The Second Stiefel-Whitney Class of Small Covers

Zhangmin Huang

Chinese Annals of Mathematics, Series B ›› 2020, Vol. 41 ›› Issue (2) : 163 -176.

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Chinese Annals of Mathematics, Series B ›› 2020, Vol. 41 ›› Issue (2) : 163 -176. DOI: 10.1007/s11401-020-0192-y
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The Second Stiefel-Whitney Class of Small Covers

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Abstract

Let π: M nP n be an n-dimensional small cover over P n and λ: ℱ(P n) → ℤ2 n be its characteristic function. The author uses the symbol c(λ) to denote the cardinal number of the image Im(λ). If c(λ) = 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.

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Small cover / Spin structure / Simple polytope

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Zhangmin Huang. The Second Stiefel-Whitney Class of Small Covers. Chinese Annals of Mathematics, Series B, 2020, 41(2): 163-176 DOI:10.1007/s11401-020-0192-y

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