Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\mathbb Z)$ and $GL_n(\mathbb Z)$

Avner Ash; Jeremy Miller; Peter Patzt

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arXiv · 2404.13776

Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\mathbb Z)$ and $GL_n(\mathbb Z)$

Avner Ash·Jeremy Miller·Peter Patzt

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Abstract

We prove that the direct sum of all homology groups of the integral general linear groups with Steinberg module coefficients form a commutative Hopf algebra, in particular a free graded commutative algebra. We use this to construct new infinite families of unstable cohomology classes of $SL_n(\mathbb Z)$.

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BibTeX RIS

Avner Ash, Jeremy Miller, Peter Patzt. 2024-04-21. Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\mathbb Z)$ and $GL_n(\mathbb Z)$. https://arxiv.org/abs/2404.13776

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