On fillable contact structures up to homotopy

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.