Evan M Fink

Let (S1 x D2, G) be a solid torus equipped with longitudinal sutures. We show that SFH(S1 x D2, G; Z2) can be equipped with a topology, such that we have the following: x in SFH(S1 x D2, G; Z2) is a contact element if and only if the set consisting of 0 and x is closed. This topology arises fairly naturally from the constructions in a previous paper, where we identify this Sutured Floer homology group with a certain exterior algebra.

The idea of a sutured topological quantum field theory was introduced by Honda, Kazez and Matic. A sutured TQFT associates a group to each sutured surface and an element of this group to each dividing set on this surface. The notion was originally introduced to talk about contact invariants in Sutured Floer Homology. We provide an elementary example of a sutured TQFT, which comes from taking exterior algebras of certain singular homology groups. We show that this sutured TQFT coincides with that of Honda-Kazez-Matic using Z2-coefficients. The groups in our theory, being exterior algebras, naturally come with the structure of a ring with unit. We give an application of this ring structure to understanding tight contact structures on solid tori.