Hopf algebras in the cohomology of $\mathcal{A}_g$, $\mathrm{GL}_n(\mathbb{Z})$, and $\mathrm{SL}_n(\mathbb{Z})$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929431557177344 |
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| author | Brown, Francis Chan, Melody Galatius, Søren Payne, Sam |
| author_facet | Brown, Francis Chan, Melody Galatius, Søren Payne, Sam |
| contents | We describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties. By relating the primitives for the coproduct to graph cohomology, we deduce that $\dim H^{2g+k}_c(\mathcal{A}_g)$ grows at least exponentially with $g$ for $k = 0$ and for all but finitely many positive integers $k$. Our proof relies on a new result of independent interest; we use a filtered variant of the Waldhausen construction to show that Quillen's spectral sequence abutting to the cohomology of $BK(\mathbb{Z})$ is a spectral sequence of Hopf algebras. From the same construction, we also deduce that $\dim H^{\binom{n}{2} - n - k}(\mathrm{SL}_n(\mathbb{Z}))$ grows at least exponentially with $n$, for $k = -1$ and for all but finitely many non-negative integers $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11528 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hopf algebras in the cohomology of $\mathcal{A}_g$, $\mathrm{GL}_n(\mathbb{Z})$, and $\mathrm{SL}_n(\mathbb{Z})$ Brown, Francis Chan, Melody Galatius, Søren Payne, Sam Algebraic Geometry Algebraic Topology Number Theory 14K10 (Primary) 14T20, 19D99, 20G10 (Secondary) We describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties. By relating the primitives for the coproduct to graph cohomology, we deduce that $\dim H^{2g+k}_c(\mathcal{A}_g)$ grows at least exponentially with $g$ for $k = 0$ and for all but finitely many positive integers $k$. Our proof relies on a new result of independent interest; we use a filtered variant of the Waldhausen construction to show that Quillen's spectral sequence abutting to the cohomology of $BK(\mathbb{Z})$ is a spectral sequence of Hopf algebras. From the same construction, we also deduce that $\dim H^{\binom{n}{2} - n - k}(\mathrm{SL}_n(\mathbb{Z}))$ grows at least exponentially with $n$, for $k = -1$ and for all but finitely many non-negative integers $k$. |
| title | Hopf algebras in the cohomology of $\mathcal{A}_g$, $\mathrm{GL}_n(\mathbb{Z})$, and $\mathrm{SL}_n(\mathbb{Z})$ |
| topic | Algebraic Geometry Algebraic Topology Number Theory 14K10 (Primary) 14T20, 19D99, 20G10 (Secondary) |
| url | https://arxiv.org/abs/2405.11528 |