Spectra of subrings of cohomology generated by characteristic classes for fusion systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917905251172352 |
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| author | Leary, Ian J. Semeraro, Jason |
| author_facet | Leary, Ian J. Semeraro, Jason |
| contents | If $\mathcal{F}$ is a saturated fusion system on a finite $p$-group $S$, we define the Chern subring $Ch(\mathcal{F})$ of $\mathcal{F}$ to be the subring of the mod-$p$ cohomology $H^*(S)$ of $S$ generated by the Chern classes of $\mathcal{F}$-stable representations of $S$. We show that $Ch(\mathcal{F})$ is contained in $H^*(\mathcal{F})$ and apply a result of Green and the first author to describe its spectrum in terms of a certain category of elementary abelian subgroups of $S$. We obtain similar results for various related subrings, including those generated by characteristic classes of $\mathcal{F}$-stable $S$-sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_10701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectra of subrings of cohomology generated by characteristic classes for fusion systems Leary, Ian J. Semeraro, Jason Group Theory 20J06, 20D20 If $\mathcal{F}$ is a saturated fusion system on a finite $p$-group $S$, we define the Chern subring $Ch(\mathcal{F})$ of $\mathcal{F}$ to be the subring of the mod-$p$ cohomology $H^*(S)$ of $S$ generated by the Chern classes of $\mathcal{F}$-stable representations of $S$. We show that $Ch(\mathcal{F})$ is contained in $H^*(\mathcal{F})$ and apply a result of Green and the first author to describe its spectrum in terms of a certain category of elementary abelian subgroups of $S$. We obtain similar results for various related subrings, including those generated by characteristic classes of $\mathcal{F}$-stable $S$-sets. |
| title | Spectra of subrings of cohomology generated by characteristic classes for fusion systems |
| topic | Group Theory 20J06, 20D20 |
| url | https://arxiv.org/abs/2404.10701 |