Strong G-fibrations and orbit projections

Topology and its Applications 163 (2014) 46–65

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Topology and its Applications www.elsevier.com/locate/topol

Strong G-fibrations and orbit projections Alexander Bykov ∗ , Aura Lucina Kantún Montiel Benemérita Universidad Autónoma de Puebla, Av. San Claudio y Rio Verde, Ciudad Universitaria, Colonia San Manuel, CP 72570, Puebla, Pue., Mexico

a r t i c l e

i n f o

MSC: 54C55 54C56 54H15 Keywords: G-space G-ANR G-fibration Orbit map

a b s t r a c t By a strong G-fibration we mean a G-map which has the right lifting property with respect to all G-SSDR-maps (these maps represent an equivariant version of SSDR-maps in the sense of F. Cathey). We show that for any compact Hausdorff group G the following natural projections are strong G-fibrations: G/K → G/H for closed subgroups H and K of G such that K ⊂ H and G/K is metrizable; the K-orbit map E → E/K if E is a metrizable G-space with only one orbit type and K is a closed normal subgroup of G. © 2013 Elsevier B.V. All rights reserved.

0. Introduction Following, for instance, [16, p. 53] we use the term “G-fibration” for the G-maps which have the equivariant homotopy lifting property with respect to any G-space; in other words, the G-maps which have the equivariant right lifting property (ERLP) with respect to all the G-embeddings X × 0 → X × [0, 1]. One can notice that the G-fibrations of G-ANR-spaces have somewhat stronger lifting property: they have the ERLP for a much more wide class of closed G-embeddings called G-SSDR-maps; the notion of an SSDR-map was introduced by F. Cathey in [11]. According to the general approach to the concepts of “fibration” and “fibrant object” (see e.g., [12]), we define a strong G-fibration as a G-map which has the ERLP with respect to all the G-SSDR-maps; consequently, a G-fibrant space is defined as a G-space E for which the constant map E → ∗ is a strong G-fibration. It follows from the definition that every G-ANR-space is G-fibrant and, moreover, the inverse limit of any inverse sequence of G-ANR’s bonded by G-fibrations is G-fibrant too. The class of G-fibrant spaces is much wider than one of G-ANR’s, but G-fibrant spaces still have such property of G-ANR’s as the equivariant homotopy extension property with respect to all closed G-pairs (X, A) of metrizable spaces. It was proved in [6] that the quotient space G/H is G-fibrant, provided that H is a closed subgroup of the compact metrizable group G. The proof of this statement is based, in fact, on the following well-known theorem: any compact Hausdorff group G is a pro-Lie group (i.e., it is an inverse limit of some inverse system * Corresponding author. E-mail addresses: [email protected] (A. Bykov), [email protected] (A.L. Kantún Montiel). 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.10.006

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of compact Lie groups bonded by continuous epimorphisms). It follows that every compact metrizable group G and, even more generally, every metrizable quotient space G/H can be represented as an inverse limit of some inverse sequence of G-ANR’s bonded by G-fibrations. In this context one can say that G/H has a natural fibrant structure. In this paper we show that the natural projection of the quotient spaces p : G/K → G/H, provided that G/K is metrizable, has also a natural fibrant structure in the sense that p can be represented as an inverse limit of some inverse sequence of G-fibrations of G-ANR’s bonded by “G-fibrations in the category of maps” (called G-fibered squares in the present paper, see Section 4). As a consequence, we get our first main result: p is a strong G-fibration (Theorem 5.1). At the end of Section 5, we prove the second main result: the K-orbit map π : E → E/K is a strong G-fibration provided that E is a metrizable G-space with only one orbit type and K is a closed normal subgroup of G (Theorem 5.5). This result is also a consequence of the natural fibrant structure of the quotient spaces, but requires more work which is done in the proof of Lemma 5.3. Clearly our main results can be considered as generalizations (to the case of a compact non-Lie group G) of the following well-known fact: p and π are G-fibrations under the condition that G is a compact Lie group (see e.g., [16, p. 54]). 1. Preliminaries The letter G will always denote a compact Hausdorff group. All the topological spaces are supposed to be Tychonoff, though the main results concern metrizable spaces. The basic definitions and facts of the theory of G-spaces can be found in [10], [13] and [16]. We recall some of them below for the convenience of the reader. By a G-space we mean a topological space X together with a fixed continuous action · : G × X → X, (g, x) → g ·x, of G on X. It is used to write simply gx instead of g ·x. Let X and Y be G-spaces. A continuous map f : X → Y is said to be a G-map or an equivariant map if f (gx) = gf (x) for every (g, x) ∈ G × X. If a G-map f is a homeomorphism, it is called a G-homeomorphism. We say that X and Y are G-equivalent if there exists some G-homeomorphism X → Y . A homotopy F : X × I → Y , where I = [0, 1], is called a G-homotopy if it is a G-map considering X × I with the action g(x, t) = (gx, t). For a given closed subgroup H of G, the quotient space G/H, that is, the set of left cosets {gH | g ∈ G}, is a G-space with the action g · g  H = gg  H. Let X be a G-space. For x ∈ X, the subgroup Gx = {g ∈ G | gx = x} is called the isotropy group at x and the G-space G(x) = {gx | g ∈ G} is called the G-orbit of x. The conjugacy class (H) of any closed subgroup H of G will be called a G-orbit type (in [10] it is called G-isotropy type). The reason for this terminology is the following: if X is a G-space, G(x) is the orbit of x ∈ X and H = Gx is the isotropy group at x, then       (H) = gHg −1  g ∈ G = Gy  y ∈ G(x) and the orbit G(x) is G-equivalent to G/Gy for each y ∈ G(x). A G-space X has only one orbit type (H) if (Gx ) = (H) for each point x ∈ X; in this case every orbit is G-equivalent to G/H. Given a G-space X, the set of its G-orbits, endowed with the quotient topology, is called the G-orbit space of X and is denoted by X/G. The natural projection πX : X → X/G (defined by πX (x) = G(x)) is called the G-orbit map or the G-orbit projection of X. If f : X → Y is a G-map, then there exists a unique continuous map f /G : X/G → Y /G, called the map induced by f , such that the diagram

