M392C: Bordism: Old and New
Lecture notes from the last two lectures are now posted.
Professor: Dan Freed, RLM 9.162
Class Meetings: TTh 2:00-3:30, RLM
Discusion/Office Hours: W 2:00-4:00, RLM
For more details, see the First Day Handout
Lecture 1: Introduction to bordism
Lecture 2: Orientations, framings, and the
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
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
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
and paper from a talk "The Cobordism Hypothesis"
I gave at Current Events Bulletin, Joint Mathematics Meeting, January,
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
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
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
Galatius-Madsen-Tillmann-Weiss on the homotopy
type of cobordism categories.
Madsen-Weiss on the Mumford's
Binz-Fischer on a model of BDiff from
Moore-Segal on 2d theories. See Appendix A.1
for the standard case discussed in lecture.
Lurie on the cobordism hypothesis.