Sutured TQFT, torsion, and tori

TL;DR

This paper employs sutured TQFT to classify contact elements in sutured Floer homology of specific manifolds, providing new proofs for the vanishing of contact invariants in cases with Giroux torsion and isolating dividing sets.

Contribution

It introduces a classification of contact elements using sutured TQFT and offers new proofs for the vanishing of contact invariants in certain torsion and dividing set scenarios.

Findings

Contact invariant vanishes for structures with Giroux torsion.

Contact invariant vanishes for structures with isolating dividing sets.

Classification of contact elements in specific sutured manifolds.

Abstract

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with $\Z$ 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 $\Z$ 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.