Mod 2 cohomology of 2-local finite groups of low rank

TL;DR

This paper computes the mod 2 cohomology of classifying spaces of free loop groups for certain compact groups and shows their isomorphism to the cohomology of related Chevalley groups over finite fields, revealing deep algebraic structures.

Contribution

It provides explicit calculations of mod 2 cohomology for specific free loop groups and establishes isomorphisms with Chevalley groups' cohomology over finite fields, advancing understanding of their algebraic topology.

Findings

Cohomology of free loop groups for Spin(7), Spin(8), Spin(9), and F_4 computed.

Isomorphisms established between these cohomologies and those of Chevalley groups G(q).

Cohomology of free loop space over BDI(4) shown to match that of BSol(q).

Abstract

We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra.