A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold \mathbf{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 \mathbb{Z} is a fiber bundle over \mathbf{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 nonfree abelian fundamental group, which are not fiber bundles over \mathbf{S}^1.

Publication:

Transactions of the American Mathematical Society (2024)

https://doi.org/10.1090/tran/9003

Author:

Lizhen Qin

Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing, Jiangsu, People’s Republic of China

Yang Su

HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, People’s Republic of China

Email: suyang@math.ac.cn

Botong Wang

Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706