On invariants of foliated sphere bundles
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929443337928704 |
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| author | Nariman, Sam |
| author_facet | Nariman, Sam |
| contents | Morita showed that for each power of the Euler class, there are examples of flat $\mathbb{S}^1$-bundles for which the power of the Euler class does not vanish. Haefliger asked if the same holds for flat odd-dimensional sphere bundles. In this paper, for a manifold $M$ with a free torus action, we prove that certain $M$-bundles are cobordant to a flat $M$-bundle and as a consequence, we answer Haefliger's question. We show that the powers of the Euler class and Pontryagin classes $p_i$ for $i\leq n-1$ are all non-trivial in $H^*(\text{BDiff}^δ_+(\mathbb{S}^{2n-1});\mathbb{Q})$. In the appendix, Nils Prigge corrects a claim by Haefliger about the vanishing of certain classes in the smooth group cohomology of $\text{Diff}_+(\mathbb{S}^3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16310 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On invariants of foliated sphere bundles Nariman, Sam Algebraic Topology Geometric Topology Morita showed that for each power of the Euler class, there are examples of flat $\mathbb{S}^1$-bundles for which the power of the Euler class does not vanish. Haefliger asked if the same holds for flat odd-dimensional sphere bundles. In this paper, for a manifold $M$ with a free torus action, we prove that certain $M$-bundles are cobordant to a flat $M$-bundle and as a consequence, we answer Haefliger's question. We show that the powers of the Euler class and Pontryagin classes $p_i$ for $i\leq n-1$ are all non-trivial in $H^*(\text{BDiff}^δ_+(\mathbb{S}^{2n-1});\mathbb{Q})$. In the appendix, Nils Prigge corrects a claim by Haefliger about the vanishing of certain classes in the smooth group cohomology of $\text{Diff}_+(\mathbb{S}^3)$. |
| title | On invariants of foliated sphere bundles |
| topic | Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2308.16310 |