Bimodules in bordered Heegaard Floer homology

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.

153 pages, 29 figures; v4: Address referee comments

References in corpus (7)

• Higher-Dimensional Algebra V: 2-Groups
• Lectures on open book decompositions and contact structures
• Fukaya categories and deformations
• Heegaard Floer homology as morphism spaces
• Hopfological algebra and categorification at a root of unity: the first steps
• Link homology and categorification
• A faithful linear-categorical action of the mapping class group of a surface with boundary

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