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X

f

πX

X/G

Y πY

f /G

Y /G

commutes. Clearly, f /G is defined by (f /G)(G(x)) = G(f (x)). If N is a closed normal subgroup of G and X is a G-space, the N -orbit space X/N is a G/N -space with the action (gN ) · (N (x)) = N (gx). Every G-map f : X → Y induces a G/N -map f /N : X/N → Y /N given by (f /N )(N (x)) = N (f (x)). Note that X/N , being a G/N -space, can be also regarded as a G-space (with the action g · N (x) = N (gx)), and the N -orbit map X → X/N can be considered as a G-map. In particular, πX : X → X/G is a G-map if X/G is taken with the trivial action of G. 2. G-ANR’s and large subgroups By a G-ANR or a G-ANR-space we mean, of course, a G-equivariant absolute neighborhood retract for the class of all metrizable G-spaces (see, for instance, [1–3] and [13] for the equivariant theory of retracts). Let us recall some important facts concerning G-ANR’s which will be used in this paper. Proposition 2.1 ([3, Theorem 1.1]). Let G be a compact Hausdorff group and X a G-ANR. Then for every closed normal subgroup N of G, the N -orbit space X/N is a G/N -ANR. In particular, X/G is an ANR. Proposition 2.2 ([13, Corollary 1.6.7]). Let H be a closed subgroup of a compact Lie group G. Then G/H is a G-ANR. The property of the quotient space G/H to be a G-ANR deserves special attention; it is characterized in [1] and [4] as follows: Proposition 2.3. Let H be a closed subgroup of a compact group G. Then the following conditions are equivalent: (1) (2) (3) (4)

G/H G/H G/H there

is a G-ANR, is locally connected and dim G/H < ∞, is a smooth manifold, exists a closed normal subgroup N of G such that N ⊆ H and G/N is a Lie group.

Definition 2.4 (S. Antonyan [1, Definition 2]). A closed subgroup H of a compact group G is called large if it satisfies the condition (4) (and hence all the equivalent conditions (1)–(4)) of Proposition 2.3. From the definition we have the following: (i) If H and K are closed subgroups of G such that H is large and H ⊂ K, then K is large. (ii) Let N be a closed normal subgroup of G. Then N is large iff the quotient group G/N is a Lie group. Proposition 2.5. Let H be a closed subgroup of a compact group G. If the quotient space G/H is metrizable then there exists a decreasing sequence {Ni }i∈N of large normal subgroups of the group G such that  i∈N (Ni H) = H. Therefore  j lim ←− G/(Ni H), qi = G/H, where qij : G/(Nj H) → G/(Ni H), j  i, are the natural projections.

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 Proof. Let (Ui )i∈N be a decreasing sequence of neighborhoods of {eH} in G/H such that i∈N Ui = {eH} i = q −1 (Ui ), i ∈ N, where q : G → G/H is the projection. According to [10, 0.4.4], we choose, for and let U i and G/N  is a Lie group. The sequence (N  )i∈N each i, a closed normal subgroup Ni such that Ni ⊂ U i i  can be substituted by a decreasing one (Ni )i∈N using simple induction: N1 := N1 , Ni+1 := Ni ∩ Ni+1 . Note that all G/Ni are Lie groups as well as G/Ni (see e.g., [14, Chapter 8, Section 46(A)]). Since q(Ni H) =   i = H. Thus  (Ni H) = H. Using this fact (and q(Ni ) ⊂ q(Ni ) ⊂ Ui , we have H ⊂ i∈N (Ni H) ⊂ i∈N U i∈N j {G/(N H), q } = G/H. 2 compactness of G/H), it is easy to check that lim i i ←− Definition 2.6. It will be said that a sequence (Ni )i∈N of subgroups of a compact group G is a pro-Lie sequence for G/H if it satisfies the conditions of Proposition 2.5. 3. G-fibrations and strong G-fibrations By a G-fibration we mean a natural equivariant version of a Hurewicz fibration: a G-map p : E → B is called a G-fibration if it has the equivariant homotopy lifting property (EHLP) with respect to every G-space X (see [16, p. 53]). Recall that, in case of metrizable E and B, it suffices to show that p has the EHLP with respect to all metrizable G-spaces in order to state that p is a G-fibration (see Remark A.1). One can note that the G-fibrations arise in the theory of G-spaces in a quite natural way due to the following statement. Proposition 3.1. Let G be a compact Lie group and H a closed subgroup of G. Then any G-map E → G/H is a G-fibration, provided that E is a metrizable G-space. This proposition seems to be well-known; it can be found, for instance, in [16, Exercise 7, p. 54] (even without the restriction for E to be metrizable). But we could not find any adequate reference to its proof and, since it is important for the present work, we give a detailed proof of Proposition 3.1 in Appendix A. In order to define the notion of a strong G-fibration, we shall use an equivariant version of SSDR-map (in the sense of the work of F. Cathey [11]). The explicit definition of a G-SSDR-map can be found in [8]. Here we point out only the following description of such maps (see [8, Theorem 2.1(c)]): a G-map s : A → X of metrizable G-spaces is a G-SSDR-map if and only if it is a closed G-embedding such that for every G-fibration p : E → B of G-ANR-spaces and for every commutative diagram of G-maps A

f

p

s

X

E

F

(3.1)

B

there exists a filler F : X → E (i.e., a G-map F : X → E such that pF = F and Fs = f ). Of course, every G-SDR-map s : A → X (that is, the G-map which embeds A in X as a strong deformation G-retract) of metrizable G-spaces is a G-SSDR-map. Definition 3.2. A G-map p : E → B of metrizable G-spaces is called a strong G-fibration if for any diagram (3.1) of G-maps, where s : A → X is a G-SSDR-map, there exists a G-map F : X → E as a filler. Definition 3.3. A metrizable G-space E is called an equivariant fibrant space or a G-fibrant space if the G-map E → ∗ is a strong G-fibration (where ∗ is the one-point set with the trivial action of G).

