Loop space homology associated to the mod 2 Dickson invariants

TL;DR

This paper computes the mod 2 loop space homology of classifying spaces related to G_2(q) and BSol(q), revealing algebraic structures over the Steenrod algebra and spectral sequence behaviors.

Contribution

It provides the first detailed calculations of mod 2 loop space homology for these classifying spaces, linking cohomology invariants to algebraic topology structures.

Findings

Computed mod 2 loop space homology as Steenrod algebra modules

Analyzed Bockstein spectral sequences for these spaces

Connected cohomology invariants to algebraic structures

Abstract

The spaces BG_2 and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has for q odd the classifying spaces of the finite groups BG_2(q)and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article compute the mod 2 loop space homology of the 2-completed classifying space of G_2(q) and of BSol(q) for all odd primes q, as algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.