G-fibrant extensions and twisted products

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G-fibrant extensions and twisted products Alexander Bykov ∗ , Jorge Alberto Sánchez Martínez Benemérita Universidad Autónoma de Puebla, Av. San Claudio y Rio Verde, Ciudad Universitaria, Colonia San Manuel, CP 72570 Puebla, Puebla, Mexico

a r t i c l e

i n f o

Article history: Received 16 December 2014 Accepted 7 May 2015 Available online xxxx MSC: 54C55 54C56 54H15 55R91

a b s t r a c t Let G be a compact metrizable group and H its closed subgroup. We prove that every compact metrizable G-space over G/H admits a G-fibrant extension over G/H. This implies that the functor of twisted product G ×H − takes H-fibrant extensions to G-fibrant extensions. In particular, the twisted product G ×H S is a G-fibrant space when S is a compact H-fibrant space. © 2015 Elsevier B.V. All rights reserved.

Keywords: G-fibration G-ANR G-fibrant space Orbit space

0. Introduction By a G-fibrant extension, where G is a topological group, we mean an equivariant version of the corresponding notion of a fibrant extension introduced by F. Cathey in [11]; it was used by him to define the strong shape category of metric compacta. In [9] one can find an analogous construction of the equivariant strong shape category. This is one of the reasons to study equivariant fibrant extensions as well as the related notions of a G-fibrant space and of a strong G-fibration. The notion of a G-fibrant space can be regarded as a generalization of the concept of a G-ANR: every G-ANR 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 G-fibrant spaces as well as G-ANR’s have equivariant homotopy extension property with respect to all G-pairs (X, A) of metrizable spaces, where A ⊂ X is closed in X. Of particular interest for the theory of G-spaces is the following fact proved in [7]: for any closed subgroup H of a compact metrizable group G the quotient space G/H is G-fibrant. In the present paper we prove * Corresponding author. E-mail addresses: [email protected] (A. Bykov), [email protected] (J.A. Sánchez Martínez). http://dx.doi.org/10.1016/j.topol.2015.12.034 0166-8641/© 2015 Elsevier B.V. All rights reserved.

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a more general statement: the twisted product G ×H S is a G-fibrant space provided that S is a compact H-fibrant space (Corollary 4.6). This statement is an easy corollary of Theorem 4.5 about fibrant extensions of twisted products. Theorem 4.5, in turn, is a consequence of our main result, Theorem 4.3, which can be formulated as follows: every compact metrizable G-space over G/H admits a G-fibrant extension over G/H. By a G-space over G/H we mean, of course, any G-map p : E → G/H. As a consequence of Theorem 4.3 we also obtain a condition under which p is a strong G-fibration (see Corollary 4.4). One of the tools used, in fact, in the proof of the main result is the construction of a fibrant extension applied to the category of G-maps. 1. Preliminaries The letter G will denote a compact Hausdorff group, its unit element is denoted by e. Although, in general, we work in the category G-TOP of G-spaces and G-maps, in Sections 3 and 4 we deal with the category G-M of metrizable G-spaces and, moreover, G will be supposed to be metrizable too. The basic facts of the theory of G-spaces can be found in [6,12] and [14]. For the convenience of the reader, below we recall some well-known definitions and facts as well as more special ones. Let S be an H-space where H is a closed subgroup of G. The twisted product G ×H S is the H-orbit space of the product G × S, when G × S is considered as an H-space with respect to the action h · (g, s) = (gh−1 , hs). The H-orbit of (g, s) ∈ G × S is denoted by [g, s]. Regarding G ×H S as G-space with the action g  · [g, s] = [g  g, s], we get the functor of twisted product G ×H − : H-TOP → G-TOP which takes the H-map f : S → S  to the G-map G ×H f : G ×H S → G ×H S  , [g, s] → [g, f (s)]. 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. If p : E → G/H is a G-map, then E has the natural structure of the twisted product G ×H S, where S = p−1 (eH), in view of the G-homeomorphism G ×H S → E, [g, s] → gs (see e.g., [12, Chapter I, Proposition 4.4]). 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 with respect to every G-space X (see [12, p. 53]). In other words, p is a G-fibration if and only if it has the right lifting property with respect to 0 all G-maps of the form ∂X : X → X × I, x → (x, 0) (here, as usual, I = [0, 1]), that is, for any commutative diagram of G-maps x

X

f

0 ∂X

(x, 0)

X ×I

E p

F

(1.1)

B

0 there is a filler F : X × I → E (i.e., a G-homotopy F : X × I → E such that pF = F and F ∂X = f ). It is  used to say that p is a regular G-fibration if for any commutative diagram (1.1) there is a filler F : X ×I → E which is, moreover, a G-homotopy relative to A ⊂ X when F is a G-homotopy relative to A. It is not hard to show that a G-fibration p : E → B is regular if B admits a G-invariant metric, in particular, if the group G is compact and B is metrizable. 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 [14] for the equivariant theory of retracts). A G-map s : A → X of metrizable G-spaces is called a G-SSDR-map 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

