Sutured TQFT, torsion, and tori

Abstract

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with $\mathbb{Z}$ coefficients, of certain sutured manifolds of the form $(\Sigma \times S^1,F\times S^1)$ where $\Sigma$ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with $\mathbb{Z}$ coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot's theorem that the contact invariant vanishes for a contact structure on $(\Sigma \times S^1,F\times S^1)$ described by an isolating dividing set.

29 pages, 16 figures