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EQUIVARIANT EMBEDDINGS AND COMPACTIFICATIONS OF FREE G -SPACES
NATELLA ANTONYAN Received 16 November 2001
For a compact Lie group G, we characterize freeG-spaces that admit freeG- compactifications. For suchG-spaces, a universal compact freeG-space of given weight and given dimension is constructed. It is shown that ifGis finite, then- dimensional Menger freeG-compactumµnis universal for all separable, metriz- able freeG-spaces of dimension less than or equal ton. Some of these results are extended to the case ofG-spaces with a single orbit type.
2000 Mathematics Subject Classification: 54H15, 54D35.
1. Introduction. By aG-space, we mean a triple(G, X, α), whereGis a topo- logical group, X is a topological space, andα:G×X →X is a continuous action.
In 1960, Palais proved that every TychonoffG-space can equivariantly be em- bedded into a compact HausdorffG-space providedGis a compact Lie group (see [17, Section 1.5]). This result was extended by de Vries [5] to the case of arbitrary locally compact Hausdorff groups. The local compactness is essential here; it was Megrelishvili who constructed in [14] a continuous actionαof a separable, complete metrizable groupGon a separable, metrizable spaceX such that(G, X, α)does not admit an equivariant embedding into a compact G-space. The reader can find other examples of this type in [15].
In this paper, we are mostly interested infreeG-spaces. Recall that aG-space Xis free if, for everyx∈X, the equalitygx=ximpliesg=e, the unity ofG.
In [2], it is proved that ifGis a compact Lie group, then any Tychonoff free G-space can equivariantly be embedded in a locally compact freeG-space. In this connection, it is natural to ask the following question.
Question1.1. Does every freeG-space have aG-embedding in a free com- pactG-space?
One of the purposes of the present paper is to answer this question forG a compact Lie group. Namely, we prove that each finitistic freeG-spaceXhas a freeG-compactification (Theorem 3.4). In the realm ofG-spaces that admit a freeG-compactification, we construct a universal, compact, freeG-space of given weight and given dimension (Theorem 4.1). This result is extended to the case of theG-spaces with a single orbit type (Theorem 5.2).
2. Preliminaries. Throughout the paper, all topological spaces are assumed to be Tychonoff (i.e., completely regular and Hausdorff). All equivariant orG- maps are assumed to be continuous.
The letter “G” will always denote a compact Lie group.
The basic ideas and facts of the theory ofG-spaces or topological transfor- mation groups can be found in Bredon [4] and Palais [17].
For the convenience of the reader, however, we recall some more special definitions and facts below.
Bye, we will always denote the unity of the groupG.
If X is a G-space, for any x∈X, we denote the stabilizer (or stationary subgroup) ofxbyGx= {g∈G|gx=x}.
If, for allx∈X,Gx= {e}, then we say that the action ofGisfreeandXis a freeG-space.
For a subsetS⊂Xand a subgroupH⊂G,H(S)denotes theH-saturation of S, that is,H(S)= {hs|h∈H, s∈S}. In particular,G(x)denotes theG-orbit {gx∈X|g∈G}ofx. IfH(S)=S, thenS is said to be anH-invariant set. The G-orbit space is denoted byX/G.
ByG/H, we will denote theG-space of cosets{gH|g∈G}under the action induced by left translations.
For each subgroupH⊆G, theH-fixed point setXHis defined to be the set {x∈X|H⊆Gx}.
The family of all subgroups ofGwhich are conjugate toHis denoted by(H), that is,(H)= {gHg−1|g∈G}. The set(H)is called aG-orbit type (or simply an orbit type). For two orbit types(H1)and(H2), we say that(H1)(H2)if and only ifH1⊆gH2g−1for someg∈G. If(H1)(H2)and(H1)≠(H2), then we write(H1)≺(H2). The relationis a partial ordering on the set of allG-orbit types. SinceGgx=gGxg−1, for anyx∈X,g∈G, we have(Gx)= {Ggx|g∈G}. We say that aG-spaceXis of the orbit type(H), or simply of type(H), if (Gx)=(H)for everyx∈X.
In this paper, we will consider onlyG-spaces that have a single orbit type (H).
An equivariant mapf :X→Y ofG-spaces is said to beisovariant or (G- isovariant) ifGx=Gf (x)for allx∈X.
IfX andY areG-spaces, then X×Y will always be regarded as aG-space equipped by the diagonal action ofG.
AG-compactification of aG-spaceXis a pair(bG, bGX), wherebG:X→bGX is aG-homeomorphic embedding into a compactG-spacebGXsuch that the imagebG(X)is dense inbGX. Usually,bGXalone is a sufficient denotation. By βGX, we will denote the maximalG-compactification ofX.
In the sequel, we will need the following lemma.
Lemma2.1(see [1]). Letf:X→S be an isovariant map ofG-spaces. Then, the maph:X→S×(X/G), defined byh(x)=(f (x), p(x))wherep:X→X/G is the orbit map, is aG-homeomorphic embedding.
