On fillable contact structures up to homotopy

Lisca, Paolo

doi:10.1090/S0002-9939-01-05964-0

On fillable contact structures up to homotopy
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Proc. Amer. Math. Soc. 129 (2001), 3437-3444 Request permission

Abstract:

Let $Y$ be a closed, oriented $3$–manifold. The set $\mathcal {F}_Y$ of homotopy classes of positive, fillable contact structures on $Y$ is a subtle invariant of $Y$, known to always be a finite set. In this paper we study $\mathcal {F}_Y$ under the assumption that $Y$ carries metrics with positive scalar curvature. Using Seiberg–Witten gauge theory, we prove that two positive, fillable contact structures on $Y$ are homotopic if and only if they are homotopic on the complement of a point. This implies that the cardinality of $\mathcal {F}_Y$ is bounded above by the order of the torsion subgroup of $H_1(Y;{\mathbb Z})$. Using explicit examples we show that without the geometric assumption on $Y$ such a bound can be arbitrarily far from holding.

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