Teichmüller space of negatively curved metrics on Gromov-Thurston manifolds is not contractible
type: Item
instance of: scholarly article
zbMATH Open document ID: 1298.57016
DOI: 10.1142/S1793525314500204
author: Gangotryi Sorcar
published in: Journal of Topology and Analysis
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.5658
Mathematics Subject Classification ID: 57N37
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 58D27
zbMATH DE Number: 6340431
zbMATH Keywords: Teichmüller space
zbMATH Keywords: negatively curved manifold
zbMATH Keywords: Gromov-Thurston manifolds
MaRDI profile type: Publication
arXiv ID: 1311.5658
cites work: Deforming homotopy equivalences to homeomorphisms in aspherical manifolds
cites work: The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible
cites work: Teichmüller spaces and negatively curved fiber bundles
cites work: Pinching constants for hyperbolic manifolds
cites work: Differential Topology
cites work: Hyperbolic manifolds with negatively curved exotic triangulations in dimension six
title: Teichmüller space of negatively curved metrics on Gromov-Thurston manifolds is not contractible
review text: Let \(M\) be a closed smooth Riemannian manifold. The author denotes by \(\mathcal{T}^{<0}(M)\) the space of negatively curved Riemannian metrics on \(M\) quotiented by the diffeomorphisms of \(M\) that are homotopic to the identity. He calls this space the Teichmüller space of \(M\), in analogy with the case of surfaces. For manifolds of higher dimensions, this space was extensively studied by Farrell and Ontaneda. The main result of the paper under review is that for all \(n=4k-2\), \(k\geq 2\), there exist closed \(n\)-dimensional Riemannian manifolds \(M\) with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that \(\pi_1(\mathcal{T}^{<0}(M))\) is nontrivial. Farrell and Ontaneda proved a similar result under the assumption that \(M\) is a hyperbolic manifold, cf. [\textit{F. T. Farrell} and \textit{P. Ontaneda}, Ann. Math. (2) 170, No. 1, 45--65 (2009; Zbl 1171.58003)].NEWLINENEWLINEThe proof uses surgery with exotic spheres from the work of \textit{M. A. Kervaire} and \textit{J. W. Milnor} [Ann. Math. (2) 77, 504--537 (1963; Zbl 0115.40505)]. It also uses the Gromov-Thurston manifolds, which are examples of negatively curved manifolds that do not have the homotopy type of a locally symmetric space cf. [\textit{M. Gromov} and \textit{W. P. Thurston}, Invent. Math. 89, 1--12 (1987; Zbl 0646.53037)].
reviewed by: Athanase Papadopoulos
description: scientific article; zbMATH DE number 6340431