# Equivariant bundles and isotropy representations Academic Article

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### abstract

• We introduce a new construction, the isotropy groupoid, to organize the orbit data for split \$\Gamma\$-spaces. We show that equivariant principal \$G\$-bundles over split \$\Gamma\$-CW complexes \$X\$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex \$A=\Gamma\backslash X\$ is a graph, with all edge stabilizers toral subgroups of \$\Gamma\$, 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.

• 2010