Spezialvorlesung Mannigfaltigkeiten im Sommersemester 2021

Lecturer: Prof. Dr. S. Goette

Teaching assistant: J.Schnitzer

Office hours: Wed 13:00-13:50 (S.G.), on demand (J.S.)

What, when, where

Lecture: Tue 10-12 BBB-room vGoette

Exercise class: Tue 14-16 BBB-room vGoette

Aditionally: Further information can be found at the ILIAS course Mannigfaltigkeiten.

There is a short script for the lecture, which will be updated continuously (version: 05.05.2021)


In the first three sessions, I want to give an overview. We can then discuss which subjects to delve into.
  1. Straightening and Smoothing. Existence and classification of PL and smooth structures on manifolds of dimension 5 and higher is controlled in a very similar way as orientations, spin structures or almost complex structures on manifolds. However, the proofs are far more complicated.
  2. Dimension 4. Combining analysis (Donaldson) and topology (Freedman-Quinn), one can construct 4-manifolds cannot be straightened or smoothed. Continuing along these lines, Taubes has shown that there is an uncountable family of smooth structures on ℝ4; .
  3. Exotic spheres. We study exotic spheres in dimension 7. These are not only instructive examples. Obstructions against smoothing and invariants of smooth structures are closely related to the groups of exotic spheres.