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ISSN (electronic): 1472-2739
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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


Consider the space Hom(n,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(n,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.

Lie groups, cohomology
Mathematical Subject Classification 2000
Primary: 57S99
Received: 21 November 2006
Revised: 13 February 2007
Accepted: 22 March 2007
Published: 30 May 2007
Thomas John Baird
Department of Mathematics
University of Toronto
Ontario M5S 2E4