Joining and gluing sutured Floer homology

Abstract

We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra multiplication and algebra action correspond to a new gluing map on SFH. It is defined algebraically, and is a special case of a more general "join" map.In a follow-up paper we show that this gluing map can be identified with the contact cobordism map of Honda-Kazez-Matić. The join map is conjecturally equivalent to the cobordism maps on SFH defined by Juhász.

Heegaard Floer homology as morphism spaces

“…In a different direction, one must proceed with care in adapting Theorem 1 to the case of bordered sutured manifolds [Zar09]. This extension is developed (in a slightly different language) in [Zar10]; see also Remark 5.1. 1.6.…”

“…As a tool for establishing Theorem 2, we use a Heegaard diagram discovered independently by Auroux [Aur10] and Zarev [Zar10] (see Section 4). Studying this diagram gives several algebraic results, including an algebraic Serre duality theorem (Theorem 10), and an interpretation of Hochschild cohomology as a knot Floer homology group (Corollary 11).…”

Formal Contact Categories

“…The Heegaard-Floer homology y HF ˚pM q was extended to surfaces and 3-manifolds with boundary, in the manner described above, by the authors P. Ozsváth, R. Lipshitz and D. Thurston [28]. The theory was further developed by R. Zarev [53,54]. In particular, when an oriented surface Σ sports a handle decomposition, determined by combinatorial data Z called an arc parameterization, there is a dg category Ap´Zq which is associated to the surface Σ.…”

“…In this section we show that a parameterization P " pZ, ϕ Z q of a pointed oriented surface pΣ, mq determines a canonical collection ZpZq of generators for the associated contact category KopΣ, mq. This material is motivated by a reading of R. Zarev [54]. Definition 5.12.…”

“…Remark 6.6. It follows invariance of the general Bordered Sutured theory that the dg categories associated to a surface by any two parameterizations are Morita equivalent [54]. In particular, there is an arc parameterization W n [53, Ex.…”