We also recall the well-known and important definition of a slice [17, page 27].
Definition2.2. A subsetSof aG-spaceXis called anH-slice inXif (1) SisH-invariant, that is,H(S)=S,
(2) the saturationG(S)is open inX, (3) ifg∈G\H, thengS∩S= ∅, (4) Sis closed inG(S).
The saturationG(S)will be said to be anH-tube. If, in addition,G(S)=X, then we say thatSis a globalH-slice inX.
If S is a global H-slice inX, then X is G-homeomorphic to the so-called twisted product G×HS. Recall that G×HS is just the H-orbit space of the productG×S on whichHacts by the ruleh(g, s)=(gh−1, hs), whereh∈H and(g, s)∈G×S. In turn,Gacts onG×HS by the formulag[g, s]=[gg, s], whereg ∈G,[g, s]∈G×HS(see [4, Section 4]).
One of the basic results of the theory of topological transformation groups is the Slice theorem, which asserts the following: ifXis aG-space andx∈X, then there exists aGx-sliceS⊂Xcontaining the pointx(see, e.g., [17, Theorem 1.7.18] or [4, Chapter II, Theorem 5.4]).
An important consequence of the Slice theorem is that ifXis aG-space with the orbits all of the same type, then the orbit mapX→X/Gis a locally trivial fibration [4, Chapter II, Theorem 5.8].
In what follows,Gwill mean “isG-homeomorphic.”
We writeX=X/Gfor the orbit space ofX.
The following definition is due to Jaworowski [12] even for G-spaces of finitely many orbit types.
Definition2.3. We say that aG-spaceX with a single orbit type(H)is of finite structureif the orbit map p:X→X has a finite trivializing cover, that is to say, there exists a finite open cover{U1, . . . , Un}ofXsuch that each p−1(Ui)isG-equivalent to(G/H)×Ui, that is, there exists aG-homeomorphism fi:p−1(Ui)→(G/H)×Uisuch thatπ (fi(x))=p(x)for everyx∈p−1(Ui).
Here, we remark that the claim “p:X→X has a finite trivializing cover”
is equivalent to “Xcan be covered by finitely manyH-tubes.” Namely, in this form, we will use the definition in what follows.
It is evident fromDefinition 2.3that any invariant subspace of aG-space of finite structure is again aG-space of finite structure.
3. G-compactifications of a single orbit type. Recall that the cone con(X) over a compact metric spaceXis the quotient set[0,1]×X/{0}×Xequipped with the quotient topology. This topology is metrizable too (see [10, Chapter VI, Lemma 1.1]). The image of the point(t, x)∈[0,1]×Xunder the canonical projectionp:[0,1]×X→con(X)will be denoted bytx, and we will simply
writeθ(think of zero) instead of 0x; this is the vertex of the cone. It is conve- nient to call the numbertintxthe norm oftxand denote it bytx.
IfX1, . . . , Xkare compact metric spaces, the joinX1∗ ··· ∗Xkis defined to be the subset of the product con(X1)× ··· ×con(Xk)consisting of all those points(t1x1, . . . , tkxk)for whichn
i=1ti=1. Below, we will consider the case whenX1= ··· =Xk=G/H, whereH is a closed subgroup ofG. In this case, Gacts coordinatewise on thek-fold joinG/H∗···∗G/Hby left translations;
so,G/H∗···∗G/His aG-space, which we will denote shortly by(G/H)∗k. In what follows, by a EuclideanG-space, we mean a real Euclidean spaceE on whichGacts by means of orthogonal transformations.
It is convenient to introduce the following notion that is closely related to the notion of the finite structure introduced by Jaworowski (seeSection 2).
Definition3.1. We say that aG-spaceXis of Euclidean type if there exists an isovariant mapf:X→Einto a EuclideanG-spaceE.
In [12], Jaworowski proved that each normal G-space of finite structure is of Euclidean type. Here, we need the following more precise version of Jaworowski’s result.
Lemma3.2. Any normalG-spaceX of a single orbit type(H)and of finite structure admits an isovariant map into a finite-dimensional, compact, metriz- ableG-spaceDof type(H).
Proof. It is known that, under the conditions of the lemma, the orbit map p:X→Xis a locally trivial fibration (see [4, Chapter II, Theorem 5.8]).
Let {U1, U2, . . . , Uk} be a finite open cover of the orbit spaceX such that, for every 1≤n≤k, p−1(Un)is equivariantly homeomorphic to the product G/H×Un, where the groupGacts on the left onG/Hand acts trivially onUn. Further, for eachn≥1, the first projection of the productp−1(Un)=G/H× Ungives an isovariant mapϕn:p−1(Un)→G/H.
