Existence of spin structures on double branched covering spaces over four-manifolds (Q1581886)

The following theorem is proved: Let \(X\) be a connected closed smooth 4-manifold with \(H_1(X,Z_2) = 0 \) and \(\widetilde X\) the double covering space over \(X\) branched along a connected closed surface \(F\) smoothly embedded in \(X\). Then \(\widetilde X\) is spin if and only if \(F\) is orientable and the modulo 2 reduction of \([F]/2 \in H_2(X,Z)\) coincides with the Poincaré dual of the second Stiefel-Whitney class \(w_2(X)\) of \(X\) for a fixed orientation of \(F\). The result generalizes some known facts for \(X = S^4\) and \(X=\mathbb{C}\mathbb{P}^2\).