Self-covering, finiteness, and fibering over a circle

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 M with free abelian fundamental group fibers over a circle under mild assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group Z \mathbb Z is a fiber bundle over S 1 S^1 , except for the 4 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 nonfree abelian fundamental group, which are not fiber bundles over S 1 S^1 .