The contact invariant in sutured Floer homology (Q1030526)

The article describes an invariant of a contact 3-manifold with convex boundary as an element of Sutured Floer Homology. The invariant generalizes the contact invariant developed by Ozsváth and Szabó for closed 3-manifolds using Heegaard Floer Homology. Sutured Manifolds were introduced by \textit{D. Gabai} [J. Differ. Geom. 18, 445--503 (1983; Zbl 0533.57013)] to study and construct taut foliations. A \textit{sutured manifold} \((M,\Gamma)\) is a compact oriented 3-manifold with boundary, together with a compact surface \(\Gamma= A(\Gamma)\cup T(\Gamma)\subset \partial M\), where \(A(\Gamma)\) is the union of pairwise disjoint annuli and \(T(\Gamma)\) is an union of tori. A \textit{balanced sutured manifold} has no closed components, \(\pi_{0}(A(\Gamma))\rightarrow\pi_{0}(\partial M)\) is surjective, and \(\chi(R_{+}(\Gamma))=\chi(R_{-}(\Gamma))\) on every component of \(M\), where \(R_{\pm}\) are open surfaces of \(\partial M-\Gamma\). The authors assume the sutured manifolds used in the article are balanced sutured manifolds. An embedded surface \(F\subset (M,\xi)\) is convex if there is a contact vector field \(X\) which is transverse to \(F\). In the setting of contact structures, a natural condition to be imposed on a contact 3-manifold \((M,\xi)\) with boundary is to require \(\partial M\) to be convex. The dividing set associated to the convex surface \(F\) and the contact vector field \(X\) is \(\Gamma_{F}(X)=\{x\in F\mid X(x)\in \xi(x)\}\), usually denoted \(\Gamma_{F}\). If \((M,\Gamma)\) is a sutured manifold and \(\xi\) is a contact structure on \(M\) with convex boundary, so that the dividing set \(\Gamma_{\partial M}\) on \(\partial M\) equals \(\Gamma\), then such manifold will be denoted by \((M,\Gamma,\xi)\). In the papers [Algebr. Geom. Topol. 6, 1429--1457 (2006; Zbl 1129.57039) and Geom. Topol. 12, No. 1, 299--350 (2008; Zbl 1167.57005)], \textit{A. Juhász} generalized the hat versions of Heegaard Floer Homology defined by \textit{P. Ozsváth} and \textit{Z. Szabó} in [Ann. Math. (2) 159, No. 3, 1027--1158 (2004; Zbl 1073.57009) and ibid. 1159--1245 (2004; Zbl 1081.57013)] by assigning a Floer homology group \(SFH(M,\Gamma)\) to a balanced sutured manifold \((M,\Gamma)\). One of the important properties of the group \(SFH(M)\) is the following: if \((M,\Gamma)\overset{T}{\rightsquigarrow} (M',\Gamma')\) is a sutured decomposition along a cutting surface \(T\), then \(SFH(M',\Gamma')\) is a direct summand of \(SFH(M,\Gamma)\). {\textbf{Main Theorem}} Let \((M,\Gamma)\) be a balanced sutured manifold, and let \(\xi\) be a contact structure on \(M\) with convex boundary, whose dividing set on \(\partial M\) is \(\Gamma\). Then there exists an invariant \(EH(M,\Gamma,\xi)\) of the contact structure which lives in \(SFH(-M,-\Gamma)/{\pm1}\) (the authors use \(\mathbb{Z}\)-coefficients). {\textbf{Properties of the Contact Class} \(EH(M,\Gamma,\xi)\):} (1) If \((M,\Gamma,\xi)\) is overtwisted, then \(EH(M,\Gamma,\xi)=0\). (2) If \((M,\xi)\) is closed, then \(EH(M-B^{3},\Gamma=S^{1},\xi\mid_{m-B^{3}})=c(M,\xi)\), where \(c(M,\xi)\) is the Ozsváth-Szabó contact invariant living in the Heegard Floer Homology, and \((B^{3},\Gamma_{\partial B^{3}},\xi\mid_{B^{3}})\) is a standard contact 3-ball. In particular, \(EH(M-B^{3},\Gamma=S^{1},\xi\mid_{M-B^{3}})\neq 0\) if \((M,\xi)\) is Stein fillable. (3) Let \((M,\xi)\) be a closed contact 3-manifold and \(N\subset M\) be a compact submanifold (without any closed componentes) with convex boundary and dividing set \(\Gamma\) on \(\partial N\). If \(c(M,\xi)\neq 0\), then \(EH(N,\Gamma,\xi\mid_{N})\neq 0\). In order to obtain the main theorem, the authors present a relative version of \textit{E. Giroux}'s theorem [Invent. Math. 141, No. 3, 615--689 (2000; Zbl 1186.53097)], which gives the equivalence between isotopy classes of contact structures and open book decompositions modulo positive stabilization. In particular, they define the notion of a partial open book decomposition adapted to a contact manifold \((M,\Gamma,\xi)\). Consider \(S\) a compact oriented surface with nonempty boundary (page), \(P\subset S\) a subsurface and \(h:P\rightarrow S\) a partially-defined monodromy map. Thus, they prove the following {\textbf{Theorem}} (relative version of Giroux's result [loc. cit.]) Every contact structure on a manifold \(M\) with non-empty convex boundary is carried by a partial open book decomposition \((S,h:P\rightarrow S)\). Any two partial open book decompositions representing the same contact structure become isotopic after performing a sequence of positive stabilizations to each. Using the properties of sutured Floer homology, the authors manage to prove that whenever \((M,\Gamma,\xi)\) is a contact structure obtained from \((M',\Gamma',\xi')\) by gluing along a \(\partial\)-parallel \((T,\Gamma_{T})\), then \(EH(M',\Gamma',\xi')\) is mapped to \(EH(M,\Gamma,\xi)\) under the inclusion of \(SFH(-M',-\Gamma')\) as a direct summand of \(SFH(-M,-\Gamma)\). Therefore, \(EH(M,\Gamma,\xi)\neq 0\) if and only if \(EH(M',\Gamma',\xi')\neq 0\). At end, they describe some examples where the computations are manageable.