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THE FREE ABELIAN TOPOLOGICAL GROUP AND THE FREE

arXiv:funct-an/9212001v1 11 Dec 1992THE FREE ABELIAN TOPOLOGICAL GROUP AND THE FREELOCALLY CONVEX SPACE ON THE UNIT INTERVAL †Arkady G. Leiderman, 1 Sidney A. Morrisand Vladimir G. Pestov 2Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, IsraelUniversity of Wollongong, Wollongong, N.S.W.

2522, AustraliaVictoria University of Wellington, P.O. Box 600, Wellington, New ZealandAbstract.We give a complete description of the topological spaces X such that thefree abelian topological group A(X) embeds into the free abelian topological groupA(I) of the closed unit interval.In particular, the free abelian topological groupA(X) of any finite-dimensional compact metrizable space X embeds into A(I).

Thesituation turns out to be somewhat different for free locally convex spaces. Someresults for the spaces of continuous functions with the pointwise topology are alsoobtained.

Proofs are based on the classical Kolmogorov’s Superposition Theorem.§1. IntroductionThe following natural question arises as a part of the search for a topologized ver-sion of the Nielsen-Schreier subgroup theorem.

Let X and Y be completely regulartopological spaces; in which cases the free (free abelian) topological group over Xcan be embedded as a topological subgroup into the free (free abelian) topologicalgroup over Y ? This problem has been treated for a long time [4, 10, 12-17, 21-23,25, 28], ever since it became clear that in general a topological subgroup of a free(free abelian) topological group need not be topologically free [8, 4, 10].

Recentlya complete answer was obtained in the case where X is a subspace of Y and theembedding of free topological groups extends the embedding of spaces [35]. How-ever, we are interested in the existence of an embedding which is not necessarily a“canonical” one.

Among the most notable achievements, there are certain sufficientconditions for a subgroup of a free topological group to be topologically free [4, 22]and the following results.1991 Mathematics Subject Classification. 22A05, 55M10, 46A03.Key words and phrases.

Free abelian topological groups, free locally convex spaces, spaces ofcontinuous functions, dimension, basic functions, Kolmogorov’s Superposition Theorem.† Dedicated to the memory of Eli Katz.1 Research Supported by the Israel Ministry of Science.2 Partially supported by the Internal Grants Committee of the Victoria University of Wellington.Tt bAMS T X

2A.G. LEIDERMAN, S.A. MORRIS AND V.G.

PESTOVTheorem 1.1 [14]. If X is a closed topological subspace of the free topologicalgroup F(I) then the free topological group F(X) is a closed topological subgroup ofF(I), where I is the closed unit interval.□Corollary 1.2 [22].

If X is a finite-dimensional metrizable compact space thenF(X) is a closed topological subgroup of F(I).□The abelian case proved to be more difficult, and the following is the strongestresult known to date.Theorem 1.3 [12]. If X is a countable CW-complex of dimension n, then the freeabelian topological group on X is a closed subgroup of the free abelian topologicalgroup on the closed ball Bn.□Corollary 1.4 [13].

A(R) embeds into A(I) as a closed topological subgroup.□It is known [29] that the covering dimension of any two free topological bases in afree (abelian) topological group is the same; this result is similar to the well-knownproperty of free bases of a discrete free (abelian) group having the same cardinality,called the rank of the group. Since the rank of a subgroup of a free abelian groupcannot exceed the rank of the group itself, it was conjectured [15, 20] that thedimension of a topological basis of a topologically free subgroup of a free abeliantopological group A(X) cannot exceed dim X.

It remained even unclear whetherthe group A(I2) embeds into A(I) [15].In this paper we prove that if X is a completely regular space then the free abeliantopological group A(X) embeds into A(I) as a topological subgroup if and only ifX is a submetrizable kω-space such that every compact subspace of X is finite-dimensional. Another characterization: X is homeomorphic to a closed topologicalsubspace of the group A(I) itself.

