Cobordism Invariance of the Homotopy Type of the Space of Positive Scalar Curvature Metrics

TL;DR

This paper proves that the homotopy type of the space of positive scalar curvature metrics on a smooth manifold is invariant under surgeries of codimension at least three, extending known results in geometric topology.

Contribution

It provides a rigorous proof of cobordism invariance of the homotopy type of positive scalar curvature metric spaces, confirming an unpublished result by Chernysh.

Findings

Homotopy type remains unchanged after surgery in codimension ≥ 3

Supports the stability of positive scalar curvature metrics under certain topological modifications

Extends the understanding of the topology of scalar curvature metric spaces

Abstract

We show that the homotopy type of the space of metrics of positive scalar curvature on a smooth manifold remains unchanged, after application of surgery in codimension at least three to the underlying manifold. This result is originally due to V. Chernysh, but remains unpublished.