Modular characteristic classes for representations over finite fields

TL;DR

This paper introduces a new system of characteristic classes for finite field representations, enabling the construction of explicit nontrivial cohomology elements in degrees linear in n, advancing understanding of these complex groups.

Contribution

It develops a novel framework of characteristic classes for finite field representations, producing explicit nontrivial cohomology elements in previously unexplored degrees.

Findings

Constructed nontrivial cohomology elements in degrees linear in n.

Produced nontrivial elements in mod p homology and cohomology of automorphism groups of free groups.

Enhanced understanding of the unstable range of cohomology for these groups.

Abstract

The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood.