Four-manifolds defined by vector-colorings of simple polytopes

We consider (non-necessarily free) actions of subgroups H⊂ Z₂ᵐ on the real moment-angle manifold RZP over a simple n-polytope P. The orbit space N(P,H)= RZP/H has an action of Z₂ᵐ/H. For general n we introduce the notion of a Hamiltonian C(n,k)-subcomplex generalizing the three-dimensional notions of a Hamiltonian cycle, theta- and K₄-subgraphs. Each C(n,k)-subcomplex C⊂ ∂ P corresponds to a subgroup HC such that N(P,HC) Sⁿ. We prove that in dimensions n 4 this correspondence is a bijection. Any subgroup H⊂ Z₂ᵐ defines a complex C(P,H)⊂ ∂ P. We prove that each Hamiltonian C(n,k)-subcomplex C⊂ C(P,H) inducing H corresponds to a hyperelliptic involution τC∈ Z₂ᵐ/H on the manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to Sⁿ) and in dimensions n 4 this correspondence is a bijection. We prove that for the geometries X= S⁴, S³× R, S²× S², S²× R², S²× L², and L²× L² 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 Z₂ᵐ/H, and for X= R⁴, L⁴, L³× R and L²× R² there are no such polytopes.