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For instance, every G-ANR is a G-fibrant space. Moreover, the inverse limit of every inverse sequence {Ei , qij } of G-ANR-spaces Ei and G-fibrations qij is a G-fibrant space [6, Corollary 3.6] (see also [11, Examples (2.2)] for the non-equivariant case). Clearly, strong G-fibrations are G-fibrations. On the other hand, it follows immediately from the description of G-SSDR-maps and Definition 3.2 that every G-fibration of G-ANR’s is a strong G-fibration. Moreover, it can be shown that every G-fibration of G-fibrant spaces is a strong G-fibration (this fact is proved in Appendix A.3). Note also that if p : E → B is a strong G-fibration such that B is a G-fibrant space, then E is a G-fibrant space too. Throughout the paper we will make use of the following fact already mentioned in Section 1: for a given closed normal subgroup N of a group G, every G/N -space X can also be considered as a G-space; if ∗ is the action of G/N on X, then the action · of G on X is defined by g · x = gN ∗ x. Consequently, every G/N -map of G/N -spaces always may be regarded as a G-map. Proposition 3.4. Let N be a closed normal subgroup of a compact group G. Then: (a) If E is a G/N -ANR, then E is a G-ANR. (b) If p : E → B is a G/N -fibration, then p is also a G-fibration. (c) If s : A → X is a G-SSDR-map, then the induced map s/N : A/N → X/N is a G/N -SSDR-map. (d) If p : E → B is a strong G/N -fibration, then p is also a strong G-fibration. (e) If E is a G/N -fibrant space, then E is a G-fibrant space. The proof of the assertions (a)–(c) and (e) is given in [9, Proposition 2.1]; (d) follows immediately from (c) and the definition of a strong G-fibration. Also note that by (e), any fibrant space (in the sense of [11]) can be regarded as a G-fibrant space with the trivial action. For the purposes of the paper we shall need one more kind of fibrations given by the following definition. Definition 3.5. A G-map p : E → B of metrizable G-spaces will be called a regular G-fibration if for any closed G-subset A of a metrizable G-space X and any diagram of G-maps f

X × {0} ∪ A × I

E p

X ×I

F

B

there exists a G-homotopy X × I → E as a filler. Clearly, every regular G-fibration is a G-fibration. On the other hand, every strong G-fibration is a regular G-fibration. In particular, every G-fibration of G-ANR’s is a regular G-fibration. It follows from the fact that the inclusion i : X × {0} ∪ A × I → X × I is a G-SSDR-map. The last statement can be easily checked. Indeed, suppose that V is an invariant neighborhood of T = X × {0} ∪ A × I in X × I. Then (by compactness of I and G) there is an open invariant neighborhood U of A in X such that U × I ⊂ V . Since X is metrizable, we can find a G-map ϕ : X → I (in this case it means that ϕ(gx) = ϕ(x) for all g ∈ G and x ∈ X) satisfying ϕ(X \ U ) = 0 and ϕ(A) = 1. The G-map

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D : (X × I) × I → X × I, defined by D((x, t), τ ) = (x, ϕ(x)tτ + t − tτ ), deforms X × I into V leaving the points of T fixed. Hence i is a G-SSDR-map according to the explicit definition given in [8]. The regular G-fibrations have the following local characterization (cf., [16, Exercise 5, p. 53]). Proposition 3.6. Let p : E → B be a G-map of metrizable G-spaces. Suppose that there is a covering U of B by open G-sets such that the restriction of p, p−1 (U ) → U is a regular G-fibration for each U ∈ U. Then p is a regular G-fibration. The proof of this proposition is given in Appendix A.2. The notion of a large subgroup allows us to present the following results, which are well-known for a compact Lie group G, in a slightly more general form. Proposition 3.7. Let H and K be closed subgroups of a compact group G such that K ⊂ H. If K is a large subgroup then the natural projection G/K → G/H is a strong G-fibration. Proof. Let N be a closed normal subgroup of the group G such that G/N is a Lie group and N ⊂ K. Then the projection p : G/K → G/H can be considered as a G/N -map because N acts trivially on G/K and G/H and we have the following commutative diagram of G/N -maps ϕ

G/K

(G/N )/(K/N ) p

p

G/H

ψ

(G/N )/(H/N )

where ϕ and ψ are the natural G/N -equivariant homeomorphisms defined by ϕ(gK) = (gN )(K/N ) and ψ(gH) = (gN )(H/N ) respectively, and p is the natural projection. Since G/N is a compact Lie group, the G/N -map p is a G/N -fibration by Proposition 3.1, and thus so is p. Therefore p is also a G-fibration (see Proposition 3.4(b)). Moreover, it is a strong a G-fibration, because G/K and G/H are G-ANR’s. 2 The next proposition is an immediate consequence of [10, Theorem II.5.8]. Proposition 3.8. Let H be a large subgroup of a compact group G. Suppose that E is a Tychonoff G-space with all orbits of type (H). Then the G-orbit map π : E → E/G satisfies the following property: for each point b ∈ E/G, there exists an open neighborhood U of b in E/G and a G-homeomorphism ϕU : π −1 (U ) → G/H × U such that the diagram ϕU

π −1 (U )

G/H × U pr U

π

(3.2)

U commutes. Proof. Let N be a closed normal subgroup of G such that N ⊂ H and G/N is a Lie group. Since N ⊂ H and all the isotropy groups are conjugate to H, N acts trivially on E and hence one can write E/N = E and consider E as a G/N -space. Since G/N is a Lie group, for each point b ∈ E/(G/N ) = E/G, there exists an open neighborhood U of b in E/G and a G/N -homeomorphism π −1 (U ) → (G/N )/(H/N ) × U over U (by [10, II.5.8]) which can also be regarded as the G-homeomorphism ϕU in the diagram (3.2). 2

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4. G-fibered squares One of the technical tools used in this paper is the notion of G-fibered square. One can notice that the property which we utilize in its definition is analogous to the property used in [12] in order to characterize the fibrations in a pro-category. Definition 4.1. A commutative diagram of G-maps f

E

E

p

(4.1)

p f

B

B

is called a G-fibered square if in the commutative diagram E f q p

P

f

E

p

B

(4.2)

p f

B

where the internal square is a pull-back diagram, the induced map q is a G-fibration. Suppose that all the G-spaces in (4.2) are metrizable. If the G-map q in the diagram (4.2) is a strong G-fibration we shall say that the diagram (4.1) is a strongly G-fibered square. Similarly, if q is a regular fibration, the diagram (4.1) is called a regular G-fibered square. Clearly, every strongly G-fibered square is regular. Note also that if in the (strongly, regular) G-fibered square (4.1) the G-map p is a (strong, regular) G-fibration, then p is also a (strong, regular) G-fibration. We omit a routine proof of the next simple proposition which clarifies the reason to introduce the notion of a fibered square. Proposition 4.2. Let {pi } : E → B be a level map of the inverse sequences E = {Ei , qij } and B = {Bi , rij } of G-spaces and G-maps such that the diagram Ei

qii+1

pi+1

pi

Bi

Ei+1

rii+1

Bi+1

is a (strongly) G-fibered square for every i. If p1 is a (strong) G-fibration then p = lim ←− {pi } : lim ←− E → lim ←− B is a (strong) G-fibration.

