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Theory of Families of Polytopes: Fullerenes and Pogorelov Polytopes

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Abstract

The paper is a review of the results of the eponymous cycle of author’s works marked by the I.I. Shuvalov I degree prize 2018 for scientific research and more recent studies. The families of three-dimensional simple polytopes defined by the condition of cyclic \(k\)-edge-connectivity are investigated. They include, for instance, flag polytopes and Pogorelov polytopes as well as related families of fullerenes and ideal right-angled hyperbolic polytopes. The methods are described for constructing families by cutting off edges and connected sum along faces and fullerenes by growth operations, for constructing cohomologically rigid families of three-dimensional and six-dimensional manifolds, and for Thurston’s geometrization of orientable three-dimensional manifolds corresponding to polytopes.

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ACKNOWLEDGMENTS

The author thanks V.M. Buchstaber for constant attention and fruitful joint work, T.E. Panov for pointing out that almost Pogorelov polytopes must, due to the Andreev theorem, correspond to the rectangular polytopes of finite volume in \(\mathbb{L}^{3}\) and the incompressibility of hypersurfaces may be proved using retraction of moment-angle manifold, A.Yu. Vesnin, A.A. Gaifullin, and O.V. Shvartsman for discussing aspects of hyperbolic geometry, and V.A. Shastin for discussing aspects of geometry of three-dimensional manifolds.

Funding

The work is supported by the Russian Foundation for Basic Research, project no. 20-01-00675.

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Correspondence to N. Yu. Erokhovets.

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Translated by E. Oborin

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Erokhovets, N.Y. Theory of Families of Polytopes: Fullerenes and Pogorelov Polytopes. Moscow Univ. Math. Bull. 76, 83–95 (2021). https://doi.org/10.3103/S0027132221020042

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