$B$-rigidity of the property to be an almost Pogorelov polytope

TL;DR

This paper investigates the $B$-rigidity property of certain 3D polytopes, including almost Pogorelov polytopes, showing that specific geometric and combinatorial properties are uniquely determined by their cohomology rings.

Contribution

It proves that the properties of being an almost Pogorelov or ideal almost Pogorelov polytope are $B$-rigid, extending known results to new classes of polytopes and their associated moment-angle manifolds.

Findings

$B$-rigidity of almost Pogorelov polytopes established

$B$-rigidity of ideal almost Pogorelov polytopes proved

$3$-dimensional associahedron and permutohedron are $B$-rigid

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 $P$. We study the notion of $B$-rigidity. A property of a polytope $P$ 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 it also has this property. We study families of $3$-dimensional polytopes defined by their cyclic $k$-edge-connectivity. These families include flag polytopes and Pogorelov polytopes, that is polytopes realizable as bounded right-angled polytopes in Lobachevsky space $\mathbb L^3$. Pogorelov polytopes include fullerenes -- simple polytopes with only pentagonal and hexagonal faces. It is known that the properties to be flag and Pogorelov polytope are $B$-rigid. We focus on almost Pogorelov polytopes, which are strongly cyclically $4$-edge-connected polytopes. They correspond to right-angled polytopes of finite volume in $\mathbb L^3$. There is a subfamily of ideal almost Pogorelov polytopes corresponding to ideal right-angled polytopes. We prove that the properties to be an almost Pogorelov polytope and an ideal almost Pogorelov polytope are $B$-rigid. As a corollary we obtain that $3$-dimensional associahedron $As^3$ and permutohedron $Pe^3$ are $B$-rigid. We generalize methods known for Pogorelov polytopes. We obtain results on $B$-rigidity of subsets in $H^*(\mathcal{Z}_P,\mathbb Z)$ and prove an analog of the so-called separable circuit condition (SCC). As an example we consider the ring $H^*(\mathcal{Z}_{As^3},\mathbb Z)$.