Let G be a locally compact Hausdorff group. We study equivariant absolute (neighborhood) extensors (G-AE's and G-ANE's) in the category G-ℳ of all proper G-spaces that are metrizable by a G-invariant metric. We first solve the linearization problem for proper group actions by proving that each X ∈ G-ℳ admits an equivariant embedding in a Banach G-space L such that L∖{0} is a proper G-space and L∖{0} ∈ G-AE. This implies that in G-ℳ the notions of G-A(N)E and G-A(N)R coincide. Our embedding result is applied to prove that if a G-space X is a G-ANE (resp., a G-AE) such that all the orbits in X are metrizable, then the orbit space X/G is an ANE (resp., an AE if, in addition, G is almost connected). Furthermore, we prove that if X ∈ G-ℳ then for any closed embedding X/G ↪ B in a metrizable space B, there exists a closed G-embedding X ↪ Z (a lifting) in a G-space Z ∈ G-ℳ such that Z/G is a neighborhood of X/G (resp., Z/G = B whenever G is almost connected). If a proper G-space X has metrizable orbits and a metrizable orbit space then it is metrizable (by a G-invariant metric).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-2-3,
author = {Sergey Antonyan},
title = {Proper actions of locally compact groups on equivariant absolute extensors},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {117-145},
zbl = {1184.54035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-2-3}
}
Sergey Antonyan. Proper actions of locally compact groups on equivariant absolute extensors. Fundamenta Mathematicae, Tome 205 (2009) pp. 117-145. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-2-3/