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

TL;DR

This paper demonstrates that the homology groups of general linear groups with Steinberg module coefficients form a Hopf algebra, leading to new unstable cohomology classes for special linear groups.

Contribution

It establishes a Hopf algebra structure on these homology groups and constructs new unstable cohomology classes for $SL_n(\mathbb Z)$.

Findings

Homology groups form a commutative Hopf algebra.

The algebra is free and graded commutative.

New infinite families of unstable cohomology classes are constructed.

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 $SL_n(\mathbb Z)$.