On distinct finite covers of 3-manifolds

TL;DR

This paper classifies certain 3-manifolds based on the property that all their connected covers of the same degree are homeomorphic, extending the understanding of covering space uniqueness in 3-manifold topology.

Contribution

It provides a complete classification of compact 3-manifolds with the property that all same-degree connected covers are homeomorphic, linking topological and group-theoretic aspects.

Findings

Classified all such 3-manifolds with boundary conditions.

Connected covers of these manifolds are uniquely determined by degree.

Discussed implications for related group-theoretic questions.

Abstract

Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic questions.