Proper actions on topological groups: Applications to quotient spaces
aa r X i v : . [ m a t h . GN ] A ug PROPER ACTIONS ON TOPOLOGICAL GROUPS:APPLICATIONS TO QUOTIENT SPACES
SERGEY A. ANTONYAN
Abstract.
Let X be a Hausdorff topological group and G a locally compactsubgroup of X . We show that the natural action of G on X is proper inthe sense of R. Palais. This is applied to prove that there exists a closedset F ⊂ X such that F G = X and the restriction of the quotient projection X → X/G to F is a perfect map F → X/G . This is a key result to prove thatmany topological properties (among them, paracompactness and normality)are transferred from X to X/G , and some others are transferred from
X/G to X . Yet another application leads to the inequality dim X ≤ dim X/G + dim G for every paracompact topological group X and a locally compact subgroup G of X having a compact group of connected components. Introduction
By a G -space we mean a completely regular Hausdorff space together with afixed continuous action of a Hausdorff topological group G on it.The notion of a proper G -space was introduced in 1961 by R. Palais [23] with thepurpose to extend a substantial portion of the theory of compact Lie group actionsto the case of noncompact ones.A G -space X is called proper (in the sense of Palais [23, Definition 1.2.2]), ifeach point of X has a, so called, small neighborhood, i.e., a neighborhood V suchthat for every point of X there is a neighborhood U with the property that the set h U, V i = { g ∈ G | gU ∩ V = ∅} has compact closure in G .Clearly, if G is compact, then every G -space is proper.Many important problems in the theory of proper actions are conjugated (see[18], [1], [2], [8], [7]) to the following major open problem: Conjecture 1.
Let G be a locally compact group. Then the orbit space X/G ofany paracompact proper G -space X is paracompact. This conjecture is open even if X is metrizable; in this case it is equivalent (see[8]) to the following old problem going back to R. Palais [23]: Conjecture 2.
Let G be a locally compact group and X a metrizable proper G -space. Then the topology of X is metrizable by a G -invariant metric. Due to Palais [23], it is known that Conjecture 2 is true for a separable metrizableproper G -space X provided the acting group G is Lie. Other special cases are . 22A05; 22F05; 54H11; 54H15; 54F45. Key words and phrases . Proper G -space; orbit space; locally compact group; dimension.The author was supported in part by grants discussed in [20] and [8]. In particular, in [8], Palais’ result is extended to the caseof a locally compact separable G and a metrizable locally separable X .In this paper we prove Conjecture 1 in an important special case, namely, when X is a topological group endowed with the natural action of its locally compactsubgroup G (see Corollary 1.5). We first show that X is a proper G -space andthen we establish a more general result (Theorem 1.2) which has many interestingapplications in the theory of topological groups.Below all topological groups are assumed to satisfy the Hausdorff separationaxiom. Theorem 1.1.
Let X be a topological group and G a locally compact subgroup of X . Then the action of G on X given by the formula g ∗ x = xg − , g ∈ G , x ∈ X ,is proper. Recall that a subset S of a proper G -space X is called G -fundamental , or just fundamental , if S is a small set and the saturation G ( S ) = { gs | g ∈ G, s ∈ S } coincides with X .Here is the key result of the paper: Theorem 1.2.
Let X be a topological group and G a locally compact subgroup of X . Then there exists a closed G -fundamental set in X . It is easy to prove (see [2, Proposition 1.4]) that in every proper G -space X , therestriction of the orbit map p : X → X/G to any closed small set is perfect (i.e., isclosed and has compact fibers). In combination with Theorem 1.2 this yields thefollowing:
Corollary 1.3.
Let X be a topological group, G a locally compact subgroup of X ,and X/G the quotient space of all right cosets xG = { xg | g ∈ G } , x ∈ X . Thenthere exists a closed subset F ⊂ X such that the restriction p | F : F → X/G is aperfect surjective map.
This fact has the following immediate corollary about transfer of properties from X to X/G : Corollary 1.4.
