Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

Christoforos Neofytidis

Abstract

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is virtually trivial. As another application, we derive the first examples of non-vanishing numerical invariants that are monotone with respect to the mapping degree on non-trivial circle bundles over aspherical manifolds with hyperbolic fundamental groups in any dimension.

On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

Abstract

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is virtually trivial. As another application, we derive the first examples of non-vanishing numerical invariants that are monotone with respect to the mapping degree on non-trivial circle bundles over aspherical manifolds with hyperbolic fundamental groups in any dimension.

Paper Structure

This paper contains 12 sections, 11 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Applications of the main result
  3. Infinite sequences of coverings
  4. Self-maps of aspherical manifolds with hyperbolic fundamental groups
  5. Fundamental group and finite covers
  6. Reduction to fiberwise covering maps
  7. Distinguishing between trivial and non-trivial bundle
  8. Groups infinite index presentable by products
  9. An infinite index presentation by products
  10. The Hopf property and the strong version of Hopf's problem
  11. The Hopf property
  12. Infinite decreasing sequences and Problem \ref{['Hopfstrong']}

Key Result

Theorem 1.3

An oriented circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial.

Figures (1)

  • Figure 1: The Hopf property for $\pi_1(M)$.

Theorems & Definitions (22)

  • Theorem 1.3
  • Example 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 2.1
  • Remark 2.2: The Borel conjecture: From homotopy equivalences to homeomorphisms
  • Corollary 2.3
  • Conjecture 2.4
  • Proposition 3.1
  • proof
  • ...and 12 more