Excision (part 1)
In algebraic topology, the excision theorem is an extremely useful tool for computing homology. In case your algebraic topology class is ancient history, let's recall the statement: if $U \subset V \subset X$ are a sequence of spaces with $\overline{U} \subset V^\circ$, then $$H_*(X - U, A - U) \cong H_*(X, A).$$In other words, relative homology is unaffected by "cutting" out $U$. In Floer homology, we also have excision, which involves a different kind of cutting. In this post I want to discuss some aspects of Floer homology in excision. In the next post, I'll give proof of a similar statement in annular Khovanov homology. Simplest version of excision: First I want to state and prove an easy version of Floer excision. I'm going to write this in a way that applies in most Floer theories, so I'll use $F$ to denote an arbitrary Floer homology. Let $Y$ be a closed connected oriented three-manifold. Suppose that $x$ is...