Bimodules in bordered Heegaard Floer homology

Abstract

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface $F$ an algebra $\mathcal{A}\left(F\right)$ and to a three-manifold $Y$ with boundary identified with $F$ a module over $\mathcal{A}\left(F\right)$. In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between $F$ and the boundary of $Y$ tensors the bordered invariant with a suitable bimodule over $\mathcal{A}\left(F\right)$. These bimodules give an action of a suitably based mapping class group on the category of modules over $\mathcal{A}\left(F\right)$. The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of $\mathcal{A}\left(F\right)$. We also prove a duality theorem relating the two versions of the $3$–manifold invariant. Finally, in the case of a genus-one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.