Wednesday, February 9th, 2011
2:45pm - 3:45pm, in McCormack 2-116

Katrin Wehrheim

MIT

How to construct topological invariants via decompositions and the symplectic category

Abstract: The goal of the talk will be to explain most words in the following abstract: A Lagrangian correspondence is a Lagrangian submanifold in the product of two symplectic manifolds. This generalizes the notion of a symplectomorphism and was 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 (generalized) Lagrangian correspondences are composable morphisms. We extend it to a 2-category by extending Floer homology to generalized Lagrangian correspondences. This is based on counts of 'holomorphic quilts' - a collection of holomorphic curves in different manifolds with 'seam values' in the Lagrangian correspondences. A fundamental isomorphism of Floer homologies ensures that our constructions are compatible with the geometric composition of Lagrangian correspondences. This provides 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 - decompose morphisms into simple morphisms (e.g. by cutting between critical levels of a Morse function) - associate to the objects and simple morphisms smooth symplectic manifolds and Lagrangian correspondences between them (e.g. using moduli spaces of bundles or representations) - check that the moves between different decompositions are associated to (good) geometric composition of Lagrangian correspondences.