Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics

Abstract

Let X and Y be a pair of smooth manifolds, each obtainable from the other by surgery in codimension at least three. We show that the corresponding spaces Riem(+)(X) and Riem(+)(Y), respectively consisting of Riemannian metrics of positive scalar curvature on X and Y, are homotopy equivalent. This result is originally due to V. Chernysh but remains unpublished.