Sutured TQFT, torsion and tori (Q2854050)

A sutured manifold \((M,\gamma)\) consists of a compact oriented \(3\)-manifold \(M\) with boundary, along with a subset \(\gamma\subset\partial M\) which is a union of annuli \(A(\gamma)\) and tori \(T(\gamma)\). The interior of each component of \(A(\gamma)\) contains a suture, that is, a homologically nontrivial oriented simple closed curve. The pair \((M,\gamma)\) is called balanced if \(M\) has no closed components, \(\chi(R_+(\gamma))=\chi(R_-(\gamma))\), and the inclusion \(A(\gamma) \hookrightarrow \partial M\) induces a surjective map on \(\pi_0\), where \(R_+(\gamma)\) are those components of \(R(\gamma)=\partial M-\text{Int}(\gamma)\) whose normal vectors point out of \(M\) and similarly \(R_-(\gamma)\) are the components of \(R(\gamma)\) with inward pointing normal vectors. The sutured Floer homology group \(SFH(M,\gamma)\) associated to a balanced sutured manifold \((M,\gamma)\) is a direct sum of groups \(SFH(M,\gamma, \mathfrak{s})\), where \(\mathfrak{s}\) ranges through the set of Spin\(^c\)-structures on \((M,\gamma)\). Let \((M,\xi)\) be a contact \(3\)-manifold with a contact invariant or contact element \(c(\xi)\) in the Heegaard Floer homology.NEWLINENEWLINEIn this paper, the author gives new proofs of important facts about contact topology and Heegaard Floer homology. Making use of the sutured topological quantum field theory (TQFT), he classifies 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. The author gives a new proof that if \(\xi\) is a contact structure with \((2\,\pi)\)-torsion on a closed \(3\)-manifold \(M\) or a balanced sutured \(3\)-manifold \((M,\gamma)\), with contact invariant \(c(\xi)\subset \widehat{HF}(-M)\) or \(SFH(-M,-\gamma)\), respectively (with \(\mathbb Z\) coefficients), then \(c(\xi)=\{0\}\). Also, a new proof of Massot's theorem is given. It is shown that for a contact structure \(\xi\) on \((\Sigma\times S^1,F\times S^1)\) described by dividing sets \(\Gamma\) on \((\Sigma,F)\) the following are equivalent in sutured Floer homology over \(\mathbb Z\) coefficients: (i)\,\(c(\xi)\neq 0\), (ii)\,\(c(\xi)\) is primitive, and (iii)\, \(\Gamma\) is not isolating.