Since the orbit spaceXis normal, there exists a closed shrinking{F1, . . . , Fk} for{U1, U2, . . . , Uk}inX, that is, Fn⊂Unfor all 1≤n≤kandk
n=1F=X [9, Theorem 1.7.8].
Let ψn :X→[0,1] be a continuous function such that Fn ⊂ψn−1(1) and
X\Un⊂ψn−1(0). Now, we define a mapfn:X→con(G/H)by the formula
fn(x)=
θ ifx∉p−1 Un
, ψn
p(x)
ϕn(x) ifx∈p−1 Un
. (3.1)
It is clear thatfnis an equivariant map, and that its restriction top−1(Fn)co- incides withϕnand is, therefore, isovariant. We consider the diagonal product
f= kn=1fn:X →
con(G/H)k
. (3.2)
Then,f is an equivariant map. Sincek
n=1fn(x) =1, for allx∈X, we conclude thatf (x)belongs, in fact, to thek-fold join(G/H)∗k.
Ifx∈Xandx∈p−1(Fn), then
Gf (x)=
∞ k=1
Gfk(x)⊂Gfn(x)=Gx. (3.3)
On the other hand,Gx⊂Gf (x) sincef is equivariant. Consequently,Gx= Gf (x), that is,f is an isovariant map.
Now, we defineDto be the closure off (X)in(G/H)∗k. Then,Dis finite- dimensional, compact and metrizable. It remains to see thatDhas the orbit type(H). Letd∈D be an arbitrary point. Since each orbit type in(G/H)∗k is(H), we see that(Gd)≤(H). On the other hand, sincef (X)is dense in Dand f (X)is of type (H), it follows from the Slice theorem [4, Chapter II, Corollary 5.5] that(H)(Gd). Thus,(Gd)=(H), and hencef:X→Dis the desired map.
Theorem3.3. For aG-spaceXof a single orbit type(H), the following are equivalent:
(1) Xhas aG-compactification of type(H),
(2) Xhas an isovariant map in a finite-dimensional, compact metrizableG- spaceDof type(H),
(3) Xis of Euclidean type,
(4) Xhas an isovariant map in a compactG-space of type(H),
(5) Xhas aG-compactification of type(H)and of the same weightwX.
Proof. (1)⇒(2). LetbGXbe aG-compactification ofXof type(H). ByLemma 3.2, there is an isovariant mapϕ:bGX→Din some finite-dimensional, com- pact metrizableG-spaceDof type(H). The restrictionϕ|Xis the desired map.
(2)⇒(3). Letϕ:X→Dbe an isovariant map in a finite-dimensional, compact metrizableG-spaceDof type(H). Since there exists an equivariant embedding i: D→E in a Euclidean G-space E (see [4, Chapter II, Theorem 10.1]), the compositionf=ϕimapsXisovariantly intoE.
(3)⇒(4). Letψ:X→Ebe an isovariant map in a EuclideanG-spaceE. Then, by Lemmas3.2and3.6, there exists an isovariant mapj:ψ(X)→Din a finite- dimensional, compact metrizableG-spaceDof type(H). Then, the composi- tionjψ:X→Dis the required map.
(4)⇒(1). Letf:X→Y be an isovariant map in a compactG-spaceY of type (H).
Letp:X→X/Gbe the orbit map. ByLemma 2.1, the diagonal producti= ϕ∆p:X→Y×(X/G)is an equivariant embedding.
LetB be any compactification of the orbit space X/G. Then,Xcan be re- garded as an invariant subset of the compactG-spaceY×B, whereGacts on Btrivially. Now, the closureXofXinY×Bis aG-compactification ofX. Since
Y is of type(H), we see thatY×Bis also of type(H). Hence,bGX=X is a G-compactification ofXof type(H).
(2)⇒(5) can be proved like the implication (4)⇒(1) using D instead ofY. In that case, if we choose the compactificationB of X/Gsuch that w(B)= w(X/G)[8, Theorem 3.5.2], then the G-compactification bGX will have the weightw(bGX)=w(X/G)becausew(D)= ℵ0. It remains only to observe that w(X)=w(X/G).
(5)⇒(1) is evident.
Recall that a paracompact spaceXis said to befinitisticif every open cover ofXhas a refinementωof a finite order, that is, there is a natural numbern such that any pointx∈Xcan belong at most tonelements ofω(see [19]).
Evidently, each compact space, as well as each paracompact finite-dimen- sional space, is finitistic.
A wide class ofG-spaces that admitG-compactifications of a single orbit type is provided by the following theorem.
Theorem3.4. Every finitisticG-spaceXof type(H)has aG-compactification bGXof the same type(H)and of the same weightwX.
For the proof, we need the following result, which was established first by Milnor for finite-dimensional spaces (cited in [17, Theorem 1.8.2]).
Lemma 3.5. Let X be a finitistic space and let {Uα}be an open covering ofX. Then, there exist a natural numbernand an open covering {Viβ}β∈Bi, i=0, . . . , n, ofX refining{Uα}such thatViβ∩Viβ = ∅wheneverβ≠β and 0≤i≤n.
