##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Justin Roberts

#### UCSD

## Introductory Meeting

##### Abstract:

This term's plan is to read Jacob Lurie's new preprint:``On the classification of Topological Field Theories'', which is available on his MIT homepage. As usual, seminar participants will give the talks, and we'll try to parcel them out at the first meeting on April 7th. But everyone is welcome - we won't force you to speak if you don't want to! \\ \noindent In 1989 Atiyah (inspired by Segal and Witten) defined a TFT to be a monoidal functor from the category of (n+1)-dimensional cobordisms to the category of vector spaces. That is, it assigns a vector space to each closed n-manifold, and linear maps between these to each (n+1)-dimensional cobordism (that is, an (n+1)-dimensional manifold whose boundary is divided into "ingoing" and "outgoing" parts), satisfying natural composition laws. The idea comes from quantum field theory, in which each slab of spacetime between "past" and "future" spacelike hypersurfaces should define a unitary map between their corresponding Hilbert spaces of states. The difference is that in QFT, the metrics on such spacetime cobordisms matter, whereas in TFT the linear maps depend only on the underlying topology of the cobordisms. The general formalism of QFT suggests that one should be able to extend this algebraic structure into lower dimensions, assigning a category to each (n-1)-dimensional manifold, a 2-category to each (n-2)-dimensional manifold, and so on, ultimately assigning some kind of n-category to the point: this n-category ought to determine the whole TFT structure. Many attempts to formulate this sort of thing were made in the early 90s, but because of the lack of a solid definition of ``n-category'', made little progress. One can also extend into higher dimensions: k-parameter families of manifolds can be added into the picture, leading to theories in which the topology of diffeomorphism groups of manifolds enters naturally. A theory of this sort in 2 dimensions was worked out by Kevin Costello a few years ago under the name ``Topological Conformal Field Theory''. Lurie's new paper provides a complete formulation of TFTs incorporating all of the above features. He provides a solid definition of n-categories in the spirit of algebraic topology, and proves many foundational results about them. Then he shows how TFTs can be characterised using this language. In particular, he proves the remarkable ``Baez-Dolan cobordism hypothesis'', which states that the n-category of n-dimensional cobordisms is the free n-categ

### April 7, 2009

### 10:30 AM

### AP&M 7218

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