K′ also denote the cochains obtained by integrating K and K′ over each simplex in Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)).

8.72. Lemma. The forms K and K′ agree up to the differential of a naturally defined form K̃. In particular, the cochains K and K′ on Wh^{diff,h,{0,1}}(M_{r}(ℂ), U(r)) agree up to an exact cochain.

Proof. We want to construct polynomials Q′_{2k} ∈ C^{∗} and R′_{2k+1} ∈ C^{′∗} with the following properties. (1) K′_{2k} = Φ(Q′_{2k}).

(2) δQ′_{2k} = \frac{k!}{2 (2k+1)!} o^{2k+1} + R′_{2k+1} − ιR′_{2k+1}.

(3) R′_{2k+1} ≡ \frac{k!}{2 (2k+1)!} p^{2k+1} modulo F^{′1}.

Given Q′_{2k} and R′_{2k+1} as above, we proceed as follows. First, using (8.64), (8.70), and (3), we find an even formal power series R̃ ∈ F^{′1} ⊂ C^{′∗} such that

R − R′ = δR̃ .

From (8.69) and (2) we conclude that δ(Q − Q′ − R̃ + ιR̃) = 0. We let ( ·)^{(w>0)} denote the components of colour weight ≠ 0 and define

K̃ = (2πi)^{− \frac{1}{2}} Φ(\frac{b}{w} (Q − Q′ − R̃ + ιR̃)^{(w>0)}) .

By (8.61), (8.67), (8.71) and (1), and because o^{2k} = 0 ∈ C^{∗} for all k > 0 and C^{′∗}, ιC^{′∗} ⊂ ker Φ by (8.68), we find that

dK̃ = Φ(Q − Q′ − R̃ + ιR̃) = Φ(Q − Q′) = K − K′ ,

which proves the lemma.

It remains to construct Q′, R′ satisfying (1–3). Our final formula for Q′ will be the sum of two formal power serieses ^{1}Q′ + ^{2}Q′, such that

(8.73)  Φ(^{1}Q′) = K′  and  Φ(^{2}Q′) = 0 ,

which implies (1). To construct ^{1}Q′, we work in Ω^{∗}([0, 1]^{2}; C^{∗}). As in (8.31), we construct a superconnection on the trivial bundle with fibre (C^{∗})^{2} by

(8.74)  ^{3}Ã′ = ( \begin{matrix}d + δ − (1 − r) o − rv p & −2v (d + δ − (1 − r) o − r p)^{2} \\ v/2 & −d − δ + (1 − rv) o + rv p \end{matrix} ) ,

where d denotes the exterior differential on [0, 1]^{2}. Again, the signs in the lower right corner make sense if we agree that the total degree in the second copy of C^{∗} is shifted by 1. We can now translate parts of the proof of Proposition 8.18 (5).

We set

1Q′ = ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} str(e^{−^{3}Ã^{′2}}) .

The first equation in (8.73) follows from Proposition 8.18 (3), (8.14), (8.66), (8.74). By Stokes’ theorem,

δ^{1}Q^{′∗} = ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} str([^{3}Ã′, e^{−^{3}Ã^{′2}}]) − ∫_{(r,v)∈[0,1]^{2}} \frac{1}{2} d str(e^{−^{3}Ã^{′2}})

= ∫_{r=0}^{1} \frac{1}{2} str(e^{−^{3}Ã^{′2}|_{v=0}} − e^{−^{3}Ã^{′2}|_{v=1}}) − ∫_{v=0}^{1} \frac{1}{2} str(e^{−^{3}Ã^{′2}|_{r=0}} − e^{−^{3}Ã^{′2}|_{r=1}}) .

(8.75)

As in (7.5), we have

3Ã^{′2}|_{v=0} = ( \begin{matrix}d + δ − (1 − r) o & 0 \\ 0 & −d − δ + o \end{matrix} )^{2} = ( \begin{matrix}−r(1 − r) o^{2} − o dr \\ & 0 \end{matrix} ) .

Thus,

(8.76)  ∫_{r=0}^{1} \frac{1}{2} str(e^{−^{3}Ã^{′2}|_{v=0}}) = ∫_{r=0}^{1} \frac{o}{2} e^{r(1−r) o^{2}} dr .

At v = 1, we find

3Ã^{′2}|_{v=1} = ( \begin{matrix}d + δ − (1 − r) o − r p & −2 (d + δ − (1 − r) o − r p)^{2} \\ 1/2 & −d − δ + (1 − r) o + r p \end{matrix} )^{2} = 0

by Lemma 8.7, so there is no contribution from this side.

The formal connection d+δ −o is flat either by (8.60) or because it corresponds to a_{0}^{#}∇^{\overline{V}^{1}}. This implies

3Ã^{′2}|_{r=0} = ( \begin{matrix}d + δ − o & 0 \\ v/2 & −d − δ + o \end{matrix} )^{2} = \binom{0}{dv/2 0} ,

so

(8.77)  ∫_{r=0}^{1} \frac{1}{2} str(e^{−^{3}Ã^{′2}|_{v=1}}) = ∫_{v=0}^{1} \frac{1}{2} str(e^{−^{3}Ã^{′2}|_{r=0}}) = 0 .

In the proof of Proposition 8.18 (5), the face r = 1 did not contribute because the corresponding curvature had coefficients in a nilpotent subalgebra of End V. We do not have this argument here. Instead, the contribution to δ^{1}Q′ at r = 1 will be cancelled by the second polynomial ^{2}Q′. To that end, we define yet another superconnection on V × [0, 1]^{2} by

4Ã = \frac{v}{2} a_{0} + (∇^{V^{0}} + v a_{1}^{0}) ⊕ (∇^{V^{1}} + v a_{1}^{1}) + 2v(1 − w) a_{2} ,

and as its preimage under Φ, we take

(8.74)  ^{4}Ã′ = ( \begin{matrix}d + δ − v p & −2v(1 − w) (d + δ − p)^{2} \\ v/2 & −d − δ + (1 − v) o + v p \end{matrix} ) .

We set

2Q′ = ∫_{(w,v)∈[0,1]^{2}} \frac{1}{2} str(e^{−^{4}Ã^{′2}}) .

Note that the definition of ^{4}Ã only involves the coefficients a_{0}, a_{1}^{0}, a_{1}^{1} and a_{2}, which are all strictly triangular with respect to h^{V}. This gives the second equation in (8.73).

The analogue of (8.75) holds for δ^{2}Q′. One easily checks that

3Ã′|_{r=1} = ^{4}Ã′|_{w=0} ,

so the contributions to δ^{1}Q′ at r = 1 and to δ^{2}Q′ at w = 0 cancel as announced above.

We recall that by (8.60), the formal connection d + δ − p has curvature q. At v = 1, we thus have

4Ã^{′2} = ( \begin{matrix}d + δ − p & −2(1 − w) q \\ 1/2 & −d − δ + v p \end{matrix} )^{2} = ( \begin{matrix}w q & ∗ \\ 0 & w q \end{matrix} ) .

Because this matrix is triangular and the diagonal terms do not involve dw, there will be no contribution to δ^{2}Q′ from the side v = 1. At v = 0, things are even simpler because ^{4}Ã′ is evidently flat. So, we find

(8.78)  ∫_{w=0}^{1} \frac{1}{2} str(e^{−^{4}Ã^{′2}|_{v=0}}) = ∫_{w=0}^{1} \frac{1}{2} str(e^{−^{4}Ã^{′2}|_{v=1}}) = 0 .

Finally, at w = 1, we compute

\begin{aligned}Ã^{′2}^{4} & = ( \begin{matrix}d + δ − v p & 0 \\ v/2 & −d − δ + o + v (p − o) \end{matrix} )^{2} \\ & = ( \begin{matrix}−v(1 − v) p^{2} + v q + p dv & 0 \\ ∗ & −v(1 − v) (p − o)^{2} + v q + (p − o) dv \end{matrix} ) . \end{aligned}

Let us define

R′ = ∫_{0}^{1} \frac{p}{2} e^{r(1−r) p^{2}−r q} dr ,

then by (8.65), we find

(8.79)  ∫_{v=0}^{1} \frac{1}{2} str(e^{−^{4}Ã^{′2}|_{w=1}}) = −R′ + ιR′ .

Combining equations (8.75)–(8.77), (8.78), (8.79), we obtain (2). Moreover, R′ clearly satisfies (3). This finishes the proof of Lemma 8.72. □

9. The unitary Whitehead space and its torsion class

We generalise the constructions of Section 6 to allow family Thom-Smale complexes such that the fibrewise cohomology carries a unitarily flat metric. In Section 9.a, we contruct a unitary Whitehead space Wh^{u}(R, G) that naturally covers the full Whitehead space Wh(R, G). The cohomology bundle H → Wh^{u}(R, G) carries a natural metric g^{H}. In Section 9.b, we extend the universal torsion class defined in Section 2 and studied in Section 8 to a class τ(R, G) ∈ H^{even} Wh^{u}(R, G).

