Abstract

In the first part of the talk, I’ll discuss Floer homology for 3-manifolds. The Seiberg-Witten invariants of a smooth 4-manifold M are defined by counting solutions to a certain PDE on M. The theory of Floer homology for 3-manifolds was developed to understand these invariants using cut and paste topology: we split a given 4-manifold M in half along a 3-manifold Y and express the invariants of M in terms of relative invariants of the two halves. The relative invariants live in a vector space (the Floer homology) associated to Y. There’s a related invariant (knot Floer homology) which assigns a vector space to a knot inside a 3-manifold.

A similar thing happens when we split Y along a surface S, but now the relative invariant is an object in a category. The second half of the talk will focus on what happens when one of the two pieces is a solid torus containing a knot. This is an extension of previous joint work with Hanselman and Watson.