Orientable small covers over the product of 2-cube with \(n\)-gon (Q2877957)

A small cover of an \(n\)-dimensional Polyhedron \(P\) is a locally standard action of \(\mathbb Z^n_2\) on an \(n\)-manifold whose quotient is \(P\). The authors determine the number of equivalence classes of orientable small covers of \(I^2\times P_k\) (square times \(k\)-gon) with respect to two distinct notions of equivalence. The Problem is translated into a purely algebraic one. According to \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)], a small cover of \(P\) is described by a mapping from the set of facets of \(P\) into \(\mathbb Z_2^n\) with a certain property (facets with non-empty intersection have linearly independent images). Orientability is expressed by an additional algebraic property of this map ([\textit{H. Nakayama} and \textit{Y. Nishimura}, Osaka J. Math. 42, No. 1, 243--256 (2005; Zbl 1065.05041)]), and also the equivalences are captured algebraically. For one of the equivalence relations, the resulting numbers of classes are described by a formula using two recursive functions of Fibonacci type. In the other case (equivalence up to equivariant homeomorphism), the formula contains a recursion over integers dividing \(k\) and Euler's \(\varphi\)-function. For \(k = 4\), this number is explicitly stated as 12180.