Suppose that p: M → B is a family equipped with a fibre-wise Morse function h and a flat vector bundle F → M. Then the lifts ξ^{u} of the associated map ξ = ξ_{p,h,F}: B → Wh(R, G) to Wh^{u}(R, G) are in one-to-one correspondence with the parallel metrics on the fibre-wise cohomology H = H^{∗}(M/B; F) → B. The class (ξ^{u})^{∗}τ(R, G) does not depend on the choice of the lift in degree ≥ 2, and we recover the unitary Morse torsion class T (M/B; F, h) of Definition 2.64 as (ξ^{u})^{∗}τ(R, G)^{[≥2]}. Because all our constructions are compatible with stabilisation, we can generalise Igusa’s constructions in [I2] to arbitrary families of manifolds equipped with unitarily flat vector bundles such that the fibrewise cohomology carries a parallel metric.

Finally, we give a natural map π: Wh^{u}(R, G) → Wh^{h}(R, G), such that τ(R, G) is preserved by π^{∗}. This shows in particular that we recover Igusa’s torsion τ(M/B; F), which is defined similarly under the assumption that H → B is trivial or has nilpotent structure group.

9.a. The unitary Whitehead space. We construct the unitary Whitehead space Wh^{u}(R; G) as a covering of the Whitehead space Wh(R, G). Note that the unitary Whitehead space should be viewed as a generalisation of Igusa’s acyclic Whitehead space Wh^{h}(R; G).

We have seen that the higher torsion forms of Section 2 can be used to construct cohomology classes if both flat bundles F |_{C} → C and H → B carry parallel metrics. Therefore, we need a variant of the Whitehead space that contains information about metrics on the tautological bundle V → Wh(R; G) and the universal cohomology bundle H → Wh(R; G).

The bundle F |_{C} admits a parallel metric iff we can choose its discrete structure group to be subgroup of the unitary group, so it is sufficient to choose a group G ⊂ R^{×} and a complex representation ρ: G → GL_{n}(ℂ) such that ρ(G) ⊂ U(r). On the other hand, if we regard ρ(G) ⊂ U(r), then the complexified bundle V → Wh(R, G) carries a natural Hermitian metric induced by the standard metric on ℂ^{r}.

All fibre-wise metrics on H form a fibre bundle over Wh(R, G) with a natural flat bundle connection for any R and G. We can take the étale space of the sheaf of horizontal sections of this bundle, which become parallel metrics on H, as a classifying space for Morse vector bundles with a given parallel metric on the fibre-wise cohomology. In fact, we will refine this concept just a little by fixing a parallel Hermitian basis of H over each simplex of our unitary Whitehead space Wh^{u}(R, G).

To keep things simple, we will take the point of view that unitary higher torsion is defined on Wh^{u}(M_{r}(ℂ), U(r)). For arbitrary R and G, we obtain the unitary Whitehead space Wh^{u}(R, G) and its universal torsion class τ(R, G) ∈ H^{even}(R, G) from a pullback construction

Wh^{u}(R, G) ⟶ Wh^{u}(M_{r}(ℂ), U(r))
↓  ↓
Wh(R, G) ⟶ Wh(M_{r}(ℂ), U(r)) ,

where the lower right arrow is induced by a representation ρ: R → M_{r}(ℂ) with ρ(G) ⊂ U(r), the space Wh^{u}(R, G) is constructed as a pullback bundle, and τ(R, G) is the pullback of τ = τ(M_{r}(ℂ), U(r)) by the upper right arrow.

For convenience, we give a simplicial model for Wh^{u}(M_{r}(ℂ), U(r)). Recall that the ℤ-graded flat vector bundles V → Wh(M_{r}(ℂ), U(r)) and H → Wh(M_{r}(ℂ), U(r)) were described in Remark 6.11.

9.1. Definition. The unitary simplicial pre-Whitehead category Wh^{′simp,u}(M_{r}(ℂ), U(r)) is defined as follows.

(1) Objects of Wh_{k}^{simp,u}(M_{r}(ℂ), U(r)) are given by triples (P, a, e), where (P, a) is an object of Wh_{k}^{simp}(M_{r}(ℂ), U(r)) and e = (e^{p}(i))_{p,i} is a family of complex bases e_{1}^{p}(i), …, e_{rk H^{p}}^{p}(i) on the spaces H^{p}((ℂ^{r})^{P} , a(i)) that are compatible with the natural isomorphisms

(id +a(i, j))_{∗}: H^{∗}((ℂ^{r})^{P} , a(j)) ⟶ H^{∗}((ℂ^{r})^{P} , a(i))

for 0 ≤ i ≤ j ≤ k.

(2) Morphisms from (P, a, e) to (Q, b, e′) ∈ Wh^{simp,u} are given by pairs (f, g) such that (f, g) is a Wh^{simp}(M_{r}(ℂ), U(r))-morphism from (P, a) to (Q, b), and the induced isomorphisms

(f, g)_{∗}: H^{∗}((ℂ^{r})^{P} , a(i)) ⟶ H^{∗}((ℂ^{r})^{Q}, b(i))

are given in the bases e and e′ by unitary matrices for one (and thus for all) 0 ≤ i ≤ k.

(3) Generalised face operators are given by re-indexing as in Definition 6.10. The simplicial unitary pre-Whitehead space Wh^{′simp,u}(M_{r}(ℂ), U(r)) is the geometric realisation of the simplicial unitary pre-Whitehead category.

We stabilise the unitary Whitehead space as in Definition 6.14. Note that if (P^{[l]}, a^{[l]}) is the elementary object of vertical degree l, then we naturally identify

H^{∗}((ℂ^{r})^{P} , a(j)) ≅ H^{∗}((ℂ^{r})^{P ∪P^{[l]}}, (a ⊕ a^{[l]})(j)) .

In particular, we can compare bases on both sides.

9.2. Definition. The (stabilised) simplicial unitary Whitehead category Wh^{simp,u}(M_{r}(ℂ), U(r)) has the same objects and face operators as Wh^{′simp,u}(M_{r}(ℂ), U(r)). Its morphisms are iterated formal compositions of Wh^{′simp,u}(M_{r}(ℂ), U(r))-morphisms and elementary expansions

(P, a, e) ⟶ (P ∪ P^{[j]}, a ⊕ a^{[j]}, e) .

The simplicial unitary Whitehead space Wh^{simp,u}(M_{r}(ℂ), U(r)) is the geometric realisation of the simplicial unitary Whitehead category.

9.3. Remark. We collect some properties of the definition above.

(1) By its construction in Remark 6.11 (1), the bundle V → Wh′(M_{r}(ℂ), U(r)) inherits a parallel metric g^{V} from the standard metric on ℂ^{r}. A similar construction yields a unitarily flat bundle (V, g^{V}) → Wh^{′simp,u}(M_{r}(ℂ), U(r)). We also let H denote fibre-wise cohomology as in Remark 6.11 (2), which is a flat vector bundle that is compatible with stabilisation, so we get H → Wh^{simp,u}(M_{r}(ℂ), U(r)).

(2) The bundle H → Wh^{simp,u}(M_{r}(ℂ), U(r)) carries a unique parallel metric g^{H} for which the bases e are unitary.

(3) Recall that Wh^{simp,h}(M_{r}(ℂ), U(r)) is the full subcategory of Wh^{simp}(M_{r}(ℂ), U(r)) consisting of those objects (P, a) ∈ Wh^{simp}(M_{r}(ℂ), U(r)) for which a(i) is acyclic for one (and thus for all) i. Then Wh^{simp,h}(M_{r}(ℂ), U(r)) is a full subcategory of Wh^{simp,u}(M_{r}(ℂ), U(r)), and the forgetful functor to Wh^{simp,h}(M_{r}(ℂ), U(r)) is just the identity. This way, we may view Wh^{simp,u}(M_{r}(ℂ), U(r)) as a generalisation of the category Wh^{simp,h}(M_{r}(ℂ), U(r)) considered in [I2].

(4) There are similar definitions for Wh^{diff,u}(M_{r}(ℂ), U(r)), Wh^{mix,u}(M_{r}(ℂ), U(r)) and their unstabilised variants if R is an ℝ-algebra. Again, Wh^{diff,u}(M_{r}(ℂ), U(r)) is the classifying space for Morse vector bundles (V, A′, h^{V}) → B whose fibre-wise cohomology H → B is equipped with a parallel metric g^{H}. It is now easy to check that all results of Section 6.d carry over to unitary Whitehead categories, so we have homotopy equivalences

Wh^{′simp,u}(M_{r}(ℂ), U(r)) ⟵_{∼}^{Φ_{1}^{u}} Wh^{′mix,u}(M_{r}(ℂ), U(r)) ⟶_{∼}^{Φ_{2}^{u}} Wh^{′diff,u}(M_{r}(ℂ), U(r))
and  Wh^{simp,u}(M_{r}(ℂ), U(r)) ⟵_{∼}^{Φ_{1}^{u}} Wh^{mix,u}(M_{r}(ℂ), U(r)) ⟶_{∼}^{Φ_{2}^{u}} Wh^{diff,u}(M_{r}(ℂ), U(r)) .

In particular, we may talk about “the” unitary pre-Whitehead space Wh^{′u}(M_{r}(ℂ), U(r)), which is well-defined up to homotopy equivalence and carries unitarily flat vector bundles (V, ∇^{V}, g^{V}) and (H, ∇^{H}, g^{H}).

Let us examine the homotopy type of the unitary Whitehead space. We denote by G(H) the space of Hermitian metrics on H → Wh(M_{r}(ℂ), U(r)). We topologize G(H) as the étale space of the sheaf of locally parallel metrics on H. In other words, the map G(H) → Wh(M_{r}(ℂ), U(r)) is a covering, and local sections are given precisely by parallel metrics. A simplicial model for G(H) can be constructed as in Definition 9.1, but using parallel metrics g^{H} on H^{∗}((ℂ^{r})^{P} , a′_{0}) over each simplex of Wh(M_{r}(ℂ), U(r)) instead of parallel bases e of this bundle.

9.4. Proposition. There is a natural map Wh^{u}(M_{r}(ℂ), U(r)) → G(H) given by assigning to a local parallel basis e of H the Hermitian metric for which e is unitary. This map is a fibre bundle with contractible fibres.

In particular, if ξ: B → Wh(M_{r}(ℂ), U(r)) is a map and (V, ∇^{V} , A′, g^{V}) = ξ^{∗}(V, ∇^{V}, A′, g^{V}), then homotopy classes of lifts ξ^{u}: B → Wh^{u}(M_{r}(ℂ), U(r)) are in bijection with globally parallel metrics g^{H} on H = H^{∗}(V, a′_{0}) = ξ^{∗}H → B.

In other words, we could have defined Wh^{u}(M_{r}(ℂ), U(r)) simply as the covering space G(H) → Wh(M_{r}(ℂ), U(r)). Nevertheless, the extra information about a local framing on H will be helpful later on.

Proof. Over each simplex σ ∈ Wh^{simp,u}, the fibre of the map Wh^{u}(M_{r}(ℂ), U(r)) → G(H) is the geometric realisation of the category of all unitary basis and unitary base changes with respect to the tautological metric g^{H} on the pullback of H to G(H). But this fibre is just the classifying bundle E(U(H|_{σ})) of the unitary group acting on the fibre of H|_{σ}, which is contractible.

The second claim follows because globally parallel metrics g^{H} on H → B correspond precisely to lifts of ξ to the space G(H). □

9.b. The universal torsion class on the unitary Whitehead space. We generalise our definition of T (M_{r}(ℂ), U(r)) ∈ H^{even}(Wh^{h}(M_{r}(ℂ), U(r))) in Definition 7.29 to the unitary setting. Because we have seen in Theorem 8.1 that T (M_{r}(ℂ), U(r)) = τ(M_{r}(ℂ), U(r)) on Wh^{h}(M_{r}(ℂ), U(r)), we will name our new class τ(M_{r}(ℂ), U(r)) ∈ H^{even}(Wh^{u}(M_{r}(ℂ), U(r))), or just τ. The letter T will be reserved for torsion forms associated directly to superconnections as in Section 2.d. Note that in general, the forms T (∇^{V}, A′, g^{V}) are not closed, so they do not define cohomology classes. However, since g^{V} and g^{H} are unitarily flat, the form T (∇^{V}, A′, g^{V}) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H}) is closed, and defines a class τ on Wh^{u}(M_{r}(ℂ), U(r)) by Stokes theorem. Finally, we show that τ is compatible with respect to stabilisation in r and thus extends to a form on Wh^{u}(M(ℂ), U) with M(ℂ) = lim_{→} M_{r}(ℂ) and U = lim_{→} U(r).

9.5. Definition. Define τ = τ(M_{r}(ℂ), U(r)) ∈ H^{even}(Wh^{′u}(M_{r}(ℂ), U(r)), ℝ) to be the class represented by the cocycle which assigns to every simplex Δ in Wh^{′diff,u}(M_{r}(ℂ), U(r)) the integral

∫_{Δ}(T (∇^{V}, A′, g^{V}) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})) .

Then τ^{[0]} measures the difference of the Franz-Reidemeister metric on det H and the metric g^{det H} induced by g^{H}. Let (V, A′, h^{V}) be a family Thom-Smale complex over B such that ∇^{V} is unitarily flat and such that H → B admits a parallel metric. If S is a simplicial structure on B, there exists a “classifying functor” Ξ: S → Wh^{′diff,u}(M_{r}(ℂ), U(r)), inducing a classifying map ξ: B → Wh^{′u}(M_{r}(ℂ), U(r)) up to homotopy, such that

(V, ∇^{V} , A′, g^{V} ) ≅ ξ^{∗}(V, ∇^{V}, A′, g^{V})  and  (H, ∇^{H}, g^{H}) ≅ ξ^{∗}(H, ∇^{H}, g^{H}) .

We have an immediate generalisation of Corollary 7.30.

9.6. Corollary. Let S be a simplicial complex on B and let Ξ: S → Wh^{′diff,u}(M_{r}(ℂ), U(r)) be a simplicial map. Let (V, ∇^{V} , A′, g^{V}) = |Ξ|^{∗} (V, ∇^{V}, A′, g^{V}) and (H, ∇^{H}, g^{H}) = |Ξ|^{∗} (H, ∇^{H}, g^{H}), then

(1)  T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H}) = |Ξ|^{∗} τ ∈ H^{even}(|S| , ℝ) .

In particular, if p: M → B is a proper submersion with a fibrewise Morse-function h: M → ℝ, if (F, ∇^{F} , g^{F}) → M is a unitarily flat vector bundle, such that the bundle H = H^{∗}(V, a_{0}) → B admits a parallel metric, and if ξ^{u}: B → Wh^{′u}(M_{r}(ℂ), U(r)) is constructed as in Remark 9.3 (3), then

(2)  T (M/B, F, h) = ξ^{u∗}τ^{[≥2]} .

This follows from the same kind of arguments as Corollary 7.30. Note here that since the parallel metric on H can be varied, we cannot define the ordinary Franz-Reidemeister torsion in degree 0. On the other hand, ξ^{u∗}τ^{[≥2]} is independent of the particular lift ξ^{u} by Corollary 7.27. For this reason, the constant function ξ^{u∗}τ^{[0]} is explicitly excluded from (2).

9.7. Proposition. The unitary torsion class τ = τ(M_{r}(ℂ), U(r)) ∈ H^{even}(Wh^{′u}(M_{r}(ℂ), U(r)), ℝ) is compatible with stabilisation and descends to Wh^{u}(M_{r}(ℂ), U(r)).

Proof. We have to show that τ is compatible with the elementary expansions in Wh^{diff,u}. Therefore, let P^{[l]} and a^{′^{[l]}} = ( _{0}^{0 1} ) as in Definition 6.28. We take the trivial metric on (ℂ^{r})^{P^{[l]}}. Then by Corollary 7.24,

T (d, d + a^{′[l]}, g^{triv}) = T^{BL}(d + a^{′[l]}, g^{triv}) = 0 .

It follows that the explicit cocycle for τ in Definition 9.5 takes the same value on (P, a, g^{H}) as on (P ∪ P^{[l]}, a′ ⊕ a^{′[l]}, g^{H}). The proposition follows from the definition of the stabilised differential unitary Whitehead space. □

We will use the universal torsion class τ ∈ H^{even}(Wh^{u}(M_{r}(ℂ), U(r))) in Section 10 to extend Igusa’s higher Franz-Reidemeister torsion to smooth fibre bundles with unitarily flat fibre-wise cohomology.

For all r, r′ ≥ 0, there are well-known natural homomorphisms

M_{r}(ℂ) ⟶ M_{r+r′}(ℂ)  and  U(r) ⟶ U(r + r′)

given by extending a matrix in M_{r}(ℂ) by zero, and an element g ∈ U(r) to g ⊕ id_{ℂ^{r′}}. Let

M(ℂ) = lim_{r→∞} M_{r}(ℂ)  and  U = lim_{r→∞} U(r)

denote the direct limits with respect to these maps.

The maps M_{r}(ℂ) → M_{r+r′}(ℂ) and U(r) → U(r + r′) induce a functor

(9.8)  Wh(M_{r}(ℂ), U(r)) ⟶ Wh(M_{r+r′}(ℂ), U(r + r′)) .

This gives rise to a pullback diagramme

V ⊕ ℂ^{r′ rk V/r} ⟶  V
↓  ↓
Wh(M_{r}(ℂ), U(r)) ⟶ Wh(M_{r+r′}(ℂ), U(r + r′)) .

