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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2203.03414 |
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| _version_ | 1866911780297506816 |
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| author | Ebert, Johannes Reinhold, Jens |
| author_facet | Ebert, Johannes Reinhold, Jens |
| contents | We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller--Morita--Mumford classes.
Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, (c) use pseudoisotopy theory and algebraic $K$-theory to get at actual diffeomorphism groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_03414 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some rational homology computations for diffeomorphisms of odd-dimensional manifolds Ebert, Johannes Reinhold, Jens Algebraic Topology We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller--Morita--Mumford classes. Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, (c) use pseudoisotopy theory and algebraic $K$-theory to get at actual diffeomorphism groups. |
| title | Some rational homology computations for diffeomorphisms of odd-dimensional manifolds |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2203.03414 |