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

Abstract

Let X X and Y 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 R i e m + ( X ) {\mathcal R}{\mathrm i}{\mathrm e}{\mathrm m}^{+}(X) and R i e m + ( Y ) {\mathcal R}{\mathrm i}{\mathrm e}{\mathrm m}^{+}(Y) , respectively consisting of Riemannian metrics of positive scalar curvature on X X and Y Y , are homotopy equivalent. This result is originally due to V. Chernysh but remains unpublished.