The additional trivial bundle ℂ^{r′ rk V/r} carries the trivial connection d, and since all coefficients a′_{∗} of A′ are extended trivially, the pull-back of A′ acts as d on ℂ^{r′ rk V/r}. Thus, we have another pullback diagramme

(9.9)  \begin{matrix}H ⊕ ℂ^{r′ rk V/r} & ⟶ & H \\ ↓ & & ↓ \\ Wh(M_{r}(ℂ), U(r)) & ⟶ & Wh(M_{r+r′}(ℂ), U(r + r′)) & . \end{matrix}

We extend the functor (9.8) to a functor

(9.10)  Wh^{u}(M_{r}(ℂ), U(r)) ⟶ Wh^{u}(M_{r+r′}(ℂ), U(r + r′))

by specifying the standard basis on the additional summand ℂ^{r′ rk V/r} in (9.9).

9.11. Proposition. The universal unitary higher torsion class τ is compatible with stabilisation

Wh^{u}(M_{r}(ℂ), U(r)) ⟶ Wh^{u}(M_{r+r′}(ℂ), U(r + r′)) .

Proof. We use the differential model of the Whitehead space. On a single simplex σ = (P, a′, e) ∈ Wh^{diff,u}(M_{r}(ℂ), U(r)), the pullback of τ(M_{r+r′}(ℂ), U(r + r′)) is given by

∫_{Δ}(T (∇^{V} ⊕ d, A′ ⊕ d, g^{V} ⊕ g^{triv}) + c̃h^{o}(∇^{H} ⊕ d, g^{H} ⊕ g^{triv}, g_{V}^{H} ⊕ g^{triv}))

= ∫_{Δ}(T (∇^{V}, A′, g^{V}) + T^{BL}(d, g^{triv}) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H}))

= ∫_{Δ}(T (∇^{V}, A′, g^{V}) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})) = τ(M_{r}(ℂ), U(r))(σ) ,

because the Bismut-Lott torsion vanishes for a trivial superconnection. □

9.12. Definition. We define the universal unitary Whitehead category by

Wh^{u}(M(ℂ), U) = lim_{r→∞} Wh^{u}(M_{r}(ℂ), U(r))

using the functors (9.10), and let the universal unitary Whitehead space Wh^{u}(M(ℂ), U) be its geometric realisation. We still denote the universal unitary higher torsion class by

τ ∈ H^{even}(Wh^{u}(M(ℂ), U)) .

9.c. The cone construction. In this section, let (V, A′, h^{V}) → B be an arbitrary family Thom-Smale complex and let g^{V} be an arbitrary adapted metric on V which need not be parallel. We construct a fibre-wise acyclic family Thom-Smale complex (V̂, Â′, hV̂ ) on the vector bundle V̂ = V ⊕H. In the next subsection, we will compare the torsion classes T (∇^{V} , A′, g^{V}) and T (∇V̂ , Â′, gV̂ ).

We fix an adapted metric g^{V} on V in the sense of Definition 2.37 and let a′′_{0} denote the adjoint of a′_{0} with respect to g^{V}. By finite-dimensional Hodge theory, we have a g^{V} -orthogonal splitting

V = ker(a′_{0} + a′′_{0}) ⊕ im(a′_{0}) ⊕ im(a′′_{0}) ,

which is preserved by the combinatorial Laplacian [a′_{0}, a′′_{0}]. Let ι: H → ker(a′_{0} + a′′_{0}) ⊂ V denote the identification of H with the harmonic elements, which we extend Ω^{∗}(B)-linearly to a map ι: Ω^{∗}(B; H) → Ω^{∗}(B; V).

We can find a smooth function ε: B → ℝ_{>} such that |λ| > ε for all non-zero eigenvalues λ of [a′_{0}, a′′_{0}]. Then we define a generalised spectral projection Π on Ω^{∗}(B; V) by

Π = \frac{1}{2πi} ∫_{εS^{1}} \frac{dλ}{λ − (A′ + a′′_{0})^{2}} .

Similar operators were already considered in [BG], Definition 10.13. The analogue of Theorem 10.9 in [BG] implies in particular that Π^{2} = Π, that Π is of total degree 0, and that Π commutes with A′, a′′_{0} and Ω^{∗}(B). Moreover, the projection Π^{[0]} in degree 0 with respect to N^{B} coincides with the orthogonal projection onto the harmonic elements of V.

Let W ⊂ Ω^{∗}(B; V) be the image of Π, then W is an A′-invariant Ω^{∗}(B)-submodule of Ω^{∗}(B; V) and Π ∘ ι: Ω^{∗}(B; H) → W is an isomorphism. Let

(9.13)  A^{′H} = (Π ∘ ι)^{−1} ∘ A′ ∘ (Π ∘ ι)

be the pullback superconnection on H.

9.14. Lemma. The superconnection A^{′H} takes the form

(1)  A^{′H} = ∇^{H} + a^{′H}  with a^{′H,[k]} ∈ Ω^{≥2}(B; End H) .

Moreover, Π ∘ ι induces an Ω^{∗}(B)-linear isomorphism of cochain complexes

(2)  Π ∘ ι: (Ω^{∗}(B; H), A^{′H}) ⟶ (W, A′) ⊂ (Ω^{∗}(B; V), A′) .

We will see later that (1) means that the difference between A^{′H} and ∇^{H} is almost negligible.

Proof. Both Π and ι are Ω^{∗}(B)-linear. Because Π^{[0]} coincides with the projection onto the (a′_{0}+a′′_{0})- harmonic elements of V, it is clear that we obtain an isomorphism. Because Π commutes with A′, it is clear that A′ acts on W. It remains to show that the restriction of A′ to W pulls back to a flat superconnection on H, which starts with ∇^{H} in degree 1.

We write the superconnection A^{′H} on H of (9.13) as

A^{′H} = ∇^{H} + ∑_{i≥0} a_{i}^{′H}

with a_{i}^{′H} ∈ Ω^{i}(B; End H). Let η ∈ Ω^{0}(B; ker(a′_{0} + a′′_{0})) represent a section of H, then clearly

a_{0}^{′H}η = (Π ∘ ι ∘ A^{′H})^{[0]} η = (A′ ∘ ι ∘ Π)^{[0]} η = a′_{0}η = 0 .

To calculate a_{1}^{′H}, note that since a_{0}^{′H} = 0, we have

ι (∇^{H} + a_{1}^{′H}) η = (Π ∘ ι ∘ A^{′H})^{[1]} η = (A′ ∘ Π ∘ ι)^{[1]} η
= (∇^{V} + a′_{1}) η + a′_{0} Π^{[1]} η ∈ Ω^{1}(B; ker(a′_{0} + a′′_{0})) .

Since the result is in Ω^{1}(B; ιH) whereas im(a′_{0}) ⊥ ιH, we also get

ι (∇^{H} + a_{1}^{′H}) η = Π^{[0]} (∇^{V} + a′_{1}) η ,

but Π^{[0]} (∇^{V} + a′_{1}) just represents the Gauß-Manin connection ∇^{H} on H, so a_{1}^{′H} = 0 as well. □

As a consequence, we can now define a flat superconnection on V̂ = V ⊕ H^{[−1]} of the form

(9.15)  Â′ = ( \begin{matrix}A′ & Π ∘ ι \\ 0 & A^{′H} \end{matrix} ) ,

where H^{[−1]} denotes the ℤ-graded bundle H with degree shifted down by one. The superconnection Â′ is apparently flat and of total degree 1. By assumption, A′ − ∇^{V} takes values in a nipotent subalgebra of End V. Similarly, A^{′H} − ∇^{H} takes values in the nilpotent subalgebra of End H that increases the vertical degree. We conclude that Â′ −∇V̂ also takes values in a nilpotent subalgebra. Moreover, the cohomology H of (V, a_{0}) is killed by H^{[−1]}, so the fibre-wise operator â′_{0} = a′_{0}+Π^{[0]}∘ι is acyclic.

9.16. Definition. Let (V, A′, ∇^{V} , g^{V}) be a family Thom-Smale complex with an adapted metric, and let g^{H} be a metric on the vector bundle H = H^{∗}(V, a′_{0}) → B. Then we call (V̂, Â′, ∇V̂ , gV̂ ) the acyclic cone of (V, A′, ∇^{V}).

9.17. Remark.

(1) The acyclic cone construction as a fibre-wise acyclic family Thom-Smale-complex is up to homotopy independent of the choice of metrics g^{V} and g^{H}. This follows from a standard argument involving an extension of metrics from B × {0, 1} to B × [0, 1] on the pullbacks of V and H via the projection B × [0, 1] → B.

(2) If g^{V} and g^{H} are unitarily flat, then so is gV̂ = g^{V} ⊕ g^{H}.

(3) We would like to turn the acyclic cone construction into a functor from the universal unitary Whitehead category Wh^{u}(M(ℂ), U) to a suitable acyclic Whitehead category. Although the whole construction above is natural, we encounter a technical problem. We start in Wh^{u}(M_{r}(ℂ), U(r)) for some fixed r. The dimension of the fibre-wise cohomology H → Wh^{u}(M_{r}(ℂ), U(r)) depends on the connected component of Wh^{u}(M_{r}(ℂ), U(r)) and is in general not equal to r. This means that Wh^{h}(M_{r}(ℂ), U(r)) is not a good receptacle for a universal cone construction. Taking the limit lim_{r→∞} Wh^{h}(M_{r}(ℂ), U(r)) is difficult because our stabilisation procedure in Definition 9.12 destroys acyclicity. Similarly, if we pass to lim_{r→∞} Wh^{u}(M(ℂ), U), then the dimension of the bundle V̂ still depends on r. To solve this problem, we would need a generalisation of the acyclic Whitehead space that allows to take the limit r → ∞ in a natural way.