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Lemma 4.3. Let H, K and N be closed subgroups of a compact group G such that H ⊂ K∩N and KN = N K. If H is large, then the diagram G/H

αN

G/N

αK

G/K

(4.3)

βK

βN

G/KN

is a strongly G-fibered square. The condition KN = N K is necessary and sufficient in order to have KN as a subgroup of G. Of course, this condition holds if one of the groups K or N is normal. Proof of Lemma 4.3. This lemma is a simple consequence of the following claim: Claim. Under the conditions of the lemma the diagram of the natural projections G/(K ∩ N )

 βN

G/N

 βK

G/K

βK

βN

G/KN

is a pull-back diagram (even if the subgroup H is not large). Proof of the Claim. Let P be the pull-back of the projections βK and βN given by    P = (xK, yN ) ∈ G/K × G/N  xKN = yKN with the diagonal action of G. We must show that the map ϕ : G/(K ∩ N ) → P defined by ϕ(x(K ∩ N )) = (xK, xN ) is a G-homeomorphism. Clearly, ϕ is a well-defined continuous G-map   and βN . It is also obvious that ϕ is injective, because being a unique map induced by the projections βK −1 the equality (xK, xN ) = (yK, yN ) means that y x ∈ K ∩ N so that x(K ∩ N ) = y(K ∩ N ). Let us check that ϕ is surjective. Suppose that (xK, yN ) ∈ P . Note that the equality xKN = yKN implies that xK ∩ yN = ∅. Indeed, if xK ∩ yN = ∅, then xKN ∩ yN = ∅, but on the other hand we have xKN = yKN ⊇ yN ; a contradiction. Therefore there is z ∈ xK ∩ yN and we have z = xk = yn for some k ∈ K and n ∈ N . This means that ϕ(z(K ∩ N )) = (xK, yN ), because zK = xkK = xK and zN = ynN = yN . Thus the G-map ϕ is a continuous bijective map of compact spaces. Hence ϕ is a G-homeomorphism, and we have proved the Claim. Now let us notice that the natural projection q : G/H → G/(K ∩ N ) is a unique G-map which exists, by the pull-back property, for the projections αK : G/H → G/K and αN : G/H → G/N (i.e., such that   q and αN = βN q). Thus the diagram (4.3) is strongly G-fibered because q is a strong G-fibration αK = βK by Proposition 3.7. 2 Let us point out two simple corollaries of Lemma 4.3 which will be used in the proof of the main theorems.

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Corollary 4.4. Let H, K, M , N be closed subgroups of a compact group G such that K ⊂ H, M ⊂ N . If the subgroups M and N are normal and, moreover, M is large, then the diagram G/KM

G/KN

G/HM

G/HN

is a strongly G-fibered square. Proof. Since M ⊂ KM , the subgroup KM is large as well as M . Obviously, KM ⊂ HM ∩ KN and HM · KN = HK · M N = HN = KH · N M = KN · HM. Hence the diagram is a strongly G-fibered square by Lemma 4.3.

2

Similarly the next corollary can be easily verified: Corollary 4.5. Let H, K and N be closed subgroups of a compact group G such that N H = HN , KH = HK and KN = N K. If H is a large subgroup then the diagram of the natural projections G/H

G/N H

G/KH

G/KN H

is a strongly G-fibered square. In the next section we shall use that the regular G-fibered squares can be characterized “locally” as well as regular G-fibrations: Proposition 4.6. Suppose that it is given the commutative diagram (4.1) of G-maps of metrizable G-spaces. Let U be a covering of the G-space B by open G-sets. If for each U ∈ U the restriction of the diagram (4.1) f

−1  −1

p



(U )

f

p−1 (U )

p

f −1 (U )

p f

U

is a regular G-fibered square, then the whole diagram (4.1) is a regular G-fibered square.  = f−1 (p−1 (U )) for every U ∈ U. Note that q −1 (U ) = Proof. Consider the diagram (4.2) and let U  −1 −1 f (p (U )) and, in the commutative diagram

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  f  −1 p−1 (U ) f q

p−1 (U )

 U

p

p

f

−1

(U )

f

U

the internal square is a pull-back (being a restriction of the pull-back square of the diagram (4.2)) so that  , is a regular G-fibration by the hypothesis. Since the family {U  }U ∈U the restriction of q, f  −1 (p−1 (U )) → U  is an open covering of P , we can apply Proposition 3.6 in order to state that the G-map q : E → P is a regular G-fibration. 2 5. Orbit maps as strong G-fibrations In this section we prove our main results: Theorems 5.1 and 5.5. The first theorem is an easy consequence of Corollary 4.4 and Proposition 2.5 proved earlier, while the proof of the second theorem needs one more result about G-fibered squares which will be given in Lemma 5.3. Theorem 5.1. Let H and K be closed subgroups of a compact group G such that K ⊂ H and G/K is metrizable. Then the projection p : G/K → G/H is a strong G-fibration. Proof. Let (Ni )i∈N be a pro-Lie sequence (of subgroups of G) for G/K (see Proposition 2.5 and Defini  tion 2.6). Since i∈N Ni ⊂ i∈N (Ni K) = K ⊂ H, it is easily seen that (Ni )i∈N will also be a pro-Lie sequence for G/H. Hence, for p : G/K → G/H, we have   i+1 i+1  p = lim , ri , ←− p/Ni , qi where qii+1 , rii+1 and p/Ni are the natural projections in the commutative diagram G/KNi

qii+1

p/Ni+1

p/Ni

G/HNi

G/KNi+1

rii+1

G/HNi+1

It is a strongly G-fibered square by Corollary 4.4. Since p/N1 is a strong G-fibration in virtue of Proposition 3.7, the limit map p is also a strong G-fibration by Proposition 4.2. 2 Let us mention one obvious corollary of Theorem 5.1; it slightly improves the result of [6]. Corollary 5.2. Let H be a closed subgroup of a compact group G such that the quotient space G/H is metrizable. Then G/H is a G-fibrant space. Proof. G/H is G-fibrant, because we have the strong G-fibration G/H → G/G, where the one-point set G/G is trivially G-fibrant. 2

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Lemma 5.3. Let K, N and H be closed subgroups of the compact group G such that H is large and the subgroups K and N are normal. Let E be a metrizable G-space with all orbits of one type (H). Then the commutative diagram of the orbit maps πN

