#### Volume 7, issue 2 (2007)

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Cohomology of the space of commuting $n$–tuples in a compact Lie group

### Thomas John Baird

Algebraic & Geometric Topology 7 (2007) 737–754
 arXiv: math.AT/0610761
##### Abstract

Consider the space $Hom\left({ℤ}^{n},G\right)$ of pairwise commuting $n$–tuples of elements in a compact Lie group $G$. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of $Hom\left({ℤ}^{n},G\right)$, which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in $G$ and the action of the Weyl group. This is an application of a general theorem concerning $G$–spaces for which every element is fixed by a maximal torus.

##### Keywords
Lie groups, cohomology
Primary: 57S99