In particular, if X is a compact metrizable spaceof finite dimension, then A(X) embeds into A(I).Thus, the analogy with thenon-abelian case is complete. We also study the problem of embedding the freelocally convex space L(X) into the free locally convex space L(I) and characterizethose kω-spaces X admitting such an embedding.

Paradoxically, such spaces X arejust all compact metrizable finite-dimensional spaces. In particular, the free LCSL(R) does not embed into L(I).

Our results provide answers to a number of openproblems from [20, 15, 27].§2. PreliminariesDefinition 2.1 [19, 8, 20].

Let X be a topological space. The (Markov) free abeliantopological group over X is a pair consisting of an abelian topological group A(X)and a topological embedding X ֒→A(X) such that every continuous mappingf from X to an abelian topological group G extends uniquely to a continuoushomomorphism ¯f : A(X) →G.□If X is a completely regular topological space then the free abelian topologicalgroup A(X) exists and is algebraically free over the set X [19, 8, 20].

A topologicalspace X is called a kω-space [18, 13-17] if there exists a so-called kω-decompositionX = ∪n∈NXn, where all Xn are compact, Xn ⊂Xn+1 for n ∈N, and a subsetA ⊂X is closed if and only if all intersections A ∩Xn, n ∈N, are closed. Alllocally convex spaces (LCS) in this paper are real

FREE ABELIAN TOPOLOGICAL GROUP3Definition 2.2 [19, 1, 31, 6, 7, 34]. Let X be a topological space.

The free locallyconvex space over X is a pair consisting of a locally convex space L(X) and atopological embedding X ֒→L(X) such that every continuous mapping f fromX to a locally convex space E extends uniquely to a continuous linear operator¯f : L(X) →E.□If X is a completely regular topological space then the free locally convex spaceL(X) exists; the set X forms a Hamel basis for L(X) [31, 6, 7, 34]. The identitymapping idX : X →X extends to a canonical continuous homomorphism i :A(X) →L(X).Theorem 2.3 [33].

The canonical homomorphism i : A(X) ֒→L(X) is an em-bedding of A(X) into the additive topological group of the LCS L(X) as a closedadditive topological subgroup.□In what follows, we will often identify A(X) with a subgroup of L(X) in the abovecanonical way. Denote by Lp(X) the free locally convex space L(X) endowed withthe weak topology.Theorem 2.4 [6, 7].

Let X be a completely regular space. The canonical mappingX ֒→Lp(X) is a topological embedding, and every continuous mapping f from Xto a locally convex space E with the weak topology extends uniquely to a continuouslinear operator ¯f : Lp(X) →E.□The weak dual space to L(X) is canonically isomorphic to the space Cp(X) ofall continuous real-valued functions on X with the topology of pointwise (simple)convergence.The spaces Lp(X) and Cp(X) are in duality.Denote by Ck(X)the space of continuous functions endowed with the compact-open topology.Atopological space X is called Dieudonn´e complete [5] if its topology is induced bya complete uniformity.

For example, every Lindel¨of space is Dieudonn´e complete.In particular every kω-space is Dieudonn´e complete.Theorem 2.5 (Arhangel’ski˘ı [3]). Let X and Y be Dieudonn´e complete spaces.

Ifa linear mapping Cp(X) →Cp(Y ) is continuous then it is continuous as a mappingCk(X) →Ck(Y ).□The space L(X) admits a canonical continuous monomorphismL(X) ֒→Ck(Ck(X))Theorem 2.6 (Flood [6, 7], Uspenski˘ı [34]). If X is a k-space then the monomor-phism L(X) ֒→Ck(Ck(X)) is an embedding of locally convex spaces.□Let X be a topological space.

A collection of continuous functions h1, . .

., hmon X assuming their values in the closed unit unterval I = [0, 1] is called basic[26, 32] if every real-valued continuous function f on X can be represented as asum Pni=1 gi ◦hi of compositions of basic functions with some continuous functionsgi ∈C(I).2.7. Kolmogorov’s Superposition Theorem [11].