Proof. AsX is finitistic, there are a natural numbernand a refinement {Wµ}of{Uα}such that the order of the cover{Wµ}is at mostn. Let{ϕµ}be a locally finite partition of unity withϕ−µ1((0,1])⊂Wµ. For every 0≤i≤n, letBi be the set of all subsetsβof the indexing set of the cover{Wµ}with cardinality|β| =i+1. Givenβ=(µ0, . . . , µi)∈Bi, we set
Viβ=
x∈X|ϕµj(x) >0 andϕµ(x) < ϕµj(x)∀0≤j≤i, µ∉β
. (3.4)
As in a neighborhood of any pointx, only a finite number ofϕµis not identi- cally zero, and it follows that eachViβis open.
Let us check thatViβ∩Viβ = ∅ifβ≠β. Indeed, since|β| =i+1= |β|and β≠β, we infer that there areµ∈β\β andµ ∈β \β. Now, ifx∈Viβ∩Viβ, it then follows thatϕµ(x)< ϕµ(x) < ϕµ(x), a contradiction.
Check that{Viβ}is a covering forX. Ifx∈Xandµ0, . . . , µmare all the indices withϕµk(x) >0 so arranged that
ϕµ0(x)=ϕµ1(x)= ··· =ϕµi(x) > ϕµi+1(x)≥ ··· ≥ϕµm(x), (3.5)
then, evidently,x∈Vmβ, whereβ= {µ0, . . . , µm}. Since
x∈
m j=0
suppϕµj⊂
m j=0
Wµj (3.6)
and {Wµ}has order ≤n, it follows thatm≤n. Consequently, i≤n, and, clearly,x∈Vi{µ0,...,µi}. Thus,{Viβ}β∈Bi,i=0, . . . , n, is an open cover ofX, and sinceViβ⊂Wµ for everyµ∈i, we see that{Viβ}refines the cover{Wµ}, and hence, the original cover{Uα}. Thus,{Viβ}is the desired cover.
Lemma3.6. Every finitisticG-spaceXhaving only one orbit type is of finite structure.
Proof. Let (H)be the only orbit type of X. Let {Sα}be a family of H- slices inXsuch thatX=
G(Sα). Then,G(Sα)G(G/H)×p(Sα)and the sets p(G(Sα))=p(Sα)constitute an open cover of the orbit spaceX/G. Now, by [6], X/Gis also finitistic, so, by the preceding lemma, we can find a natural number nand an open cover{Uiβ}β∈Bi,i=0, . . . , nofX/Gwhich refines{p(Sα)}and is such thatUiβ∩Uiβ = ∅ifβ≠β. Then, the setUiβ=p−1(Uiβ)is anH-slice inUiβ, that is,G(Uiβ)G(G/H)×Uiβ[17, Proposition 1.7.2], andUiβ=p(Uiβ).
It then follows that the unionUi=
β∈BiUiβis anH-slice overUi=
β∈BiUiβ
(see [17, Proposition 1.7.3]). Thus,G(Ui)G(G/H)×Ui, and hence{Ui}ni=1is a finite trivializing cover forX/G.
Proof ofTheorem3.4. It follows from Lemmas3.6and3.2thatXis of Euclidean type. Now, the claim follows fromTheorem 3.3.
Proposition3.7. If aG-spaceX of type(H)admits aG-compactification bGXof the same type(H), then its maximalG-compactificationβGXis also of the same type(H).
Proof. Indeed, there exists aG-mapf:βGX→bGX. Hence,(Gt)(Gf (t))= (H)for everyt∈βGX. On the other hand, sinceXis dense inβGXandXis of type(H), it follows from the Slice theorem that(H)(Gt)for everyt∈βGX (see [4, Chapter II, Corollary 5.5]). Thus,(Gt)=(H)for everyt∈βGX.
The following is an example of a freeZ2-action on the Hilbert cube with a removed point, which does not have a freeZ2-compactification.
Example3.8(see [12]). LetX=[−1,1]∞\{0}, where 0=(0,0, . . .)∈[−1,1]∞, andG=Z2, the cyclic group of order two. So,Xis the Hilbert cube with a re- moved point. Consider the free action of Z2 on X defined by the standard involution{xi} → {−xi}. It turns out that the freeZ2-spaceXdoes not have a freeZ2-compactification.
Assume the contrary. Then, byTheorem 3.3, there exists an isovariant map f:X→Ein a EuclideanZ2-space. SinceXis a freeZ2-space,f−1(0)= ∅, where 0 denotes the origin ofE. Clearly, the radial retractionr:E\{0} →Sonto the
unit sphere ofEis an isovariant map. Hence, the compositionϕ=r f:X→S is isovariant too.
