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

Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities