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Proper actions on topological groups: Applications to quotient spaces

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F of X such that FG=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 that many topological properties (among them, paracompactness and normality) are transferred from X to ferred from X/G to X. Yet another application leads to the inequality dim X<= dim X/G + dim G for every paracompact group X and its locally compact subgroup G.


Introduction
By a G-space we mean a completely regular Hausdorff space together with a fixed continuous action of a Hausdorff topological group G on it.
The notion of a proper G-space was introduced in 1961 by R. Palais [24] with the purpose to extend a substantial portion of the theory of compact Lie group actions to the case of noncompact ones.
A G-space X is called proper (in the sense of Palais [24, Definition 1.2.2]), if each point of X has a, so called, small neighborhood, i.e., a neighborhood V such that for every point of X there is a neighborhood U with the property that the set U, V = {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 [19], [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 of any paracompact proper G-space X is paracompact.
In this paper we prove Conjecture 1 in an important particular case, namely, when X is a topological group endowed with the natural action of its locally compact subgroup G. (see Corollary 1.5). We first show that X is a proper G-space and then we establish a more general result (Theorem 1.2) which has many interesting applications in the theory of topological groups.
Below all topological groups are assumed to satisfy the Hausdorff separation axiom.
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 −1 , 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, the restriction of the orbit map p : X → X/G to any closed small set is perfect (i.e., is closed and has compact fibers). In combination with Theorem 1.2 this yields: 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 lateral classes xG = {xg | g ∈ G}, x ∈ X. Then there exists a closed subset F ⊂ X such that the restriction p| F : F → X/G is a perfect 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 also inherited 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 also has the property P.
Among such properties P we single out just some of those which provide new results in Corollary 1.4; these are: paracompactness, countable paracompactness, weak paracompactness, normality, perfect normality,Čech-completeness, being a kspace (see [16, §5.1, §5.2, §5.3, §1.5, §3.9, §3.3]) and stratifiability (see [14]). Thus, we get the following positive solution of Conjecture 1 in an important special case: Corollary 1.5. Let X be a paracompact topological group and G a locally compact subgroup of X. Then the quotient space X/G is paracompact.
In this connection it is in order to recall the following remarkable result of A. V. Arhangel'skii [9]: every topological group is the quotient of a paracompact zero-dimensional group. Hence, the local compactness of G is essential in Corollary 1.5.
We recall that a locally compact group is called almost connected, if its space of connected components is compact. Corollary 1.6. Let P be a topological property stable under open perfect maps and also inherited by closed subsets. Assume that X is a paracompact group with the property P and let G be an almost connected subgroup of X. Then the quotient space X/G also has the property P.
Among properties stable under open perfect maps and also inherited by closed subsets we highlight strong paracompactness and realcompactness (see [16,Exercises 5.3.C(c), 5.3H(d), and Theorem 3.11.4 and Exercises 3.11.G]). Thus, we get the following Corollary 1.7. Let X be a strongly paracompact (resp., paracompact and realcompact) topological group and G an almost connected subgroup of X. Then the quotient space X/G is strongly paracompact (resp., paracompact and realcompact).
We note that the converse of the first statement in this corollary is not true. Namely, it is known that the Baire space B(ℵ 1 ) of weight ℵ 1 (which is, in fact, a commutative topological group) is strongly paracompact and its product with the aditive 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 1 ) and its subgroup G = R × {0} provide the desired counterexample, answering negatively a question from [11]. Furtheremore, [12, Open Problem 3.2.1, p.151] asks whether every locally strongly paracompact group is strongly paracompact? The same group R × B(ℵ 1 ) provides a negative answer to this question too. Indeed, since the product of a compact space and a strongly paracompact space is strongly paracompact (see [16,Exercise 5.3H(a)]), we infer that R × B(ℵ 1 ) is locally strongly paracompact.
Combining our Corollary 1.5 with a result of Abels [1, Main Theorem], we obtain the following Corollary 1.8. 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 ≥ 1.
