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

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.