Heegaard Floer homology and contact structures. (English) Zbl 1083.57042

Ozsváth, Peter; Szabó, Zoltán

Duke Math. J. 129, No. 1, 39-61 (2005).

In [Ann. Math. (2) 159, No. 3, 1027–1158 (2004; Zbl 1073.57009)], the authors of the paper under review defined several Floer homology groups for closed oriented 3-manifolds \(Y\) with a Spin\(^c\) structure \(s\) associated to \(Y\) via a Heegaard diagram for \(Y\) and the Lagrangian Floer homology construction.
J. Martinet [Proc. Liverpool Singularities-Symp. II, Dept. Pure Math. Univ. Liverpool 1969–1970, 142–163 (1971; Zbl 0215.23003)] proved that every \(3\)-manifold admits a contact structure. Y. Eliashberg [Invent. Math. 98, No. 3, 623–637 (1989; Zbl 0684.57012)] gave a classification for the overtwisted contact structures on 3-manifolds. For a closed, oriented three-manifold \(Y\) endowed with a cooriented contact structure \(\xi\), E. Giroux [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 405–414 (2002; Zbl 1015.53049)] asserted the existence of an open book decomposition adapted to \(\xi\). For a contact structure on a closed, oriented three-manifold \(Y\), the paper under review uses Giroux’s open book decomposition to describe an invariant that takes values in the three-manifold’s Floer homology \(\widehat{HF}\), which vanishes for overwisted contact structures and is nonzero for Stein-fillable ones.

Reviewer: Guang-Cun Lu (Beijing)

MSC:

57R58	Floer homology
53D10	Contact manifolds (general theory)
57R17	Symplectic and contact topology in high or arbitrary dimension

Keywords:

contact structure; three-manifold; Floer homology

Citations:

Zbl 1073.57009; Zbl 0215.23003; Zbl 0684.57012; Zbl 1015.53049

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