# M392C: Bordism: Old and New

## Announcements

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

Homework #1 due October 18

Homework #2 due November 6

Homework #3 due November 20

Homework #4 due December 6

## Readings

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.