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A

f

X

E p

s

F

3

(1.2)

B

there exists a filler F : X → E. Clearly, every G-SDR-map s : A → X of metrizable G-spaces (that is, the G-map which embeds A in X as a strong deformation G-retract) is a G-SSDR-map. The next definition gives us a much more general example of a G-SSDR-map. Let A be an invariant closed subset of a G-space X. An infinite strong G-deformation of X on A is a G-map D : X × [0, ∞) → X satisfying: (a) D(x, 0) = x for all x ∈ X, (b) D(a, t) = a for all a ∈ A and t ∈ [0, ∞), (c) for each invariant neighborhood U of A in X there is λ ∈ [0, ∞) such that D(X × [λ, ∞)) ⊆ U . A closed G-embedding s : A → X will be called a G-ISDR-map if there is an infinite strong G-deformation of X on s(A). It can be easily seen that every G-ISDR-map of metrizable G-spaces is a G-SSDR-map. Let us point out two simple facts about G-SSDR-maps (cf. [11, Corollary 1.6]). Let A be a closed G-subset of a metrizable G-space X. Then: (∗) X × 0 ∪ A × I → X × I is a G-SSDR-map. (∗∗) If the embedding A → X is a G-SSDR-map, then so is the embedding X × {0, 1} ∪ A × I → X × I (see [8, Proposition A.3] for the proof). A G-space E is called an equivariant fibrant space or a G-fibrant space if for every G-SSDR-map s : A → X and every G-map f : A → E, there exists a G-map F : X → E such that F ◦ s = f . For instance, every G-ANR is a G-fibrant space.  is such that E  is a G-fibrant space, then it is said that s is a G-fibrant If a G-SSDR-map s : E → E extension of E. Every compact metrizable G-space E admits a G-fibrant extension. In fact, one can find a  into some G-fibrant space E  (see [9, Proposition 3.5]). G-ISDR-map i : E → E  It is easy to give an example of a G-SSDR-map which is not a G-ISDR-map. Nevertheless, if s : E → E is a G-fibrant extension of a compact metrizable G-space E, then s must be a G-ISDR-map. To prove this  of E for which there exists an infinite strong G-deformation D of E  take a G-fibrant extension i : E → E    onto E. Since E is G-fibrant and s is a G-SSDR-map, we can find a G-map ϕ : E → E such that ϕ ◦ s = i.  → E.  By (∗∗) there is a G-homotopy H : id   ψ ◦ ϕ rel. s(E) Similarly, ψ ◦ i = s for some G-map ψ : E E  is G-fibrant (this proves, by the way, that because idE |s(E) = ψ ◦ ϕ|s(E) while s is a G-SSDR-map and E any two G-fibrant extensions are homotopically equivalent). The required infinite strong G-deformation  of E  : E  × [0, ∞) → E  onto s(E) is defined as follows: D(x,  t) = H(x, t) for t ∈ [0, 1] and D(x,  t) = D   ψ ◦ D ϕ(x), t − 1 for t ∈ [1, ∞). A G-map p : E → B of metrizable G-spaces is called a strong G-fibration if it has the right lifting property with respect to all G-SSDR-maps, that is, if for any diagram (1.2) of G-maps, where s : A → X is a G-SSDR-map, there exists a G-map F : X → E as a filler. Clearly, strong G-fibrations are G-fibrations. On the other hand, it follows immediately from the definition of G-SSDR-maps 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 [8, Proposition A.2]. 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. More facts about strong G-fibrations and G-fibrant spaces can be found in [8] and [10].

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2. Some facts about the category of G-maps In this section and in the rest of the paper we deal with the category Map(G-TOP) whose objects are G-maps of G-spaces and morphisms p → p are pairs (f  , f ) of G-maps such that the diagram E

f

p

B

E p

f

(2.1)

B

commutes. In this category one can introduce, in an obvious way, the concepts analogous to the notions of a G-homotopy, a strong G-deformation retract, a G-fibration, etc., known for the category G-TOP (see [13] and [4] for fiberwise equivariant homotopy theory and retract theory). Our purpose is Theorem 2.7 which represents the analog of the well-known fact that any continuous map can be factored into a composition of an SDR-map and a fibration. 2.1. Fibrations and deformations in the category of G-maps Let α = (f  , f ) : p → p be a morphism represented by the diagram (2.1). If A ⊂ B and F ⊂ E are G-subsets such that p(F ) ⊂ A, then one can consider the restriction q : F → A of p; in this case we write q ⊂ p. The symbol α−1 (q) will denote the restriction f  −1 (F ) → f −1 (A) of p , α−1 (q) ⊂ p . Similarly, if q  ⊂ p , q  : F  → A , then α(q  ) is the restriction f  (F  ) → f (A ) of p. Let q ⊂ p. By a strong G-deformation retraction of p onto q we mean a pair of usual strong G-deformation retractions D : E × I → E and D : B × I → B of E and B onto F and A, respectively, which form a morphism p × idI → p in Map(G-TOP), that is, related by the equality p ◦ D = D ◦ (p × idI ) (where p × idI : E × I → B × I takes (x, t) to (p(x), t)). Let σ = (s , s) : q → p, be a morphism such that s and s are G-embeddings. If there is a strong G-deformation retraction of p onto σ(q), we will say that the inclusion σ : q → p is a G-SDR-morphism. Definition 2.1. A morphism α : p → p will be called G-fibered if, for any G-map h : X → Y , α has the right lifting property with respect to the morphism (∂0X , ∂0Y ) : h → h × idI . Similarly, α : p → p (in Map(G-M)) is said to be a strongly G-fibered morphism when it has the right lifting property with respect to all morphisms (s , s) : h → h such that both s and s are G-SSDR-maps. Let id∗ be the identity map of a one-point set {∗} onto itself (with the trivial action of the group G). Then one can easily see that: (i) p → id∗ is a G-fibered morphism ⇔ p is a G-fibration. (ii) p → id∗ is a strongly G-fibered morphism ⇔ p is a strong G-fibration of G-fibrant spaces ⇔ p is a G-fibration of G-fibrant spaces (recall that the last “⇔” is [8, Proposition A.2]). Some more simple facts about G-fibered morphisms are indicated further in Propositions 2.2, 2.3 and 2.5, the standard proofs of which will be omitted. Proposition 2.2. Suppose that p = {pi , βij }i≥0 is an inverse sequence in Map(G-TOP) and let p be its inverse limit with the projections βi : p → pi , that is, (p, {βi }) = lim p. If p0 is a (strong) G-fibration and βii+1 ←−

is a (strongly) G-fibered morphism for each i, then p is a (strong) G-fibration and each βi is a (strongly) G-fibered morphism. Note that for any commutative diagram (2.1) one can always consider the commutative diagram

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E f u

P

p

E

f p

(2.2)

p f

B

B

where the internal square is pull-back, that is, P (together with f and p) is the pull-back of the triad f

p

B  −→ B ←− E. Of course, the G-map u is uniquely determined by the diagram (2.1). Proposition 2.3. Let α : p → p be a morphism given by the diagram (2.1). Then α is a (strongly) G-fibered morphism if and only if the G-maps f , f  and u in the diagram (2.2) are (strong) G-fibrations. Remark 2.4. In [8] a morphism α : p → p given by the diagram (2.1) is called a G-fibered square if the G-map u in diagram (2.2) is a G-fibration. Therefore one can say that α = (f  , f ) is a G-fibered morphism iff it is a G-fibered square and both f and f  are G-fibrations. We will also say that a morphism α : p → p given by the diagram (2.1) is regular G-fibered if the G-maps u and f (and hence f  ) in the diagram (2.2) are regular G-fibrations. For instance, the G-fibered morphism α represented by the diagram (2.1) is regular if the G-spaces E, B and B  are metrizable. Proposition 2.5. Let α = (f  , f ) : p → p be a regular G-fibered morphism. Suppose that for some restriction q ⊂ p there is a strong G-deformation retraction of p onto q. Then there exists a strong G-deformation retraction of p onto q  = α−1 (q). 2.2. Mapping cocylinders in the category of G-maps 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, I are G-maps. We shall also use the symbol πB for the G-map B → B × B, ω → (ω(0), ω(1)). Now let f : B  → B be a G-map. Recall that the mapping cocylinder of f , denoted by coCyl(f ), is defined f

π0

B as the pull-back of the triad B  −→ B ←− B I . Thus we have the pull-back diagram

f

coCyl(f )  0 π B

B

BI 0 πB

f

(2.3)

B

   0 Explicitly, coCyl(f ) = {(b , ω) ∈ B  ×B I |f (b ) = ω(0)} with the projections π B : coCyl(f ) → B , (b , ω) → b and f : coCyl(f ) → B I , (b , ω) → ω. The notion of a cocylinder can be defined in Map(G-TOP) in the standard way: the cocylinder coCyl(α) of a given morphism α = (f  , f ) : p → p is the G-map

coCyl(α) : coCyl(f  ) → coCyl(f ), (e , ω  ) → (p (e ), p ◦ ω  ).

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It is easy to see that we have the following pull-back diagram in the category Map(G-TOP): α 

coCyl(α)

pI

0 π p

(2.4)

0 0 πp0 =(πE ,πB )

p

α

p

0  0 0 = (π where pI : E I → B I is given by pI (ω  ) = p ◦ ω  , α  = (f , f), π p E , πB ). The cocylinders in Map(G-TOP) have properties which are analogous to the corresponding properties in the categories TOP and G-TOP. We will consider some of them.

Lemma 2.6. Let p : E → B be a G-fibration. Then the morphism (πE , πB ) : pI → p × p is G-fibered. If, moreover, E and B are G-fibrant spaces, then (πE , πB ) is a strongly G-fibered morphism. Proof. To apply further Proposition 2.3 consider the commutative diagram of G-maps EI pI p

BI

W

πE

πB

E×E

p×p

B×B

where the internal square is pull-back, so that W = {(y1 , y2 , ω) ∈ E × E × B I | ω(0) = p(y1 ), ω(1) = p(y2 )} and the G-map p : E I → W is defined by p(ω) = (ω(0), ω(1), p ◦ ω). 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

p

¯i

X ×I

E

H

B

A

 h

p

i

X

EI

 H

(2.5)

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 (2.5) is equivalent to the existence of a filler F ∗ : X → E I in the right diagram (2.5) (F and F ∗ are related by F ∗ (x)(t) = F (x, t)). Now let Z be an arbitrary G-space. Putting A = Z ×{0} and X = Z ×I in the diagrams (2.5), we observe that a filler F : (Z × I) × I → E, for the left diagram, exists because p is a G-fibration and the inclusion

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¯i : Z × (I × {0, 1} ∪ {0} × I) → Z × I × I 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 for any G-fibration p : E → B. In particular, taking the trivial case when B is a one-point set {∗}, we have πE = p, so that πE is a G-fibration for any G-space E. Thus the first statement of the lemma is proved in view of Proposition 2.3. To prove the second statement, when E and B are supposed to be G-fibrant spaces, note first that p is a strong G-fibration [8, Proposition A.2]. Therefore there exists a filler F : X × I → E in the left diagram (2.5) for every G-SSDR-map i : A → X (because ¯i is a G-SSDR-map by the assertion (∗∗) of Preliminaries). This implies the existence of a filler F ∗ : X → E I in the right diagram (2.5), that is, this shows that p is a strong G-fibration. As above, the particular case B = {∗} gives us πE = p and, since p : E → {∗} is a strong G-fibration (because E is G-fibrant), we conclude that πE is a strong G-fibration for any G-fibrant space E. 2 0 Recall that the G-fibration πB : B I → B has the section sB : B → B I , b → cb , where cb : I → B is the constant map taking t to b. Moreover, there is the strong G-deformation retraction D : B I × I → B I of B I onto sB (B) given by D(ω, t) = ωt , where ωt : I → B is such that ωt (τ ) = ω(τ (1 − t)). Since the diagram    0 (2.3) is pull-back, we have a similar fact for π B : the G-map sf : B → coCyl(f ), b → (b , cf (b ) ), is a section 0 while the map for π B

Df : coCyl(f ) × I → coCyl(f ),

((b , ω), t) → (b , ωt ),

is a strong G-deformation retraction of coCyl(f ) onto sf (B  ). Passing to coCyl(α), we get the morphisms σα = (sf  , sf ) and δα = (Df  , Df ) which satisfy the analogous conditions. In particular, the morphism σα = (sf  , sf ) : p → coCyl(α),

(2.6)

sf (b ) = (b , cf (b ) ), b ∈ B  , sf  (y  ) = (y  , cf  (y ) ), y  ∈ E  ,

(2.7)

given by

is a G-SDR-morphism. Now we can formulate and prove the following theorem resuming the above discussion: Theorem 2.7. Every morphism α = (f  , f ) : p → p in Map(G-TOP) can be factored into the composition σ

α 

α p −→ coCyl(α) −→ p, represented by the commutative diagram

coCyl(f  ) sf 

f˜ f

E



E coCyl(α)

p

coCyl(f )

B

p f˜

sf f

B

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where σα = (sf  , sf ) is a G-SDR-morphism and α  = (f , f). Moreover, (i) if p and p are G-fibrations, then the morphism α  is G-fibered, and hence the G-map coCyl(α) is a G-fibration;  is strongly G-fibered, and hence (ii) if p and p are G-fibrations of G-fibrant spaces, then the morphism α the G-map coCyl(α) is a G-fibration of G-fibrant spaces. Proof. We define α  = (f , f) : coCyl(α) → p by α  = πp1 ◦ α . Then for the G-SDR-morphism σα , given by 1 1 the formulas (2.6) and (2.7), we have α = α  ◦ σα . Indeed, f ◦ sf = πB ◦ f ◦ sf = πB ◦ sB ◦ f = f and,   similarly f ◦ sf  = f .  : h × idI → Let us prove the statement (i). Suppose that h : X → Y is any G-map. We must find a filler γ coCyl(α) for the commutative diagram β

h

coCyl(α)

0 0 ι=(∂X ,∂Y )

(2.8)

α 

h × idI

p

γ

From the diagrams (2.4) and (2.8) we obtain the following commutative diagram α◦β 

h

0 ◦β π p

πp0

α

p

ι

pI

πp1

p

(2.9)

β γ

h × idI

p

where the morphism β  has been added using the fact that p is a G-fibration. Now, by Lemma 2.6, we can find a morphism β : h × idI → pI such that the diagram h ι

h × idI

α◦β 

β

(α◦β  ;γ)

pI (2.10)