LetSkbe a sphere of arbitrary dimensionk >0, considered as aG-space with the antipodal action ofZ2.
Claim 1. Each sphere Sk can Z2-equivariantly be embedded into the Z2- spaceX.
Indeed, it suffices to show that the Z2-maps from Sk to [−1,1] separate points ofSk. Leta, b∈Sk,a≠b. Ifb= −a, then we first choose a continuous mapf:Sk→[−1,1]withf (a)=1 andf (b)= −1 and then definef (x)= (f (x)−f (−x))/2,x∈Sk. Clearly,f is aZ2-map withf (a)=1 andf (b)=
−1. Ifb≠−a, then we first choose a continuous mapf :Sk→[−1,1] with f (a)=f (−b)=1 andf (b)=f (−a)= −1 and then definef (x)=(f (x)− f (−x))/2,x∈Sk. Clearly,f is aZ2-map withf (a)=1 andf (b)= −1.
Now, by Claim 1there exists a G-embeddingi:SkX. The composition q=ϕi:Sk→Sis then an equivariant (i.e., an antipodal) map. But, according to the classical Borsuk-Ulam theorem (see, e.g., [18, Chapter 5, Section 8, Corollary 8]), there is no such a map fork >dimS.
This example also has the following interesting property in spirit of Douwen’s paper [20].
Corollary 3.9. Let f :X→X be the standard involution on the Hilbert cube with a removed point (Example 3.8). Then, the Stone- ˇCech compactification βf:βX→βXhas a fixed point.
Proof. Indeed, otherwiseβXis a freeZ2-compactification ofX, which con- tradicts the claim ofExample 3.8.
4. Universal finite-dimensional compact freeG-spaces. In this section, we prove the following theorem.
Theorem4.1. For every infinite cardinal numberτand for every nonneg- ative integern≥dimG, there exists a compact freeG-spaceᏲτnwithw(Ᏺnτ)= τ, dim(Ᏺnτ)=n which is universal in the following sense:Ᏺnτ contains a G- homeomorphic copy of any freeG-spaceXof Euclidean type withwX≤τand dimβGX≤n. In particular,Ᏺnτ contains aG-homeomorphic copy of each para- compact freeG-spaceXwithwX≤τanddimX≤n.
We notice that a similar result for the nonfree case was established earlier in [13].
Before proceeding with the proof, we will establish the following lemma.
Lemma4.2. LetX be a paracompact freeG-space. Then, the following two properties are fulfilled:
(1) dimX=dim(X/G)+dimG;
(2) dimβGX=dimX.
Proof. (1) Letp:X→X/Gbe the orbit map. It is well known [4, Chap- ter II, Theorem 5.8] thatpis a locally trivial fibration with fibers homeomor- phic toG. Let{Uα}be an open trivializing cover of the orbit spaceX/G, that is,p−1(Uα)GG×Uα. By compactness ofG, the mapp is closed, and by a theorem of E. Michael [8, Theorem 5.1.13], the orbit spaceX/G is paracom- pact, too. Then, there exists a locally finite closed cover {Fα}of X/G such that Fα ⊂Uα for each index α. It follows that p−1(Fα)G G×Fα and the family{p−1(Fα)}constitute a locally finite closed cover ofX. Then, accord- ing to the Sum theorem [9, Theorem 3.1.10], dimX=maxα{dimp−1(Fα)}. But dimp−1(Fα)=dim(G×Fα). Being a closed subset of a paracompact space,Fαis itself paracompact. On the other hand,Gis a polyhedron. Hence, Morita’s the- orem [16] is applicable here and, accordingly this logarithmic low holds true:
dim(G×Fα)=dimG+dimFα. Thus, we have dimX=dimG+maxα{dimFα}. Applying once more the sum theorem, we get dim(X/G)=maxα{dimFα}. Con- sequently, dimX=dim(X/G)+dimG.
(2) We will use the formulaβ(X/G)=(βGX)/G(see [3]). Consider two cases.
(a) Let dimX <∞. Then,Xhas finite structure (Lemma 3.6) and thenβGXis a freeG-space (Proposition 3.7). Applying twice the equality established in the previous step, we get
dimβGX=dim βGX
/G+dimG
=dimβ(X/G)+dimG
=dim(X/G)+dimG
=dimX.
(4.1)
(b) Let dimX= ∞. ByClaim 1, we have dimX=dimG+dim(βGX)/G, which implies that dim(βGX)/G= ∞. But the orbit map does not rise dimension [6];
in particular,
dimβGX=dim βGX
/G= ∞ =dimX. (4.2) The following lemma in the nonfree case was proved by Megrelishvili [13]
even for noncompact acting groups.
Lemma4.3. Letf:X→Y be aG-map of a compact free G-spaceX into a compactG-spaceY. Then, there exist a compact freeG-space Z andG-maps ϕ:X→Z,ψ:Z→Y such thatf=ψϕanddimZ≤dimX,wZ≤wY.
