Non-zero degree maps and surface bundles over \(S^ 1\)
type: Item
instance of: scholarly article
title: Non-zero degree maps and surface bundles over \(S^ 1\)
zbMATH Open document ID: 0868.57029
author: Michel Boileau
author: Shicheng Wang
published in: Journal of Differential Geometry
publication date: 1996-12-08T00:00:00Z
review text: The authors study non-zero degree maps \(f:M\to N\) between 3-dimensional compact orientable irreducible \(\partial\)-irreducible manifolds. One may ask whether there exists a map of non-zero degree between given manifolds of the same dimension, and can think of the existence of such a map as defining a partial ordering on the set of homeomorphism classes of compact connected manifolds of a given dimension. As suggested by Gromov, this partial order can be defined as follows: say that \(M\) dominates \(N\), denoted by \(M\geq N\), if there is a non-zero degree allowable map from \(M\) to \(N\). Each closed hyperbolic orientable \(n\)-manifold with \(n\neq 3\) dominates only finitely many closed orientable hyperbolic \(n\)-manifolds. The authors show that this result fails in dimension 3. The authors apply the results to study a conjecture of Thurston which says that any complete finite volume hyperbolic 3-manifold is finitely covered by a surface bundle over \(S^1\).
reviewed by: Karl Heinz Dovermann
Mathematics Subject Classification ID: 57R35
zbMATH DE Number: 955313
zbMATH Keywords: homeomorphism classes
zbMATH Keywords: hyperbolic 3-manifold
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214458532
OpenAlex ID: W1577838117
Wikidata QID: Q115177214
DOI: 10.4310/JDG/1214458532
description: scientific article; zbMATH DE number 955313