E

E/N

πK

(5.1)

pK

E/K

pN

E/KN

is a strongly G-fibered square in which the projections πK , πN , pK and pN are strong G-fibrations. In the proof of this lemma it will be used the following result proved by the authors in [7]. Lemma 5.4 ([7, Lemma 4.3]). Let H be a large subgroup of a compact group G. If a metrizable G-space E has only one orbit type (H), then there is a closed G-embedding E → U such that U is a G-ANR-space with all orbits of the same type (H). Proof of Lemma 5.3. Let us denote by π, qK , qN and qKN the natural projections E → E/G, E/K → E/G, E/N → E/G and E/KN → E/G, respectively. Since H is large, we can choose an open covering U of E/G such that for each U ∈ U there is a G-homeomorphism ϕU : π −1 (U ) → G/H × U for which the diagram ϕU

π −1 (U )

G/H × U pr U

π

U commutes (see Proposition 3.8). The G-homeomorphism ϕU induces the G-homeomorphisms (of the corresponding orbit spaces) −1 ϕU /K : qK (U ) → G/KH × U,

−1 ϕU /N : qN (U ) → G/N H × U,

−1 ϕU /KN : qKN (U ) → G/KN H × U

such that the following diagram commutes πN

π −1 (U )

pK

πK pN

−1 (U ) qK

G/H × U

ϕU /K

−1 (U ) qN

−1 qKN (U )

αN ×id

ϕU /N

G/N H × U

αK ×id

G/KH × U

βN ×id

βK ×id

G/KN H × U

It follows from Corollary 4.5 that the lower square of this diagram is strongly G-fibered. Hence the upper −1 square is strongly G-fibered too. In particular, it is a regular G-fibered square. Since {qKN (U )}U ∈U is an

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57

open covering of E/KN , we conclude that the diagram (5.1) is a regular G-fibered square in virtue of Proposition 4.6. To see that the K-orbit map πK : E → E/K is a (regular) G-fibration, just put, for a moment, N = G in the regular G-fibered square (5.1). Then pK = idE/G is (trivially) a regular G-fibration, and so is πK . Consequently, the maps pK and pN (and, of course, πN ) in the square (5.1) are G-fibrations. Indeed, for instance, pK can be regarded as the KN/N -orbit map E/N → E/N/(KN/N ) of the metrizable G/N -space E/N . Note that all the G/N -orbits of E/N are of the type (HN/N ) and HN/N is a large subgroup of G/N . Therefore pK is a G/N -fibration (by the already proved fact) and hence it is also a G-fibration. By the way, the same argument shows that the projections πK , πN , pK and pN will be strong G-fibrations if we prove that the square (5.1), as a matter of fact, is strongly G-fibered. According to Lemma 5.4 we can assume that E is an invariant closed subset of a G-ANR-space U whose orbits are of the same type (H). Let us consider the following two similar commutative diagrams E

U π N

πN q

q

πK

π K

E/N

P

P

U/N pK

pK

E/K

pN

E/KN

U/K

pN

U/KN

where the internal squares are pull-back diagrams and the G-maps π K , π N , pK and pN , in the second diagram, are orbit maps which are, of course, extensions of the corresponding maps in the first diagram. We have already shown above, that all the maps in these diagrams, in particular, q and q are G-fibrations (recall that the external squares are regular G-fibered). To finish the proof of the lemma we must only show that q is a strong G-fibration. Since U is a G-ANR, the orbit spaces U/K, U/N and U/KN are G-ANR’s too by Proposition 2.1. Hence the G-space P is also a G-ANR because the internal square in the second diagram is pull-back and the map pK is a G-fibration (see e.g., [9, Proposition 2.4]). Therefore we can state that q is a strong G-fibration being a G-fibration of G-ANR’s. Finally, since E ⊂ U , E/K ⊂ U/K and E/N ⊂ U/N , we can regard the G-space P as a G-subspace of P˜ such that the diagram E q

P

U q˜

P˜

−1 −1 commutes and, moreover, q−1 (P ) = E, because π N (E/N ) = π K (E/K) = E. In other words, this diagram is pull-back and hence q is a strong G-fibration as well as q˜. 2

Theorem 5.5. Let K be a closed normal subgroup of a compact group G. If E is a metrizable G-space with only one orbit type, then the K-orbit projection π : E → E/K is a strong G-fibration.

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58

Proof. Assume that all the orbits of E have the type (H). Then the quotient space G/H is metrizable being G-equivalent to every G-orbit of E. Let (Ni )i∈N be a pro-Lie sequence (of subgroups of G) for G/H.   We have i∈N (Ni H) = H by Proposition 2.5, and therefore N = i∈N Ni ⊂ H. Since each orbit of E is G-equivalent to G/H, the normal subgroup N acts on each G-orbit (and hence on the whole space E) trivially. This implies that  i+1 i+1   π = lim , ri , ←− π/Ni , qi i+1 i+1 because lim } = E/N = E and lim } = E/N K = E/K, where qii+1 , rii+1 are the ←− {E/Ni , qi ←− {E/Ni K, ri natural projections in the commutative diagram

E/Ni

qii+1

E/Ni+1 π/Ni+1

π/Ni

E/Ni K

E/Ni+1 K

rii+1

By Lemma 5.3, this diagram is a strongly G/Ni+1 -fibered (and hence strongly G-fibered) square. Since π/N1 is a strong G-fibration by the same lemma, we conclude that π is also a strong G-fibration (see Proposition 4.2). 2 Corollary 5.6. Let E be a metrizable G-space with only one orbit type. If the orbit space E/G is fibrant, then E is a G-fibrant space. Acknowledgement The authors would like to thank the referee for helpful remarks and suggestions. Appendix A A.1. Proof of Proposition 3.1 Since E and G/H are metrizable, in order to prove that p is a G-fibration, we must show that for a commutative diagram of G-maps f

X × {0}

E p

X ×I

F

(A.1)

G/H

there exists a filler F : X × I → E, provided that X is a metrizable G-space. Let A = F −1 (eH), where e is the identity element of G. Then A is an H-space and X × I = GA (more precisely, X × I can be identified with the twisted product G ×H A by means of the correspondence ga ↔ [g, a], which defines a G-homeomorphism; see [10, Chapter II, Proposition 3.2]). Therefore every (x, t) ∈ X × I is represented as (x, t) = g(a, t) = (ga, t) for some g ∈ G and (a, t) ∈ A. Recall that ga = g  a iff g  = gh−1 and a = ha for some h ∈ H.

