Self-covering, finiteness, commutativity, and fibering over tori

TL;DR

This paper investigates the structure of self-covering manifolds, showing under certain conditions they fiber over tori or circles, and provides examples illustrating the complexity when these conditions are not met.

Contribution

It establishes new results on fibering properties of self-covering manifolds with abelian fundamental groups and constructs examples demonstrating the complexity beyond these cases.

Findings

Self-covering manifolds with abelian fundamental groups fiber over tori.

High-dimensional, free abelian fundamental groups imply fibering over a circle.

Nonabelian fundamental groups lead to more complex, non-fibering examples.

Abstract

A topological space is called self-covering if it is a nontrivial cover of itself. We prove that, under mild assumptions, a closed self-covering manifold with an abelian fundamental group fibers over a torus in various senses. As a corollary, if its dimension is above $5$ and its fundamental group is free abelian, then it is a fiber bundle over a circle. We also construct non-fibering examples when these assumptions are not fulfilled. In particular, one class of examples illustrates that the structure of self-covering manifolds is more complicated when the fundamental groups are nonabelian, and the corresponding fibering problem encounters significant difficulties.