## Algebraic & Geometric Topology

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

Thomas John Baird

#### Abstract

Consider the space $Hom(ℤn,G)$ 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 $Hom(ℤn,G)$, 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 2 (2007), 737-754.

Dates
Revised: 13 February 2007
Accepted: 22 March 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796703

Digital Object Identifier
doi:10.2140/agt.2007.7.737

Mathematical Reviews number (MathSciNet)
MR2308962

Zentralblatt MATH identifier
1163.57026

Subjects
Primary: 57S99: None of the above, but in this section

Keywords
Lie groups cohomology

#### Citation

Baird, Thomas John. Cohomology of the space of commuting $n$–tuples in a compact Lie group. Algebr. Geom. Topol. 7 (2007), no. 2, 737--754. doi:10.2140/agt.2007.7.737. https://projecteuclid.org/euclid.agt/1513796703

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