DIMENSIONS OF TOPOLOGICAL GROUPS CONTAINING THE BOUQUET OF TWO CIRCLES

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1 proceedings of the american mathematical society Volume 114, Number 4, APRIL 1992 DIMENSIONS OF TOPOLOGICAL GROUPS CONTAINING THE BOUQUET OF TWO CIRCLES TAKASHI KIMURA (Communicated by Dennis Burke) Abstract. In this paper we prove the following: If a topological group G contains the bouquet S'vS1, then dim G > 2 holds. This is a counterexample to a question of Bel'nov in the one-dimensional case. 1. Introduction In this paper we assume that all spaces are Tychonoff. Bel'nov [1] proved that every space X can be embedded into a homogeneous space Gx such that indgx = indx, IndGx IndX, dimct*- = dimz in the case when the corresponding dimension of X is finite. The space Gx is a free group over X with continuous right translations. Bel'nov (see [5]) asked whether every space X can be embedded into a topological group G with dim G < dim X. Shakhmatov [5] proved that if n 0, 1, 3, 7, then the «-dimensional sphere Sn cannot be embedded into an «-dimensional topological group. Thus the answer to the above question is no. However, Shakhmatov also proved that in case dimx = 0 the answer to this question is yes. Hence it is natural to ask whether every space X can be embedded into a topological group G with dimc7 < dim A" in case dimx =1 or 3 or 7. In this paper we prove that the answer to this question is no in case dim X 1. Namely, we prove that if a topological group G contains the bouquet Sx v S1, then dim G > 2 holds. Obviously, dim(sl V51) = 1. Hence the bouquet Sl V Sl is a counterexample to a question of Bel'nov in the one-dimensional case. 2. Preliminaries lemmas We denote by R, Q, 7, Sl the real line, the rational space, the closed unit interval [0, 1], the 1-sphere, respectively. We put dl2 = (7x{0, 1})U ({0, 1} x 7). We denote by d the Euclidean metric on R3. For two continuous Received by the editors January 16, 1990, in revised form, November 5, Mathematics Subject Classification (1985 Revision). Primary 54F45, 54C25; Secondary 05C10, 22A05. Key words phrases. Topological group, dimension, embedding, bouquet, graph American Mathematical Society /92 $ $.25 per page

2 1110 TAKASHIKIMURA mappings / g of a space X into R3 we put d(f, g) = sup{d(f(x), g(x)) :x X}. For a subset X of R3 for any e > 0 we put U(X, e) = {a R3 : d(a, X) < e}. For a subset X of R3 we denote by ix the inclusion mapping of X into R3. For a mapping /: X > y for a subset ^ of I we denote by fia the restriction of / to A. Let TV,3 be the subspace of R3 consisting of all points which have at most one rational coordinate. The following lemma is needed in Lemma. Let X be a compact metrizable space with dimx < 1. Then for every continuous mapping f:x R3 for any e > 0 there exists an embedding g: X -> R3 such that d(f, g) < s g(x) c TV3. Proof. See the proof of [2, ]. Let us set L = {(x,y,0) R3:x2+y2 = 9}, M = {(x,0, z) R3: (x-2)2 + z2 = 1}, C = {(x, y, 0) R3 : x2 + y2 = 4}, Z = {(0,0,z) R3: T2 = {a Ri:d(a, z R}, C) = I}. Then, obviously, T2 is a torus. For every 6, 0 < 6 < 2%, we put Fe = {(rcoso, rsind, z) R3: r > 0 z R]. Let K be a polyhedral 1-sphere satisfying the following two conditions (*) : KcU(C, l)n(((rxq)u(qxr))x{0}) W \Fe n K\ = 1 for every d, 0 < 6 < 2n. We shall construct a continuous mapping pk : R3 - (Tí UZ) > T2. For every 6, 0 < 6 < 2n, let pe : Fe - {xg} Fe n T2 be the projection of Fe - {xe} from the point xe onto Fe n T2, where {xe} = Fe n K. Let pk : R3 - (K U Z) -> T2 be the mapping defined by Pic(rcosd, rsind, z) = pe(rcos6, rsind, z) for every (rcosd, rsind, z) R3 - (K u Z). Then /7/r is continuous. The proof of the following lemma is easy, so we omit the proof Lemma. For every e>0 there exists a polyhedral l-sphere K with property (*) such that d(iu{t2te),pklu{t2te]) < 2e. Let k : I2 T2 be the natural quotient such that X(x, 0) = k(x, 1 ) = (3 cos 2nx, 3 sin 27r.x, 0), A(0, x) = X( 1, x) = (2 + cos 2nx, 0, sin 2nx), A(l/2, 1/2) = (-1,0,0).

