Geodesic
Isovariant extensors and the characterization of equivariant homotopy equivalences
Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 857-880

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We extend the well-known theorem of James–Segal to the case of an arbitrary family $\mathcal{F}$ of conjugacy classes of closed subgroups of a compact Lie group $G$: a $G$-map $f\colon\mathbb{X}\to\mathbb{Y}$ of metric $\operatorname{Equiv}_{\mathcal{F}}$-$\mathrm{ANE}$-spaces is a $G$-homotopy equivalence if and only if it is a weak $G$-$\mathcal{F}$-homotopy equivalence. The proof is based on the theory of isovariant extensors, which is developed in this paper and enables us to endow $\mathcal{F}$-classifying $G$-spaces with an additional structure.
Keywords: classifying $G$-spaces, isovariant absolute extensor, weak equivariant homotopy equivalence. 
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     author = {S. M. Ageev},
     title = {Isovariant extensors and the characterization of equivariant homotopy equivalences},
     journal = {Izvestiya. Mathematics },
     pages = {857--880},
     publisher = {mathdoc},
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     number = {5},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2012_76_5_a0/}
}
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S. M. Ageev. Isovariant extensors and the characterization of equivariant homotopy equivalences. Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 857-880. http://geodesic.mathdoc.fr/item/IM2_2012_76_5_a0/