Download this article
 Download this article For screen
For printing
Recent Issues

Volume 24, 1 issue

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

Christoforos Neofytidis

Algebraic & Geometric Topology 23 (2023) 3205–3220
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 nonzero 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 nontrivial covering and the bundle is virtually trivial. As another application, we derive the first examples of nonvanishing numerical invariants that are monotone with respect to the mapping degree on nontrivial circle bundles over aspherical manifolds with hyperbolic fundamental groups in any dimension.

Keywords
Hopf property, degree of self-map, homotopy equivalence, aspherical manifold, circle bundle, fundamental group, hyperbolic group
Mathematical Subject Classification
Primary: 55M25
References
Publication
Received: 14 July 2021
Revised: 13 February 2022
Accepted: 27 May 2022
Published: 26 September 2023
Authors
Christoforos Neofytidis
Department of Mathematics
The Ohio State University
Columbus, OH
United States

Open Access made possible by participating institutions via Subscribe to Open.