The Second Stiefel-Whitney Class of Small Covers

Abstract

Let π: M ⁿ → P 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.

Keywords

Small cover / Spin structure / Simple polytope