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Proper actions of locally compact groups on equivariant absolute extensors

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Antonyan S.

Fundamenta Mathematicae

2009

tom 205

nr 2

117-145

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).

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Antonyan S.

1 2009

Wyniki wyszukiwania

Proper actions of locally compact groups on equivariant absolute extensors