Proper actions of locally compact groups on equivariant absolute extensors

Sergey Antonyan. "Proper actions of locally compact groups on equivariant absolute extensors." Fundamenta Mathematicae 205.2 (2009): 117-145. <http://eudml.org/doc/282650>.

@article{SergeyAntonyan2009,
abstract = {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).},
author = {Sergey Antonyan},
journal = {Fundamenta Mathematicae},
keywords = {proper G-space},
language = {eng},
number = {2},
pages = {117-145},
title = {Proper actions of locally compact groups on equivariant absolute extensors},
url = {http://eudml.org/doc/282650},
volume = {205},
year = {2009},
}

TY - JOUR
AU - Sergey Antonyan
TI - Proper actions of locally compact groups on equivariant absolute extensors
JO - Fundamenta Mathematicae
PY - 2009
VL - 205
IS - 2
SP - 117
EP - 145
AB - 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).
LA - eng
KW - proper G-space
UR - http://eudml.org/doc/282650
ER -