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

Baird, Thomas John

doi:10.2140/agt.2007.7.737

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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

DOI: 10.2140/agt.2007.7.737

arXiv: math.AT/0610761

Abstract

Consider the space $Hom (ℤ^{n}, G)$ 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 $Hom (ℤ^{n}, G)$ , 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.

Keywords

Lie groups, cohomology

Mathematical Subject Classification 2000

Primary: 57S99

References

Publication

Received: 21 November 2006

Revised: 13 February 2007

Accepted: 22 March 2007

Published: 30 May 2007

Authors

Thomas John Baird
Department of Mathematics University of Toronto Toronto Ontario M5S 2E4 Canada

Volume 7, issue 2 (2007)