Existence of spin structures on cyclic branched covering spaces over four-manifolds (Q2765977)

Let \(\widetilde X\) be an \(m\)-fold branched covering over an oriented closed connected smooth 4-manifold \(X\) with \(H_1(X,{\mathbb Z}_2)\), branched along an oriented connected closed surface \(F\) smoothly embedded into \(X\). The article proves that if \(m\) is odd, then \(\widetilde X\) is spin if and only if \(X\) is spin. NEWLINENEWLINENEWLINEIf \(m\) is even, the author obtains the following condition for \(\widetilde X\) to be spin. Recall that \([F]\in H_2(M,{\mathbb Z})\) is divisible by \(m\). Let \([F]/m\) be an element in \(H_2(M,{\mathbb Z})\) with \(m([F]/m)=[F]\). For \(m\) even, the author shows that \(\widetilde X\) is spin if and only if the modulo \(2\) reduction of \([F]/m\) coincides with the second Stiefel-Whitney class \(w_2(X)\) of \(X\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00014].