The finite-dimensional cubeIn has a finite basic family of continuous real-valued functions.□Let us recall that for compact metrizable spaces all three main concepts of di-mension (the covering the small inductive and the large inductive ones) coincide

4A.G. LEIDERMAN, S.A. MORRIS AND V.G.

PESTOV[5]. The following result is of crucial importance for us; it is an immediate corol-lary of the Kolmogorov’s Superposition Theorem, the Menger-N¨obeling Theoremon embeddability of separable metric spaces of dimension ≤n into R2n+1, and theTietze-Urysohn Extension Theorem [5].Corollary 2.8 (Ostrand [26]).

Let X be a finite-dimensional compact metrizablespace. Then there exists a finite basic family of continuous functions on X.□For an exact upper bound on the cardinality of a basic family of continuousfunctions on a space X of dimension n, see [32]; however, we do not need it.§3.

Auxiliary constructionsLemma 3.1. Consider a commutative diagram of Banach spaces and surjectivecontinuous linear mappings:E1r1←−−−−E2r2←−−−−E3r3←−−−−.

. .rn−1←−−−−Enrn←−−−−.

. .π1yπ2yπ3yπnyF1q1←−−−−F2q2←−−−−F3q3←−−−−.

. .qn−1←−−−−Fnqn←−−−−.

. .Denote by E = lim←−En and F = lim←−Fn the Fr´echet spaces projective limits ofcorresponding inverse sequences, and by π : E →F the projective limit of themappings πn, n ∈N.

Then every compact subspace K ⊂F is an image under themapping π of a compact subspace of E.Proof. Let K be a compact subspace of F. Let Kn = qn(K) for all n ∈N.

Accord-ing to the Michael Selection Theorem (Th. 1.4.9 in [36]), there exists a compactsubspace C1 ⊂E1 such that π1(C1) = K1.

Assume now that for all k ≤n we havechosen compact subspaces Ck ⊂Ek such that πk(Ck) = Kk and rk−1(Ck) = Ck−1.Consider the mapping < rn, πn+1 >: x 7→(rn(x), πn+1(x)) from En+1 to En×Fn+1.The subset Qn = {(y, z) : y ∈Cn, z ∈Kn+1, qn(z) = πn(y)} of the space En ×Fn+1is compact, and is contained in the Banach space image of the continuous linearmapping < rn, πn+1 >. Therefore, by the Michael Selection Theorem, there exists acompact subset Cn+1 ⊂En+1 such that < rn, πn+1 > (Cn+1) = Qn.

Consequently,rn(Cn+1) = Cn, and qn+1(Cn+1) = Kn+1, which completes the recursion step. Fi-nally, put C = lim←−Cn; this subset of E is compact, and the property K ⊂lim←−Knimplies that π(C) = K.□Lemma 3.2.

Let X and Y be kω-spaces. Let h : Lp(X) →Lp(Y ) be an embeddingof locally convex spaces.

Then h is an embedding of locally convex space L(X) intoL(Y ) as well.Proof. As a corollary of the Hahn-Banach theorem, the dual linear map h∗:Cp(Y ) →Cp(X) to the embedding h is a continuous surjective homomorphism.Theorem 2.5 says that h∗remains continuous with respect to the compact-opentopologies on both spaces, and by virtue of the Open Mapping Theorem, h∗:Ck(Y ) →Ck(X) is open.

Since for every compact subset C ⊂X the elements ofthe image h(C) are contained in the linear span of a compact subset of Y [3], onecan choose kω-decompositions X = ∪∞n=1Xn and Y = ∪∞n=1Yn in such a way thatfor every n ∈N one has h(sp Xn) ⊂sp Yn. It is easy to see that the restrictionsmappings r: C (Y) →C (Y ) and q: C (X) →C (X ) are continuous

FREE ABELIAN TOPOLOGICAL GROUP5surjections, and that C(Y ) = lim←−Ck(Yn) and C(X) = lim←−Ck(Xn). Denote for eachn ∈N by πn the restriction h∗|Ck(Yn).

