Equivariant Bundles and Isotropy Representations

Ian Hambleton; Jean-Claude Hausmann

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arXiv · 0704.2763

Equivariant Bundles and Isotropy Representations

Ian Hambleton·Jean-Claude Hausmann

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Abstract

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 $A=Γ\backslash X$ 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 and Goresky-Kottwitz-MacPherson.

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BibTeX RIS

Ian Hambleton, Jean-Claude Hausmann. 2009-06-21. Equivariant Bundles and Isotropy Representations. https://arxiv.org/abs/0704.2763

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