9.d. Unitary and acyclic higher torsion. We show that the analytic torsion of a family Thom-Smale complex (V, ∇^{V} , A′) is related to the torsion of its acyclic cone (V̂, ∇V̂ , Â′, gV̂ ). If both V and H carry parallel metrics, this allows us to reduce the calculation of the unitary torsion to the acyclic case.

Let (V, ∇^{V} , A′) be a family Thom-Smale complex with an adapted metric g^{V}, and let (V̂, ∇V̂ , Â′) be the acyclic cone of Definition 9.16. We have a natural filtration

(9.18)  (V̂, Â′) ⊃ (V, A′) ⊃ 0 ,

to which gV̂ = g^{V} ⊕ g^{H} is adapted in the sense of Definition 7.35. The metric gV̂ is parallel iff g^{V} and g^{H} are.

9.19. Theorem. Let (V, ∇^{V} , A′) be a family Thom-Smale complex, let (V̂, ∇V̂ , Â′) be its acyclic cone with respect to the metric g^{V}, and let gV̂ = g^{V} ⊕ g^{H}. Then modulo exact forms, we have

T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H}) = T (∇V̂ , Â′, gV̂ ) .

Proof. The bundle V̂ has a two-step filtration (9.18) and carries an adapted metric gV̂ = g^{V} ⊕ g^{H}. Moreover, (V̂, â′_{0}) is acyclic by construction. In other words, the spectral sequence associated to (9.18) degenerates at E_{2} = 0.

The E_{0}-term of the spectral sequence is the direct sum

(E_{0}, ∇^{E_{0}}, D′_{0}, g^{E_{0}}) = (V, ∇^{V} , A′, g^{V} ) ⊕ (H^{[−1]}, ∇^{H}, A^{′H}, g^{H}) ,

and its torsion is given by

T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) = T (∇^{V} , A′, g^{V} ) − T (∇^{H}, A^{′H}, g^{H}) .

We extend A^{′H} to a flat superconnection on \overline{H} = H × (0, 1] by

\overline{A}^{′H} = s^{N^{H}} (A^{′H} + \frac{∂}{∂s} ds) s^{−N^{H}}

and set

∇\overline{H} = (\overline{A}^{′H})^{[1]} = ∇^{H} + (\frac{∂}{∂s} − \frac{N^{H}}{s} ) ds .

Then we still have

\overline{A}^{′H} = ∇\overline{H} + ∑_{i≥2} ā_{i}^{′H}

with ā_{i}^{′H} ∈ Ω^{i}(B × (0, 1], End \overline{H}).

We note that \overline{H} is its own fibre-wise cohomology, and if g\overline{H} is a metric on \overline{H}, then the induced metric on the fibre-wise cohomology is again g\overline{H}, so

dT (∇\overline{H}, \overline{A}^{′H}, g\overline{H}) = ch^{o}(∇\overline{H}, g\overline{H}) − ch^{o}(∇\overline{H}, g\overline{H}) = 0 .

We may thus take g\overline{H} to be the pullback of g^{H}. It is now easy to see that

lim_{s→0} T (∇\overline{H}, \overline{A}^{′H}, g\overline{H})|_{B×\{s\}} = T^{BL}(∇^{H}, g^{H}) = 0 ,

and this is cohomologous to T (∇^{H}, A^{′H}, g^{H}). We conclude that

(9.20)  T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) = T (∇^{V} , A′, g^{V} ) ∈ Ω^{even}(B)/dΩ^{odd}(B) .

The E_{1} term of our spectral sequence is given by

(E_{1}, ∇^{E_{1}}, D′_{1}, g^{E_{1}}) = (H ⊕ H^{[−1]}, ∇^{H⊕H}, D′_{1}, g_{V}^{H} ⊕ g^{H}) ,

where the superconnection D′_{1} on H ⊕ H^{[−1]} in matrix notation equals

D′_{1} = \binom{∇^{H} id}{∇^{H}} .

By the appendix of [BL], we conclude that

\begin{aligned}T^{BL}(D′_{1}, g^{E_{1}}) & = T^{BL}(D′_{1}, g_{V}^{H} ⊕ g^{H}) \\ & = c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H}) . \end{aligned} (9.21)

By Theorem 7.37, we now have

\begin{aligned}T (∇V̂ , Â′, gV̂ ) & = T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) + T^{BL}(∇^{E_{1}}, D′_{1}, g^{E_{1}}) \\ & = T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H}) . □ \end{aligned}

9.22. Remark. If we assume that both g^{V} and g^{H} are parallel, then Theorem 9.19 gives us an identity of cohomology classes on B. It shows that the unitary torsion can be completely reduced to the torsion in the acyclic case. This is another justification for our definition of τ on Wh^{u}(M(ℂ), U).

We recall that the torsion on the acyclic Whitehead space Wh^{h}(M_{r}(ℂ), U(r)) is completely determined by its restriction to the two-index Whitehead space Wh^{h,{0,1}}(M_{r}(ℂ), U(r)), which is a deformation retract of Wh^{h}(M_{r}(ℂ), U(r)) by Igusa’s two-index theorem in [I2], Theorem 5.7.1. It would be nice to know that τ ∈ H^{even}(Wh^{u}(M(ℂ), U)) is also determined by its restriction to Wh^{h,{0,1}}(M_{r}(ℂ), U(r)). Unfortunately, for the reasons explained in Remark 9.17 (3), we do not obtain this conclusion immediately.

10. The Framing Principle and a higher Cheeger-Müller Theorem

We use Igusa’s framing principle to establish Theorem 0.3 for fibre bundles with fibrewise Morse functions and fibrewise acyclic flat vector bundles. We establish a cohomological analogue of Igusa’s splitting lemma. Using this result and Theorem 0.3, we prove the generalised second statement of Theorem 0.2. Throughout this chapter, we will use the complex matrix ring R = M(ℂ) = lim_{→} M_{r}(ℂ) and the unitary group G = U = lim_{→} U(r). We let τ = τ(M(ℂ), U) denote the universal unitary torsion class on Wh^{u}(M(ℂ), U).

10.a. Igusa’s framed function theorem. We recall the definition of generalised fibre-wise Morse functions and framed functions in the sense of [I1]. Framed functions always exist by Igusa’s framed function theorem. By [I2], Chapter 4, each generalised Morse function gives rise to a classifying map ξ_{p,h,F}: B → Wh(M(ℂ), U). Suppose that this map lifts to Wh^{u}(M(ℂ), U), then we can define a torsion class τ(M/B; F, h) by pulling back the universal torsion class τ(M(ℂ), U) ∈ Wh^{u}(M(ℂ), U).

Let p: M → B be a fibre-bundle with n-dimensional fibres, and let h: M → ℝ be a smooth function. We denote by C ⊂ M the set of fibre-wise singularities, i.e., the zero set of dh|_{T X}. The subset of fibre-wise Morse singularities will be denoted C_{Morse}.

10.1. Definition. A point p ∈ C_{b} ⊂ X_{b} is called a birth-death singularity of index k if there is a neighbourhood V ⊂ B of b, a function f_{0}: V → ℝ, a local trivialisation of p^{−1}V ≅ B × X with coordinates b_{1}, …, b_{m} of V and x_{0}, …, x_{n} on X_{b} of X around p, such that

f(q) = f_{0}(b_{1}, …, b_{m}) − \frac{x_{1}^{2}}{2} − ⋯ − \frac{x_{k}^{2}}{2} + \frac{x_{k+1}^{2}}{2} + ⋯ + \frac{x_{n−1}^{2}}{2} + \frac{x_{n}^{3}}{3} − b_{1} x_{n}

for all q ∈ p^{−1}V near p. Let C_{b-d} ⊂ C denote the set of all birth-death singularities of h.

10.2. Definition. Let M → B be a fibre bundle with fibre-wise boundary ∂M = M_{0} ∪ M_{1}. A generalised fibre-wise Morse function is a function h: M → [0, 1] such that 0 and 1 are regular values with h^{−1}(i) = M_{i} for i ∈ {0, 1}, and such that h has only fibre-wise Morse singularities and birth-death singularities, i.e., a function with the property that C = C_{Morse} ∪ C_{b-d}.

Then as in [I2], Chapter 4, each proper generalised fibre-wise Morse function h: M → ℝ with finitely many singularities on each fibre gives rise to a classifying map ξ_{p,h,F}: B → Wh(M(ℂ), U) up to homotopy. Assume that g^{F} and g^{H} are parallel metrics on F → M and H → B, then we also have a lift ξ_{p,h,F}^{u}: B → Wh^{u}(M(ℂ), U). We can now combine Igusa’s definition of the higher Franz-Reidemeister torsion with our definition of Morse torsion forms.

