Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics (Q2839318)

If \(X\) is a smooth compact manifold of dimension \(n\), then \(\text{Riem}(X)\) is the space of all Riemannian metrics on \(X\) equipped with its standard smooth topology. An embedding \(i: S^p\hookrightarrow X\) with trivial normal bundle can be extended to an embedding \(\bar i:S^p\times D^{q+1}\hookrightarrow X\) with \(p + q + 1 = n\). By removing an open neighborhood of \(S^p\) a manifold \(X\setminus \bar i\left(S^p\times \text{Int}(D)^{q+1}\right)\) with boundary \(\bar i\left(S^p\times S^q\right)\) is obtained, where \(\text{Int}(D)^{q+1}\) is the interior of the disc \(D^{q+1}\). As the handle \(D^{p+1}\times S^q\) has diffeomorphic boundary, the map \(\bar i_{|S^p\times S^q}\) can be used to glue the manifolds \(X\setminus \bar i\left(S^p\times \text{Int}(D)^{q+1}\right)\) and \(D^{p+1}\times S^q\) together by identifying their boundaries and consequently obtain the manifold \(Y=\left(X\setminus \bar i\left(S^p\times \text{Int}(D)^{q+1}\right)\right)\cup (D^{p+1}\times S^q)\). The manifold \(Y\) is said to be obtained from \(X\) by surgery on the embedding \(\bar i\). If \(X\) admits a Riemannian metric of positive scalar curvature, a natural and intriguing question arises of what can be said about the space \(\text{Riem}^{+}(X)\) consisting of all such metrics.NEWLINENEWLINEIn this paper, the author proves that if \(X\) is a smooth compact manifold of dimension \(n\) and \(Y\) is obtained from \(X\) by surgery on a sphere \(i:S^p\hookrightarrow X\) with \(p+q+1=n\) and \(p,q\geq 2\), then the spaces \(\text{Riem}^{+}(X)\) and \(\text{Riem}^{+}(Y)\) are homotopy equivalent. This result is originally due to V.~Chernysh but remains unpublished.