Proof. We will first prove the claim in case whenY is a freeG-space, too.
Consider the induced mapf :X/G→Y /G. By Mardeši´c’s factorization theo- rem [9, Theorem 3.3.2], there exist a compact spaceZ and continuous maps ϕ :X/G→Z, ψ :Z →Y /Gsuch thatf =ψϕ and dimZ ≤dim(X/G), wZ ≤w(Y /G).
Denote bypthe orbit mapY→Y /G. It is well known [11, Chapter IV, Propo- sition 4.1] that we have the following (pullback) commutative diagram:
Z
π ψ
Y
p
Z ψ Y /G
(4.3)
whereZ is a compactG-space withZ/G=Z ,π :Z→Z —the orbit map and ψ—an equivariant map that induces the mapψ. In fact,Zis theG-invariant subset ofZ ×Y defined as follows:Z= {(z, y)|ψ(z)=p(y)}, whereGacts onZ ×Y byg(z, y)=(gz, y)forg∈Gand (z, y)∈Z ×Y. Thus,Z is a compact freeG-space andψ:Z→Y is the restriction of the second projection Z ×Y→Y.
Now, we defineϕ:X→Z byϕ(x)=(ϕq(x), f (x)), whereq:X→X/Gis the orbit map. It is easy to check thatf=ψϕ.
On the other hand,wZ=wZ ≤w(Y /G)=wY.
Let us check that dimZ≤dimX. AsZis a paracompact freeG-space, we can applyLemma 4.2, according to which dimZ=dimZ +dimG≤dim(X/G)+ dimG=dimX.
Now we pass to the general case. ByLemma 3.2, there is an isovariant map h:X→D to a compact freeG-spaceD. Consider the productT =h(X)×Y and the map r :X →T defined byr (x)=(h(x), f (x)), x∈ X. Since X is free and his isovariant, we infer that T is a freeG-space. It is clear thatr is equivariant andwT=wY. Now, we apply the preceding case, according to which there exist a compact G-space Z and G-mapsϕ:X→Z, ψ1:Z →T such that dimZ≤dimX, wZ≤wT and r =ψ1ϕ. Observe that wT =wY becausewh(X)= ℵ0; so,wZ≤wY. Putψ=π2ψ1, whereπ2:T →Y is the second projection. Then,ψ:Z→Y is aG-map such thatf=ψϕ. It remains to observe thatZis a freeG-space; this is immediate from the equivariance of ψ1and from the freeness ofT.
Proof ofTheorem4.1. LetBτbe a universal TychonoffG-cube of weight τ(see [3]), that is,Bτis aG-space homeomorphic to the Tychonoff cube[0,1]τ and contains a G-homeomorphic copy of every G-space of weight≤τ. Let {Yt}t∈T be the family of all invariant free subsetsYt⊂Bτ of Euclidean type such that dimβGYt ≤n. This family is nonempty because the group G be- longs to it. For eacht∈T, we denote byitthe identical embedding ofYtinto Bτ. Consider the discrete sumY =
t∈TβGYt, which naturally becomes aG- space. ByProposition 3.7, each βGYt is a freeG-space. Consequently,Y is a paracompact freeG-space. As dimβGYt≤nfor allt∈T, then, by the Sum the- orem [9, Theorem 3.1.10], we have dimY ≤n. Consequently, byLemma 4.2, dimβGY=dimY≤n.
Next, each mapit:Yt→Bτcan be extended to aG-mapit:βGYt→Bτ (see [17, Section 5]); so, a mapi:Y→Bτarises defined byi(y)=it(y)fory∈βGYt. Applying once more [17, Section 5], we extendito aG-mapj:βGY →Bτ. As Y has finite structure according toProposition 3.7,βGY is a compact freeG- space. By virtue of Lemma 4.3, there exist a compact freeG-space Ᏺnτ and G-mapsϕ:βGY →Ᏺnτ,ψ:Ᏺnτ→Bτsuch thati=ψϕand dimᏲnτ≤n,wᏲnτ≤ wBτ=τ. We claim thatᏲnτ is the desiredG-space.
Indeed, letXbe an arbitrary freeG-space such that dimX≤nandwX≤τ. SinceXis equivariantly embeddable inBτ, there exists at∈T such thatYt is G-homeomorphic toX. As the restriction ofionYtis a homeomorphism, the restrictionϕ|Yt is also a homeomorphism. Besides,ϕ|Yt is equivariant. Thus, Xis equivariantly embeddable inᏲnτ.
IfXis paracompact, then, byLemma 4.2, dimβGX=dimX≤n, and hence Xcan be embedded equivariantly inᏲnτ.
