Cohomology of Commuting Varieties of Connected Compact Reductive Lie Groups

TL;DR

This paper computes the rational cohomology of the commuting variety of a compact reductive Lie group by relating it to a simpler variety and using topological theorems to establish an isomorphism.

Contribution

It introduces a method to determine the cohomology of commuting varieties by connecting them to more accessible varieties and analyzing fiber structures.

Findings

Rational cohomology of the commuting variety is explicitly calculated.

A new approach using the Vietoris-Begle theorem is demonstrated.

The cohomology map between related varieties is shown to be an isomorphism.

Abstract

We calculate the rational cohomology of the commuting variety $X_{G, n}$ consisting of $n$-tuples of commuting elements of a compact reductive group $G$. This is done by studying a map from a related variety $Y_{G, n}$, which has easily calculated cohomology. The proof studies the fibers of the map and uses the Vietoris-Begle theorem to prove that the induced map on rational cohomology is an isomorphism.