Bordism Old and New

Summer 2013, Central Lecture of the Graduiertenkolleg 1150 (Homotopy and Cohomology). Thur 11-13:00 (SemR 0.011) (University link). First lecture: April 11th. This website will continue to be updated as the course approaches.

Lecture Notes and Exercises

Full lecture notes (in one pdf, last update July 17, 2013)

Abstract

A basic problem in topology is to gain an understanding of the structure of both manifolds and maps between them. A powerful tool for this is the theory of bordism, as initiated in the mid-20th century by René Thom and Lev Pontryagin. This theory provides a powerful feedback loop between manifolds on the one hand and maps between them, on the other.

Topological field theories are a modern extension of bordism invariants. They  provide a further bridge linking topology and algebra together. The classification of topological field theories, which combines methods and ideas from differential topology, homotopy theory, and  higher category theory, has lead to a greater understanding of this back-and-forth interplay.  

In this lecture course we will take stroll along these bridges. We will focus on some of the techniques and ideas which play a part in the classification of TFTs, beginning with the classical theory of bordism and progressing into more modern developments.

Prerequisites: Algebraic topology (homology, cohomology, and homotopy theory). Smooth manifolds, tangent bundles. Classifying spaces. Exposure to characteristic classes, spectra, or spectral sequences would be helpful.

Homework: The best way to learn is by doing, and so I will provide (optional) homework problems every week.

Text: There is no single text. I will give some references and I am hoping to provide lecture notes. I am drawing from lectures given by:

A Preliminary List of Topics/Lectures…  subject to change

  1. Overview
  2. Morse Theory and Handles, the second stable stem.
  3.  Thom Transversality, “Transversality unlocks the secrets of manifolds”
  4. The Pontryagin-Thom construction
  5. Weak fibrations and h-principles (Segal categories?)
  6. The Hirsch-Smale theorem (a.k.a. Immersion theory)
  7. Configuration Spaces, scanning,  and more.
  8. The third stable stem, via geometry.
  9. The Cobordism Hypothesis, part 1: ideas and statements
  10. The Galatius-Madsen-Tillmann-Weiss theorem
  11. The Cobordism Hypothesis, part 2: overview of the proof
  12. leftovers and applications.
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One response to “Bordism Old and New

  1. Pingback: Summer Time | Chris Schommer-Pries

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