##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Justin Roberts

#### UCSD

## "The blob complex" by Kevin Walker and Scott Morrison

##### Abstract:

The blob complex is a chain complex which can be associated to any pair ($n$-dimensional manifold, $n$-category). It has very nice properties under cutting and pasting manifolds along lower-dimensional pieces (with ``corners" of arbitrary dimension) and so amounts to an $n$-dimensional topological quantum field theory which makes sense ``all the way down to the point". It is a generalisation and unification of several ideas. Well-known TQFTs such as Turaev-Viro theory and Chern-Simons theory can be recovered from the cases $n=2$ and $n=3$, with appropriate kinds of $2$-category (spherical tensor category) and $3$-category (modular category) respectively. What is novel is that the construction is a homotopy-invariant or``derived" construction, which allows for much more general input categories, and hence new kinds of TQFT. In the simplest case - when $n=1, M$ is the circle, and $C$ is any associative algebra (viewed as a $1$-category with just one object) - it is equivalent to the Hochschild chain complex of $C$, which can be thought of as the ``derived cocentre" of $C$ (this example is very closely related to Costello's work on topological conformal field theories.) It seems very likely that the blob complex formulation is general enough to allow $4$-dimensional gauge theories, with their exact triangles and other homological baggage, to fall into place as TQFTs which can be encoded using algebra and combinatorics. A large part of the idea consists of giving a suitable definition of n-category. The approach here is very natural in this geometric context, and seems to be relatively easy to understand in comparison with the other current approach (via multisimplicial sets) by Lurie, whose ``topological chiral homology" is presumably just a different way of saying the same thing.

### April 5, 2011

### 10:30 AM

### AP&M 7218

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