Lisca, Paolo; Stipsicz, András I. On the existence of tight contact structures on Seifert fibered 3-manifolds. (English) Zbl 1233.57013 Duke Math. J. 148, No. 2, 175-209 (2009). This paper is about the construction of positive tight contact structures on 3-dimensional Seifert manifolds. In a previous paper [“Ozsváth-Szabó invariants and tight contact three-manifolds. II.”, J. Differ. Geom. 75, No. 1, 109–141 (2007; Zbl 1112.57005)] the same authors had proved that the Seifert manifolds \(M_n\) obtained as \((2n-1)\)-surgery along the torus knot \(T_{2,2n-1}\) do not carry positive tight contact structures. In the paper under review it is proved however that every other 3-dimensional Seifert manifold does admit a positive tight contact structure. The proof requires many case distinctions but the basic new ingredient is a nonvanishing criterion for the contact Ozsváth-Szabó invariant. This invariant was introduced by P. Ozsváth and Z. Szabó in [“Heegard Floer homology and contact structures”, Duke Math. J. 129, No. 1, 39–61 (2005; Zbl 1083.57042)] as an invariant of contact manifolds with values in the Heegard-Floer homology (which is associated to the spin\(^c\) structure determined by the contact field). Nonvanishing of the Ozsváth-Szabó invariant implies tightness of the contact structure. In Theorem 3.3 of the paper under review the authors give a sufficient criterion for nonvanishing of the Ozsváth-Szabó invariant for contact structures given by certain surgery diagrams. In the main body of the paper this is then applied to the Seifert manifolds in question. Reviewer: Thilo Kuessner (Seoul) Cited in 1 ReviewCited in 21 Documents MSC: 57R17 Symplectic and contact topology in high or arbitrary dimension 57R58 Floer homology 57R65 Surgery and handlebodies 53D35 Global theory of symplectic and contact manifolds Keywords:contact structures; Seifert manifolds; Ozsváth-Szabó invariant; Heegard-Floer homology Citations:Zbl 1112.57005; Zbl 1083.57042 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link References: [1] V. Colin, E. Giroux, and K. Honda, “On the coarse classification of tight contact structures” in Topology and Geometry of Manifolds (Athens, Ga., 2001) , Proc. Sympos. Pure Math. 71 , Amer. Math. Soc., Providence, 2003, 109–120. · Zbl 1052.57036 [2] C. Caubel, A. NéMethi, and P. Popescu –.Pampu, Milnor open books and Milnor fillable contact \(3\)-manifolds , Topology 45 (2006), 673–689. · Zbl 1098.53064 · doi:10.1016/j.top.2006.01.002 [3] F. Ding and H. 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