Abstract
We consider (not necessarily free) actions of subgroups \(H\subset\mathbb Z_2^m\) on the real moment–angle manifold \(\mathbb R\mathcal Z_P\) corresponding to a simple convex \(n\) -polytope \(P\) with \(m\) facets. A criterion for the orbit space \(\mathbb R\mathcal Z_P/H\) to be a topological manifold (perhaps with boundary) can be extracted from results by M. A. Mikhailova and C. Lange. For any dimension \(n\) we construct a series of manifolds \(\mathbb R\mathcal Z_P/H\) homeomorphic to \(S^n\) and a series of manifolds \(M^n=\mathbb R\mathcal Z_P/H\) admitting a hyperelliptic involution \(\tau\in\mathbb Z_2^m/H\) , that is, an involution \(\tau\) such that \(M^n/\langle\tau\rangle\) is homeomorphic to \(S^n\) . For any simple \(3\) -polytope \(P\) we classify all subgroups \(H\subset\mathbb Z_2^m\) such that \(\mathbb R\mathcal Z_P/H\) is homeomorphic to \(S^3\) . For any simple \(3\) -polytope \(P\) and any subgroup \(H\subset\mathbb Z_2^m\) we classify all hyperelliptic involutions \(\tau\in\mathbb Z_2^m/H\) acting on \(\mathbb R\mathcal Z_P/H\) . As a corollary we show that a three-dimensional small cover has three hyperelliptic involutions in \(\mathbb Z_2^3\) if and only if it is a rational homology \(3\) -sphere and if and only if it corresponds to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them.
