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 \geq 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\geq 5 and d\neq 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) .