Finite Chevalley groups and loop groups

TL;DR

This paper establishes a relationship between the mod ell cohomology of finite Chevalley groups over finite fields and the loop groups of connected compact Lie groups, revealing a deep connection in algebraic topology.

Contribution

It proves an isomorphism between the mod ell cohomology of classifying spaces of finite Chevalley groups and loop groups under certain conditions, extending understanding of their topological properties.

Findings

Cohomology of finite Chevalley groups matches that of loop groups for specific field sizes.

Existence of an integer b linking the cohomologies for q=p^{ab}.

Provides new insights into the structure of classifying spaces in algebraic topology.

Abstract

Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is isomorphic to the mod ell cohomology of the classifying space of the loop group LG for q=p^{ab}, a>0.