Self-Covering, finiteness, and fibering over a circle

TL;DR

This paper investigates the conditions under which self-covering manifolds with abelian fundamental groups fiber over a circle, providing a complete classification for free abelian groups and constructing counterexamples for non-free cases.

Contribution

It establishes criteria for when self-covering manifolds with free abelian fundamental groups fiber over a circle, and introduces an algebraic criterion for module finite generation.

Findings

Self-covering manifolds with free abelian fundamental groups fiber over S^1 under certain conditions.

Complete classification for manifolds with fundamental group Z regarding fibering over S^1.

Examples of self-covering manifolds with non-free abelian fundamental groups that do not fiber over S^1.

Abstract

A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold $M$ with free abelian fundamental group fibers over a circle under certain assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group $\mathbb Z$ is a fiber bundle over $S^1$, except for the $4$-dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with non-free abelian fundamental group, which are not fiber bundles over $S^1$