Combinatorial 3-manifolds with transitive cyclic symmetry

TL;DR

This paper develops criteria and constructions for transitive cyclic combinatorial 3-manifolds, extending classifications and introducing new infinite families, including neighborly lens spaces with diverse topologies.

Contribution

It provides combinatorial criteria for infinite families of transitive cyclic manifolds and extends classification of such 3-manifolds up to 22 vertices.

Findings

Extended classification of 3-manifolds with transitive cyclic symmetry

Constructed new infinite families of such manifolds

Identified a family of neighborly combinatorial lens spaces with diverse topologies

Abstract

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.