Manifolds realized as orbit spaces of non-free ${\mathbb{Z}}_2^k$-actions on real moment-angle manifolds

Abstract

We consider (non-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. The criterion when the orbit space $\mathbb{R}{\mathcal{Z}}_P/H$ is a topological manifold (perhaps with a boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For any dimension $n$ we construct series of manifolds $\mathbb{R}{\mathcal{Z}}_P/H$ homeomorphic to $S^n$ and 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 $M^n/⟨\tau ⟩$ 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 obtain that a $3$-dimensional small cover has $3$ hyperelliptic involutions in ${\mathbb{Z}}_2^3$ if and only if it is a rational homology $3$-sphere and if and only if it correspond to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them.

52 pages, 19 figures. In the 2nd version some references are added, some misprints are corrected. Results on $4$-manifolds are announced