10.3. Definition. Assume that F → M and H → B admit parallel metrics. Let h: M → ℝ be a proper generalised fibre-wise Morse function with finitely many fibre-wise singularities on each fibre, and let ξ_{p,h,F}^{u}: B → Wh^{u}(M(ℂ), U) be the corresponding classifying map. Then we define the higher Franz-Reidemeister torsion

τ(M/B; F, h) = (ξ_{p,h,F}^{u} )^{∗}τ(M(ℂ), U)^{[≥2]} ∈ H^{even,≥2}(B; ℝ) .

We could also define τ(M/B; F, h)^{[0]} = τ(M/B; F, h, g^{V} , g^{H})^{[0]} in degree 0, however, it seems better to work with Franz-Reidemeister metrics instead.

Note that τ(M/B; F, h) still depends on the choice of h. To define an independent invariant of p: M → B and F → M only, Igusa further restricts the space of admissible functions h. Up to homotopy, there is a unique splitting of the vertical tangent bundle

T X|_{C_{Morse}} = T_{Morse}^{s}X ⊕ T_{Morse}^{u}X ⟶ C_{Morse}

such that d^{2}h is positive (negative) definit on T_{Morse}^{s}X (T_{Morse}^{u}X). There is a similar splitting

T X|_{C_{b-d}} = T_{b-d}^{s}X ⊕ T_{b-d}^{u}X ⊕ T_{b-d}^{c}X ⟶ C_{b-d} ,

where T_{b-d}^{c} = ker(d^{2}h) ≅ ℝ × C_{b-d} is the unique cubic direction corresponding to the coordinate x_{n} in Definition 10.1, which is oriented by the third derivative of h.

In order to obtain a somewhat unique generalised fibre-wise Morse function, Igusa considers functions h such that the bundles T_{Morse}^{u}X and T_{b-d}^{u}X admit compatible trivialisations. In the coordinates of Definition 10.1, let

C^{±} = \{ q | x_{1}(q) = ⋯ = x_{n−1}(q) = 0, b_{1} > 0, and x_{n} = ±\sqrt{b_{1}} \} ⊂ C_{Morse} ,

then C^{+} has fibre-wise Morse index k + 1 and C^{−} has index k.

10.4. Definition. Trivialisations T_{Morse}^{u}X ≅ C_{Morse} × ℝ^{ind^{h}} and T_{b-d}^{u}X ≅ C_{b-d} × ℝ^{ind^{h}} are called compatible if near each point p ∈ C_{b-d}, the trivialisation of T_{b-d}^{u}X near p extends continously to the trivialisation of T_{Morse}^{u}X|_{C^{−}}, and the trivialisation of T_{b-d}^{u}X ⊕T_{b-d}^{c}X extends continously to the trivialisation of T_{Morse}^{u}X|_{C^{+}}. A framed function is a generalised fibre-wise Morse function on M with fixed compatible trivialisations of T_{Morse}^{u}X and T_{b-d}^{u}X.

With this definition, we can now state a version of Igusa’s framed function theorem ([I1]).

10.5. Fact (Framed Function Theorem). Let p: M → B be a fibre bundle with fibre-wise boundary ∂M = M_{0} ∪ M_{1}. If dim M ≥ 2 dim B, then M carries a framed function h: M → [0, 1] with h^{−1}(i) = M_{i} for i = 0, 1. Let p: M → B be a fibre bundle with dim M > 2 dim B, and let h_{0}, h_{1}: M → ℝ be framed functions as above, then there exists a framed function h on M × [0, 1] → B × [0, 1] with the same properties that restricts to the given functions and framings on M × {0, 1}.

This implies that for dim M > 2 dim B, the homotopy class of ξ_{p,h,F}: B → Wh(M(ℂ), U) is independent of the choice of the framed function h. Of course, the same holds for any lift ξ_{p,h,F}^{u}: B → Wh^{u}(M(ℂ), U) with respect to a fixed parallel metric on H. Igusa defines a unique higher Franz-Reidemeister torsion for unitarily flat bundles F → M such that H^{∗}(M/B; F) → B is trivial, cf. [I2], Theorem 5.7.5. We extend Igusa’s definition as follows.

10.6. Definition. Assume that M → B is a smooth fibre bundle with dim M > 2 dim B, with compact fibres, and with fibre-wise boundary ∂M = M_{0}∪M_{1} → B. Assume that 0 and 1 are regular values of the framed function h: M → [0, 1] with h^{−1}(0) = M_{0} and h^{−1}(1) = M_{1}. Let F → M be a unitarily flat vector bundle such that the fibre-wise cohomology H^{∗}(M/B; F) → B admits a parallel metric. Let ξ_{p,h,F}^{u}: B → Wh^{u}(M(ℂ), U) be the classifying map defined by these data. Then the unitary higher Franz-Reidemeister torsion of (M, M_{0}) → B and F is defined as

τ(M/B, M_{0}/B; F) = τ(M/B, M_{0}/B; F, h) = (ξ_{p,h,F}^{u} )^{∗}τ; .

If M → B is as above but with dim M ≤ 2 dim B, then for N sufficiently large, define

τ(M/B, M_{0}/B; F) = τ(M × ℝP^{2N}/B , M_{0} × ℝP^{2N}/B; F ) .

It is an easy consequence of Fact 10.5 that τ is well-defined. For suppose that h_{0}: M × ℝP^{2N_{0}} → [0, 1] and h_{1}: M ×ℝP^{2N_{1}} → [0, 1] are framed functions with fibre-wise compact sublevels and finitely many critical points on each fibre. Then both functions can be extended to framed functions on h̄_{0,1}: M × ℝP^{2N_{0}} × ℝP^{2N_{1}} → [0, 1] by adding a fixed Morse function ℝP^{2N_{i}} → (−ε, ε) on the additional factor ℝ^{N_{1}} or ℝ^{N_{0}} and correcting near the boundary components. The functions h̄_{0} and h̄_{1} can then be joined by a framed function on the bundle

(M × ℝP^{2N_{0}} × ℝP^{2N_{1}}) × [0, 1] ⟶ B × [0, 1] .

Thus, we clearly have

τ((M × ℝP^{N_{0}}, M_{0} × ℝP^{N_{0}})/B; F, h_{0})

= τ((M × ℝP^{2N_{0}} × ℝP^{2N_{1}}, M_{0} × ℝP^{2N_{0}} × ℝP^{2N_{1}})/B; F, h̄_{0})

= τ((M × ℝP^{2N_{0}} × ℝP^{2N_{1}}, M_{0} × ℝP^{2N_{0}} × ℝP^{2N_{1}})/B; F, h̄_{1})

= τ((M × ℝP^{2}N_{1}, M_{0} × ℝP^{N_{1}})/B; F, h_{1}) .

10.7. Remark. Assume that M → B is a fibre-bundle with closed fibres and that h: M → (0, 1) is a fibre-wise Morse function. Then by Theorem 6.20 and Theorem 8.1, we clearly have

τ_{ch}(M/B; F, h) = (ξ_{p,h,F}^{u} )^{∗}τ(M(ℂ), U) .

If h is framed, then

τ(M/B; F) = T (M/B; F, h) .

Nevertheless, Igusa’s Definition is much more general than Definition 2.64. One need not assume that h is fibre-wise Morse, so τ(M/B; F) is defined for all bundles with compact fibres by Fact 10.5, and one can work with fibers that have a boundary.

10.b. Igusa’s framing principle. We recall Igusa’s framing principle, and we use it to compare the higher analytic torsion of Bismut-Lott with Igusa’s higher Franz-Reidemeister torsion if there exists a fibre-wise Morse function and a fibre-wise acyclic flat vector bundle. At this point, the identity of Igusa’s and our higher torsion on Wh^{h}(M_{r}(ℂ), U(r)) proved in Section 8 is crucial, since we need both torsions within a single equation.

Recall that the additive characteristic class ^{0}J(E) was defined in [BG] by

(10.8)  ^{0}J(T^{u}X) = \frac{1}{2} ∑_{k=1}^{∞} ζ′(−2k) ch(E)^{[4k]} ∈ H^{4∗}(M; ℝ)

for an arbitrary vector bundle E over a manifold M. Let now h be a fibre-wise Morse function. The pushdown or “transfer” map p̂_{∗}: H^{∗}(C; ℝ) → H^{∗}(B; ℝ) produces a class

p̂_{∗}((−1)^{ind h} ^{0}J(T^{u}X)) ∈ H^{4∗}(B; ℝ) .

This class can in fact be extended to the situation where h is a generalised fibre-wise Morse function, cf. [I4].

The following theorem has been proved in [I2], Chapter 6 and [I4] in the case that H → B is a trivial bundle, but for generalised fibre-wise Morse functions. We will recall Igusa’s proof and show that it extends to the unitary case. For simplicity, we restrict our attention to true fibre-wise Morse functions and closed fibres.

10.9. Theorem. Assume that F → M and H → B carry parallel metrics. Let h be a fibre-wise Morse function. Then

τ(M/B; F, h) = τ(M/B; F) + 2p̂_{∗}((−1)^{ind h} ^{0}J(T_{Morse}^{u}X)) rk F .

