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

Avner Ash; Jeremy Miller; Peter Patzt

paper

Hopf algebras, Steinberg modules, and the unstable cohomology of $S L_{n} (Z)$ and $G L_{n} (Z)$

arXiv:2404.13776

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 $S L_{n} (Z)$ .

11 pages

Hopf algebras, Steinberg modules, and the unstable cohomology of SLn​(Z) and GLn​(Z)

Abstract

Hopf algebras, Steinberg modules, and the unstable cohomology of $S L_{n} (Z)$ and $G L_{n} (Z)$