## 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

### Keywords:

double branched covering space; spin structure; involution

### Citations:

Zbl 0989.53029; Zbl 1075.57010
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