πp =(πp0 ;πp1 )=(πE ,πB )

p×p

commutes. By the commutativity of this diagram we have πp0 ◦ β = α ◦ β  , and hence β and β  define a  0 ◦ γ unique morphism γ  : h × idI → coCyl(α) such that α ◦γ  = β and π p  = β (see the diagram (2.4)). One 0 ◦ β and 0 ◦ γ can easily check that γ  is the required filler of the diagram (2.8). Indeed, π  ◦ ι = β ◦ ι = π p

p

α ◦γ  ◦ ι = β ◦ ι = α  ◦ β, so that γ  ◦ ι = β. On the other hand, α ◦γ  = πp1 ◦ α ◦γ  = πp1 ◦ β = γ. The proof of the statement (ii) is quite similar. Just repeat the same argument replacing the morphism ι : h → h × idI in the diagrams (2.8), (2.9) and (2.10) by a pair of G-SSDR-maps (s, s ). Note also that, for the existence of a morphism β  in the diagram (2.9), the condition “p is a strong G-fibration of G-fibrant spaces” will be used. 2

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3. Fibrant extensions in the category of G-maps In this paper we are interested only in G-ANR-resolutions of compact metrizable G-spaces. Therefore the general definition of a G-resolution, given in [5], can be reduced to a simpler one which looks as follows. Definition 3.1. Let E be a compact G-space and let E = {Ei , qij } be an inverse sequence of G-spaces and G-maps. A mapping q = {qi } : E → E (that is, a family of G-maps {qi : E → Ei } such that qij ◦ qj = qi for all i ≤ j) is called a G-resolution of E if: (1) (E, q) = lim E, ←−

(2) for every i and any invariant open neighborhood U of qi (E) in Ei there exists j ≥ i such that qij (Ej ) ⊆ U . When all the G-spaces Ei are G-ANR’s, it will be said that q : E → E is a G-ANR-resolution of E. In the obvious way, this definition can be extended to the category of G-maps. Definition 3.2. Let p : E → B be a G-map of compact G-spaces. Let p = {pi , βij } be an inverse sequence of G-maps and G-morphisms, where βij = (qij , rij ) is represented by the commutative diagram of G-maps Ei

qij

Ej (3.1)

pj

pi

Bi

rij

Bj

A mapping {βi } = {(qi , ri )} : p → p (that is, a family of G-morphisms {βi = (qi , ri ) : p → pi } such that βij ◦ βj = βi for all i ≤ j) is called a G-resolution of p if the mappings {qi } : E → {Ei , qij } and {ri } : E → {Bi , rij } are G-resolutions of E and B, respectively. Let p = {pi , βij } be an inverse sequence of G-maps and G-morphisms. In view of Theorem 2.7, we can construct inductively the following commutative diagram of G-morphisms p0

β01

β12

σ1

σ0 =id

p0

p1

β01

p1

p2

pn

...

σ2 β12

n+1 βn

σn

...

p2

pn

n+1 βn

pn+1

... (3.2)

σn+1

pn+1

...

factoring each G-morphism σn ◦ βnn+1 : pn+1 → pn into the composition σn+1

βn+1

n pn+1 −→ pn+1 −→ pn

where σn+1 is a G-SDR-morphism and pn+1 = coCyl(σn ◦ βnn+1 ).  = {  , that Thus p produces the inverse sequence p pi , βij }. We will denote by F(p) the inverse limit of p  . Now suppose that (p, {βi }) = lim p. Then the inclusions {σi } induce a unique is, (F(p), {βi }) = lim p ←−

←−

inclusion σ : p → F(p) such that βi ◦ σ = σi ◦ βi for all i.

Theorem 3.3. Let p : E → B be a G-map of compact metrizable spaces. Suppose that p → p = {pi , βij } is a G-resolution of p such that each pi is a G-fibration of metrizable G-spaces. Then the induced inclusion σ : p → F(p) satisfies the following conditions:

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(1) F(p) is a G-fibration, (2) there is a strong infinite G-deformation of F(p) onto σ(p). If, moreover, each pi is a G-fibration of G-fibrant spaces (in particular, of G-ANR’s), then F(p) is a G-fibration of G-fibrant spaces. Proof. Since all the G-maps pi are G-fibrations, all the G-maps pi in the diagram (3.2) are G-fibrations too and the bonding morphisms βii+1 are G-fibered by Theorem 2.7(i). This implies that F(p) is a G-fibration and every projection βi : F(p) → pi is G-fibered by Proposition 2.2. Similarly, if all pi are G-fibrations of G-fibrant spaces then, using Theorem 2.7(ii) and Proposition 2.2, we conclude that F(p) is also a G-fibration of G-fibrant spaces. To prove the statement (2) we use the notation assumed in Definition 3.2, in particular, the morphisms βij are represented by the diagram (3.1). Suppose also that the maps and the morphisms in the diagram (3.2) i → B i , βj = ( →B  and let βi = ( are given by pi : E qij , rij ), σi = (si , ui ). Let p = F(p) : E qi , ri ) : p → pi be i  and u : B → B.  the projections. The inclusion σ = (s, u) : p → p consists of some embeddings s : E → E By the construction of the diagram (3.2), for each i, σi is a G-SDR-morphism, that is, there exists a strong G-deformation retraction of pi onto σi (pi ) which can be lifted, by Proposition 2.5, to some strong G-deformation retraction (Di , Fi ): ×I E

Di

p×id  I

×I B

 E p

Fi

(3.3)

 B

of p onto βi−1 (σi (pi )) because βi is a regular G-fibered morphism (recall that all the G-spaces are supposed  onto Xi = qi −1 (si (Ei )), that to be metrizable). In particular, Di is a strong G-deformation retraction of E  and Di (x, t) = x for all x ∈ Xi and t ∈ I. Note that is, we have Di (x, 0) = x, Di (x, 1) ∈ Xi for all x ∈ E  = X0 ⊃ X1 ⊃ . . . ⊃ Xi ⊃ Xi+1 ⊃ ..., (a) E  there is j such that Xj ⊂ U . (b) for every G-neighborhood U of s(E) in E (a) follows from the commutativity of the diagram (3.2). To prove (b) we first find, using the compactness k such that q−1 (V ) ⊂ U . Then s−1 (V ) is a of s(E), an index k and a G-neighborhood V of qk (s(E)) in E k k j G-neighborhood of qk (E) in Ek and, since {qi } : E → {Ei , qi } a G-resolution, we have qkj (Ej ) ⊂ s−1 k (V ) for some j ≥ k. Therefore Xj = qj −1 (sj (Ej )) ⊂ qk −1 ( qkj sj (Ej )) = qk −1 (sk qkj (Ej )) ⊂ qk−1 (V ) ⊂ U.  be a map defined as follows:  :E  × [0, ∞) → E Let D  D(x, t) = Di (Di+1 (x, t − i), 1), when t ∈ [i, i + 1].  is a well-defined continuous map. Clearly, it is G-equivariant. Moreover, D(x,  0) = It is easily checked that D  (recall that X0 = E).  Since each Di is a G-homotopy D0 (D1 (x, 0), 1) = D0 (x, 1) = D0 (x, 0) = x for all x ∈ E   E  × ([i, ∞)) = Xi relative to s(E) ⊂ Xi , we have that D(x, t) = x for all x ∈ s(E) and t ∈ [0, ∞). Finally, D(   is an infinite because Dj (E × 1) = Xj ⊂ Xi for j ≥ i. Therefore, in virtue of (b), we can conclude that D  onto s(E). strong G-deformation of E   × [0, ∞) → B  by Similarly, defining F : B

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F(x, t) = Fi (Fi+1 (x, t − i), 1), when t ∈ [i, i + 1],  = F ◦(  onto u(B). It is easy to see that p◦ D we get an infinite strong G-deformation of B p ×id[0,∞) ) because   of commutativity of the diagram (3.3) for each i. Thus (D, F ) is the required infinite strong G-deformation of p onto σ(p). 2 4. Fibrant extensions of the twisted product G ×H S Definition 4.1. A pro-Lie sequence of subgroups of a given compact metrizable group G is a decreasing sequence {Ni }i∈N of closed normal subgroups of G such that G/Ni is a Lie group for every i and Ni = {e} i∈N

(and hence G = lim{G/Ni , πij }, ←−

where πij : G/Nj → G/Ni , j ≥ i, are the natural projections). It is well known that for every compact metrizable group G exists a pro-Lie sequence (see, e.g., [15, §46]). Proposition 4.2. Let p : E → G/H be a G-map, where G is a compact metrizable group and H is its closed subgroup. If E is a compact metrizable G-space, then for any pro-Lie sequence {Ni}i∈N of subgroups of G there exists a G-ANR-resolution p → p = {pi , αij } of p consisting of the G-fibrations pi : Ei → G/HNi . Proof. Let S = p−1 (eH). Then E can be identified with the twisted product G ×H S by means of the canonical G-homeomorphism G ×H S → E, given by [g, s] → gs. We can consider the H-space S as a closed invariant subset of some H-AR-space M (see e.g., [1]). Since S is compact it has a countable neighborhood base {Wi }i∈N in M consisting of open H-subsets Wi such that Wi+1 ⊆ Wi for every i. For a given pro-Lie sequence {Ni }i∈N of subgroups of G let Vi = G ×H Wi and define Ei as the Ni -orbit space of Vi , that is, Ei = Vi /Ni for each i. Note that by the choice of {Wi } we get the decreasing sequence of open G-spaces {Vi } in G ×H M which form a neighborhood base of G ×H S in G ×H M . Defining, for j ≥ i, the G-map qij : Ej → Ei by qij (Nj (vj )) = Ni (vj ), we obtain the inverse sequence E = {Ei , qij }. It is not hard to see that E = G ×H S = lim E ←−

with the natural projections qi : E → Ei given by qi ([g, s]) = Ni ([g, s]). Moreover, the mapping {qi } : E → E is a G-resolution. To see this, observe that for a given i the family {Vj /Ni }j≥i is an open neighborhood base of qi (E) = (G ×H S)/Ni in Ei = Vi /Ni . But Vj /Ni = qij (Ej ) for j ≥ i. This proves that the condition (2) of Definition 3.1 holds for {qi }. Since there is a natural G-homeomorphism (see e.g., [2, Proposition 2]) (G ×H Wi )/Ni → G/Ni ×HNi /Ni Wi /(Ni ∩ H),