Let P be a topological property stable under perfect maps and alsoinherited by closed subsets. Assume that X is a topological group with the property P and let G be a locally compact subgroup of X . Then the quotient space X/G alsohas the property P . Among such properties P we single out just some of those which provide newresults in Corollary 1.4; these are: paracompactness, countable paracompactness,weak paracompactness, normality, perfect normality, ˇCech-completeness, being a k -space (see [16, § § § § § § Corollary 1.5.
Let X be a paracompact topological group and G a locally compactsubgroup of X . Then the quotient space X/G is paracompact.
In this connection it is in order to recall the following remarkable result ofA. V. Arhangel’skii [9]: every topological group is the quotient of a paracompactzero-dimensional group. Hence, the local compactness of G is essential in Corol-lary 1.5. PPLICATIONS OF PROPER ACTIONS 3
We recall that a locally compact group is called almost connected if its space ofconnected components endowed with the quotient topology is compact.
Corollary 1.6.
Let P be a topological property stable under open perfect maps andalso inherited by closed subsets. Assume that X is a paracompact group with theproperty P and let G be an almost connected subgroup of X . Then the quotientspace X/G also has the property P . Among properties stable under open perfect maps and also inherited by closedsubsets we highlight strong paracompactness and realcompactness (see [16, Exer-cises 5.3.C(c), 5.3H(d), and Theorem 3.11.4 and Exercises 3.11.G]). Thus, we getthe following:
Corollary 1.7.
Let X be a strongly paracompact (resp., paracompact and real-compact) topological group and G an almost connected subgroup of X . Then thequotient space X/G is strongly paracompact (resp., paracompact and realcompact).
Remark 1.8.
We note that the converse of the first statement in this corollary isnot true. Namely, it is known that the Baire space B ( ℵ ) of weight ℵ (which is,in fact, homeomorphic to a commutative metrizable topological group) is stronglyparacompact and its product with the additive group R of the reals is not (see [16,Exercises 5.3F(a) and 5.3F(b)] ). Hence, the direct product X = R × B ( ℵ ) andits subgroup G = R × { } provide the desired counterexample, answering negativelya question from [11] . Furthermore, [12, Open Problem 3.2.1, p.151] asks whetherevery locally strongly paracompact group is strongly paracompact? The same group R × B ( ℵ ) provides a negative answer to this question too. Indeed, since the productof a compact space and a strongly paracompact space is strongly paracompact (see [16, Exercise 5.3H(a)] ), we infer that R × B ( ℵ ) is locally strongly paracompact. Combining our Corollary 1.5 with a result of Abels [1, Main Theorem], we obtainthe following:
Corollary 1.9.
Let X be a paracompact group, G an almost connected subgroup of X , and K a maximal compact subgroup of G . Then there exists a K -invariant subset S ⊂ X such that X is K -homeomorphic to the product G/K × S . In particular, X is homeomorphic to R n × S for some n ≥ . In [10] A. V. Arhangel’skii has studied properties which are transferred in theopposite direction, i.e., from
X/G to X . The next corollary is a unified result ofthis sort which implies many of those in [10] as well as provides some new ones. Corollary 1.10.
Let P be a topological property invariant and inverse invariantof perfect maps, and also stable under multiplication by a locally compact group.Assume that X is a topological group and let G be a locally compact subgroup of X such that the quotient space X/G has the property P . Then the group X also hasthe property P . Among such properties we mention just some: paracompactness, being a k -space, ˇCech-completeness (see [16, § § § SERGEY A. ANTONYAN
Corollary 1.11.
Let P be a topological property invariant and inverse invariant ofopen perfect maps, and also stable under multiplication by a locally compact group.Assume that X is a paracompact group and let G be an almost connected subgroupof X such that the quotient space X/G has the property P . Then the group X alsohas the property P . Among such properties we highlight realcompactness (see [16, Theorem 3.11.14and Exercise 3.11.G, and also take into account that every locally compact groupis realcompact]).Coorollary1.5 is further applied to prove the following Hurewicz type formula:
Theorem 1.12.
Let X be a paracompact topological group and G an almost con-nected subgroup of X . Then dim X ≤ dim X/G + dim G. Remark 1.13 ([24]) . If in this theorem X is a locally compact group then, in fact,the equality holds: dim X = dim X/G + dim G. All the proofs are given in section 3.2.
