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 C 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.