Ozsváth-Szabó invariants and fillability of contact structures

Abstract

Abstract. Recently Ozsváth and Szabó defined an invariant of contact structures with values in the Heegaard-Floer homology groups. They also proved that a version of the invariant with twisted coefficients is non trivial for weakly symplectically fillable contact structures. In this article we show that their non vanishing result does not hold in general for the contact invariant with untwisted coefficients. As a consequence of this fact Heegaard-Floer theory can distinguish between weakly and strongly symplectically fillable contact structures.

Infinitely many universally tight contact manifolds with trivial Ozsváth–Szabó contact invariants

“…This fact raised the hope that these invariants could be non trivial for any tight contact structure. However in [6] we showed that the untwisted contact invariant reduced modulo 2 can vanish even for weakly symplectically fillable contact structures. Those examples however left open the question whether the twisted invariants were non trivial for every tight contact structures.…”

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

“…If no such cover of the base exists (and V is not a Lens space) then its base B is a sphere with exceptional points of order (2, 2, n), (2,3,3), (2,3,4) or (2, 3, 5) (see [31,Theorem 13.3.6]). In each case B is covered by S 2 and all curves in the regular locus of B bounds a disk whose pre-image in S 2 is disconnected so ξ is virtually overtwisted according to [7, Proposition 4.1 and Lemma 4.7].…”

Comultiplicativity of the Ozsv{á}th-Szab{ó} contact invariant

“…Honda, Kazez, and Matić have since shown this to be the case for all such open books [7], [9], [8]. However, the precise relationship between c(ξ) and the tightness of ξ is still unknown -there are tight contact structures with vanishing contact invariant [4]. In fact, Ghiggini, Honda, and Van Horn-Morris recently showed that the contact invariant vanishes for contact structures with positive Giroux torsion [5].…”