Groups, Geometry, and Dynamics

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Volume 4, Issue 1, 2010, pp. 127–162
DOI: 10.4171/GGD/77

Published online: 2009-12-23

Equivariant bundles and isotropy representations

Ian Hambleton[1] and Jean-Claude Hausmann

(1) McMaster University, Hamilton, Canada

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

Hambleton Ian, Hausmann Jean-Claude: Equivariant bundles and isotropy representations. Groups Geom. Dyn. 4 (2010), 127-162. doi: 10.4171/GGD/77