Hyperbolic 3-manifolds admitting no fillable contact structures

Li, Youlin; Liu, Yajing

doi:10.1090/proc/13870

Hyperbolic 3-manifolds admitting no fillable contact structures
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by Youlin Li and Yajing Liu PDF

Proc. Amer. Math. Soc. 147 (2019), 351-360 Request permission

Abstract:

Given an irreducible 3-manifold $M$, Eliashberg asked whether $M$ admits a tight or fillable contact structure. When $M$ is a Seifert fibered space, this question is completely solved. However, it has not been answered for any hyperbolic 3-manifold $M$, even for the existence question of fillable contact structures. In this paper, we find infinitely many hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that some of these manifolds do admit tight contact structures.

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