Let Y be a closed, oriented 3-manifold. The set 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 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 F_Y is bounded above by the order of the torsion subgroup of H_1(Y;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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