Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres

TL;DR

This paper surveys known results on centrally symmetric polytopes and manifolds, and introduces new enumerations and infinite series of symmetric spheres with few vertices, highlighting their geometric and combinatorial properties.

Contribution

It provides a comprehensive survey and introduces new enumerations of nearly neighborly centrally symmetric spheres and products of spheres, including an infinite series of vertex-transitive examples.

Findings

Enumerated nearly neighborly centrally symmetric spheres with few vertices

Presented an infinite series of vertex-transitive nearly neighborly centrally symmetric 3-spheres

Analyzed symmetry properties of polytopes and manifolds

Abstract

The aim of this paper is to give a survey of the known results concerning centrally symmetric polytopes, spheres, and manifolds. We further enumerate nearly neighborly centrally symmetric spheres and centrally symmetric products of spheres with dihedral or cyclic symmetry on few vertices, and we present an infinite series of vertex-transitive nearly neighborly centrally symmetric 3-spheres.