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Self-Covering, finiteness, and fibering over a circle

Lizhen Qin, Yang Su, Botong Wang

TL;DR

The paper addresses when closed self-covering manifolds over a circle necessarily fiber over $S^1$, focusing on CAT manifolds with free abelian $igl( ext{pi}_1(M)igr)$. It develops a rigorous algebraic foundation (via a Noetherian criterion) to deduce finiteness properties of infinite cyclic covers and leverages this to prove a fibering theorem for dimension $ preccurlyeq 5$ with $ ext{pi}_1(M) i Z$; it also derives homological periodicity phenomena for the monodromy and provides counterexamples using algebraic $K$-theory to show that self-coverings do not always yield fibering, including a 5-manifold smooth counterexample. The results unify geometric topology with commutative algebra, yielding a near-complete classification in dimensions $ eq 4$ and offering a framework for further exploration in the non-abelian setting and lower dimensions. The work highlights the intricate interplay between infinite cyclic covers, finiteness obstructions, and Whitehead torsion in determining fibering over the 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 $M$ with free abelian fundamental group fibers over a circle under certain 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 $S^1$, 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 non-free abelian fundamental group, which are not fiber bundles over $S^1$

Self-Covering, finiteness, and fibering over a circle

TL;DR

The paper addresses when closed self-covering manifolds over a circle necessarily fiber over , focusing on CAT manifolds with free abelian . It develops a rigorous algebraic foundation (via a Noetherian criterion) to deduce finiteness properties of infinite cyclic covers and leverages this to prove a fibering theorem for dimension with ; it also derives homological periodicity phenomena for the monodromy and provides counterexamples using algebraic -theory to show that self-coverings do not always yield fibering, including a 5-manifold smooth counterexample. The results unify geometric topology with commutative algebra, yielding a near-complete classification in dimensions and offering a framework for further exploration in the non-abelian setting and lower dimensions. The work highlights the intricate interplay between infinite cyclic covers, finiteness obstructions, and Whitehead torsion in determining fibering over the 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 with free abelian fundamental group fibers over a circle under certain assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group is a fiber bundle over , except for the -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 non-free abelian fundamental group, which are not fiber bundles over
Paper Structure (7 sections, 40 theorems, 100 equations)

This paper contains 7 sections, 40 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Algebraic Foundation
  3. CW Complex: Homological Finiteness
  4. CW Complex: Homotopy Finiteness and Homological Periodicity
  5. Manifold: Fibering Theorems
  6. Manifold: Non-Fibering Examples
  7. Further Questions

Key Result

Theorem 1

Let $X$ be defined as above. Suppose there exists a homotopy equivalence $h: X \rightarrow X_{k}$ with $k>1$ and $h_{\sharp} (G \times 0) = G \times 0$, where $h_{\sharp}: \pi_{1} (X) \rightarrow \pi_{1} (X_{k})$ is the isomorphism induced by $h$. If $X$ is homotopy equivalent to a CW complex of fin

Theorems & Definitions (88)

  • Theorem 1
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 2
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 78 more