Seminar on the s-cobordism theorem and surgery theory

Seminar delayed to the winter term due to the corona crisis

Content of the Seminar

In the first talks we will study handlebody decompositions of bordisms and we will define Whitehead groups. This will allow us to formulate and prove the s-cobordism theorem, which is a generalization of the h-cobordism theorem to non-simply-connected manifolds.

This will lead us to Whitehead torsion and Reidemeister torsion which will be studied both from the geometric and algebraic point of view.

The remaining part of the seminar is dominated by the question whether a given CW-comples is homotopy equivalent to a topological manifold, and if it is, whether it is even homotopy equivalent to a smooth manifold.

These questions lead to a long exact sequence, the surgery exact sequence. This sequence exists in several versions, depending on whether we treat smooth, piecewise linear or topological manifolds.

Literature

• Diarmuid Crowley, Wolfgang Lück, Tibor Macko, a book project with the title: Surgery Theory: Foundations
• Wikipedia page on the Surgery exact sequence
• John Milnor: Lectures on the h-cobordism theorem

Requirements

Time and Place

Program

Registration

Related web pages

• Announcement in the Commented List of Courses (Kommentiertes Vorlesungsverzeichnis) of the mathematics department in Regensburg