To complete the proof, it remains to see that dimᏲnτ=nandwᏲnτ=τ. AsᏲnτ
contains an equivariant homeomorphic copy of then-dimensional, compact freeG-spaceG×Ikwithk=n−dimG, we infer that dimᏲnτ=n. On the other hand, the discrete sumZofτmany copies ofGis a metrizable freeG-space of weightwZ=τ, and henceᏲnτ contains an equivariant homeomorphic copy ofZ. This yields thatwᏲnτ=τ.
FromTheorem 4.1, the following corollary follows immediately.
Corollary 4.4. Any paracompact freeG-spaceX has a free G-compact- ificationbGXof weightw(bGX)≤wXand of dimensiondimbGX≤dimX.
Corollary4.5. LetGbe a finite group. Then, for any integern≥0, there is a free action ofGon the Menger compactumµnsuch that every separable, metrizable, freeG-spaceX withdimX≤nadmits an equivariant embedding intoµn.
Proof. By the preceding corollary,X has a compact, metrizable, freeG- compactificationbGXof dimbGX≤dimX. It remains to apply Dranishnikov’s result [7, Corollary and Theorem 3] to the effect that there is a unique free action ofGon the Menger compactumµnsuch thatµncontains an equivariant homeomorphic copy of each compact, metrizable, freeG-space of dimension less than or equal ton.
5. The case ofG-spaces of a single orbit type. In this section, we generalize Theorem 4.1to the case ofG-spaces of Euclidean type that may not be free, but have a single orbit type.
LetHbe a closed subgroup ofGandXbe aG-space of type(H). LetN(H)be the normalizer ofHinGandW (H)=N(H)/H, the Weyl group. Below, for any n∈N(H), we denote bynthe lateral classnH. The groupW (H)acts freely onXH, theH-fixed point set ofX. At the same time,W (H)acts onG/Hby the
formula
n∗gH=gn−1H, n∈W (H), gH∈G/H. (5.1) The twisted product(G/H)×W (H)XHis just theW (H)-orbit space of the prod- uctG/H×XH endowed with the diagonal action of W (H). It is well known (see [4, Chapter II, Corollary 5.11]) thatXisG-homeomorphic to theG-space (G/H)×W (H)XH, equipped with the action ofGgiven by the formula
g ∗[gH, x]=[ggH, x], g ∈G, [gH, x]∈(G/H)×W (H)XH. (5.2) Lemma5.1. IfHis a closed subgroup ofGandYis a freeW (H)-space, then the twisted productT =(G/H)×W (H)Y has only one orbit type(H). Besides, wT=wY anddimT=dimY+dim(G/N(H)).
Proof. Indeed, let[gH, x]be a point of(G/H)×W (H)Y fixed under an el- ementg ∈G. Then, [ggH, x]=[gH, x]or, equivalently,(ggH, x)=(n∗ gH,nx), for some n∈N(H). Then,ggH=gn−1Handx=nx. Since W (H) acts freely onY, the equalityx=nx implies thatn∈H. The equalityggH= gn−1H yields thatg =gn−1hg−1 for someh∈H, and hence,g ∈gHg−1. Consequently, the stabilizer of[gH, x]is just the groupgHg−1, and hence, theG-space(G/H)×W (H)Y has only one orbit type(H).
Sincew(G/H)≤ ℵ0, we see that w
(G/H)×W (H)Y
≤wY . (5.3)
On the other hand,Y is a subset ofT, sowY≤wT.
For the second equality, byLemma 4.2and by the above quoted Morita’s theorem [16], we have
dimT=dim
(G/H)×W (H)Y
=dim
(G/H)×Y
−dimW (H)
=dim(G/H)+dimY−
dimN(H)−dimH
=dimY+dimG−dimH−dimN(H)+dimH
=dimY+dimG−dimN(H)
=dimY+dim
G/N(H) .
(5.4)
Theorem5.2. For every closed subgroupH⊂G, every infinite cardinal num- ber τ and for every nonnegative integern≥dimG, there exists a compact G-spaceᏲnτ(H)of type(H)withw(Ᏺnτ(H))=τ,dim(Ᏺτn(H))=nwhich is uni- versal in the following sense:Ᏺτn(H)contains aG-homeomorphic copy of any G-spaceXof Euclidean type and of the single orbit type(H)such thatwX≤τ anddimβGX≤n. In particular,Ᏺnτ(H)contains aG-homeomorphic copy of each paracompactG-spaceXof type(H)withwX≤τ anddimX≤n.
Proof. Letk=n−dim(N(H)/H). Then, we have k=n−dim
G/N(H)
=n−dimG+dimN(H)
≥dimN(H)−dimH
=dim
N(H)/H
=dimW (H).
(5.5)
Hence, byTheorem 4.1, there exists a universal compact freeW (H)-space Ᏺτkof dimensionkand weightτ.
SetᏲnτ(H)=(G/H)×W (H)Ᏺkτ. ByLemma 5.1,Ᏺnτ(H)is a compactG-space of the single orbit type(H). We claim that it is the required one.
