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 $\mathbb RZ_P$ with a canonical action of $\mathbb Z_2^m=(\mathbb Z/2\mathbb Z)^m$. We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on $\mathbb RZ_P$. The orbit space $N(P,H)=\mathbb RZ_P/H$ has an action of $\mathbb Z_2^m/H$. 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 $C\subset \partial P$ 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 $n\leqslant 4$ this correspondence is a bijection. Any subgroup $H\subset \mathbb Z_2^m$ defines a complex $C(P,H)\subset \partial P$. We prove that each Hamiltonian $C(n,k)$-subcomplex $C\subset C(P,H)$ inducing $H$ corresponds to a hyperelliptic involution $\tau_C\in\mathbb Z_2^m/H$ on the manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to $S^n$) and in dimensions $n\leqslant 4$ 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 $\mathbb Z_2^m/H$, 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.