[Paper Review] $B$-rigidity of ideal almost Pogorelov polytopes

Николай Юрьевич Ероховец|arXiv (Cornell University)|May 15, 2020

Commutative Algebra and Its Applications25 references4 citations

TL;DR

This paper proves that ideal almost Pogorelov polytopes—obtained by truncating all ideal vertices of a right-angled ideal polytope in hyperbolic 3-space—are $B$-rigid, meaning their moment-angle complex cohomology ring uniquely determines their combinatorial type. The result establishes cohomological rigidity for three families of manifolds: moment-angle complexes, canonical quasitoric manifolds, and small covers over these polytopes, all constructed as pullbacks from a linear model.

ABSTRACT

Toric topology assigns to each $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold $\mathcal{Z}_P$ with an action of a compact torus $T^m$ such that $\mathcal{Z}_P/T^m$ is a convex polytope of combinatorial type $P$. A simple $n$-polytope is called $B$-rigid, if any isomorphism of graded rings $H^*(\mathcal{Z}_P,\mathbb Z)= H^*(\mathcal{Z}_Q,\mathbb Z)$ for a simple $n$-polytope $Q$ implies that $P$ and $Q$ are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial $3$-polytope obtained by cutting off all the ideal vertices of an ideal right-angled polytope in the Lobachevsky (hyperbolic) space $\mathbb L^3$. These polytopes are exactly the polytopes obtained from any, not necessarily simple, convex $3$-polytopes by cutting off all the vertices followed by cutting off all the "old" edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We prove that any ideal almost Pogorelov polytope is $B$-rigid. This produces three cohomologically rigid families of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canonical $6$-dimensional quasitoric manifolds and canonical $3$-dimensional small covers, which are "pullbacks from the linear model".

Motivation & Objective

• To establish $B$-rigidity for a new class of 3-dimensional simple polytopes arising from ideal right-angled polytopes in hyperbolic 3-space.
• To investigate whether the integral cohomology ring of the moment-angle complex $Χ_P$ determines the combinatorial type of the polytope $P$.
• To extend the known families of $B$-rigid polytopes by proving that ideal almost Pogorelov polytopes are $B$-rigid.
• To demonstrate that canonical quasitoric manifolds and small covers over these polytopes inherit cohomological rigidity from the underlying polytope.

Proposed method

• Define ideal almost Pogorelov polytopes as those obtained by truncating all ideal vertices and edges of an ideal right-angled polytope in hyperbolic 3-space $\mathbb{L}^3$, resulting in a combinatorial 3-polytope with specific face structure.
• Use the barycentric subdivision of the boundary of the original polytope to characterize the dual of the ideal almost Pogorelov polytope.
• Construct the moment-angle complex $\mathcal{Z}_P$ as a $T^m$-manifold over $P$, with $\mathcal{Z}_P / T^m \cong P$, and analyze its integral cohomology ring.
• Apply a 3-coloring of the facets of $P$ to define a $\mathbb{Z}_2^3$-action on a double cover $\boldsymbol{R}(P)$, which decomposes into two hyperbolic manifolds of finite volume.
• Use the action of the kernel of a homomorphism $\varphi: G(Q) \to \mathbb{Z}_2^2$ induced by the coloring to construct a hyperbolic manifold $\widehat{\boldsymbol{R}_i}$ as a quotient $\mathbb{L}^3 / \mathrm{Ker}\,\varphi$, homeomorphic to a part of $\boldsymbol{R}(P)$.
• Prove that the cohomology ring isomorphism $H^*(\mathcal{Z}_P, \mathbb{Z}) \cong H^*(\mathcal{Z}_Q, \mathbb{Z})$ implies combinatorial equivalence of $P$ and $Q$, establishing $B$-rigidity.

Experimental results

Research questions

• RQ1 Are ideal almost Pogorelov polytopes $B$-rigid, i.e., does their moment-angle complex cohomology ring determine their combinatorial type?
• RQ2 Can the cohomological rigidity of moment-angle complexes be extended to polytopes derived from ideal right-angled polytopes in hyperbolic 3-space?
• RQ3 Do canonical quasitoric manifolds and small covers over ideal almost Pogorelov polytopes inherit cohomological rigidity from the base polytope?
• RQ4 Is the hyperbolic structure of the quotient manifold $\mathbb{L}^3 / \mathrm{Ker}\,\varphi$ compatible with the topological structure of the moment-angle complex?
• RQ5 How does the barycentric subdivision of the original polytope’s boundary relate to the structure of the ideal almost Pogorelov polytope?

Key findings

• Any ideal almost Pogorelov polytope is $B$-rigid, meaning that if $H^*(\mathcal{Z}_P, \mathbb{Z}) \cong H^*(\mathcal{Z}_Q, \mathbb{Z})$ as graded rings, then $P$ and $Q$ are combinatorially equivalent.
• The moment-angle complex $\mathcal{Z}_P$ over an ideal almost Pogorelov polytope has a cohomology ring that uniquely determines the polytope’s combinatorial type.
• Canonical 6-dimensional quasitoric manifolds over ideal almost Pogorelov polytopes are cohomologically rigid, inheriting $B$-rigidity from the base polytope.
• Canonical 3-dimensional small covers over these polytopes are also cohomologically rigid, as they arise as pullbacks from a linear model.
• The construction of $\boldsymbol{R}(P)$ as a double of two hyperbolic manifolds of finite volume, each homeomorphic to $\mathbb{L}^3 / \mathrm{Ker}\,\varphi$, supports the topological rigidity of the system.
• The $\mathbb{Z}_2^3$-action on $\boldsymbol{R}(P)$, induced by a 3-coloring of the facets, realizes each quadrilateral face as a flat 2-torus, and the entire structure is preserved under cohomological equivalence.