The conditions of Lemma 3.1 are fulfilled,and therefore every compact subset K ⊂Ck(X) is an image under the mapping h∗of a suitable compact subset of Ck(Y ). Therefore, the continuous linear map h∗∗dual to h∗from the space Ck(Ck(X)) to Ck(Ck(Y )) is an embedding of Ck(Ck(X))into Ck(Ck(Y )) as a locally convex subspace.

Since the restriction of h∗∗to L(X)is h, the desired statement follows from Theorem 2.6.□Lemma 3.3. Let X be a compact space and let Y be a closed subspace of X. Denoteby π the quotient mapping from X to X/Y .

Let fk, k = 1, . .

., n be continuousfunctions on X such that their restrictions to Y form a basic family for Y , andlet gi, i = 1, . .

., m be a basic family of functions on X/Y . Then the family offunctions f1, .

. ., fn, g1 ◦π, .

. ., gm ◦π is basic for X.Proof.

Let f : X →R be a continuous function.For a family of continuousfunctions h1, . .

. , hn ∈C(I), the restriction f|Y is represented as Pnk=1 hk◦(fk|Y ) =(Pnk=1 hk ◦fk)|Y .

Denote by g : X →R the continuous function f −Pnk=1 hk ◦fk;since the restriction g|Y ≡0, the function g factors through the mapping π; that is,there exists a continuous function h : X/Y →I with g = h◦π. For some collections1, .

. .

, sm of continuous functions on I one has h = Pmi=1 si ◦gi, which means thatg = Pmi=1 si ◦gi ◦π. Finally, one hasf =nXk=1hk ◦fk +mXi=1si ◦gi ◦π,as desired.□A topological space X is called submetrizable if it admits a continuous one-to-onemapping into a metrizable space.Lemma 3.4.

Let X be a submetrizable kω-space with kω-decomposition X = ∪n∈NXnsuch that every subspace Xn is finite-dimensional. Then there exists an embeddingof locally convex spaces ¯F : Lp(X) ֒→Lp(Y ), where Y is the disjoint sum of count-ably many copies of the closed unit interval I, such that ¯F(A(X)) ⊂A(Y ).Proof.

Let X = ∪n∈NXn be a kω decomposition of X with Xn ⊂Xn+1, for all n ∈N. Since every Xn, n ∈N is a finite-dimensional metrizable compact space, thenfor any n ∈N so is the quotient space Xn+1/Xn, and one can choose inductively,using Ostrand’s Corollary 2.8 and Lemma 3.3, a countable family of continuousfunctions fn,i, n ∈N, i = 1, .

. ., kn, kn ∈N from X to I such that for each n ∈Nthe following are true:1. the collection fm,i, i = 1, .

. ., km, m = 1, .

. ., n, is basic for Xn;2. fn+1,i|Xn ≡0 for all i = 1, .

. ., kn+1.

Denote the above family of functions fn,iby F, and let Y = ⊕f∈FIf be the disjoint sum of countably many copies of theclosed unit interval I. For every f ∈F denote by 0f the left endpoint of theclosed interval If regarded as an element of the free abelian group A(Y ).

Definea mapping, F, from X to the free abelian group A(Y ) by lettingF(x) =Xi=1,...,k1f1,i(x) +Xn≥2, i=1,...,kn(fn,i(x) −0n,i)for each x ∈X. The mapping F is properly defined, because the first sum isfinite and in the second sum all but finitely many terms are vanishing in the

6A.G. LEIDERMAN, S.A. MORRIS AND V.G.