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59

Let d be an invariant metric on X and ρ be a left invariant metric on G, which are compatible with topologies of X and G. We define a new metric d on X × I as follows: if (x1 , t1 ) = (g1 a1 , t1 ) and (x2 , t2 ) = (g2 a2 , t2 ) for some (a1 , t1 ), (a2 , t2 ) ∈ A, then       d (x1 , t1 ), (x2 , t2 ) = inf ρ g1 , g2 h−1 + d(a1 , ha2 ) + |t1 − t2 |. h∈H

It is not hard to verify that d is indeed a well defined (i.e., it does not depend on the choice of gi and ai for xi ) invariant metric. Moreover, it is compatible with the topology of X ×I being equivalent to the metric d on X × I given by d ((x1 , t1 ), (x2 , t2 )) = d(x1 , x2 ) + |t1 − t2 |. Let us check this equivalence. Let (x0 , t0 ) = (g0 a0 , t0 ) ∈ X × I for (a0 , t0 ) ∈ A and g0 ∈ G. Suppose first that {(xn , tn )} is a sequence in X × I converging to (x0 , t0 ) with respect to the metric d : lim d ((xn , tn ), (x0 , t0 )) = 0. Let us take, as a compatible metric with the topology of G × A, the following one:           d g, (a, t) , g  , a , t = ρ g, g  + d (a, t), a , t . Now let ε > 0. Since the map θ : G × A → GA, (g, a) → ga, is open (in fact, it is the orbit projection G × A → G ×H A), the image under θ of the open ε-ball about (g0 , (a0 , t0 )) is a neighborhood of (x0 , t0 ). Therefore there is an integer m such for any n > m we can choose gn ∈ G and an ∈ X such that (an , tn ) ∈ A, gn (an , tn ) = (xn , tn ) and     d gn , (an , tn ) , g0 , (a0 , t0 ) = ρ(gn , g0 ) + d(an , a0 ) + |tn − t0 | < ε.  n , tn ), (x0 , t0 )) < ε for all n > m. Consequently, lim d((x  n , tn ), (x0 , t0 )) = 0. This implies that d((x  Conversely, suppose that lim d((xn , tn ), (x0 , t0 )) = 0 for some sequence {(xn , tn )}. Let, for all n, (xn , tn ) = (gn an , tn ), where (an , tn ) ∈ A. By compactness, there is hn ∈ H for each n such that     + d(a0 , hn an ) + |tn − t0 |. d (x0 , t0 ), (xn , tn ) = ρ g0 , gn h−1 n −1 Hence lim ρ(g0 , gn h−1 n ) = 0, lim d(a0 , hn an ) = 0 and lim |tn −t0 | = 0. Since θ is continuous, lim d(g0 a0 , gn hn · hn an ) = 0; that is

lim d(x0 , xn ) = lim d(g0 a0 , gn an ) = 0. Thus     lim d (x0 , t0 ), (xn , tn ) = lim d(x0 , xn ) + |t0 − tn | = 0, and we have proved the equivalence of the metrics d and d .  → A for an H-invariant neighborhood U  of A in By [2, Proposition 4.1], there is an H-retraction R : U  = U A for some neighborhood U of H in X × I. More precisely, this neighborhood can be represented as U G and the retraction R can be defined as R(ub) = τ (u)b, u ∈ U , b ∈ A, for some retraction τ : U → H (see the proof of Proposition 4.1 in [2] for details). This representation of R will be used in the rest of the proof. Since H is compact there exists ε > 0 such that g ∈ U , provided that ρ(g, h) < ε for some h ∈ H. It ˜  whenever d((x, follows that (x, t) ∈ U t), (a , t )) < ε for some (a , t ) ∈ A. Indeed, if (x, t) = (ga, t) for (a, t) ∈ A and           d˜ (x, t), a , t = inf ρ g, h−1 + d a, ha + t − t  < ε, h∈H

 then ρ(g, h−1 0 ) < ε for some h0 ∈ H. By the choice of ε, we have g ∈ U and hence (x, t) = g(a, t) ∈ U .

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Now choose a positive integer n such that 1/n < ε and let   Ak = A ∩ X × [0, k/n] ,

k = 0, 1, . . . , n.

Every Ak is an H-space; in particular, A0 = A∩(X ×{0}) and An = A. Note that for each (x, t) ∈ Ak+1 \Ak   because d((x, the point (x, k/n) ∈ U k/n), (x, t)) = t − k/n  1/n < ε. Observe also that, in this case, R((x, k/n)) ∈ A ∩ (X × {k/n}) ⊂ Ak . Indeed, if (x, k/n) = u(a, k/n) for some u ∈ U and (a, k/n) ∈ A, then R((x, k/n)) = τ (u)(a, k/n) = (τ (u)a, k/n). Therefore the map rk : Ak+1 → Ak given by   rk (x, t) =



(x, t)

if (x, t) ∈ Ak ,

R((x, k/n))

if (x, t) ∈ Ak+1 \ Ak ,

is obviously a well defined H-retraction for k = 0, . . . , n − 1. Hence the composition r = r0 r1 . . . rn−1 , r : A → A0 , is an H-retraction. Finally, define F : X × I → E as follows: if (x, t) = (ga, t) for (a, t) ∈ A, then F(ga, t) = gf r(a, t). It is easy to check that F is a well defined continuous G-map. Moreover,     F(x, 0) = gf r(a, 0) = gf (a, 0) = f (ga, 0) = f (x, 0) (here it was used that (a, 0) ∈ A0 and hence r(a, 0) = (a, 0)) and       pF(x, t) = p gf r(a, t) = gpf r(a, t) = gF r(a, t) = gH = gF (a, t) = F (x, t), because r(a, t) ∈ A0 , (a, t) ∈ A and A = F −1 (eH). Thus F is the required filler of the diagram (A.1). 2 A.2. This subsection is devoted to the proof of Proposition 3.6. Some ideas of the proof of [15, Chapter 2, Section 7, Theorem 12] will be used, but the main idea of our argument is to reduce the proof eventually to the corresponding case of the fiberwise G-AE-spaces considered in [5, Lemma 3.5]. For this we have to recall some facts. For a given G-space B, by B I we denote the space of all the continuous paths ω : I → B (with the compact-open topology). Clearly B I is a G-space with the action · given by (g · ω)(t) = gω(t), t ∈ I. Note 0 1 0 1 that the maps πB : B I → B and πB : B I → B, defined by πB (ω) = ω(0) and πB (ω) = ω(1) respectively, are G-maps. Now let p : E → B be a G-map and consider the following commutative diagram of G-maps EI pI q

0 πE

 E

p

(A.2)

0 πB

t

E

BI

p

B

 is where the internal square is a pull-back and the G-map pI is defined by pI (ω) = pω. The G-space E I known as the cocylinder of p and will be denoted by coCyl(p); coCyl(p) = {(x, ω) ∈ E × B | p(x) = ω(0)}.

