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On spin structures of double branched covering spaces. (English) Zbl 1317.57021

Let \(\widetilde{X}\rightarrow X\) be a double branched covering map over a closed smooth oriented \(n\)-manifold, \(n\geq 4\), branched along a closed submanifold \(F\subset X\) of codimension 2. In two previous papers, [Osaka J. Math. 37, No. 2, 425–440 (2000; Zbl 0989.53029)] and [Kobe J. Math. 20, No. 1–2, 39–51 (2003; Zbl 1075.57010)], the author has considered whether \(\widetilde{X}\) is spin or not on the assumption that \(H_1(X;\mathbb{Z}_2))=0\), and obtained that \(\widetilde{X}\) is spin if and only if \(F\) admits an orientation such that \([F]\in H_{n-2}(X;\mathbb{Z})\) is twice a characteristic homology class. In the present paper the author first studies the similar problem for the spin\(^c\) case. As a corollary it is obtained that if \(X\) is a spin\(^c\) manifold, then \(\widetilde{X}\) also is a spin\(^c\) manifold. Then the author gives a necessary and sufficient condition for \(\widetilde{X}\) to be spin without any assumption on \(H_\ast(X)\). Moreover, a necessary and sufficient condition for a double covering space to admit a spin structure which is preserved by the covering transformation map of odd type is given. A conjecture in the context is also formulated.
Reviewer: Ioan Pop (Iaşi)

MSC:

57R91 Equivariant algebraic topology of manifolds
57M10 Covering spaces and low-dimensional topology
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