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

Ebert, Johannes; Wiemeler, Michael

Abstract

Let M^d be a simply connected spin manifold of dimension d≥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≥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).