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 $P$.
Definition 1. 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 $3$-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).
Theorem. Any ideal almost Pogorelov polytope is $B$-rigid.
Definition 2. A family of manifolds is called cohomologically rigid over the ring $R$, if for any two manifolds $M$ and $N$ from the family any isomorphism of graded rings $H^*(M,R)\simeq H^*(N,R)$ implies that $M$ and $N$ are diffeomorphic.
Any ideal almost Pogorelov polytope $P$ has a canonical colouring of facets in $3$ colours corresponding to vertices, edges and facets of the polytope that gives $P$ via cutting off vertices and “old” edges. This colouring produces the $6$-dimensional quasitoric manifold $M(P)$ and the $3$-dimensional small cover $N(P)$, which are known as “pullbacks from the linear model”.
Corollary. The families $\{\mathcal{Z}_P\}$, $\{M(P)\}$, and $\{N(P)\}$ indexed by the ideal right-angled hyperpolic $3$-polytopes are cohomologically rigid over $\mathbb Z$, $\mathbb Z$ and $\mathbb Z_2$ respectively.
We also plan to discuss the geometry of the $3$-manifolds $N(P)$.
[E20]
N. Erokhovets,
$B$-rigidity of ideal almost Pogorelov polytopes, arXiv:2005.07665v3.
[E21]
Nikolai Yu. Erokhovets,
$B$-rigidity of the property to be an almost Pogorelov polytope, Contemporary Mathematics, 772, 2021, 107–122.