# Katrin Wehrheim : The symplectic category: correspondences, quilts, and topological applications

A 'correspondence' between two manifolds is a submanifold in the product. This generalizes the notion of a map (whose graph is a correspondence) ... and is of little use in general since the composition of correspondences, though naturally defined, can be highly singular.

Lagrangian correspondences between symplectic manifolds however are highly useful (and will be defined carefully). They were introduced by Weinstein in an attempt to build a symplectic category that has morphisms between any pair of symplectic manifolds (not just symplectomorphic pairs).

In joint work with Chris Woodward we define such a cateory, in which all Lagrangian correspondences are composable morphisms. We extend it to a 2-category by constructing a Floer homology for generalized Lagrangian correspondences. One of the applications is a general prescription for constructing topological invariants. We consider e.g. 3-manifolds or links as morphisms (cobordisms or tangles) in a topological category. In order to obtain a topological invariant from our generalized Floer homology, it suffices to

(i) decompose morphisms into simple morphisms (e.g. by cutting between critical levels of a Morse function)

(ii) associate to the objects and simple morphisms smooth symplectic manifolds and Lagrangian correspondences between them (e.g. using moduli spaces of bundles or representations)

(iii) check that the moves between different decompositions are associated to 'good' geometric composition of Lagrangian correspondences

**Category**: Geometry and Topology**Duration**: 01:34:46**Date**: March 3, 2009 at 4:25 PM**Views**: 114-
**Tags:**seminar, Geometry/topology Seminar

## 0 Comments