On fillable contact structures up to homotopy (Q2731942)

Let \(Y\) be a closed oriented 3-manifold carrying metrics with positive scalar curvature. The author proves that two fillable contact structures on \(Y\) are homotopic if and only if they induce the same \(\text{Spin}^c\) structure on \(Y\) (or equivalently, they are homotopic on the complement of a point). One direction is obvious. For the other direction, one needs to know how many properties a \(\text{Spin}^c\) structure on \(Y\) can determine. Through Seiberg-Witten theory on a symplectic 4-manifold with contact boundary, one finds that a certain topological quantity is determined by the Riemannian structure and the \(\text{Spin}^c\) structure on \(Y\). This topological quantity happens to determine the homotopy class of contact structures. As a corollary, the number of homotopy classes of fillable contact structures on \(Y\) is bounded above by the order of the torsion subgroup of the first integral homology of \(Y\). The author also shows by examples that if one drops the geometric assumption on Y, both the theorem and the corollary fail.