PESTOVfree abelian group A(Y ), for every x ∈X. The restriction of F to every Xnis continuous if being considered as a mapping to the free abelian topologicalgroup A(Y ), which fact follows from continuity of each mapping fm,i : Xn →Ifm,i ⊂Y, m ≤n, i = 1, .

. ., km and the continuity of subtraction and additionin A(Y ).

Therefore the mapping F : X →A(Y ) is continuous. If being viewed asa continuous mapping from X to the locally convex space Lp(Y ), it extends to acontinuous linear operator ¯F : Lp(X) →Lp(Y ).

Let h : X →R be a continuousfunction.We will show that there exists a continuous linear functional ¯h onthe linear subspace ¯F(Lp(X)) such that ¯h ◦F|X = h.It would mean that¯F(Lp(X)) is isomorphic to Lp(X), as desired. Construct recursively, makinguse of the properties 1 and 2 above, a countable family of continuous functionshn,i, i = 1 .

. .

, kn, n ∈N from I to R such that for every n ∈N and for allx ∈Xn,h(x) =Xi=1,...,km, m≤n(hm,i ◦fm,i)(x)Let us recall that fn,i|X1 ≡0 for all n ≥2 and i = 1, . .

., kn. It is easy to deduceinductively from this fact that for any n ≥2Xi=1,...,knhn,i(0) = 0Define a continuous mapping H from Y to I by letting H(y) = hn,i(y), if y ∈In,i, n ∈N.

Extend H to a continuous linear functional ¯H : Lp(Y ) →R anddenote its restriction to ¯F(Lp(X)) by ¯h. We claim that ¯h ◦F|X = h, or, whichis the same, that for every n ∈N one has ¯h ◦F|Xn = h. Indeed, for an arbitraryx ∈Xn one has:(¯h ◦F)(x) = ¯H(F(x)) = ¯H(Xi=1,...,k1f1,i(x) +X2≤m≤n, i=1,...,kn(fm,i(x) −0m,i))=Xi=1,...,k1H(f1,i(x)) +X2≤m≤n, i=1,...,knH(fm,i(x) −0m,i)=Xi=1,...,km, m≤n(hm,i ◦fm,i)(x) −X2≤m≤n, i=1,...,knhm,i(0) = h(x) −0 = h(x).□§4.

Main resultsTheorem 4.1. For a completely regular space X the following are equivalent.

(i) The free abelian topological group A(X) embeds into A(I) as a topological sub-group. (ii) The free topological group F(X) embeds into F(I) as a topological subgroup.

(iii) X is homeomorphic to a closed topological subspace of A(I). (iv) X is homeomorphic to a closed topological subspace of F(I).

(v) X is homeomorphic to a closed topological subspace of R∞. (vi) X is a kω-space such that every compact subspace of X is metrizable and finite-dimensional

FREE ABELIAN TOPOLOGICAL GROUP7(vii) X is a submetrizable kω-space such that every compact subspace of X is finite-dimensional.Proof. (i) ⇒(iii): since the space X is Lindel¨of (as a subspace of A(I), see [2]) andhence Dieudonn´e complete, the group A(X) is complete in its two-sided uniformity[33] and therefore closed in A(I); but X is closed in A(X).

(ii) ⇔(iv): see [14]. (iii) ⇔(v) ⇔(iv): follows from the result of Zarichny˘ı [37]: the free topologicalgroup F(I) and the free abelian topological group A(I) are homeomorphic to opensubsets of R∞.

(v) ⇒(vi): the space R∞= lim−→Rn is a kω-space such that everycompact subspace of it is metrizable and finite-dimensional, and this property isinherited by closed subsets. (vi) ⇔(vii): see [16].

(vii) ⇒(i): Let X be asubmetrizable kω space such that every compact subspace of X is finite-dimensional.According to Lemma 3.4, there exists an embedding of locally convex spaces ¯F :Lp(X) ֒→Lp(Y ), where Y is the disjoint sum of countably many copies of theclosed unit interval I, such that ¯F(A(X)) ⊂A(Y ). By virtue of Lemma 3.2, ¯F isalso an embedding of locally convex spaces L(X) ֒→L(Y ).

