Abstract:
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.
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