On the topology of the space of pinched negatively curved metrics with finite volume and identical ends (Q2812005)

Let \(M\) be a connected, noncompact smooth manifold and \(\mathcal{M}(M)\) be the space of all Riemannian metrics on \(M\), endowed with the compact-open \(C^{\infty}\)-topology. The author considers the subspace \(\mathcal{M}^{<0}(M)\) of \(\mathcal{M}(M)\) consisting of all complete Riemannian metrics on \(M\) with finite volume and whose sectional curvatures are all bounded by two negative numbers. Let \(\mathcal{M}_{\infty}^{<0}(M,g)\) be the subspace (with direct limit topology) of \(\mathcal{M}^{<0}(M)\) consisting of all Riemannian metrics \(g'\in \mathcal{M}^{<0}(M)\) such that \(g'\) and \(g\) agree on \(M-K\) for some compact subset \(K\subset M\). The main result of this article: Let \(M\) be a noncompact manifold and assume that \(\mathcal{M}^{<0}(M)\) is nonempty. Then, for any \(g\in \mathcal{M}^{<0}(M)\): (1) \(\mathcal{M}_{\infty}^{<0}(M,g)\) has infinitely many path connected components, provided \(\dim M\geq 10\). (2) If \(\mathcal{K}\) is any component of \(\mathcal{M}^{<0}_{\infty}(M,g)\), then \(\pi_1(\mathcal{K})\) contains a subgroup isomorphic to \((\mathbb{Z}/2)^{\infty}\), provided \(\dim M~\geq~ 14\), and (3) For each odd prime \(p\), \(\pi_{2p-4}(\mathcal{K})\) contains a subgroup isomorphic to \((\mathbb{Z}/p)^{\infty}\), provided \(\frac{\dim M-10}{2}> 2p-4\).