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61

Suppose that A is a closed G-subset of a G-space X. Consider the following commutative diagrams of G-maps f

A

EI

f

X × {0} ∪ A × I

E

q

X

F

coCyl(p)

p

X ×I

F

(A.3)

B

mutually related by the equations: F (x, t) = ( pF )(x)(t), f (x, 0) = (tF )(x) and f (a, t) = f (a)(t) for x ∈ X, a ∈ A, t ∈ I. Observe that the existence of a filler X → E I in the left diagram (A.3) is equivalent to the existence of a filler X × I → E in the right diagram (A.3). As a consequence of this observation we get the following assertion: p is a regular G-fibration if and only if E I is a G-AE over coCyl(p) (with respect to q). Remark A.1. Putting X = coCyl(p), A = ∅ and F = id in (A.3), we obtain: p has the EHLP with respect to coCyl(p) ⇒ q has a G-equivariant section ⇒ p is a G-fibration. If E and B are metrizable then so is coCyl(p), and hence p has the EHLP with respect to all metrizable G-spaces ⇔ p is a G-fibration. Further we deal with the category Map(G-M) whose objects are G-maps of metrizable G-spaces and morphisms p → p are pairs (f  , f ) of G-maps such that the diagram (4.1) commutes. The morphism (f  , f ) : p → p will be called a G-AE-square if E  is G-AE over P in the diagram (4.2). It is not hard to notice that a morphism p → p is a G-AE-square if and only if it has the right lifting property with respect to all the inclusions (s, idX ) : s → idX , where s : A → X is a closed G-embedding of metrizable G-spaces. Using this fact, it is easy to see, for instance, that the composition of G-AE-squares is also a G-AE-square. 0 0 , πB ) : pI → p Now the above assertion can be restated as follows: p : E → B is a regular G-fibration iff (πE is a G-AE-square. It is denoted by pU the restriction p−1 (U ) → U of p for an open G-subset U of B. We define the 0 1 ) and ι ◦ (πp1−1 (U ) , πU ) respectively, where morphisms αU , βU : pIU → p as the compositions ι ◦ (πp0−1 (U ) , πU ι : pU → p is the inclusion. Since ι represents a pull-back diagram, we can slightly generalize the above statement: pU is a regular G-fibration if and only if αU : pIU → p is a G-AE-square. Now we are ready to prove Proposition 3.6.  Using the Lebesgue Covering Lemma, we can easily find an Proof of Proposition 3.6. Let (x, ω0 ) ∈ E. integer n and a sequence of open G-subsets U1 , . . . , Un of B such that pUi is a regular G-fibration and ω0 ([(i − 1)/n, i/n]) ⊂ Ui for each i = 1, . . . , n. Then the set W =

   i − 1 i   , ⊂ Ui , i = 1, . . . , n ω ∈ BI  ω n n

 = p−1 (W ). The proof will be finished if we is an open G-invariant neighborhood of ω0 and (x, ω0 ) ∈ W −1  I −1   show that V = q (W ) = (p ) (W ) is a G-AE over W with respect to the restriction qW  : V → W of q. I  with respect to Indeed, in this case, due to arbitrary choice of (x, ω), we can state that E is G-AE over E q by [5, Lemma 3.5], and hence p will be a regular G-fibration.

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Note that in the following restriction  I −1 (W ) p pIW

qW 

 W

0 πV

pW

0 πW

tW  p

E

(A.4)

W

B

0 of the diagram (A.2) the internal square is a pull-back. Thus we need only show that the morphism (πV0 , πW ): pIW → p is a G-AE-square. Let us construct the following commutative diagram

p1,n αn

p1,n−1

p12

p2,n

...

p2,n−1

pIU2

...

p3,n

α2

pIU1 αU1

αU2 βU1

p

pIUn−1 αUn

βU2

p

pIUn

βUn

p

...

p

where every square is a pull-back. It is easy to see that the morphisms α2 , α3 , . . . , αn are G-AE-squares because they are obtained from the G-AE-squares αU2 , αU3 , . . . , αUn respectively by taking pull-backs. Therefore the composition α = αU1 α2 . . . αn , α : p1,n → p is also a G-AE-square. The G-map p1,n : V → W and the morphism α = (πV , πW ) can be described directly as follows: let i = p−1 (Ui ), i = 1, . . . , n, then U    I × · · · × U  I  ω1 (1) = ω2 (0), . . . , ωn−1 (1) = ωn (0) , V = (ω1 , . . . , ωn ) ∈ U 1 n    W = (ω1 , . . . , ωn ) ∈ U1I × · · · × UnI  ω1 (1) = ω2 (0), . . . , ωn−1 (1) = ωn (0) , p1,n (ω1 , . . . , ωn ) = (pω1 , . . . , pωn ),

πV (ω1 , . . . , ωn ) = ω1 (0),

πW (ω1 , . . . , ωn ) = ω1 (0).

The G-spaces V and W are naturally G-equivalent to the G-spaces V and W respectively by means of the correspondences  h, h : ω → (ω1 , . . . , ωn ), given by ωi (t) = ω( i−1+t n ), i = 1, . . . , n. I 0  The composition of (h, h) : pW → p1,n and α : p1,n → p is exactly the morphism (πV0 , πW ) : pIW → p, and  therefore it is a G-AE-square as well as α and (h, h). 2

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A.3.