(4.1)

which takes Ni ([g, w]) to [gNi , (Ni ∩ H)(w)], we can state that each Ei is a G-ANR. Indeed, each Wi / (Ni ∩ H) is an H/(Ni ∩ H)-ANR by [3, Theorem 1.1], because Wi is an H-ANR as an open H-subset of the H-AR-space M . Clearly, Wi /(Ni ∩ H) can be also regarded as an HNi /Ni -space because the group H/(Ni ∩ H) is isomorphic to the subgroup HNi /Ni of G/Ni (by means of the correspondence h(Ni ∩ H) → hNi ). Thus Wi /(Ni ∩ H) is an HNi /Ni -ANR and, since G/Ni is a Lie group, the twisted product in (4.1), which can be identified with Ei = (G ×H Wi )/Ni , is a G/Ni -ANR by [2, Proposition 8] and hence it is a G-ANR.

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Now let us define pi : Ei → G/(HNi ) by pi (Ni ([g, wi ])) = gHNi . In other words, pi is the map of the Ni -orbit spaces induced by the natural map G ×H Wi → G/H, [g, wi ] → gH. Obviously pi is a G/Ni -map (and hence a G-map) such that pi ◦ qi = π i ◦ p and pi ◦ qij = π ji ◦ pj for j ≥ i, where π i : G/H → G/(HNi ) and π ji : G/(HNj ) → G/(HNi ) are projections. Thus we have the morphisms αi = (qi , π i ) : p → pi and αij = (qij , π ji ) : pj → pi . It is easy to see that (G/H, {π i }) = lim{G/(HNi ), π ji } ←−

and hence {π i } : G/H → {G/(HNi ), π ji } is a G-resolution (because the projections π i are surjective). Since G/Ni is a Lie group for each i, all the quotient spaces G/(HNi ) are G-ANR’s by, for instance, [3, Proposition 2.2]. Thus we can conclude that {αi } : p → {pi , αij } is a G-ANR-resolution. Finally, every pi can be regarded as a G/Ni -map Ei → (G/Ni )/(HNi /Ni ) which is, since G/Ni is a compact Lie group, a G/Ni -fibration (see e.g.,[8, Proposition 3.1]). Thus pi is also a G-fibration by [10, Proposition 2.1]. 2 Theorem 4.3. Let H be a closed subgroup of a compact metrizable group G. Let p : E → G/H be a G-map,  together with where E is a compact metrizable G-space. Then E admits a G-fibrant extension i : E → E the commutative diagram i

E p

 E (4.2)

p

G/H  :E  × [0, ∞) → E  where p is a strong G-fibration for which there exists an infinite strong G-deformation D    of E onto i(E) over G/H, that is, satisfying the equality p ◦ D(y, t) = p(y) for all (y, t) ∈ E × [0, ∞). If S = p−1 (eH) and S = p−1 (eH), then the restriction i : S → S of the embedding i is an H-fibrant extension of S. Proof. By Proposition 4.2 we can find a G-ANR-resolution p = {pi , αij } of p such that each pi is a G-fibration of G-ANR’s. Now applying Theorem 3.3 we have a G-extension p of p represented by the commutative diagram of G-maps E p

G/H

 E p

(4.3)

 B

 D) : p × id[0,∞) → p where p is a strong G-fibration. Moreover, there is a infinite strong G-deformation (D,  = p−1 (G/H) and p = p|  , we obtain the diagram (4.2) as a restriction of (4.3). of p on p. Putting E E  t) = D(  and Since D(b, t) = b for all b ∈ G/H and t ∈ [0, ∞), we have pD(y, p(y)) = p(y) for all y ∈ E  and one can define the desired infinite strong G-deformation D  E  × [0, ∞)) ⊂ E,  t ∈ [0, ∞). Therefore D(  as a restriction of D. To prove the last statement of the theorem, first note that S is H-fibrant because the constant map p, being a strong G-fibration, is also a strong S → {eH} is a strong H-fibration as a restriction of p (  H-fibration by [7, Proposition 4.1(d)]). Moreover, we have D(S × [0, ∞)) ⊂ S and therefore the restriction  to S × [0, ∞) gives us a an infinite strong H-deformation of S onto i(p−1 (eH)) = i(S). Thus i : S → S of D is an H-SSDR-map and hence an H-fibrant extension. 2

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Corollary 4.4. Let H be a closed subgroup of a group G and let E be a compact metrizable G-space. Then a G-map p : E → G/H is a strong G-fibration if and only if p−1 (eH) is an H-fibrant space. Proof. The part “only if” is obvious (just repeat the argument in the proof of Theorem 4.3 which shows that S is H-fibrant). Now suppose that S = p−1 (eH) is an H-fibrant space. By Theorem 4.3 (and in its terms), we have the  For the map idS : S → S, since S is H-fibrant and i is an H-SSDR-map, H-fibrant extension i : S → S.  with the twisted there exists an H-map r : S → S such that r ◦ i = idS . Identifying, as usual, E and E   products G ×H S and G ×H S, respectively, we have i = G ×H i (indeed, i([g, s]) = i(gs) = gi(s) = [g, i(s)]).  → E, given by r = G ×H r, we have r ◦ i = idE and, moreover, p ◦ r = p (because Then for the map r : E  onto E over G/H. This implies easily p ◦ r(g s) = p(g r( s)) = gH = p(g s)), that is, r is a G-retraction of E that p is a strong G-fibration as well as p. 2 Theorem 4.5. Let H be a closed subgroup of a compact metrizable group G. If j : S → S is an H-fibrant extension of a compact metrizable H-space S, then G ×H j : G ×H S → G ×H S is a G-fibrant extension of G ×H S. Proof. First we shall indicate an H-fibrant extension of S for which the assertion of the theorem holds a priori. Let E = G ×H S. Clearly, we can assume that S is an H-subset of E (by the identification s ≡ [e, s] ∈ E ), so that E = GS. Then for the natural projection p : E → G/H, gs → gH, we have  because the S = p−1 (eH). Applying Theorem 4.3 to p we obtain the desired H-fibrant extension i : S → S,   G-map G ×H i can be identified with the G-fibrant extension i : E → E given by this theorem. Now let j : S → S be any H-fibrant extension. Then G ×H j is a G-SSDR-map by [7, Proposition 4.1(c)]. Thus we need only show that G ×H S is a G-fibrant space. For this we shall use the existence of H-maps ϕ : S → S and ψ : S → S such that ψ ◦ ϕ  idS (see Preliminaries). Let s : A → X be a G-SSDR-map and let f : A → G ×H S be a G-map. Of course, we may assume that A ⊂ X. To complete the proof we must find a G-map f : X → G ×H S for which f ◦ s = f . Since  = G ×H S is G-fibrant, there is a G-map F : X → G ×H S such that the diagram E A

f

G ×H S

G×H ϕ

G ×H S

s F

X  and X  = F −1 (S)  we get the commutative diagram of H-maps commutes. Putting A = f −1 (S) A s

f

S

ϕ

S

F

X where f  and s are the restrictions of f and s, respectively, to A and F  is the restriction of F to X  . For the H-map ψ ◦ F  : X  → S we have (ψ ◦ F  )|A = ψ ◦ F  ◦ s = ψ ◦ ϕ ◦ f   f  . Since S is an H-fibrant space, the H-pair (X  , A ) has the homotopy extension property with respect S and therefore f  can be extended to some H-map f : X  → S so that f ◦ s = f  . Then the map f = G ×H f is the desired extension of f . 2

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Corollary 4.6. Let H be a closed subgroup of a compact metrizable group G. Let S be a compact metrizable H-space. If S is an H-fibrant space, then G ×H S is a G-fibrant space. Proof. Since S is an H-fibrant space, the identity map idS is trivially an H-fibrant extension of S. By Theorem 4.5, G × idS = idG×H S : G ×H S → G ×H S is a G-fibrant extension. In particular, G ×H S is a G-fibrant space. 2 References [1] S.A. Antonyan, Equivariant embeddings into G-AR’s, Glas. Mat. 22 (42) (1987) 503–533. [2] S.A. Antonyan, Extensorial properties of orbit spaces of proper group actions, Topol. Appl. 98 (1999) 35–46. [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, R. Jiménez, S. de Neymet, Fiberwise retraction and shape properties of the orbit space, Glas. Mat. 35 (55) (2000) 191–210. [5] S.A. Antonyan, S. Mardešić, Equivariant shape, Fundam. Math. 127 (1987) 213–224. [6] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972. [7] A. Bykov, The homogeneous space G/H as an equivariant fibrant space, Topol. Appl. 157 (2010) 2604–2612. [8] A. Bykov, A.L. Kantún Montiel, Strong G-fibrations and orbit projections, Topol. Appl. 163 (2014) 46–65. [9] A. Bykov, M. Texis, Equivariant strong shape, Topol. Appl. 154 (2007) 2026–2039. [10] A. Bykov, A. Torres Juan, Fibrant extensions of free G-spaces, Topol. Appl. 159 (2012) 1179–1186. [11] F. Cathey, Strong shape theory, in: Shape Theory and Geometric Topology, in: Lecture Notes Math., Springer, Berlin, 1981, pp. 216–239. [12] T. Dieck, Transformation Groups, Walter de Gruyter, Berlin-New York, 1987. [13] I.M. James, General Topology and Homotopy Theory, Springer-Verlag, 1984. [14] R.S. Palais, The classification of G-spaces, Mem. Am. Math. Soc. 36 (1960). [15] L.S. Pontrjagin, Topological Groups, Princeton Univ. Press, 1939.