UC Riverside Seminar on Cobordism and Topological Field Theories

In the last quarter of 2009, there is a seminar at UC Riverside on Cobordism and Topological Field Theories run by Julie Bergner. The seminar webpage is here. The goal is to work through recent notes of Lurie giving an outline of his proof of the Cobordism Hypothesis, relating cobordism classes of manifolds and topological field theories. This work brings together several areas of recent mathematical interest: topological field theories, cobordisms of manifolds, and homotopical approaches to higher categories. The basic definitions and examples of all of the above will be covered and then we’ll work towards understanding Lurie’s proof.

The main reference is

J. Lurie, *On the Classification of Topological Field Theories*

John Baez kicked off the seminar with an introduction to the cobordism hypothesis, how it began its life, and what it means in dimension $n=1$. Already here an issue of “framing” comes into play concerning the first Reidemeister move.

The definition of cobordism, and the category nCob of closed oriented $(n-1)$-manifolds and diffeomorphism classes of oriented cobordisms between them.

The basics of symmetric monoidal categories and functors and the definition of topological field theory.

Introduces 2-dimensional topological field theories and commutative Frobenius algebras.

Today’s story should involve lots of people, but the stars are Sir Michael Atiyah and Edward Witten. It begins with a paper Witten wrote in 1982, called “Supersymmetry and Morse Theory”. In this paper, Witten shows how to use ‘supersymmetric quantum mechanics’ to compute the de Rham cohomology of a compact manifold, $M$, via Morse theory. This was perhaps the first instance of using quantum theory to find topological invariants.

A sketch of a $2$-extended TFT illustrated by chopping up manifolds into pieces and considering the problems encountered when one considers TFT’s of dimension higher than $2$.

- Notes by Alex Hoffnung.
- Blog entry
- (These are notes for both October 30 and November 6)

Introduces $2$-categories of cobordisms and suggests the need for $(\mathrm{\infty},2)$-categories and gives a definition for extended $C$-valued TFT’s.

All pictures were drawn by Christopher Walker?. Otherwise these notes are mostly the unpolished version from seminar.

Revised on May 29, 2012 22:04:00
by Andrew Stacey
(129.241.15.200)