Linearization of proper group actions

Fundamenta Mathematicae, 2009

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).

Topology and its Applications, 2014

a r t i c l e i n f o a b s t r a c t MSC: 22F05 57N20 54C55 54C25 54H15 46B99

Topology and its Applications, 2017

Let G be a matrix Lie group. We prove that a proper Gspace X of finite structure, which is metrizable by a G-invariant metric, is a G-ANR (resp., a G-AR) iff for any compact subgroup H ⊂ G the Hfixed point set X H is an ANR (resp., an AR). An equivariant embedding result for proper G-spaces of finite structure is also obtained. Problem 1.1 (Jaworowski's Problem). Let G be a compact Lie group and X a metrizable G-space that has a finitely many G-orbit types (e.g., when G is a finite group). Assume that for each compact subgroup H ⊂ G, the H-fixed point set X H = {x ∈ X | hx = x, ∀h ∈ H} is an ANR (resp., an AR). Is then X a G-ANR (resp., a G-AR)? This problem still remains open. Moreover, it is open even in a very special case when G = Z 2 , X = Q = [−1, 1] ∞ , the Hilbert cube, and X Z 2 = { * }, a sigleton. In this case the problem is equivalent to the following old problem of R.D. Anderson posed in the mid-sixties (see [39, p. 292], see also [40, p. 656] and [41, Problem 930]): Problem 1.2 (Anderson's Problem). Let α : Q → Q be a based-free involution, i.e., α 2 = 1 Q and α has a unique fixed point. Is then α conjugate

2011

We apply equivariant joins to give a new and more transparent proof of the following result: if G is a compact Hausdorff group and X a G-ANR (respectively, a G-AR), then for every closed normal subgroup H of G, the H-orbit space X/H is a G/H-ANR (respectively, a G/H-AR). In particular, X/G is an ANR (respectively, an AR).

Topology and its Applications, 1999

Extensorial properties of orbit spaces of locally compact proper group actions are investigated.

2006

Let G be a locally compact Hausdorff group. We study orbit spaces of equivariant ab neighborhood extensors ( G-ANE’s) in the category of all proper G-spaces that are metrizable by a Ginvariant metric. We prove that if a proper G-spaceX is aG-ANE(respectively, aG-ANE(n), n 0), andH a closed normal subgroup of G such that all theH -orbits inX are metrizable, then the H -orbit spaceX/H is aG/H -ANE(respectively, aG/H -ANE(n)). Other related results are also establish  2005 Elsevier B.V. All rights reserved. MSC:54C55; 54C20; 54H15

Bulletin of the American Mathematical Society, 1987

1961

International Journal of Mathematics and Mathematical Sciences, 2003

For a compact Lie groupG, we characterize freeG-spaces that admit freeG-compactifications. For suchG-spaces, a universal compact freeG-space of given weight and given dimension is constructed. It is shown that ifGis finite, then-dimensional Menger freeG-compactumμ nis universal for all separable, metrizable freeG-spaces of dimension less than or equal ton. Some of these results are extended to the case ofG-spaces with a single orbit type.

Topology and its Applications, 2005

Let G be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors (G-ANE's) in the category of all proper G-spaces that are metrizable by a Ginvariant metric. We prove that if a proper G-space X is a G-ANE (respectively, a G-ANE(n), n 0), and H a closed normal subgroup of G such that all the H-orbits in X are metrizable, then the H-orbit space X/H is a G/H-ANE (respectively, a G/H-ANE(n)). Other related results are also established.