Proper actions of locally compact groups on equivariant absolute extensors

Sergey Antonyan¹

¹Departamento de Matemáticas Facultad de Ciencias Universidad Nacional Autónoma de México 04510 México D.F., Mexico

Fundamenta Mathematicae, Tome 205 (2009) no. 2, pp. 117-145

Résumé

Let $G$ be a locally compact Hausdorff group. We study equivariant absolute (neighborhood) extensors ($G$-${\rm AE}$'s and $G$-{\rm ANE's}) in the category $G$-$\mathcal M$ 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 $X\in G$-$\mathcal M$ admits an equivariant embedding in a Banach $G$-space $L$ such that $L\setminus\{0\}$ is a proper $G$-space and $L\setminus\{0\}\in G$-AE. This implies that in $G$-$\mathcal M$ 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 $G$-${\rm ANE}$ (resp., a $G$-${\rm AE})$ such that all the orbits in $X$ are metrizable, then the orbit space $X/G$ is an ANE (resp., an ${\rm AE}$ if, in addition, $G$ is almost connected). Furthermore, we prove that if $X\in G$-$\mathcal M$ then for any closed embedding $X/G\hookrightarrow B$ in a metrizable space $B$, there exists a closed $G$-embedding $X\hookrightarrow Z$ (a lifting) in a $G$-space $Z\in G$-$\mathcal M$ 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).

Détail

DOI : 10.4064/fm205-2-3

Keywords: locally compact hausdorff group study equivariant absolute neighborhood extensors g g anes category g mathcal proper g spaces metrizable g invariant metric first solve linearization problem proper group actions proving each g mathcal admits equivariant embedding banach g space setminus proper g space setminus g ae implies g mathcal notions g a g a coincide embedding result applied prove g space g ane resp g orbits metrizable orbit space ane resp addition almost connected furthermore prove g mathcal closed embedding hookrightarrow metrizable space there exists closed g embedding hookrightarrow lifting g space g mathcal neighborhood resp whenever almost connected proper g space has metrizable orbits metrizable orbit space metrizable g invariant metric