The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible (Q2389210)

In this paper, \(M\) is a closed smooth manifold, \(\mathrm{Diff}(M)\) is the group of smooth diffeomorphisms of \(M\), \(\mathrm{Diff}_0(M)\) is the subgroup of \(\mathrm{Diff}(M)\) of smooth diffeomorphisms that are homotopic to the identity and \(\mathcal{MET}(M)\) is the space of all Riemannian metrics on \(M\) with the smooth topology, on which \(\mathrm{Diff}(M)\) acts by pull-back. The group \(\mathcal{D}(M)=\mathbb{R}^+\times \mathrm{Diff}(M)\) acts of \(\mathcal{MET}(M)\) by scaling and pull-back. \(\mathcal{D}_0(M)\) is the group \(\mathbb{R}^+\times \mathrm{Diff}_0(M)\). The authors call the quotient space \(\mathcal{T}(M)=\mathcal{MET}(M)/\mathcal{D}_0(M)\) the Teichmüller space of metrics on \(M\). Finally, for \(0\leq\epsilon\leq\infty\), \(\mathcal{MET}^{\epsilon}(M)\) denotes the space of all \(\epsilon\)-pinched negatively curved Riemannian metrics on \(M\), that is, the space of metrics for which there exists a positive real number \(\lambda\) such that \(\lambda g\) has all its sectional curvatures in the interval \([-(1+\epsilon),-1]\). The authors call the quotient space \(\mathcal{M}^{\epsilon}(M)=\mathcal{MET}^{\epsilon}(M)/\mathcal{D}(M)\) the moduli space of \(\epsilon\)-pinched negatively curved metrics on \(M\), and the space \(\mathcal{T}^{\epsilon}(M)=\mathcal{MET}^{\epsilon}(M)/\mathcal{D}_0(M)\) the Teichmüller space of \(\epsilon\)-pinched negatively curved metrics on \(M\). The main result of this paper is that if \(M\) is hyperbolic, then the natural inclusion \(\mathcal{T}^{\epsilon}(M)\hookrightarrow \mathcal{T}(M)\) is, in general, not homotopically trivial. More precisely, the authors prove the following Theorem: For any integer \(k_0\geq 1\), there is an integer \(n_0=n_0(k_0)\) such that the following holds. Given \(\epsilon >0\) and a closed hyperbolic \(n\)-manifold \(M\) with \(n\geq n_0\), there is a finite sheeted cover \(N\) of \(M\) such that, for every \(1\leq k\leq k_0\) with \(n+k \equiv 3\) mod \(4\), the map \(\pi_k(\mathcal{T}^{\epsilon}(N))\to \pi_k(\mathcal{T}(N))\) induced by the inclusion \(\mathcal{T}^{\epsilon}(N)\hookrightarrow \mathcal{T}(N)\) is nonzero. Consequently \(\pi_k(\mathcal{T}^{\epsilon}(N))\not=0\). In particular, \(\mathcal{T}^{\delta}(N)\) is not contractible for every \(\delta\) such that \(\epsilon\leq \delta\leq\infty\), provided \(k_0\geq 4\). As a consequence, the authors obtain the following: Corollary: Let \(M\) be a closed hyperbolic manifold of dimension \(n\geq 6\). Then, \(\mathcal{T}^{\epsilon}(M)\) is not contractible.