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

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