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

Avner Ash; Jeremy Miller; Peter Patzt

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

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arxiv 2404.13776 v1 pith:PPEBPXTH submitted 2024-04-21 math.AT math.KTmath.NT

Avner Ash , Jeremy Miller , Peter Patzt This is my paper · ORCID

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keywords mathbbalgebracohomologycommutativegroupshopfsteinbergunstable

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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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