Self-covering, finiteness, commutativity, and fibering over tori
Lizhen Qin, Yang Su
TL;DR
This work addresses when a closed self-covering manifold with abelian $\pi_1$ fibers over a torus and, under stronger hypotheses, becomes a torus bundle. It builds a multi-layered framework—starting from a homotopy fibration $p: M\to \mathbb{T}^n$ with a finitely dominated Poincaré fiber, then upgrading to approximate fibrations, block bundles, stable bundles, and finally bona fide bundles—using an interplay between topology and a robust algebraic finiteness theory. A key algebraic foundation guarantees finiteness of twisted modules and integral eigenvalues, enabling precise control over monodromies and torus actions; this yields positive fibering results (over $\mathbb{T}^n$ and $\mathbb{S}^1$) in high dimensions and clarifies obstructions via nonfibering examples when hypotheses fail. The paper also extends the conversation to nonabelian fundamental groups, constructing self-covering manifolds from self-covering complexes and showing substantial obstacles in achieving fibering, thereby connecting surgery theory, algebraic K-theory obstructions, and torus-affine geometry in the study of self-covering phenomena.
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.