The Second Stiefel-Whitney Class of Small Covers

Let π: Mn → Pn be an n-dimensional small cover over Pn and λ: ℱ(Pn) → ℤ2n 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 Mn is given. As a byproduct, under some special conditions, the author uses the second Stiefel-Whitney class to detect when Pn is n-colorable or (n + 1)-colorable.