Summer 2013, Central Lecture of the Graduiertenkolleg 1150 (Homotopy and Cohomology). Thur 11-13:00 (SemR 0.011) (University link). First lecture: April 11th.

Lecture Notes and Exercises

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

• Course Syllabus
• Lecture 1: Overview (April 11, 2013)
  • Exercises 1 (pdf)
  • Does anyone have a link to a good video of the “belt-trick” or something equivalent?
• Lecture 2: Pontryagin’s Construction (April 18,2013)
  • Exercises 2 (pdf) (29.4.2013 – begining of exercise 2.1 corrected)
  • Addendum (pdf) on the relationship between stable tangential framings and stable normal framings. (there are still some errors in this. 29.4.2013)
• Lecture 3: Introduction to Jet Transversality. (April 25, 2013) (update: 29.4.2013)
  • Exercises 3 (pdf) (exercise 3.2 corrected)
• Lecture 4: The Pontryagin-Thom Construction (May 2, 2013)
  • Exercises 4 (pdf) (There are some really fun ones this time!)
• (No Lecture 9.5.2013)
• Lecture 5: A General Proof of Density Theorems (May 16, 2013) (like Thom’s Jet Transversality Theorem) (update 16.5.2013: Incorrect Lemma 5.1.8 stricken, proof of Globalization Thm 5.2.3 altered slightly to avoid using the faulty lemma).
  • Exercises 5 (pdf)
• (No Lecture 23.5.2013 or 30.5.2013)
• Lecture 6: End of Proof of Jet Transversality (June 6, 2013) (see previous week for lecture notes) No exercises this week.
• Lecture 7: The h-principle, part I. (June 13,2013)
  • Exercises 7 (pdf)
• Lecture 8: The h-principle, part II: (June 20, 2013)
• Lecture 9: The h-principle, part III (June 27, 2013)
  • Notes on fibrations
  • Notes on flexible sheaves
• Lecture 10: The h-principle , part IV (July 4, 2013)
  • A surprising lemma of Émile Borel
• Lecture 11: The h-principle, part V (July 11, 2013)
  • Smooth functions satisfy the h-principle
• Lecture 12: The h-principle, part VI (July 18, 2013)
  • Gromov’s Theorem for open partial differential relations

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:

• Dan Freed, Bordism Old and New (UT Austin, Fall ’12). Yes, I stole his title.
• John Francis, The h-Principle in Topology (Northwestern, winter ’10/’11)
• Michael Weiss, Immersion Theory for Homotopy Theorists (U. of Aberdeen, winter ’04/’05)

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.