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 $τ_C\in\mathbb Z_2^m/H$ 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.

💡 Research Summary

The paper investigates the interplay between simple convex polytopes, vector colourings, and group actions to construct and classify hyperelliptic (hyper‑elliptic) four‑dimensional manifolds. Starting from toric topology, the author recalls that a simple n‑polytope P with m facets determines an n‑dimensional real moment‑angle manifold ℝZₚ = (P × ℤ₂^m)/∼, equipped with a canonical ℤ₂^m‑action whose orbit space is P itself. For any subgroup H ⊂ ℤ₂^m (rank m‑r) one obtains a homomorphism Λ: {facets of P} → ℤ_r²; the kernel of the induced epimorphism ϕ_Λ : G(P) → ℤ_r² is precisely H. When Λ is linearly independent (i.e., a vector‑colouring), the H‑action on ℝZₚ is free and the quotient N(P,Λ)=ℝZₚ/H is a smooth n‑manifold carrying an effective ℤ_r²‑action.

A central theme is the identification of those Λ for which N(P,Λ) is homeomorphic to the n‑sphere Sⁿ. The author introduces the notion of a Hamiltonian C(n,k)‑subcomplex C ⊂ ∂P, a higher‑dimensional analogue of Hamiltonian cycles, theta‑subgraphs, and K₄‑subgraphs in the 1‑skeleton of a 3‑polytope. Each such subcomplex determines a subgroup H_C ⊂ ℤ₂^m with the property that N(P,H_C) ≃ Sⁿ. The first main theorem (Theorem 3.14) proves that for dimensions n ≤ 4 this correspondence is bijective: N(P,Λ) ≃ Sⁿ if and only if the associated complex C(P,Λ) is equivariantly equivalent to the standard C(n,r) complex.

The paper then turns to hyperelliptic involutions. A closed n‑manifold M is called hyperelliptic if it admits a non‑trivial involution τ such that M/⟨τ⟩ ≃ Sⁿ. The author shows that a Hamiltonian C(n,k)‑subcomplex C ⊂ C(P,Λ) induces a canonical vector‑colouring Λ_C of rank k+1, and the resulting manifold N(P,Λ_C) carries a hyperelliptic involution τ_C belonging to ℤ_{k+1}². Moreover, for n ≤ 4 the map “hyperelliptic involution ↔ proper Hamiltonian C(n,r‑1)‑subcomplex inducing Λ” is a bijection (Theorem 4.40). This extends earlier results for n = 3, where cycles, theta‑graphs, and K₄‑subgraphs corresponded to hyperelliptic involutions.

The final part applies the theory to the ten Thurston‑type geometries that can appear as products of spheres, Euclidean spaces, and hyperbolic spaces in dimension four: X = X₁×⋯×X_k with each X_i ∈ {S^{n_i}, ℝ^{n_i}, L^{n_i}}. Using right‑angled 4‑polytopes P in each geometry, the author constructs linearly independent vector‑colourings Λ such that N(P,Λ) is a geometric manifold modelled on X and possesses a hyperelliptic involution. This succeeds for six geometries—S⁴, S³×ℝ, S²×S², S²×ℝ², S²×L², and L²×L²—while it is proved impossible for the remaining four—ℝ⁴, L⁴, L³×ℝ, and L²×ℝ². The impossibility results rely on combinatorial constraints on the existence of Hamiltonian C(4,k)‑subcomplexes in right‑angled 4‑polytopes, ultimately linked to the Four‑Color Theorem and parity arguments on 2‑faces.

In summary, the paper provides a comprehensive framework that translates combinatorial data (vector‑colourings and Hamiltonian subcomplexes) into topological and geometric information about real moment‑angle manifolds. It establishes precise criteria for when these manifolds are spheres, when they admit hyperelliptic involutions, and which four‑dimensional geometries can support such structures. The work unifies and extends previous three‑dimensional constructions, offers new examples of hyperelliptic four‑manifolds, and delineates sharp non‑existence boundaries, thereby advancing the understanding of toric topology, group actions, and geometric structures on manifolds.