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

2025/10/28 by Qin, Lizhen, Su, Yang
#55R10 #57R19 #57R65 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.24032

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.

Citations

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• Isomorphism Conjecture for Baumslag-Solitar groups
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