Preliminaries
Throughout the paper, unless otherwise is stated, by a group we shall mean atopological group G satisfying the Hausdorff separation axiom; by e we shall denotethe unity of G .All topological spaces are assumed to be Tychonoff (= completely regular andHausdorff). The basic ideas and facts of the theory of G -spaces or topologicaltransformation groups can be found in G. Bredon [15] and in R. Palais [22]. Ourbasic reference on proper group actions is Palais’ article [23]. Other good sourcesare [20], [1] and [2].For the convenience of the reader we recall, however, some more special defini-tions and facts below.By a G -space we mean a topological space X together with a fixed continuousaction G × X → X of a topological group G on X . By gx we shall denote the imageof the pair ( g, x ) ∈ G × X under the action.If Y is another G -space, a continuous map f : X → Y is called a G -map or anequivariant map, if f ( gx ) = gf ( x ) for every x ∈ X and g ∈ G .If X is a G -space, then for a subset S ⊂ X and for a subgroup H ⊂ G , the H -hull (or H -saturation) of S is defined as follows: H ( S )= { hs | h ∈ H, s ∈ S } .If S is the one point set { x } , then the G -hull G ( { x } ) usually is denoted by G ( x )and called the orbit of x . The orbit space X/G is always considered in its quotienttopology.A subset S ⊂ X is called H -invariant if it coincides with its H -hull, i.e., S = H ( S ). By an invariant set we shall mean a G -invariant set.For any x ∈ X , the subgroup G x = { g ∈ G | gx = x } is called the stabilizer (orstationary subgroup) at x .A compatible metric ρ on a metrizable G -space X is called invariant or G -invariant, if ρ ( gx, gy ) = ρ ( x, y ) for all g ∈ G and x, y ∈ X . If ρ is a G -invariant PPLICATIONS OF PROPER ACTIONS 5 metric on any G -space X , then it is easy to verify that the formula e ρ (cid:0) G ( x ) , G ( y ) (cid:1) = inf { ρ ( x ′ , y ′ ) | x ′ ∈ G ( x ) , y ′ ∈ G ( y ) } defines a pseudometric e ρ , compatible with the quotient topology of X/G . If, inaddition, X is a proper G -space then e ρ is, in fact, a compatible metric on X/G [23, Theorem 4.3.4].For a closed subgroup H ⊂ G , by G/H we will denote the G -space of cosets { gH | g ∈ G } under the action induced by left translations.A locally compact group G is called almost connected if the quotient group G/G of G modulo the connected component G of the identity is compact.Such a group has a maximal compact subgroup K , i.e., every compact subgroupof G is conjugate to a subgroup of K [1, Theorem A.5]. The corresponding classicaltheorem on Lie groups can be found in [19, Ch. XV, Theorem 3.1].In 1961 Palais [23] introduced the very important concept of a proper action ofan arbitrary locally compact group G and extended a substantial part of the theoryof compact Lie transformation groups to noncompact ones.Let X be a G -space. Two subsets U and V in X are called thin relative to eachother [23, Definition 1.1.1], if the set h U, V i = { g ∈ G | gU ∩ V = ∅} , called thetransporter from U to V , has a compact closure in G . A subset U of a G -space X iscalled G -small , or just small , if every point in X has a neighborhood thin relativeto U . A G -space X is called proper (in the sense of Palais), if every point in X hasa small neighborhood.Clearly, if G is compact, then every G -space is proper. Furthermore, if G actsproperly on a compact space, then G has to be compact as well. If G is discreteand X is locally compact, the notion of a proper action is the same as the classicalnotion of a properly discontinuous action. When G = R , the additive group of thereals, proper G -spaces are precisely the dispersive dynamical systems with regularorbit space (see [13, Ch. IV]).Important examples of proper G -spaces are the coset spaces G/H with H acompact subgroup of a locally compact group G . Other interesting examples thereader can find in [1], [2], [5], [6] and [20].In what follows we shall need also the definition of a twisted product G × K S ,where K is a closed subgroup of G , and S a K -space. G × K S is the orbit space ofthe K -space G × S on which K acts by the rule: k ( g, s ) = ( gk − , ks ). Furthermore,there is a natural action of G on G × K S given by g ′ [ g, s ] = [ g ′ g, s ], where g ′ ∈ G and[ g, s ] denotes the K -orbit of the point ( g, s ) in G × S . We shall identify S , by meansof the K -equivariant embedding s [ e, s ], s ∈ S , with the K -invariant subset { [ e, s ] | s ∈ S } of G × K S . This K -equivariant embedding S ֒ → G × K S inducesa homeomorphism of the K -orbit space S/K onto the G -orbit space ( G × K S ) /G (see [15, Ch. II, Proposition 3.3]).The twisted products are of a particular interest in the theory of transformationgroups (see [15, Ch. II, § G -space locally is a twistedproduct. For a more precise formulation we need to recall the following well knownnotion of a slice (see [23, p. 305]): Definition 2.1.