63

At the end of this subsection, we prove the following assertion:

Proposition A.2. Let p : E → B be a G-fibration of G-fibrant spaces E and B. Then p is a strong G-fibration. For a given G-map p : E → B, consider the commutative diagram of G-maps EI pI

p

(A.5)

BI

W

πE

πB

E×E

p×p

B×B

where the internal square is a pull-back diagram, so that    W = (y1 , y2 , ω) ∈ E × E × B I  ω(0) = p(y1 ), ω(1) = p(y2 ) , πE (ω) = (ω(0), ω(1)), πB (ω) = (ω(0), ω(1)), p × p(y1 , y2 ) = (p(y1 ), p(y2 )) and the G-map p : E I → W is defined by p ( ω ) = ( ω (0), ω  (1), p ◦ ω  ) for ω  ∈ EI . Now suppose that A is closed G-subset of a G-space X and consider the commutative diagrams of G-maps h

X × {0, 1} ∪ A × I

E p

¯i

X ×I

H

B

A

 h

p

i

X

EI

 H

(A.6)

W

mutually related by the equations:  h(a)(t) = h(a, t) and    H(x) = h(x, 0), h(x, 1), H ∗ (x) , where H ∗ (x)(t) = H(x, t), for x ∈ X, a ∈ A, t ∈ I. Observe that the existence of a filler F : X × I → E in the left diagram (A.6) is equivalent to the existence of a filler F ∗ : X → E I in the right diagram (A.6) (F and F ∗ are related by F ∗ (x)(t) = F (x, t)). Using the correspondence F ↔ F ∗ , we obtain the following assertion: if p is a G-fibration, then so is p . To see this, just put, for a given G-space Z, A = Z × {0} and X = Z × I in the diagrams (A.6). Then a filler F exists because the inclusion ¯i : Z × (I × {0, 1} ∪ {0} × I) → Z × I × I in the left diagram can be regarded as the inclusion (Z × I) × {0} → (Z × I) × I due to the homeomorphism 

   I × I, I × {0, 1} ∪ {0} × I ≈ I × I, I × {0} .

Therefore, in the right diagram, a filler F ∗ : Z × I → E also exists. This proves that p is a G-fibration. By the way, the proved assertion can be restated as follows: the morphism (πE , πB ) : pI → p × p is a G-fibered square, provided that p is a G-fibration. Proposition A.3. Let A be closed G-subset of a G-space X. If the embedding i : A → X is a G-SSDR-map, then so is the embedding ¯i : X × {0, 1} ∪ A × I → X × I.

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Proof. Let p : E → B be a G-fibration of G-ANR’s. To prove the proposition, we must only show that there is a filler F : X × I → E in the left diagram (A.6). Since E × E, B × B and B I are G-ANR’s, the G-space W in the diagrams (A.5) and (A.6) is a G-ANR too, because it is the pull-back of p × p and the G-fibration πB (see [9, Proposition 2.4]). By the above assertion, we have that p : E I → W is a G-fibration of AN R’s. Therefore there is a filler F ∗ : X → E I in the right diagram (A.6). But this implies the existence of a filler F : X × I → E in the left diagram (A.6). 2 Proof of Proposition A.2. Consider the commutative diagram (3.1), where s : A → X is a G-SSDR-map. Clearly, we can assume that A ⊂ X so that s(a) = a for all a ∈ A. To prove that p : E → B is a strong G-fibration, we must find a filler X → E of the diagram, that is a G-map F : X → E such that F|A = f and pF = F . Since E is a G-fibrant space, there exists a G-extension f : X → E of f : A → E, that is, f |A = f . Let h : X × {0, 1} ∪ A × I → B be a G-map defined by h(x, 0) = pf (x), h(x, 1) = F (x) and h(a, t) = F (a) for x ∈ X, a ∈ A and t ∈ I. The G-map h is well defined because F (a) = pf (a) = pf (a) for all a ∈ A. Now, using that B is a G-fibrant space and X × {0, 1} ∪ A × I → X × I is a G-SSDR-map by Proposition A.3, we can find a G-extension H : X × I → B of h. Thus we get the commutative diagram X

f

p

i0

X ×I

E

H

B

 : X × I → E as a filler of the where i0 (x) = (x, 0). Since p is a G-fibration, there exists a G-homotopy H diagram. Moreover, this G-homotopy can be chosen so that it is fixed on the points of A as well as the homotopy H (it can be shown using, for instance, the existence of an invariant metric on B; the proof itself is quite similar to one for the non-equivariant case). It is easy to check that the G-map F : X → E, defined  by F (x) = H(x, 1) for x ∈ X is the required filler of the diagram (3.1). Indeed,  1) = H(a,  0) = f (a) = f (a) F (a) = H(a,  and pF (x) = pH(x) = H(x, 1) = F (x) for a ∈ A and x ∈ X. 2 References [1] S.A. Antonyan, Existence of a slice for arbitrary compact transformation group, Mat. Zametki 56 (5) (1994) 3–9 (in Russian); English transl: Math. Notes 56 (1994) 1101–1104. [2] S.A. Antonyan, Orbit spaces of proper equivariant absolute extensors, Topol. Appl. 153 (2005) 698–709. [3] S.A. Antonyan, Compact group actions on equivariant absolute neighborhood retracts and their orbit spaces, Topol. Appl. 158 (2011) 141–151. [4] S.A. Antonyan, Equivariant extension properties of coset spaces of locally compact groups and approximate slices, Topol. Appl. 159 (2012) 2235–2247. [5] S.A. Antonyan, R. Jimenez, S. de Neymet, Fiberwise retraction and shape properties of the orbit space, Glas. Mat. 35 (55) (2000) 191–210. [6] A. Bykov, The homogeneous space G/H as an equivariant fibrant space, Topol. Appl. 157 (2010) 2604–2612. [7] A. Bykov, A.L. Kantún Montiel, Orbit projections and G-ANR-resolutions, Glas. Mat. 47 (67) (2012) 193–205. [8] A. Bykov, M. Texis, Equivariant strong shape, Topol. Appl. 154 (2007) 2026–2039. [9] A. Bykov, A. Torres Juan, Fibrant extensions of free G-spaces, Topol. Appl. 159 (2012) 1179–1186. [10] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972. [11] F. Cathey, Strong shape theory, in: Shape Theory and Geometric Topology, in: Lect. Notes Math., Springer, Berlin, 1981, pp. 216–239. [12] D.A. Edwards, H.M. Hastings, Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lect. Notes Math., vol. 542, Springer-Verlag, Berlin, 1976.

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R.S. Palais, The classification of G-spaces, Mem. Am. Math. Soc. 36 (1960). L.S. Pontryagin, Selected Works, vol. 2. Topological Groups, Class. Sov. Math., Gordon & Breach, New York, 1986. E.H. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966. T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin–New York, 1987.

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