3 DIMENSIONS OF TOPOLOGICAL GROUPS 1111 Let c7:72-{(l/2, 1/2)} -» dl2 be the projection of 72-{(l/2, 1/2)} from the point (1/2, 1/2) onto <972. It is obvious that \ÀqÀ~l(a)\ = 1 for every a T2 - {(-1, 0, 0)}. Thus we can define the mapping p : T2 - {(-1, 0, 0)} -» T2 such that {p(a)} = ÀqÀ~l(a). Then we have ^(T2 -{(-1,0, 0)}) = L U M p is continuous. The proof of the following lemma is also easy, so we omit the proof Lemma. For every e > 0 there exists ô > 0 such that d(iu(lum,<j)nt2) > / (7(LUA/\<5)nT2 > ß\U(LUM,S)nT2\) < 3. The main theorem By Lemma 2.3, we can take Ô > 0 such that d(iu(lun,s)mi> We fix ô in this section. P\u(LuM,S)m2) < Iß Lemma. Let f: I2» R3 be a continuous mapping such that d(fdi2, X\9I2) <ô/4. Then dim f(i2)> 2 holds. Proof. Suppose that dim/(72) < 1. By Lemma 2.1, we can take an embedding g: f(i2) - R3 such that d(if{p),g) < a/4 g /(72) c TV3. By Lemma 2.2, we can take a polyhedral 1-sphere K with property (*) such that d(iu{t?,â/2)» Pk\u(V,ô/2)) < & Note that 8 f(l2) n(köz) = 0, because g o /(72) c N. Since K c ((R x Q) u (Q x R)) x {0} g o f(i2) n ([-3, 0] x {(0,0)}) = 0, we have (-1,0,0) Pk g f(i2) Let v = popkogof. Then z; is a continuous mapping of 72 into T2 such that u(i2) c LöM d(v\d 2, Aa/2) < 1/3. We take a lifting v : I2 R2 of i/. Let us set yi,- = 7x {/} for i = 0, 1, 7?, = {/} x 7 for z = 0, 1. Since d(v\di1, l\an) < 1/3, we may assume that ù(ai)cv(ai,l/3) for i = 0,1, 0(Bi)c V(B, 1/3) for/ = 0, 1, where V(N, e) is the e-neighborhood of N in R2. Let us set D = R x {1/2} E = {1/2} x R. Then D is a partition in R2 between v(aq) v(a{), E is a partition in R2 between ù(bq) v(b\). Thus z/_1(d) is a partition in 72 between Aq yli, v~l(e) is a partition in 72 between 7i0 B\. Thus we have ù~l(d) n î>_1(-^) ^ 0 On the other h, since u(i2) c LuAf, we have v(i2) C p~l(lum), where p: R2 - T2 is the covering mapping. Thus we have Dn7?nz>(72) = 0. This implies that v~l(d)ni>~l(e) = 0. This is a contradiction. Hence we have dim/(72) > 2. Lemma 3.1 has been proved. Recall that the bouquet Sl V Sl is the one point union of Sl Sl with the common point. We are now in a position to establish our main theorem Theorem. If a topological group G contains the bouquet Sl V Sl, then dim G > 2 holds. Proof. Suppose that a topological group G with dim G < 1 contains the bouquet S1 V 51. Since L U M is homeomorphic to S1 V Sl since G is

