Bordered Sutured Floer Homology

Abstract

Bordered Sutured Floer Homology Rumen Zarev We investigate the relationship between two versions of Heegaard Floer homology for 3– manifolds with boundary—the sutured Floer homology of Juhasz, and the bordered Heegaard Floer homology of Lipshitz, Ozsvath, and Thurston. We define a new invariant called Bordered sutured Floer homology which encompasses these two invariants as special cases. Using the properties of this new invariant we prove a correspondence between the original bordered and sutured homologies. In one direction we prove that for a 3–manifold Y with connected boundary F = ∂Y , and sutures Γ ∈ ∂Y , we can compute the sutured Floer homology SFH(Y ) from the bordered invariant ĈFA(Y )A(F ). The chain complex SFC(Y,Γ) defining SFH is quasi-isomorphic to the derived tensor product ĈFA(Y ) ĈFD(Γ) where A(F )ĈFD(Γ) is a module associated to Γ. In the other direction we give a description of the bordered invariants in terms of sutured Floer homology. If F is a closed connected surface, then the boredered algebra A(F ) is a direct sum of certain sutured Floer complexes. These correspond to the 3–manifold (F D) × [0, 1], where the sutures vary in a finite collection. Similarly, if Y is a connected 3– manifold with boundary ∂Y = F , the module ĈFA(Y )A(F ) is a direct sum of sutured Floer complexes for Y where the sutures on ∂Y vary over a finite collection. The multiplication structure on A(F ) and the action of A(F ) on ĈFA(Y ) correspond to a natural gluing map on sutured Floer homology. (Further work of the author shows that this map coincides with the one defined by Honda, Kazez, and Matic, using contact topology and open book decompositions).