M392C: Bordism: Old and New


Lecture notes from the last two lectures are now posted.

Basic Information

Professor: Dan Freed, RLM 9.162

Class Meetings: TTh 2:00-3:30, RLM 12.166

Discusion/Office Hours: W 2:00-4:00, RLM 9.162

For more details, see the First Day Handout

Lecture Notes

Lecture 1: Introduction to bordism

Lecture 2: Orientations, framings, and the Pontrjagin-Thom construction

Lecture 3: The Pontrjagin-Thom theorem

Lecture 4: Stabilization

Lecture 5: More on stabilization

Lecture 6: Classifying spaces

Lecture 7: Characteristic classes

Lecture 8: More characteristic classes and the Thom isomorphism

Lecture 9: Tangential structures

Lecture 10: Thom spectra and X-bordism

Lecture 11: Hirzebruch's signature theorem

Lecture 12: More on the signature theorem

Lecture 13 (revised): Categories

Lecture 14: Bordism categories

Lecture 15: Duality

Lecture 16: 1-dimensional TQFTs

Lecture 17: Invertible topological quantum field theories

Lecture 18: Groupoids and spaces

Lecture 19: Gamma-spaces and deloopings

Lecture 20 (revised): Topological bordism categories

Lecture 21: Sheaves on Man

Lecture 22: Remarks on the proof of GMTW

Lecture 23: An application of Morse-Cerf theory

Lecture 24: The cobordism hypothesis

Other Notes

Background notes on fiber bundles, vector bundles, and the tangent bundle


Homework #1 due October 18

Homework #2 due November 6

Homework #3 due November 20

Homework #4 due December 6


Rene Thom's classic paper Quelques propriétés globales des variétés différentiables

John Milnor's survey article A survey of cobordism theory

Slides and paper from a talk "The Cobordism Hypothesis" I gave at Current Events Bulletin, Joint Mathematics Meeting, January, 2012.

Ralph Cohen's survey article Stability phenomena in the topology of moduli spaces. See especially the first few sections.

Mike Hopkins' ICM plenary talk Algebraic topology and modular forms. The first few sections discuss the homotopy groups of spheres.

Chapter 8 of Davis and Kirk's book Lecture Notes in Algebraic Topology.

Hatcher on Vector Bundles, from his Vector Bundles and K-Theory book in progress.

Bott and Tu on characteristic classes, Chapter IV from Differential Forms in Algebraic Topology.

Klaus and Kreck on the rational Hurewicz theorem.

Milnor-Stasheff on the linear independence of Pontrjagin numbers in the rational oriented bordism group.

Segal on simplicial sets and classifying spaces of categories.

Friedman's expository introduction to simplicial sets.

Segal on Gamma-spaces and spectra.

Bott lectures (notes by Mostow and Perchik) on characteristic classes, etc. The first several sections contain a beautiful exposition of simplicial sets and computation of cohomology.

Galatius-Madsen-Tillmann-Weiss on the homotopy type of cobordism categories.

Madsen-Weiss on the Mumford's conjecture.

Binz-Fischer on a model of BDiff from embeddings.

Moore-Segal on 2d theories. See Appendix A.1 for the standard case discussed in lecture.

Lurie on the cobordism hypothesis.