Indeed, byLemma 5.1, w
(G/H)×W (H)Ᏺkτ
=wᏲkτ=τ, dimᏲnτ(H)=dim
(G/H)×W (H)Ᏺkτ
=dimᏲkτ+dim
G/N(H)
=k+dim
G/N(H)
=n.
(5.6)
Now, ifXis aG-space with the single orbit type(H)such thatwX≤τ and dimX≤n, then, sinceX=(G/H)×W (H)XH, it follows fromLemma 5.1that w(XH)≤τand dimXH≤k.
ByTheorem 4.1, there is aW (H)-equivariant embeddingf:XHᏲkτ. Then, the map F : (G/H)×W (H)XH → (G/H)×W (H)Ᏺkτ, generated by f, is a G- equivariant embedding. We recall thatF is defined as follows:F ([gh, x])= [gH, f (x)]for every[gh, x]∈(G/H)×W (H)XH(see [17, Theorem 1.7.10]).
It remains only to recall that
X=(G/H)×W (H)XH, Ᏺτn(H)=(G/H)×W (H)Ᏺτk. (5.7) This completes the proof.
FromTheorem 5.2, the following corollary follows immediately.
Corollary5.3. Any paracompactG-spaceXof a single orbit type(H)has aG-compactificationbGXof the same orbit type(H)such thatw(bGX)≤wX anddimbGX≤dimX.
References
[1] S. A. Antonyan,Retraction properties of an orbit space, Mat. Sb. (N.S.)137(179) (1988), no. 3, 300–318 (Russian), translation in Math. USSR-Sb.65(1990), no. 2, 305–321.
[2] ,Based free compact Lie group actions on the Hilbert cube, Mat. Zametki 65(1999), no. 2, 163–174 (Russian), translation in Math. Notes65(1999), no. 1-2, 135–143.
[3] S. A. Antonyan and Yu. M. Smirnov, Universal objects and bicompact exten- sions for topological groups of transformations, Dokl. Akad. Nauk SSSR 257(1981), no. 3, 521–526 (Russian), translation in Soviet Math. Dokl.23 (1981), no. 2, 279–284.
[4] G. E. Bredon,Introduction to Compact Transformation Groups, Pure and Applied Mathematics, vol. 46, Academic Press, New York, 1972.
[5] J. de Vries,On the existence ofG-compactifications, Bull. Acad. Polon. Sci. Sér. Sci.
Math. Astronom. Phys.26(1978), no. 3, 275–280.
[6] S. Deo and H. S. Tripathi,Compact Lie group actions on finitistic spaces, Topology 21(1982), no. 4, 393–399.
[7] A. N. Dranishnikov,Free actions of0-dimensional compact groups, Izv. Akad.
Nauk SSSR Ser. Mat.52(1988), no. 1, 212–228 (Russian), translation in Math. USSR-Izv.32(1989), no. 1, 217–232.
[8] R. Engelking,General Topology, Monografie Matematyczne, vol. 60, PWN—Polish Scientific Publishers, Warsaw, 1977.
[9] ,Dimension Theory, North-Holland Mathematical Library, vol. 19, North- Holland Publishing, Amsterdam, 1978.
[10] S.-T. Hu,Theory of Retracts, Wayne State University Press, Detroit, 1965.
[11] D. Husemoller,Fibre Bundles, McGraw-Hill, New York, 1966.
[12] J. Jaworowski,An equivariant extension theorem andG-retracts with a finite struc- ture, Manuscripta Math.35(1981), no. 3, 323–329.
[13] M. G. Megrelishvili,A factorization theorem and universal compact spaces for G-spaces, Russian Math. Surveys38(1983), no. 6, 125–126.
[14] , A Tikhonov G-space that does not have compact G-extension and G- linearization, Russian Math. Surveys43(1988), no. 2, 177–178.
[15] M. G. Megrelishvili and T. Scarr,Constructing TychonoffG-spaces which are not G-Tychonoff, Topology Appl.86(1998), no. 1, 69–81.
[16] K. Morita,On the dimension of product spaces, Amer. J. Math.75(1953), 205–223.
[17] R. S. Palais,The classification ofG-spaces, no. 36, Mem. Amer. Math. Soc., 1960.
[18] E. H. Spanier,Algebraic Topology, McGraw-Hill, New York, 1966.
[19] R. G. Swan,A new method in fixed point theory, Comment. Math. Helv.34(1960), 1–16.
[20] E. K. van Douwen,βXand fixed-point free maps, Topology Appl.51(1993), no. 2, 191–195.
Natella Antonyan: Departamento de Matemáticas, Facultad de Ciencias, UNAM, Ciu- dad Universitaria, 04510, México D.F., Mexico; Departamento de Matemáticas-DIA, In- stituto Tecnolo-gico de Monterrey, Calle del Puente 222, Ejidos de Huipulco, Tlalpan, 14380 México D.F., Mexico
E-mail address:[email protected]; [email protected]