We note that generalised fibre-wise Morse functions enter through Igusa’s framed function theorem. We remark that a general version of the framing principle could also be concluded from a generalisation of Theorem 0.3 to generalised Morse functions.

Proof. We copy the proof of [I2]. Replacing M by M × ℝ and h by h̄ = (p, z) ↦ h(p) + \frac{z^{2}}{2} , we may assume that T^{s}X → C contains a trivial subbundle corresponding to the coordinate z. We note that both h and h̄ give rise to the same homotopy class of maps ξ: B → Wh^{u}(M(ℂ), U), and thus by Definition 10.6 to the same higher torsion. We continue to write M and h for the new bundle and fibre-wise Morse function.

Igusa then constructs a generalised fibre-wise Morse function h_{1}: M → ℝ with the following properties.

(1) h_{1} = h outside a neighbourhood U of C.

(2) h_{1} is related by cancelation of Morse-type singularities to a framed function h_{2} that coincides with h_{2} outside M \ U.

(3) The critical set C_{1} of h_{1} takes the form C_{1} = C ∪ C′, and no negative gradient flow line of h_{1} originating in C_{1} and leaving U or originating in C converges to C′.

By (2), the map ξ_{1}: B → Wh^{u}(M(ℂ), U) constructed from h_{1} is homotopic to the map ξ_{2}: B → Wh^{u}(M(ℂ), U) from h_{2} because the corresponding functors are homotopic up to an elementary extension. This shows that

(10.10)  τ(M/B; F) = τ(M/B; F, h_{2}) = τ(M/B; F, h_{1})

because h_{2} is framed.

Note that h_{1} is no longer a Morse function. By (3), we can construct the complex (V_{1}, ∇^{V_{1}}, A′_{1}) outside p(C′_{b-d}) such that

(V_{1}, ∇^{V_{1}}) = (V, ∇^{V} ) ⊕ (W, ∇^{W} ) ,

where the two summands correspond to C and C′, and

A′_{1} = ( \begin{matrix}B′ & ∗ \\ 0 & A′ \end{matrix} ) .

We also note that V_{1} carries a natural parallel metric because obviously F |_{U} carries a parallel metric by the assumption that g^{V} is adapted and parallel. Again by [I2], we may assume that each birth-death singularity of h_{1} locally gives rise to an elementary expansion of W in the sense of Definition 6.28. This allows us to define a filtration of (V_{1}, ∇^{V_{1}}, A′_{1}) outside p(C′_{b-d}) by

(10.11)  (V_{1}, ∇^{V_{1}}, A′_{1}) ⊃ (W, ∇^{W} , B′) ⊃ 0 ,

and this filtration is compatible with stabilisation at the birth-death singularities of h_{1}.

The subcomplex (W, ∇^{W} , B′) is fibre-wise acyclic. By Igusa’s calculations in [I2], Chapters 6 and 7, we obtain

T (∇^{W} , B′, g^{W} ) = −2 p̂_{∗}((−1)^{ind h} ^{0}J(T^{u}X)) rk F .

Note that Igusa’s calculations apply here because by Theorem 8.1, Igusa’s higher Franz-Reidemei-ster torsion is equal to the torsion of Definition 10.6 in the acyclic case.

It follows from the fibre-wise acyclicity of (W, ∇^{W} , B′) that the spectral sequence associated to (10.11) degenerates at E_{1} = H. Theorem 7.37 and (10.10) now imply that

τ(M/B; F) = τ(M/B; F, h_{1})
= τ(∇^{V} , A′, g^{V} ) + T (∇^{W} , B′, g^{W} )
= τ(M/B; F, h) − 2 p̂_{∗}((−1)^{ind h} ^{0}J(T^{u}X)) rk F . □

10.c. A Proof of Theorem 0.3. We give a short proof of Theorem 0.3 and show that it is compatible with some of Igusa’s results.

Proof of Theorem 0.3. Let M → B be a family of compact manifolds, and let F → M be a flat vector bundle. We assume that there exists a fibre-wise Morse function h: M → ℝ and that the vector bundles F → M and H^{∗}(M/B; F) → B carry parallel metrics. We construct a family Thom–smale complex (V, ∇^{V} , A′) as usual. Theorem 0.1 then reduces to

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

= p̂_{∗}((−1)^{ind h} ^{0}J(T^{s}X − T^{u}X)) rk F .

By definition of T (M/B; F) and τ(M/B; F, h) = T (M/B; F, h), we have

(10.13) (T (T^{H}M, g^{T X}, ∇^{F} , g^{F} ) − τ(A′, g^{V} , g^{V} ) − c̃h^{o}(∇^{H}, g_{L_{2}}^{H}, g_{V}^{H}))^{[≥2]}

= T (M/B; F) − τ(M/B; F, h) .

The framing principle above allows us to compute

(10.14) τ(M/B; F, h)

= τ(M/B; F) + p̂_{∗}((−1)^{ind h} ^{0}J(T X)) rk F − p̂_{∗}((−1)^{ind h} ^{0}J(T^{s}X − T^{u}X)) rk F

= τ(M/B; F) + ∫_{M/B} e(T X) ^{0}J(T X) rk F − p̂_{∗}((−1)^{ind h} ^{0}J(T^{s}X − T^{u}X)) rk F ,

where the last identity holds because the singular current (−1)^{ind h} δ_{C} is homologous to e(T X) by the family Poincaré-Hopf theorem.

Combining (10.12)–(10.14), we obtain

T (M/B; F) = τ(M/B; F) + ∫_{M/B} e(T X) ^{0}J(T X) rk F ,

which was the assertion of Theorem 0.3. □

10.d. A glueing formula for the unitary torsion. We prove a glueing formula for the unitary torsion of a family of manifolds that are glued together from two submanifolds with a common hypersurface. The main ingredient is Theorem 7.37.

Suppose that M^{+} → B and M^{−} → B are two proper submersions with fibre-wise boundary ∂M^{±} = M_{0}^{±} ∪ M_{1}^{±}. Let F^{±} → M^{±} be two flat vector bundles with parallel metric g^{F^{±}}.

We assume that there exists a diffeomorphism φ: M_{1}^{−} → M_{0}^{+} that lifts to a parallel linear isometry Φ: F^{−}|_{M_{1}^{−}} → F^{+}|_{M_{0}^{+}}, so

F^{−}|_{M_{1}^{−}} ⟶^{Φ} F^{+}|_{M_{0}^{+}}
↓  ↓
M_{1}^{−} ⟶^{φ} M_{0}^{+} .

By glueing, we obtain

(10.15)  F := F^{−} ∪_{Φ} F^{+} ⟶ M := M^{−} ∪_{φ} M^{+} .

Moreover, we put ∂M = M_{0} ∪ M_{1} with

M_{0} = M_{0}^{−}  and  M_{1} = M_{1}^{+} .

The following result is a cohomological version of Igusa’s Splitting Lemma in [I2], 5.7.8, where a similar result was obtained for maps to the Whitehead space, but with more restrictive conditions on H(M^{±}/B, M_{0}^{±}; F).

10.16. Theorem. Assume that the fibre-wise cohomology bundles H(M^{±}/B, M_{0}^{±}; F) → B carry parallel metrics. Then the fibre-wise cohomology H(M/B, M_{0}/B; F) → B also carries a parallel metric, and

τ(M/B, M_{0}/B; F) = τ(M^{−}/B, M_{0}^{−}/B; F^{−}) + τ(M^{+}/B, M_{0}^{+}/B; F^{+}) ∈ H^{even,≥2}(B; ℝ) .

Proof. We may assume that there exist framed function h^{±}: M^{±} → [0, 1] such that i = 0, 1 are regular values with (h^{±})^{−1}(i) = M_{i}^{±}. Then the function h: M → [0, 1] with

h(p) = \{ \begin{matrix}\frac{h^{−}(p)}{2} & for p ∈ M^{−}, and \\ \frac{1+h^{+}(p)}{2} & for p ∈ M^{+}. \end{matrix}

is framed and satisfies h^{−1}(i) = M_{i} for the regular values i = 0, 1.

We construct a family Thom-Smale complex (V, ∇^{V} , A′) → B as before and let ξ = ξ_{p,h,F}: M → Wh^{u}((ℂ), U) denote the corresponding map to the universal unitary Whitehead space. Then V admits a natural filtration

(V, ∇^{V} , A′) ⊃ (V^{+}, ∇^{V^{+}}, A^{+′}) ⊃ 0 .

The two steps of the associated spectral sequence take the form

E_{0} = (V^{−}, ∇^{V^{−}}, A^{−′}) ⊕ (V^{+}, ∇^{V^{+}}, A^{+′})
and  E_{1} = (H^{∗}(V^{−}, a_{0}^{−′}) ⊕ H^{∗}(V^{+}, a_{0}^{+′}) , ∇^{E_{1}} + d′_{1}) .

