March 23, 2007

Katrin Wehrheim (MIT)

Title: A symplectic category featuring Lagrangian correspondences and holomorphic quilts

Abstract: In joint work with Chris Woodward we define a symplectic category Symp whose morphisms are generalized Lagrangian correspondences. In monotone or exact settings, we extend Symp to a 2-category whose 2-morphism spaces are Floer homology groups. This induces a functor Symp --> Cat to Donaldson-Fukaya type categories and functors between them. These algebraic structures arise naturally from holomorphic quilts and all proofs can be given by pictures and a fundamental "strip shrinking" isomorphism. This provides a general machinery for constructing new topological invariants or TQFT's Top --> Cat from a "symplectization" Top --> Symp of a topological category Top. To construct the latter it suffices to associate smooth Lagrangian correspondences to "simple morphisms" (e.g. 3-cobordisms or tangles with one critical point) and to check that the Cerf moves (which connect equivalent decompositions into simple morphisms) correspond to embedded composition of Lagrangian correspondences.