Contents:
Broadly speaking, simple homotopy theory studies the combinatorial aspects of algebraic topology. More specifically, a map between finite CW complexes is called a simple homotopy equivalence if it can be written as a finite composition of elementary expansions and collapses. A natural question is whether every homotopy equivalence is of this form. In general, the answer is no, and the failure of a homotopy equivalence to be simple is measured by the Whitehead torsion. The torsion takes values in the Whitehead group, a K-theoretic object that depends only on the fundamental group.
Simple homotopy theory has been successfully applied in manifold topology. For example, work of Turaev and others shows that two orientable closed 3-manifolds are homeomorphic if and only if they are simple homotopy equivalent. In high dimensions, simple homotopy theory provides a criterion for determining when an h-cobordism is trivial, known as the s-cobordism theorem.
In this seminar, we will give an introduction to simple homotopy theory and discuss some of its applications to manifold topology. Time permitting, and depending on the background and interests of the participants, we may also discuss Waldhausen’s more modern approach to the subject.
References:
Cohen: A course in simple homotopy theory.
Luck-Macko: Surgery theory: Foundations.
Lurie: Algebraic K-theory and Manifold Topology.
More to be found in the program below.
Program:
Available here: program.
Practical information:
To pass the seminar, you will be required to give a talk presenting part of the material listed in the program. In addition, you will need to submit a written report (in PDF format) providing detailed proofs of the arguments presented in your talk. Your grade will be based primarily on the written report, but also on the quality of your presentation. The final report must be submitted by the end of July.
I strongly recommend preparing a draft of your report before giving your talk and sending it to me at that stage. This will allow me to provide more effective feedback and help you prepare your presentation.
It is also advisable to prepare notes for your talk that are not identical to the draft of your report. Since the talk will last about 90 minutes, it will not be possible to go through all proofs in full detail. Your talk notes should therefore reflect what can reasonably be covered within this time. You will be allowed to consult your notes while you give your talk.
There will also be a Repetitorium, which will be scheduled individually with each participant. The purpose of this meeting is to help clarify the material and provide suggestions for improving your talk. Ideally, we should meet at least three days before your presentation.