Четырехмерные гиперэллиптические многообразия, определяемые векторными раскрасками простых многогранников

Математический сборник. 2026. Т. 217. № 5. С. 45–89.

Ероховец Николай Юрьевич

Toric topology assigns to each simple convex n-polytope P with m facets an n-dimensional real moment-angle manifold RZP with a canonical action of Zm2=(Z/2Z)m. We consider (not necessarily free) actions of subgroups H⊂Zm2 on RZP. The orbit space N(P,H)=RZP/H carries an action of Zm2/H. For general n we introduce the notion of Hamiltonian C(n,k)-subcomplex in the boundary of an n-polytope P generalizing the notions of Hamiltonian cycle (for k=2), Hamiltonian theta-subgraph (for k=3) and Hamiltonian K4-subgraph (for k=4) in the 1-skeleton of a 3-polytope. Each C(n,k)-subcomplex C⊂∂P corresponds to a subgroup {HC⊂Zm2} such that N(P,HC)≃Sn. We prove that in dimensions n⩽4 this correspondence is a bijection. Any subgroup H⊂Zm2 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∈Zm2/H on the manifold N(P,H) (that is, an involution with 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 Zm2/H, and there are no such polytopes for X=R4, L4, L3×R and L2×R2.

Keywords: convex polytope, гиперэллиптическое многообразие, Hamiltonian subcomplex, nonfree action of a finite group, rational homology sphere, гамильтонов подкомплекс, несвободное действие конечной группы, рациональная гомологическая сфера

Publication based on the results of: Toric Topology, Hyperbolic Geometry and Applications to Data Analysis (2025)