In [10] A. V. Arhangel'skii has studied properties which are transferred in the opposite direction, i.e., from X/G to X. The next corollary is a unified result of this sort which implies many of those in [10] as well as provides some new ones. Corollary 1.9. Let P be a topological property invariant and inverse invariant of 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 has the property P.

Corollary 1.10. Let P be a topological property invariant and inverse invariant of open 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 subgroup of X such that the quotient space X/G has the property P. Then the group X also has the property P.
Among such properties we highlight realcompactness (see [16,Theorem 3.11.14 and Exercise 3.11.G, and also take into account that every locally compact group is realcompact]).
Coorollary1.5 is further applied to prove the following Hurewicz type formula: Theorem 1.11. Let G be a locally compact subgroup of a paracompact topological group X. Then 25]). If in this theorem X is a locally compact group then, in fact, the equality holds: All the proofs are given in section 3.

Preliminaries
Throughout the paper, unless otherwise is stated, by a group we shall mean a topological group G satisfying the Hausdorff separation axiom; by e we shall denote the unity of G.
All topological spaces are assumed to be Tychonoff (= completely regular and Hausdorff). The basic ideas and facts of the theory of G-spaces or topological transformation groups can be found in G. Bredon [15] and in R. Palais [23]. Our basic reference on proper group actions is Palais' article [24]. Other good sources are [21], [1] and [2].
For the convenience of the reader we recall, however, some more special definitions and facts below.
By a G-space we mean a topological space X together with a fixed continuous action G × X → X of a topological group G on X. By gx we shall denote the image of 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 an equivariant 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: 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 quotient topology.
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 (or stationary subgroup) at x.
A compatible metric ρ on a metrizable G-space X is called invariant or Ginvariant, if ρ(gx, gy) = ρ(x, y) for all g ∈ G and x, y ∈ X. If ρ is a G-invariant metric on any G-space X, then it is easy to verify that the formula defines a pseudometric ρ, compatible with the quotient topology of X/G. If, in addition, X is a proper G-space then ρ is, in fact, a metric on X/G [24, Theo- 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 space of connected components of G is compact. Such a group has a maximal compact subgroup K, i.e., every compact subgroup of G is conjugate to a subgroup of K [1, Theorem A.5]. The corresponding classical theorem on Lie groups can be found in [20, Ch. XV, Theorem 3.1].
In 1961 Palais [24] introduced the very important concept of a proper action of an arbitrary locally compact group G and extends a substantial part of the theory of 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 each other [24, Definition 1.
Clearly, if G is compact, then every G-space is proper. Furthermore, if G acts properly on a compact space, then G has to be compact as well. If G is discrete and X is locally compact, the notion of a proper action is the same as the classical notion of a properly discontinuous action. When G=R, the additive group of the reals, proper G-spaces are precisely the dispersive dynamical systems with regular orbit space (see [13,Ch. IV]). Important examples of proper G-spaces are the coset spaces G/H with H a compact subgroup of a locally compact group G. Other interesting examples the reader can find in [1], [2], [5], [6] and [21].
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 of the K-space G × S on which K acts by the rule: k(g, s) = (gk −1 , 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 means of the K-equivariant embedding s → [e, s], s ∈ S, with the K-invariant subset The twisted products are of a particular interest in the theory of transformation groups (see [15, Ch. II, § 2]). It turns out that every G-space locally is a twisted product. For a more precise formulation we need to recall the following well known notion of a slice (see [24, p. 305]): It turns out that each tubular set with a compact slicing subgroup is a twisted product. The tubular neighborhood G(S) is G-homeomorphic to the twisted product 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 this fact without a specific reference.
One of the most powerful results in the theory of topological transformation groups states (see [24,Proposition 2.3.1]) that, if X is a proper 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 longer true that a slice exists at each point of X (see [4]). Generalizing the case of Lie group actions, in [7] the following approximate version of Palais' Slice Theorem [24, Proposition 2.3.1] for non-Lie group actions was proved, which plays a key role in the proof of Theorem 1.11: Theorem 2.2 (Approximate Slice Theorem [7]). Let G be a locally compact group, X a proper G-space and x ∈ X. Then for any neighborhood O of x in X, there exist a compact large subgroup K ⊂ G with G x ⊂ K, and a K-slice S such that x ∈ S ⊂ O.

