The second Stiefel-Whitney class of small covers
\textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)] introduced the notion of a small \(n\)-dimensional cover. This is a smooth closed \(n\)-manifold \(M^n\) with a locally standard action of \(\mathbb Z_2^n\) such that the orbit space is a simple convex \(n\)-dimensional polytope \(P^n\). Associated to such a small cover is a characteristic function \(\lambda:\mathcal F(P^n)\longrightarrow \mathbb Z_2^n\) where \(\mathcal F(P^n)\) is the set of facets of \(P^n\). The combinatorics of the polytope \(P^n\) has strong connections with the algebraic topology of \(M^n\). Let \(M^n\) be a small cover, \(P^n\) the orbit polytope and \(\lambda\) the characteristic function. The paper under review obtains, under certain assumptions on the cardinality of \(\mathrm{im}(\lambda)\) and colorability of \(P^n\), necessary and sufficient conditions under which the small cover \(M^n\) is spin. The author therefore is able to use the second Stiefel-Whitney class of the small cover \(M^n\) to detect when \(P^n\) is \(n\)-colorable or \((n+1)\)-colorable. The paper is clearly written and interesting.
scientific article; zbMATH DE number 7188399
scientific article