Cohomological Rigidity of Families of Manifolds Associated with Ideal Right-Angled Hyperbolic 3-Polytopes

Nikolai Yu. Erokhovets

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 99-138

Résumé

In toric topology, every $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets is assigned 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 $P$ is said to be $B$-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 arbitrary (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. A family of manifolds is said to be cohomologically rigid over a ring $R$ if two manifolds from the family are diffeomorphic whenever their graded cohomology rings over $R$ are isomorphic. As a result we obtain three cohomologically rigid families of manifolds over ideal almost Pogorelov polytopes: moment–angle manifolds, canonical six-dimensional quasitoric manifolds over $\mathbb Z$ or any field, and canonical three-dimensional small covers over $\mathbb Z_2$. The latter two classes of manifolds are known as pullbacks from the linear model.

Détail

Keywords: ideal right-angled polytope, $B$-rigidity, cohomological rigidity, almost Pogorelov polytope, pullback from the linear model.