Hyperelliptic four-manifolds defined by vector-colorings of simple polytopes

Toric topology assigns to each simple convex $n$-polytope $P$ with $m$ facets an $n$-dimensional real moment angle manifold $\mathbb{R}{Z}_P$ with a canonical action of ${\mathbb{Z}}_2^m=(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^m$. We consider (non-necessarily free) actions of subgroups $H\subset {\mathbb{Z}}_2^m$ on $\mathbb{R}{Z}_P$. The orbit space $N(P,H)=\mathbb{R}{Z}_P\mathrm{/}H$ has an action of ${\mathbb{Z}}_2^m\mathrm{/}H$. For general $n$ we introduce the notion of a Hamiltonian $C(n,k)$-subcomplex in the boundary of an $n$-polytope $P$ generalizing the notions of a Hamiltonian cycle ($k=2$), Hamiltonian theta-subgraph ($k=3$) and Hamiltonian $K_4$-subgraph ($k=4)$ in the $1$-skeleton of a $3$-polytope. Each $C(n,k)$-subcomplex $C\subset \mathrm{\partial }P$ corresponds to a subgroup $H_C\subset {\mathbb{Z}}_2^m$ such that $N(P,H_C)\simeq S^n$. We prove that in dimensions $n⩽4$ this correspondence is a bijection. Any subgroup $H\subset {\mathbb{Z}}_2^m$ defines a complex $C(P,H)\subset \mathrm{\partial }P$. We prove that each Hamiltonian $C(n,k)$-subcomplex $C\subset C(P,H)$ inducing $H$ corresponds to a hyperelliptic involution ${\tau }_C\in {\mathbb{Z}}_2^m\mathrm{/}H$ on the manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to $S^n$) and in dimensions $n⩽4$ this correspondence is a bijection. We prove that for the geometries $\mathbb{X}={\mathbb{S}}^4$, ${\mathbb{S}}^3\times \mathbb{R}$, ${\mathbb{S}}^2\times {\mathbb{S}}^2$, ${\mathbb{S}}^2\times {\mathbb{R}}^2$, ${\mathbb{S}}^2\times {\mathbb{L}}^2$, and ${\mathbb{L}}^2\times {\mathbb{L}}^2$ there exists a compact right-angled $4$-polytope $P$ with a free action of $H$ such that the geometric manifold $N(P,H)$ has a hyperelliptic involution in ${\mathbb{Z}}_2^m\mathrm{/}H$, and for $\mathbb{X}={\mathbb{R}}^4$, ${\mathbb{L}}^4$, ${\mathbb{L}}^3\times \mathbb{R}$ and ${\mathbb{L}}^2\times {\mathbb{R}}^2$ there are no such polytopes.