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

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 $H^*(\mathcal{Z}_{As^3},\mathbb Z)$.