Let X be a G -space and K a closed subgroup of G . A K -invariantsubset S ⊂ X is called a K -kernel if there is a G -equivariant map f : G ( S ) → G/K ,called the slicing map, such that S = f − ( eK ) . The saturation G ( S ) is called atubular set and the subgroup K will be referred as the slicing subgroup. SERGEY A. ANTONYAN
If in addition G ( S ) is open in X then we shall call S a K -slice in X .If G ( S ) = X then S is called a global K -slice of X . It turns out that each tubular set with a compact slicing subgroup is a twistedproduct. The tubular neighborhood G ( S ) is G -homeomorphic to the twisted prod-uct G × K S ; namely the map ξ : G × K S → G ( S ) defined by ξ ([ g, s ]) = gs is a G -homeomorphism (see [15, Ch. II, Theorem 4.2]). In what follows we will use thisfact without a specific reference.One of the most powerful results in the theory of topological transformationgroups is Palais’ slice theorem [23, Proposition 2.3.1] which states that, if X is aproper G -space with G a Lie group, then for any point x ∈ X , there exists a G x -slice S in X such that x ∈ S . In general, when G is not a Lie group, it is no longertrue that a G x -slice exists at each point of X (see [4]). Generalizing the case ofLie group actions, in [2] and [7] (see also [4] for the case of compact non-Lie groupactions), approximate versions of Palais’ slice theorem for non-Lie group actionswere proved. Below, in the proof of Theorem 1.12, we shall need the followingglobal slice theorem established by H. Abels [1, Main Theorem]: Theorem 2.2 (Global Slice Theorem) . Let G be an almost connected group, K amaximal compact subgroup of G , and X a proper G -space with a paracompact orbitspace. Then X admits a global K -slice. On any group G one can define two natural (but equivalent) actions of G givenby the formulas g · x = gx, and g ∗ x = xg − , respectively, where in the right parts the group operations are used with g, x ∈ G .Throughout we shall consider the second action only.By U ( G ) we shall denote the Banach space of all right uniformly continuousbounded functions f : G → R endowed with the supremum norm. Recall that f iscalled right uniformly continuous, if for every ε > O of the unity in G such that | f ( y ) − f ( x ) | < ε whenever yx − ∈ O .We shall consider the induced action of G on U ( G ), i.e.,( gf )( x ) = f ( xg ) , for all g, x ∈ G. It is easy to check that this action is continuous, linear and isometric (see e.g., [3,Proposition 7]).
Proposition 2.3.
Let G be a group. Then for every f ∈ U ( G ) , the map f ∗ : G → U ( G ) defined by f ∗ ( x )( g ) = f ( xg − ) , x, g ∈ G , is a right uniformly continuous G -map.Proof. A simple verification. (cid:3) Proofs
Proof of Theorem 1.1 . Choose a neighborhood U of the identity in X such that U = U − and U ∩ G has a compact closure in G . We claim that for every x ∈ X ,the neighborhood xU is G -small. Indeed, let y ∈ X be any point. Two cases arepossible. PPLICATIONS OF PROPER ACTIONS 7
Case 1 . Assume that y ∈ xU G . Then y = xu u h with u , u ∈ U and h ∈ G . We claim that xU h is a neighborhood of y thin relative to xU . Indeed, if g ∈ h xU, xU h i , then g − h − ∈ U ∩ G . Since U ∩ G has a compact closure, wesee that so does the set ( U ∩ G ) h which contains g − . This yields that h xU, xU h i is contained in h − ( U ∩ G ) − , which also has a compact closure. Hence the trans-porter h xU, xU h i has a compact closure, as required. Case 2 . Assume that y / ∈ xU G . In this case y / ∈ xU G . Indeed, if y ∈ xU G thenthe neighborhood xU x − y of y should meet the set xU G . Then xux − y = xvh forsuitable elements u, v ∈ U and h ∈ G . Then y = xu − vh ∈ xU G , a contradiction.Hence the open set V = X \ xU G is a G -invariant neighborhood of y , and V isthin relative to xU because the transporter h xU, V i is empty in this case. (cid:3) Lemma 3.1.
