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On the homotopy type of the space of metrics of positive scalar curvature (2012.00432v2)
Published 1 Dec 2020 in math.DG, math.AT, and math.GT
Abstract: Let $Md$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}+(Md)$ the space of such metrics on $Md$. We show that $\mathcal{R}+(Md)$ is homotopy equivalent to $\mathcal{R}+(Sd)$, where $Sd$ denotes the $d$-dimensional sphere with standard smooth structure. We also show a similar result for simply connected non-spin manifolds $Md$ with $d\geq 5$ and $d\neq 8$. In this case let $Wd$ be the total space of the non-trivial $S{d-2}$-bundle with structure group $SO(d-1)$ over $S2$. Then $\mathcal{R}+(Md)$ is homotopy equivalent to $\mathcal{R}+(Wd)$.