##### Department of Mathematics,

University of California San Diego

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### Topology Learning Seminar

## Justin Roberts

#### UCSD

## Ozsvath--Szabo homology

##### Abstract:

This term's topology learning seminar will be on Ozsvath-Szabo homology. About 7 years ago, Ozsvath and Szabo invented this construction (which they call "Heegaard Floer homology") in an attempt to give a different definition of Seiberg-Witten theory. Their theory has been incredibly successful in applications to low-dimensional topology. In brief, they show how to associate a family of homology groups to a 3--manifold by choosing a Heegaard splitting and computing a suitable Lagrangian intersection Floer homology. Some of the most important features of the construction are: 1. 4--dimensional cobordisms induce maps between homology groups; the invariants of closed 4-manifolds are conjecturally equal to the Seiberg-Witten invariants. 2. There is a version of the homology for knots in $S^3$, which leads to an exact formula (not just a bound!) for the genus of knots. Consequently their homology distinguishes the unknot, and can be used to prove many old conjectures about surgery on knots. 3. The theory gives rise to powerful invariants of contact structures on 3--manifolds and can distinguish tight from overtwisted. 4. The homology for knots, unlike all earlier gauge-theoretic invariants, can actually be calculated by purely combinatorial means. There is a strong hope that this will eventually lead to a complete combinatorial calculation of the Ozsvath-Szabo/Seiberg-Witten/Donaldson invariants of 4-manifolds. The first meeting will be Tuesday 10th April, in room 7218, at 10.30am. I will give an introductory talk and then we will arrange the schedule of speakers for the rest of the term. Anyone is welcome to attend - attendance does not necessarily lead to being volunteered for a talk!

### April 10, 2007

### 10:30 AM

### AP&M 7218

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