On the homotopy type of the space of metrics of positive scalar curvature

Abstract

Let $M^d$ be a simply connected spin manifold of dimension $d\ge 5$ admitting Riemannian metrics of positive scalar curvature. Denote by ${\mathcal{R}}^{+}(M^d)$ the space of such metrics on $M^d$. We show that ${\mathcal{R}}^{+}(M^d)$ is homotopy equivalent to ${\mathcal{R}}^{+}(S^d)$, where $S^d$ denotes the $d$-dimensional sphere with standard smooth structure.

We also show a similar result for simply connected non-spin manifolds $M^d$ with $d\ge 5$ and $d\ne 8$. In this case let $W^d$ be the total space of the non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then ${\mathcal{R}}^{+}(M^d)$ is homotopy equivalent to ${\mathcal{R}}^{+}(W^d)$.