On the homotopy type of the space of metrics of positive scalar curvature
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866910671199797248 |
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| author | Ebert, Johannes Wiemeler, Michael |
| author_facet | Ebert, Johannes Wiemeler, Michael |
| contents | Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(S^d)$, where $S^d$ denotes the $d$-dimensional sphere with standard smooth structure.
We also show a similar result for simply connected non-spin manifolds $M^d$ with $d\geq 5$ and $d\neq 8$. In this case let $W^d$ be the total space of the non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(W^d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_00432 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the homotopy type of the space of metrics of positive scalar curvature Ebert, Johannes Wiemeler, Michael Differential Geometry Algebraic Topology Geometric Topology Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(S^d)$, where $S^d$ denotes the $d$-dimensional sphere with standard smooth structure. We also show a similar result for simply connected non-spin manifolds $M^d$ with $d\geq 5$ and $d\neq 8$. In this case let $W^d$ be the total space of the non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(W^d)$. |
| title | On the homotopy type of the space of metrics of positive scalar curvature |
| topic | Differential Geometry Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2012.00432 |