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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Volume 318, Pages 99–138
DOI: https://doi.org/10.4213/tm4274
(Mi tm4274)
 

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

Nikolai Yu. Erokhovets

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991 Russia
References:
Abstract: 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.
Keywords: ideal right-angled polytope, $B$-rigidity, cohomological rigidity, almost Pogorelov polytope, pullback from the linear model.
Funding agency Grant number
Russian Foundation for Basic Research 20-01-00675
This work was supported by the Russian Foundation for Basic Research, project no. 20-01-00675.
Received: March 24, 2022
Revised: May 11, 2022
Accepted: May 13, 2022
English version:
Proceedings of the Steklov Institute of Mathematics, 2022, Volume 318, Pages 90–125
DOI: https://doi.org/10.1134/S0081543822040083
Bibliographic databases:
Document Type: Article
UDC: 515.14+515.16+514.15+514.172.45
Language: Russian
Citation: Nikolai Yu. Erokhovets, “Cohomological Rigidity of Families of Manifolds Associated with Ideal Right-Angled Hyperbolic 3-Polytopes”, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Collected papers, Trudy Mat. Inst. Steklova, 318, Steklov Math. Inst., Moscow, 2022, 99–138; Proc. Steklov Inst. Math., 318 (2022), 90–125
Citation in format AMSBIB
\Bibitem{Ero22}
\by Nikolai~Yu.~Erokhovets
\paper Cohomological Rigidity of Families of Manifolds Associated with Ideal Right-Angled Hyperbolic 3-Polytopes
\inbook Toric Topology, Group Actions, Geometry, and Combinatorics. Part~2
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2022
\vol 318
\pages 99--138
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4274}
\crossref{https://doi.org/10.4213/tm4274}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4538838}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2022
\vol 318
\pages 90--125
\crossref{https://doi.org/10.1134/S0081543822040083}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85142204834}
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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