The Second Stiefel-Whitney Class of Small Covers


Zhangmin HUANG.The Second Stiefel-Whitney Class of Small Covers[J].Chinese Annals of Mathematics B,2020,41(2):163~176
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Zhangmin HUANG;
Abstract: Let $\pi: M^n\longrightarrow P^n$ be an $n$-dimensional small cover over $P^n$ and $\lambda:\mathcal{F}(P^n)\longrightarrow \mathbb{Z}_2^n$ be its characteristic function. The author uses the symbol $c(\lambda)$ to denote the cardinal number of the image ${\rm Im}(\lambda)$. If $c(\lambda)=n+1$ or $n+2$, then a necessary and sufficient condition on the existence of spin structure on $M^n$ is given. As a byproduct, under some special conditions, the author uses the second Stiefel-Whitney class to detect when $P^n$ is $n$-colorable or $(n+1)$-colorable.


Small cover, Spin structure, Simple polytope


57R19, 57R20, 52B11
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