The Vanishing of the Contact Invariant in the Presence of Torsion

We prove that the Ozsváth-Szabó contact invariant of a closed contact 3manifold with positive 2π–torsion vanishes. In 2002, Ozsváth and Szabó [OSz1] defined an invariant of a closed contact 3-manifold (M, ξ) as an element of the Heegaard Floer homology group ĤF (−M). The definition of the contact invariant was made possible by the work of Giroux [Gi3], which related contact structures and open book decompositions. The Ozsváth-Szabó contact invariant has undergone an extensive study, e.g., [LS1, LS2]. Recently Honda, Kazez and Matić [HKM2] defined an invariant of a contact 3-manifold with convex boundary as an element of Juhász’ sutured Floer homology [Ju1, Ju2]. The goal of this paper is to use this relative contact invariant to prove a vanishing theorem in the presence of torsion. Recall that a contact manifold (M, ξ) has positive nπ-torsion if it admits an embedding (T 2 × [0, 1], ηnπ) →֒ (M, ξ), where (x, y, t) are coordinates on T 2 × [0, 1] ≃ R/Z × [0, 1] and ηnπ = ker(cos(nπt)dx−sin(nπt)dy). The torsion was an essential ingredient for distinguishing tight contact structures on toroidal 3-manifolds (see for example [Gi1]), and is a source of nonfiniteness of the number of isotopy classes of tight contact structures ([CGH, Co, HKM1]). Theorem 1 (Vanishing Theorem). If a closed contact 3-manifold (M, ξ) has positive 2πtorsion, then its contact invariant c(M, ξ) in ĤF (−M) vanishes. The coefficient ring of ĤF (−M) is Z in Theorem 1. In general, we do not understand the effect of torsion when twisted coefficients are used. Theorem 1 was first conjectured in [Gh2, Conjecture 8.3], and partial results were obtained by [Gh1], [Gh2], and [LS3]. The corresponding vanishing result for the contact class in monopole Floer homology has been recently announced by Mrowka and Rollin (and is motivated by [Ga]). Theorem 1, together with a non-vanishing result of the contact invariant proved by Ozsváth and Szabó ([OSz2, Theorem 4.2]), implies that a contact manifold with positive 2π-torsion is not strongly symplectically fillable. This non-fillability result was conjectured by Eliashberg, and first proved by Gay [Ga]. Date: This version: May 27, 2007. 1991 Mathematics Subject Classification. Primary 57M50; Secondary 53C15.