The Smale conjecture for hyperbolic 3-manifolds: \(\text{Isom}(M^3)\simeq\text{Diff}(M^3)\)
type: Item
instance of: scholarly article
title: The Smale conjecture for hyperbolic 3-manifolds: \(\text{Isom}(M^3)\simeq\text{Diff}(M^3)\)
zbMATH Open document ID: 1030.57026
author: David Gabai
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The main result of the paper states that, for a closed hyperbolic 3-manifold \(M\), the inclusion of the (finite) isometry group \(\text{Isom}(M)\) into the diffeomorphism group \(\text{Diff}(M)\) is a homotopy equivalence. For Haken 3-manifolds, this follows from results of \textit{A. Hatcher} [Topology 15, 343-347 (1976; Zbl 0335.57004)] and Ivanov (the original Smale conjecture, proved by Hatcher, is for the case of the 3-sphere). By previous work of \textit{D. Gabai} [J. Am. Math. Soc. 10, 37-74 (1997; Zbl 0870.57014)] and \textit{D. Gabai, G. R. Meyerhoff} and \textit{N. Thurston} [Ann. Math. (2) 157, 335-431 (2003)], the inclusion \(\text{Isom}(M)\to \text{Diff}(M)\) induces a bijection of \(\pi_0\text{Isom}(M)\) and \(\pi_0\text{Diff}(M)\). The main result is then equivalent to the fact that the path component \(\text{Diff}_0(M)\) in \(\text{Diff}(M)\) of the identity map of \(M\) is contractible. For the proof, the contractibility of the space of Riemannian metrics on \(M\) is used, in conjunction with the insulator theory of the above-mentioned paper of Gabai, to reduce to the Haken case. It follows also that the space of hyperbolic metrics on a complete hyperbolic 3-manifold of finite volume is contractible.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 55P10
zbMATH DE Number: 1782691
zbMATH Keywords: Smale conjecture
zbMATH Keywords: hyperbolic 3-manifold
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348284
OpenAlex ID: W2154015635
Wikidata QID: Q115202657
DOI: 10.4310/JDG/1090348284
description: scientific article; zbMATH DE number 1782691