Composition and Cobordism Maps

Abstract

We study the relationship between the algebra of module homomorphisms under composition and 4-dimensional cobordisms in the context of bordered Heegaard Floer homology. In particular, we prove that composition of module homomorphisms of type-$D$ structures induces the pair of pants cobordism map on Heegaard Floer homology in the morphism spaces formulation of the latter, due to Lipshitz--Ozsv\'{a}th--Thurston. Along the way, we prove a gluing result for cornered 4-manifolds constructed from bordered Heegaard triples. As applications, we present a new algorithm for computing arbitrary cobordism maps on Heegaard Floer homology and construct new nontrivial $A_\infty$-deformations of Khovanov's arc algebras. Motivated by this last result and a K\"{u}nneth theorem for Heegaard Floer complexes of connected sums, we also prove the existence of a tensor product decomposition for arc algebras in characteristic 2 and show that there cannot be such a splitting over $\Z$.