Let U be a unity neighborhood in X such that U = U − and U ∩ G has compact closure in X . Then for every x ∈ X , the neighborhood U x is a G -smallset (under the action g ∗ x ).Proof. Take y ∈ X arbitrary. Since by Theorem 1.1, xU is a G -small set, the point xy must have a neighborhood V which is thin relative to xU . We claim that theneighborhood x − V of the point y is the desired one, i.e., it is thin relative to U x .This claim will follow from the following equality: h U x, x − V i = h xU, V i x. In its turn, this equality results from the following chain of obvious equivalences: g ∈ h U x, x − V i ⇐⇒ U xg − ∩ x − V = ∅ ⇐⇒ x (cid:0) U xg − ∩ x − V (cid:1) = ∅⇐⇒ xU xg − ∩ V = ∅ ⇐⇒ gx − ∈ h xU, V i ⇐⇒ g ∈ h xU, V i x. Thus, the closure h U x, x − V i equals to (cid:0) h xU, V i (cid:1) x . Since h xU, V i is compact andthe right translations are autohomeomorphisms of X , we conclude that (cid:0) h xU, V i (cid:1) x ,and hence, h U x, x − V i is compact. This completes the proof. (cid:3) Proposition 3.2.
Let X be a group and G a locally compact subgroup of X . Thenthere exists a locally finite covering of X consisting of G -invariant open sets of theform S i G , where each S i is an open G -small subset of X .Proof. By Theorem 1.1, X is a proper G -space, and hence, one can choose a G -small neighborhood U of the unity in X . By virtue of Markov’s theorem [12,Theorem 3.3.9], there exists a right uniformly continuous function f : X → [0 , f ( e ) = 0 and f − (cid:0) [0 , (cid:1) ⊂ U. Then, by Proposition 2.3, f induces an X -equivariant map f ∗ : X → U ( X )defined by the rule: f ∗ ( x )( g ) = f ( xg − ) , x, g ∈ X. Denote by Z the image f ∗ ( X ). Clearly, Z is the X -orbit of the point f ∗ ( e ) inthe X -space U ( X ), and the metric of U ( X ) induces an X -invariant metric on Z .We claim that(3.2) f − ∗ (Γ x,V ) ⊂ x − ∗ U = U x, for every x ∈ X, where V = [0 ,
1) and Γ x,V is the open subset { ϕ ∈ U ( X ) | ϕ ( x ) ∈ V } of U ( X ). SERGEY A. ANTONYAN
First we observe that Γ x,V = x − Γ e,V and f − ∗ (Γ e,V ) ⊂ f − ( V ) . Then (3.2) follows from (3.1) and the X -equivariance of f ∗ .Besides, since f ∗ ( x ) ∈ Γ x,V for every x ∈ X , we see that the sets Γ x,V , x ∈ X ,constitute a covering of Z .From now on we restrict ourselves only by the induced actions of the subgroup G ⊂ X , i.e., we will consider X and Z just as G -spaces.Now, it follows from (3.2) and from the G -equivariance of f ∗ that(3.3) f − ∗ (cid:0) G (Γ x,V ) (cid:1) ⊂ U xG, for every x ∈ X. Since f ∗ : X → Z is G -equivariant, it induces a continuous map e f ∗ of the G -orbitspaces, i.e., we have the following commutative diagram: X f ∗ −→ Z ↓ p ↓ qX/G f f ∗ −→ Z/G where p and q are the G -orbit maps.It follows from (3.3) that(3.4) e f ∗− (cid:0) q (Γ x,V ) (cid:1) ⊂ p ( U x ) , for every x ∈ X. Thus, the open covering { p ( U x ) | x ∈ X } of the G -orbit space X/G is refinedby the open covering { e f ∗− (cid:0) q (Γ x,V ) (cid:1) | x ∈ X } .Since the metric of Z is G -invariant, the orbit space Z/G is pseudometrizable(see Preliminaries). Hence the open covering { q (Γ x,V ) | x ∈ X } of Z/G admitsan open locally finite refinement, say { W i | i ∈ I} (see [16, Theorem 4.4.1 andRemark 4.4.2]). Then, clearly, { p − (cid:0) e f ∗− (cid:0) W i ) (cid:1) | i ∈ I} is an open locally finiterefinement of { U xG | x ∈ X } consisting of G -invariant