Homotopy groups of the observer moduli space of Ricci positive metrics

Abstract

The observer moduli space of Riemannian metrics is the quotient of the space $\mathcal{ℛ}\left(M\right)$ of all Riemannian metrics on a manifold $M$ by the group of diffeomorphisms ${Diff}_{x_0}\left(M\right)$ which fix both a basepoint $x_0$ and the tangent space at $x_0$. The group ${Diff}_{x_0}\left(M\right)$ acts freely on $\mathcal{ℛ}\left(M\right)$ provided that $M$ is connected. This offers certain advantages over the classic moduli space, which is the quotient by the full diffeomorphism group. Results due to Botvinnik, Hanke, Schick and Walsh, and Hanke, Schick and Steimle have demonstrated that the higher homotopy groups of the observer moduli space ${\mathcal{ℳ}}_{x_0}^{s>0}\left(M\right)$ of positive scalar curvature metrics are, in many cases, nontrivial. The aim in the current paper is to establish similar results for the moduli space ${\mathcal{ℳ}}_{x_0}^{Ric>0}\left(M\right)$ of metrics with positive Ricci curvature. In particular we show that for a given $k$, there are infinite-order elements in the homotopy group ${\pi }_{4k}{\mathcal{ℳ}}_{x_0}^{Ric>0}\left(S^n\right)$ provided the dimension $n$ is odd and sufficiently large. In establishing this we make use of a gluing result of Perelman. We provide full details of the proof of this gluing theorem, which we believe have not appeared before in the literature. We also extend this to a family gluing theorem for Ricci positive manifolds.