Thus, every proper G-space is covered by invariant open subsets each of which is a twisted product.
Note that for G a compact group this theorem was proved in [4]. A version of it, without requiring K to be a large subgroup, was obtained in [2].
We recall that here a closed subgroup H ⊂ G is called large, if there exists a closed normal subgroup N ⊂ G such that N ⊂ H and G/N is a Lie group.
We refer to [7] for further characterizations of large subgroups.
On any group G one can define two natural (but equivalent) actions of G given by the formulas g · x = gx, and g * x = xg −1 , 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 continuous bounded functions f : G → R endowed with the supremum norm. Recall that f is called right uniformly continuous, if for every ε > 0 there exists a neighborhood O of the unity in G such that |f (y) − f (x)| < ε whenever yx −1 ∈ O.
We shall consider the induced action of G on U (G), i.e., 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
Proof. A simple verification.

Proofs
Proof of Theorem 1.1. Choose a neighborhood U of the identity in X such that U = U −1 and U 3 ∩ 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 are possible. Case 1. Assume that y ∈ xU 2 G. Then y = xu 1 u 2 h with u 1 , u 2 ∈ U and h ∈ G. We claim that xU 2 h is a neighborhood of y thin relative to xU . Indeed, if g ∈ xU, xU 2 h , then g −1 h −1 ∈ U 3 ∩ G. Since U 3 ∩ G has a compact closure, we see that so is the set (U 3 ∩ G)h to which belongs g −1 . This yields that xU, xU 2 h is contained in h −1 (U 3 ∩ G) −1 , which also has a compact closure. Hence the transporter xU, xU 2 h has a compact closure, as required.
Case 2. Assume that y / ∈ xU 2 G. In this case y / ∈ xU G. Indeed, if y ∈ xU G then the neighborhood xU x −1 y of y should meet the set xU G. Then xux −1 y = xvh for suitable elements u, v ∈ U and h ∈ G. Then y = xu −1 vh ∈ xU 2 G, a contradiction.
Hence the open set V = X \ xU G is a G-invariant neighborhood of y, and V is thin relative to xU because the transporter xU, V is empty in this case. This completes the proof. Then, by Proposition 2.3, f induces an X-equivariant map f * : X → U (X) defined by the rule: Denote by Z the image f * (X). Clearly, Z is the X-orbit of the point f * (e) in the X-space U (X), and the metric of U (X) induces an X-invariant metric on Z. We claim that First we observe that Γ x,V = x −1 Γ e,V and . 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, 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 Since f * : X → Z is G-equivariant, it induces a continuous map f * of the G-orbit spaces, i.e., we have the following commutative diagram: It follows from (3.3) that Thus, the open covering Since the metric of Z is G-invariant, the orbit space Z/G is pseudometrizable yielding that S is a G-fundamental subset of X. Since the closure of a G-small set is G-small (see e.g., [2, Proposition 1.2(b)]), the closure S is the desired closed G-fundamental set. This completes the proof.
Proof of Corollary 1.6. Since G is almost connected, it has a maximal compact subgroup K (see Preliminaries). Since by Corollary 1.5, the quotient X/G is paracompact, 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 Preliminaries). 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 the restriction p| S : S → X/G of the G-orbit map p : X → X/G is an open and perfect surjection. Now the result follows.
Proof of Corollary 1.9. 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 is a 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, strongly paracompact [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.
Proof of Corollary 1.10. It is quite similar to the proof of Corollary 1.6.
Proof of Theorem 1.11. Consider three cases.
Casa 1. Assume that G is almost connected. In this case G has a maximal compact subgroup, say, K (see Preliminaries). Since, by Corollary 1.5, the quotient X/G is paracompact, one can apply a theorem of H. Abels [1,Main Theorem], according to which X admits a global K-slice, say S, and hence, X is G-homeomorphic to the twisted product G × K S. By another result of H. Abels [1, Theorem 2.1], X is homeomorphic (moreover, K-equivariantly homeomorphic) to the product G/K×S.