sets. It then follows thateach set p − (cid:0) e f ∗− (cid:0) W i ) (cid:1) is contained in some set U xG , x ∈ X , which yields that p − (cid:0) e f ∗− (cid:0) W i ) (cid:1) = S i G, where S i = p − (cid:0) e f ∗− (cid:0) W i ) (cid:1) ∩ U x .Now, S i , being a subset of the G -small set U x (see Lemma 3.1), is itself G -small.Thus { S i G | i ∈ I} is the desired covering. (cid:3) Proof of Theorem1.2 . Let { S i G | i ∈ I} be the locally finite open covering of X from Proposition 3.2. Then the union S = S i ∈I S i is a G -small set (see e.g., [2,Proposition 1.2(d)]). On the other hand, SG = (cid:16) [ i ∈I S i (cid:17) G = [ i ∈I S i G = X, yielding that S is a G -fundamental subset of X . Since the closure of a G -smallset is G -small (see e.g., [2, Proposition 1.2(b)]), the closure S is the desired closed G -fundamental set. (cid:3) PPLICATIONS OF PROPER ACTIONS 9
Proof of Corollary 1.6 . Since G is almost connected, it has a maximal compactsubgroup K (see Preliminaries). Since by Corollary 1.5, the quotient X/G is para-compact, due to a result of Abels [1, Main Theorem], X admits a global K -slice S , and hence, it is G -homeomorphic to the twisted product G × K S (see Pre-liminaries). Since the group K is compact, it then follows that the K -orbit map G × S → G × K S ∼ = G X is open and perfect. This yields immediately that therestriction p | S : S → X/G of the G -orbit map p : X → X/G is an open and perfectsurjection. Now the result follows. (cid:3)
Proof of Corollary 1.10 . By virtue of Corollary 1.3, X admits a closed subset F ⊂ X such that the restriction of the quotient projection p : X → X/G to F isa perfect surjection p | F : F → X/G . It then follows from the hypothesis that F has the property P . Since a locally compact group is paracompact (even, stronglyparacompact [12, Theorem 3.1.1]), then again by the hypothesis, the product G × F has the property P . Since F is a closed G -small set, the action map G × F → X is perfect (see Abels [2, Proposition 1.4]), and since F G = X we see that the map G × F → X is surjective. Then X has the property P by the hypothesis. (cid:3) Proof of Corollary 1.11 . It is quite similar to the proof of Corollary 1.6. (cid:3)
Proof of Theorem 1.12 . Since G is almost connected, it has a maximal compactsubgroup, say, K (see Preliminaries). Since, by Corollary 1.5, the quotient X/G isparacompact, one can apply the Global Slice Theorem (see Theorem 2.2), accordingto which X admits a global K -slice, say S . Then X is G -homeomorphic to thetwisted product G × K S (see Preliminaries). In turn, due to a result of H. Abels [1,Theorem 2.1], the twisted product G × K S is homeomorphic to the product G/K × S .Thus, X ∼ = G/K × S .Since G is locally compact and paracompact (see [12, Theorem 3.1.1]) and thequotient map G → G/K is open and closed, we infer that
G/K is also locallycompact and paracompact. Further, since S is paracompact, according to a theoremof Morita [21] one has:(3.5) dim ( G/K × S ) ≤ dim G/K + dim S. Since K is compact, according to a result of V. V. Filippov [17] one has theinequality:(3.6) dim S ≤ dim S/K + dim q where q : S → S/K is the K -orbit projection anddim q = sup { dim g − ( a ) | a ∈ S/K } . Further, since K acts freely on S , we see that dim q = dim K. Consequently, combining (3.5) and (3.6) one obtains:(3.7) dim (
G/K × S ) ≤ dim G/K + dim K + dim S/K.
