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.

中文翻译：

---

多面体家族理论：富勒烯和波戈列洛夫多面体

摘要

该论文是对以 2018 年 II Shuvalov I 学位奖为标志的作者作品同名循环的结果进行的回顾，以进行科学研究和最近的研究。研究了由循环\(k\) -边连通性条件定义的三维简单多面体族。例如，它们包括旗形多面体和 Pogorelov 多面体以及富勒烯和理想直角双曲多面体的相关家族。描述了通过沿面和富勒烯通过生长操作切断边和连接和来构造族的方法，用于构造三维和六维流形的上同调刚性族，以及用于对应于多面体的可定向三维流形的 Thurston 几何化.