Teichmüller space of negatively curved metrics on Gromov-Thurston manifolds is not contractible
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)].
scientific article; zbMATH DE number 6340431