Bimodules in bordered Heegaard Floer homology

TL;DR

This paper advances bordered Heegaard Floer homology by establishing naturality properties, computing algebra homology, and explicitly describing mapping class group actions, thereby deepening understanding of 3-manifold invariants and their relations to knot Floer homology.

Contribution

It introduces bimodules representing boundary diffeomorphisms, relates Hochschild homology to knot Floer homology, and explicitly computes the genus one case, enhancing the framework of bordered Floer homology.

Findings

Bimodules encode boundary diffeomorphisms in bordered Floer homology.

Hochschild homology of bimodules corresponds to knot Floer homology.

Explicit calculations for genus one surfaces and mapping class group actions.

Abstract

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). 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 A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). 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 A(F). 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.