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

The main result of this paper is that when $M_0$, $M_1$ are two simply connected spin manifolds of the same dimension $d \geq 5$ which both admit a metric of positive scalar curvature, the spaces $\mathcal{R}^+(M_0)$ and $\mathcal{R}^+(M_1)$ of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when $M_0$ and $M_1$ are also spin cobordant. We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that $d \neq 8$ in that case.