2011 Theses Doctoral

Bordered Sutured Floer Homology

Zarev, Rumen

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 = 𝛿𝑌 , and sutures 𝚪 ∈ 𝛿𝑌, we can compute the sutured Floer homology 𝑆𝐹𝐻(𝑌) from the bordered invariant 𝐶𝐹𝐴(𝑌)𝐴(𝐹). The chain complex 𝑆𝐹𝐻(𝑌,𝚪) defining 𝑆𝐹𝐻 is quasi-isomorphic to the derived tensor product 𝐶𝐹𝐴(𝑌)x𝐶𝐹𝐷(𝚪) where _𝒜(𝐹) 𝐶𝐹𝐷(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 bordered algebra 𝐴(𝐹) is a direct sum of certain sutured Floer complexes. These correspond to the 3-manifold (𝐹 \ 𝐷²;)×[0,1], where the sutures vary in a finite collection. Similarly, if 𝑌 is a connected 3-manifold with boundary 𝛿𝑌 = 𝐹, the module 𝐶𝐹𝐴(𝑌)_𝒜(𝐹) is a direct sum of sutured Floer complexes for 𝑌 where the sutures on d𝑌 vary over a finite collection. The multiplication structure on 𝒜(𝐹) and the action of 𝒜(𝐹) on 𝐶𝐹𝐴(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).