Equivalence of Contact Gluing Maps in Sutured Floer Homology

Abstract

We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description. In particular, we establish a conjecture of Zarev, that his gluing map on sutured Floer homology is equivalent to the contact gluing map.1 Homology groups are over F2-coefficients through the entirety of this paper.

Bordered invariants of pairs of sutured manifolds with torus boundary

“…Generalising the definition of ContDiffSut to bordered-sutured manifolds is somewhat fiddly in the most generality; cf. [17]. The following suffices for us: We often suppress the inclusion map i from notation.…”

“…A proof that the associated functor to the homotopy category respects triangles is somewhat non-standard: it is shown in [5] using techniques of bordered-sutured Floer homology which avoid the need for the use of the exact triangle detection lemma, and by [17] the maps involved are equivalent to those defined in [11].…”