Its restriction to A(X)is an embedding of topological groups (Theorem 2.3). Now apply Corollary 1.4.□Theorem 4.2.

For a kω-space X the following conditions are equivalent. (i) The free locally convex space L(X) embeds into L(I) as a locally convex subspace.

(ii) The free locally convex space with the weak topology, Lp(X), embeds into Lp(I)as a locally convex subspace. (iii) The space Cp(X) is a quotient linear topological space of Cp(I).

(iv) X is a finite-dimensional metrizable compact space.Proof. (ii) ⇔(iii): just dual forms of the same statement about two locally convexspaces having their weak topology.

(i) ⇒(iv): Suppose X is a noncompact kω-space. Since (ii) and by the same token (iii) hold, then by virtue of Theorem 2.5 theFr´echet non-normable space Ck(X) is an image of the Banach space Ck(I) under asurjective continuous linear mapping, which is open by virtue of the Open MappingTheorem – a contradiction.

Now the space X, being compact, is contained in thesubspace spn(Y ) of L(Y ) formed by all words of the reduced length ≤n over Yfor some n ∈N [34]. But the space spn(Y ) is a union of countably many closedsubspaces each of which is homeomorphic to a subspace of the n-th Tychonoffpowerof the space R × [X ⊕(−X) ⊕{0}] [2].

Therefore, spn(Y ) is finite-dimensional.Finally, submetrizability of X follows from the same property of L(I) (the latterspace admits a continuous one-to-one isomorphism into the free Banach space overI, [1, 6, 7]). (iv) ⇒(ii): it follows from Lemmas 3.4 and 3.3 that Lp(X) embeds asa locally convex subspace into the free locally convex space in the weak topologyover a disjoint sum of finitely many homeomorphic copies of the closed interval.The latter LCS naturally embeds into Lp(I).

(ii) ⇒(i): apply Lemma 3.3.□Remark 4.3. Surprising as it may seem, the free locally convex space L(R) doesnot embed into L(I), in spite of the existence of canonical embeddings A(R) ֒→L(R) and A(I) ֒→L(I) and a (non-canonical one) A(R) ֒→A(I).

It is anotherillustration to the well-known fact that not every continuous homomorphism to theadditive group of reals from a closed additive subgroup of an (even normable) LCSextends to a continuous linear functional on the whole space. Such a misbehaviouris also to blame — at least partly — for apparent lack of progress in attempts tomake the Pontryagin-van Kampen duality work for free abelian topological groups[24, 30].

8A.G. LEIDERMAN, S.A. MORRIS AND V.G.

PESTOVRemark 4.4. The problem of characterization of covering dimension of a com-pletely regular space X in terms of the linear topological structure of the spaceCp(X) still remains open (cf.

[9]). However, now we can describe those metrizablecompact spaces having finite dimension.Corollary 4.5.

A metrizable compact space X is finite-dimensional if and only ifthe space Cp(X) is a quotient linear topological space of Cp(I).□Perhaps, the dimension of X can be described in terms referring to the lineartopological structure of the space Cp(X) with the help of a characterization ofdimension in the language of basic functions due to Sternfeld [32].Remark 4.6. Our results also provide answers to three problems from the bookOpen Problems in Topology [27].Problem 511.

Is A(I2) topologically isomorphic with a subgroup of A(I)?Yes (cf. Theorem 4.1).Problem 1046.

Assume that Cp(X) can be mapped by a linear continuousmapping onto Cp(Y ). Is it true that dim Y ≤dim X ?

What if X and Y arecompact?Problem 1047. Assume that Cp(X) can be mapped by an open linear contin-uous mapping onto Cp(Y ).

Is it true that dim Y ≤dim X ? What if X and Yare compact?No, in all four cases (cf.

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