Next, since
X/G ∼ = ( G × K S ) /G ∼ = S/K (see Preliminaries) and dim
G/K +dim K = dim G (see Remark 1.13), it then follows from (3.7) thatdim ( G/K × S ) ≤ dim X/G + dim G, as required. (cid:3) Acknowledgement.
The author thanks the referee for useful comments.
References
1. H. Abels,
Parallelizability of proper actions, global K -slices and maximal compact sub-groups , Math. Ann. (1974), 1–19.2. H. Abels, A universal proper G -space , Math. Z. (1978), 143-158.3. S. A. Antonian, Equivariant embeddings into G -AR’s , Glasnik Matematiˇcki
22 (42) (1987),503–533.4. S. A. Antonyan,
Existence of a slice for arbitrary compact transformation groups , Matem-aticheskie Zametki (1994), 3-9: English transl. in: Math. Notes
56 (5-6) (1994),101-1104.5. S. A. Antonyan,
The Banach-Mazur compacta are absolute retracts , Bull. Acad. Polon. Sci.Ser. Math.
46, no. 2 (1998), 113-119.6. S. A. Antonyan,
Extensorial properties of orbit spaces of proper group actions , TopologyAppl. (1999), 35-46.7. S. A. Antonyan, Orbit spaces and unions of equivariant absolute neighborhood extensors ,Topology Appl. (2005), 289-315.8. S. A. Antonyan and S. de Neymet,
Invariant pseudometrics on Palais proper G -spaces ,Acta Math. Hung.
98 (1-2) (2003), 41-51.9. A. V. Arhangel’skii,
Classes of topological groups , Russian Math. Survays
36, no. 3 (1981),151-174.10. A. V. Arhangel’skii,
Quotients with respect to locally compact subgroups , Houston J. Math.
31, no. 1 (2005), 215-226.11. A. V. Arhangel’skii and V. V Uspenskij,
Topological groups: local versus global , AppliedGeneral Topology
7, no. 1 (2006), 67-72.12. A. Arhangel’skii and M. Tkachenko,
Topological Groups and Related Structures , AtlantisPress/World Scientific, Amsterdam-Paris, 2008.13. N. P. Bhatia, G. P. Szeg¨o,
Stability theory of Dynamical Systems , Springer Verlag, Berlin,1970.14. C. J. R. Borges,
On stratifiable spaces , Pacific J. Math.
17, no. 1 (1966), 1-16.15. G. Bredon,
Introduction to compact transformation groups , Academic Press, 1972.16. R. Engelking,
General Topology , PWN-Pol. Sci. Publ., Warsaw, 1977.17. V. V. Filippov,
Dimensionality of spaces with the action of a bicompact group , Math. Notes,
25, no. 3 (1979), 171-174.18. O. H´ajek,
Parallelizability revisited , Proc. Amer. Math. Soc.,
27, no. 1 (1971), 77-84.19. G. Hochschild,
The structure of Lie groups , Holden-Day Inc., San Francisco, 1965.20. J. L. Koszul,
Lectures on groups of transformations , Tata Institute of Fundamental Re-search, Bombay, 1965.21. K. Morita,
On the Dimension of Product Spaces , American Journal of Mathematics, (1953), pp. 205-223.22. R. Palais,
The classification of G -spaces , Memoirs of the AMS, (1960).23. R. Palais, On the existence of slices for actions of non-compact Lie groups , Ann. of Math. (1961), 295-323.24. E. G. Skljarenko, On the topological structure of locally bicompact groups and their quotientspaces , Amer. Math. Soc. Transl., Ser. 2, (1964), 57-82. Departamento de Matem´aticas, Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, 04510 M´exico Distrito Federal, M´exico.
E-mail address ::