Manifolds Realized as Orbit Spaces of Non-free $\mathbb Z_2^k$-Actions on Real Moment–Angle Manifolds

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.