Since G is locally compact and paracompact and the natural quotient map G → G/K is open and closed, we infer that G/K is also locally compact and paracompact. Further, since S is paracompact, according to a theorem of Morita [22] one has: Since K is compact, according to a result of V. V. Filippov [18] one has the inequality: where q : S → S/K is the K-orbit projection and dim q = sup {dim g −1 (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: Next, since X/G ∼ = (G × K S)/G ∼ = S/K (see Preliminaries) and dim G/K + dim K = dim G (see Remark 1.12), it then follows from (3.7) that dim (G/K × S) ≤ dim X/G + dim G, as required.
Case 2. Let G be totally disconnected. Then G is zero-dimensional (see [17,Theorem 1.4.5]). Hence, one has to prove that dim X ≤ dim X/G. By virtue of Approximate Slice Theorem (see Theorem 2.2), X is covered by a family of open tubular sets U i . Assume that f i : U i → G/K i is the corresponding slicing map, where each K i is a compact large subgroup of G.
Let p : X → X/G be the quotient map. Since, by Corollary 1.5, X/G is paracompact the open covering {p(U i )} of X/G has a locally finite closed shrinking {F i }, i.e., F i ⊂ p(U i ) and F i = X/G (see [16,Remark 5.1.7]). Then, putting Φ i = p −1 (F i ), we get a locally finite closed shrinking Thus, X is covered by a locally finite family of closed tubular sets Φ i . Consequently, by virtue of the locally finite sum theorem (see [17,Theorem 3.1.10]), it suffices to show that dim Φ ≤ dim X/G, for every member Φ of the covering {Φ i }.
Let ϕ : Φ → G/K be the slicing map corresponding to a tubular set Φ ∈ {Φ i }, where K is a compact large subgroup of G (of course, depending upon Φ). Since K is a large subgroup, the quotient G/K is locally connected, in fact, it is a manifold [7,Proposition 3.2]. On the other hand G/K is totally disconnected, and hence, G/K should be a discrete space. This yields that, if we put Q = f −1 (eK), then each gQ is closed and open in Φ, and Φ is the disjoint union of the sets gQ one g out of every coset in G/K. In other words, Φ is just homeomorphic to the product G/K × Q.
Next, since G/K is locally compact and paracompact, and Q is paracompact (being a closed subset of the paracompact space X), according to a theorem of Morita [22] one has: Since K is compact, according to a result of V. V. Filippov [18] one has the inequality: (3.9) dim Q ≤ dim Q/K + dim r where r : Q → Q/K is the K-orbit projection. Further, since K acts freely on Q, we see that dim r = dim K ≤ dim G = 0. Consequently, combining (3.8) and (3.9), and taking into account that dim K = 0 = dim G/K (this follows, for instance, from dim G = 0 and Remark 1.12), one obtains: (3.10) dim Φ ≤ dim Q/K.
Further, since Φ is a tubular set with the K-slice Q, one has Φ/G ∼ = Q/K (see Preliminaries). Then it follows from (3.10) that (3.11) dim Φ ≤ dim Φ/G. But Φ/G is a closed subset of X/G, and since X/G is normal, due to monotonicity of the dimension (see [17,Theorem 3.1.4]), one has dim Φ/G ≤ dim X/G. This, together with (3.11), now yields that dim Φ ≤ dim X/G, as required.
Case 3. Let G be arbitrary. By case 1, we have: where G 0 is the identity component of G. Since G/G 0 is a totally disconnected locally compact group which acts naturally on the G 0 -orbit space X/G 0 , then according to case 2, one has: (3.13) dim X/G 0 ≤ dim X/G 0 G/G 0 . Now, since X/G0 G/G0 ∼ = X/G and dim G 0 ≤ dim G, it follows from (3.12) and (3.13) that dim X ≤ dim X/G + dim G. This completes the proof.