4 1112 takashi kimura homogeneous, we may assume that e = (3, 0, 0) Lu M c G, where e is the neutral element of G. Let y>\ : I G <p2: I» G be the mappings defined by <Pi(x) = (3cos2nx, 3sin27tx, 0) 9t(x) (2 + cos 2nx, 0, sin2^x) for every x I. Let y>: I2 > G be the mapping defined by <p(x, y) = q>\(x) <p2(y) for every (x, y) I2, where is the group composition of G. Then <p(i2) is compact metrizable. Since (pi(0) = <p\(l) = y>2(0) = Ç>2(1) = (3,0,0) = e, we have y>\di2 X9Ii. Let /: (p(i2) > R3 be a continuous mapping such that f\f(ap) lum Since dimc7 < 1, we have dim <p(i2) < 1. Thus, by Lemma 2.1, we can take an embedding g: q>(i2) -» R3 such that d(f, g) < Ô/4. Then we have d(g o y>\di2, Xigp) < 5/4. Thus, by Lemma 3.1, we have dim(p(i2) = dim g o <p(i2) > 2. This is a contradiction. Theorem 3.2 has been proved. 4. Remarks questions 4.1 Remark. Vopenka ([7] or see [3, 18-11]) constructed a compact space X such that dim X = n < to ind X = oo. Shakhmatov [4] proved that the above space X cannot be embedded into a finite-dimensional, locally pseudocompact topological group. However, it is unknown whether the above space X can be embedded into a topological group with the same dimension of X Remark. In the proof of Theorem 3.2 we get a contradiction to assume that dim<p(i2) < 1. Because q>(i2) is compact metrizable, if indc7 < 1 or Ind G < 1, then we have dim<p(i2) < 1. Hence the bouquet Sl V Sl cannot be embedded into a topological group G such that ind G < 1 or Ind G < 1 or dim G < 1. It is well known that every separable metrizable space X with dimx = «can be embedded into the (2«+ l)-dimensional cube 72"+1. Obviously, 72n+1 can be embedded into the (2«+ 1)-dimensional torus T2n+1. Hence every separable metrizable space X with dimx = «can be embedded into a (2«+l)- dimensional compact metrizable topological group. It is easy to see that every «- dimensional, locally separable metrizable space can be embedded into a (2«+l)- dimensional metrizable topological group Question. Does there exist a mapping cp: œ <y such that every metrizable space X with dim X n can be embedded into a (metrizable) topological group G with dim G <(p(n)l Shakhmatov also gave some similar problems in [6]. It is obvious that the bouquet S1 VS1 can be embedded into the 2-dimensional torus T2. However, it is well known that if «> 8, then the complete graph Kn cannot be embedded into T2. Since K% contains Sl V Sl, Kn cannot be embedded into a topological group G with dim G < 1 for each «> 8. Obviously, K can be embedded into the 3-dimensional torus T3.

5 dimensions of topological groups Question. Let «> 8. Can the complete graph Kn be embedded into a topological group G with dim G < 2? A space X is atriodic if X does not contain the complete bipartite graph K\ ;3. Obviously, the bouquet Sl V Sl is not atriodic Question. Can the complete bipartite graph 7<i 3 be embedded into a topological group G with dim G = 1? If a graph X is atriodic, then every component of X is homeomorphic to S ' or 7 or the singleton. Thus an atriodic graph can be embedded into S1 x Zp, where p is the number of components of X Zp is a group consisting of p points with discrete topology. Hence if the answer to the above question is negative, then a graph X can be embedded into a topological group G with dim G = 1 if only if X is atriodic. Added in proof. H. Kato J. Kulesza, independently, constructed an n- dimensional space X such that every topological group G containing X has dim G > «+ 1. They also gave a negative answer to Question 4.5. References 1. V. K. Bel'nov, Dimension oftopologically homogeneous spaces free homogeneous spaces, Soviet Math. Dokl. 19 (1978), R. Engelking, Dimension theory, Polish Scientific, Warszawa, K. Nagami, Dimension theory, Academic Press, New York London, D. B. Shakhmatov, Closed embeddings into pseudocompact spaces preserving the covering dimension, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 43 (1988), _, Imbedding into topological groups preserving dimensions, Topology Appl. 36 (1990), _, A survey of current researches open problems in the dimension theory of topological groups, Questions Answers Gen. Topology 8 (1990), P. Vopenka, On the dimension of compact spaces, Czechoslovak Math. J. 8 (1958), Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki, 305, Japan Current address : Department of Mathematics, Faculty of Education, Saitama University, Urawa, Saitama 338 Japan