Groups, Geometry, and Dynamics

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Volume 4, Issue 1, 2010, pp. 127–162

DOI: 10.4171/GGD/77

Equivariant bundles and isotropy representations

Ian Hambleton (1) and Jean-Claude Hausmann (2)

(1) Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, L8S 4K1, HAMILTON, ONTARIO, CANADA
(2) Section de Mathématiques, Université de Genève, 2–4, rue du Lièvre, Case postale 64, 1211, GENÈVE 4, SWITZERLAND

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split Γ-spaces. We show that equivariant principal G-bundles over split Γ-CW complexes X can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex A = Γ \ X is a graph, with all edge stabilizers toral subgroups of Γ, we obtain a purely combinatorial classification of bundles with structural group G a compact connected Lie group. If G is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof–May–Segal [18] and Goresky–Kottwitz–MacPherson [10].

Keywords: Equivariant bundles