By our assumptions, the bundles H^{∗}(V^{±}, a_{0}^{±′}) ≅ H^{∗}(M^{±}/B, M_{0}^{±}; F) → B carry parallel metrics g^{H^{±}}. Let g^{H^{+}⊕H^{−}} denote the direct sum of these metrics on E_{1} = H^{∗}(V^{−}, a_{0}^{−′})⊕H^{∗}(V^{+}, a_{0}^{+′}). Moreover, the coboundary

d′_{1}: H^{∗}(V^{−}, a_{0}^{−′}) ⟶ H^{∗}(V^{+}, a_{0}^{+′})

is parallel with respect to ∇^{E_{1}} by Proposition 7.33. This implies that the total fibre-wise cohomology

H := H^{∗}(V, a′_{0}) ≅ H^{∗}(M/B, M_{0}/B; F) ⟶ B

carries a parallel metric induced by g^{H^{+}⊕H^{−}} via finite-dimensional Hodge theory, which we denote by g^{H}. The appendix of [BL] implies that

(10.17)  T^{BL}(D′_{1}, g^{H^{+}⊕H^{−}}) = T^{BL}(∇^{E_{1}} + d′_{1}, g^{H^{+}⊕H^{−}}) = 0 .

We can now apply Theorem 7.37 to find that

T (∇^{V} , A′, g^{V} ) + c̃h^{o}(∇^{H}, g^{H}, g_{V}^{H})
= T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) + T^{BL}(D′_{1}, g^{E_{1}}) + c̃h^{o}(∇^{H}, g^{H}, g^{E_{∞}})
= T (∇^{E_{0}}, D′_{0}, g^{E_{0}}) + c̃h^{o}(∇^{E_{1}}, g^{H^{+}⊕H^{−}}, g^{E_{1}}) + T^{BL}(D′_{1}, g^{H^{+}⊕H^{−}})
= T (∇^{V^{+}}, A^{+′}, g^{V^{+}}) + c̃h^{o}(∇^{H^{+}}, g^{H^{+}}, g_{V^{+}}^{H^{+}})
+ T (∇^{V^{−}}, A^{−′}, g^{V^{−}}) + c̃h^{o}(∇^{H^{−}}, g^{H^{−}}, g_{V^{−}}^{H^{−}})

modulo exact forms on B. The theorem follows. □

For proper submersions with closed fibres, Theorem 10.16 implies a cut-and-paste formula. Here, we assume that M_{0} → B and M_{1} → B are two proper submersion such that

M_{i} = M_{i}^{+} ∪_{M_{i}^{0}} M_{i}^{−}

for i = 0, 1, where M_{i}^{±}, M_{1}^{±} are families of closed submanifolds intersecting in the common boundary M_{i}^{0} = M_{i}^{+} ∩ M_{i}^{−}. Assume that F_{i}: M_{i} is a flat vector bundle with a parallel metric g^{F_{i}} for i = 0, 1 and that there exists a diffeomorphism φ: M_{0}^{0} → M_{1}^{0} which lifts to a linear isometry Φ: F_{0}|_{M_{0}^{0}} → F_{1}|_{M_{1}^{0}},

F_{0}|_{M_{0}^{0}} ⟶^{Φ} F_{1}|_{M_{1}^{0}}
↓  ↓
M_{0}^{0} ⟶^{φ} M_{1}^{0} .

Then we can construct two new families of closed submanifolds

M′_{0} = M_{0}^{−} ∪_{φ} M_{1}^{+}  and  M′_{1} = M_{1}^{−} ∪_{φ} M_{0}^{+}

that carry flat vector bundles F′_{i} → M′_{i} for i = 0, 1.

10.18. Corollary. Assume that the bundles H^{∗}(M_{i}^{−}/B; F_{i}) → B and H^{∗}(M_{i}^{+}/B, M_{i}^{0}/B; F) → B carry parallel metrics for i = 0, 1. Then H^{∗}(M_{i}/B; F_{i}) → B and H^{∗}(M′_{i}/B; F_{i}) → B also carry parallel metrics, and we have

τ(M_{0}/B; F) + τ(M_{1}/B; F) = τ(M′_{0}/B; F) + τ(M′_{1}/B; F) ∈ H^{even,≥2}(B; ℝ) .

Proof. This is an immediate consequence of Theorem 10.16. □

10.e. A proof of Theorem 0.2. We want to use Corollary 10.18 to construct some more examples of “exotic” smooth structures on fibre bundles from Hatcher’s example as described in Section 5.a. Assume that we are in the situation of Section 5.c. That is, we fix k > 0 and l > 0 sufficiently large with respect to k, and we let X be an arbitrary compact connected (2l − 1)-dimensional manifold. We decompose X along a hypersurface X^{0} into

X = X^{+} ∪_{X^{0}} X^{−} ,

such that X^{−} is diffeomorphic to the (2l − 1)-dimensional disc D^{2l−1}.

We also let M_{γ} → S^{4k} denote Hatcher’s example constructed starting with a generator γ ∈ π_{4k−1}(O) of ker J. In particular, M_{γ} admits a fibre-wise Morse function with six critical points (in fact, four would already be enough), such that the unique maximum has a trivial disc bundle neighbourhood, so we can decompose

M_{γ} = M_{γ}^{+} ∪_{M_{γ}^{0}} M_{γ}^{−} ,

where now M_{γ}^{+} → S^{4k} is diffeomorphic to D^{2l−1} × S^{4k}.

Assume that X carries a flat vector bundle F → X of rank r with parallel metric g^{F}, then F |_{X^{−}} is trivial. We fix a diffeomorphism φ: X^{0} × S^{4k} → M_{γ}^{0} and a lift Φ: F |_{X^{0}} × S^{4k} → M_{γ}^{0} × ℂ^{r} as a parallel linear isometry. We construct

F′_{γ} := (F |_{X^{+}} × S^{4k}) ∪_{Φ} (M_{γ}^{−} × ℂ^{r}) ⟶ M′_{γ} = (X^{+} × S^{4k}) ∪_{φ} M_{γ}^{−}

Glueing the other two parts gives the trivial bundle

ℂ^{r} × S^{2l−1} × S^{4k} ⟶ S^{2l−1} × S^{4k} = X^{−} ∪_{φ} M_{γ}^{+} .

By construction, the fibre-wise cohomology bundles of X^{−} × S^{4k} → S^{4k}, (X^{+}, X^{0}) × S^{4k} → S^{4k}, M_{γ}^{+} → S^{4k} and M_{γ}^{−} → S^{4k} with coefficients in F respectively in ℂ^{r} are all trivial. We may apply Theorem 0.3 and Corollary 10.18 and get

\begin{aligned}T (M′_{γ}/S^{4k}; F′_{γ}) & = τ(M′_{γ}/S^{4k}; F′_{γ}) = τ(M_{γ}/S^{4k}, ℂ^{r}) \\ & = 2^{2−k} b_{k} ζ′(−2k) u ∈ H^{4k}(S^{4k}; ℝ) . \end{aligned} (10.19)

The above construction of M′_{γ} generalises the example of Section 5.c to the case of non-trivial F. As in Section 5.d, we may glue j copies of the bundle M′_{γ} → S^{4k} to an arbitrary fibre bundle M → B with fibre X and 4k-dimensional compact oriented base B, to obtain a new bundle

p_{j}: M_{j} ⟶ B

which is homeomorphic to the original M → B as a bundle. For negative j, we glue |j| copies of M′_{−γ} → S^{4k} to M → B. We assume that F → M is a unitarily flat vector bundle, then after glueing, we have homeomorphic unitarily flat vector bundles F_{j} → M_{j} for all j ∈ ℤ.

Let g^{H} be a fixed metric on the bundle

H^{∗}(M_{j}/B; F_{j}) ≅ H = H^{∗}(M/B; F) ⟶ B .

Then as in (5.22), we compute

∫_{M_{j}/B} T (M_{j}/B; F_{j}, g^{H}) = ∫_{M/B} T (M/B; F, g^{H}) + 2^{2−k} j b_{k} ζ′(−2k) u ,

where T (M_{j}/B; F_{j}, g^{H}) ∈ Ω^{even}(B)/dΩ^{odd}(B) is defined as in (5.20) by

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

This implies the following generalisations of Theorem 5.30, where Diff(M; F) denotes the group of parallel lifts of arbitrary diffeomorphisms of M to a given flat vector bundle F, and Diff^{+}(M; F) denotes the subgroup coming from orientation preserving diffeomorphisms whenever M is orientable.

10.20. Theorem. Let p: M → B be a proper submersion and let (F, ∇^{F}) → M be a vector bundle with a parallel metric g^{F}. 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 in Section 5.e. Assume that for i, j ∈ ℤ with i ≠ j, there exists diffeomorphisms Φ ∈ Diff(M; F) 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) rk F .

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.

Clearly, Theorem 10.20 implies the generalised second statement of Theorem 0.2 by the same arguments as in Section 5.e.

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NWF 1 Mathematik, Universität Regensburg, 93040 Regensburg, Germany

   E-mail address: sebastian.goette@mathematik.uni-regensburg.de