Homotopy Invariance of the Space of Metrics with Positive Scalar Curvature on Manifolds with Singularities

In this paper we study manifolds, $X_{\Sigma}$, with fibred singularities, more specifically, a relevant space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ is homotopy invariant under certain surgeries on $X_{\Sigma}$.