Q & A in General Topology, Vol. 15 (1997)

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1 Q & A in General Topology, Vol. 15 (1997) ON MAPPINGS WITH THE ElLENBERG PROPERTY JANUSZ J. CHARATONIK AND WLODZIMIERZ J. CHARATONIK University of Wroclaw (Wroclaw, Poland) Universidad Nacional Aut6noma de Mexico (Mexico, D. F., Mexico) ABSTRACT. Mappings f : X ~ Y are considered in the paper having the following property (called the Eilenberg property): for each mapping 9 from Y to the unit circle, the condition go f,... 1 on X implies 9,... 1 on Y. Various results are collected and open questions are asked. Let lr be the real line, C be the complex plane, and S = {z E C : Izl = 1} be the unit circle. A mapping a : X ~ S of a separable metric space X is said to be inessential (writing a '" 1 on X) provided that it belongs to the same component of the space SX as the constant mapping ao : X ~ {I} C S (compare [12, Chapter 11, Part b, 5-9]). For compact spaces X the condition a '" 1 on X is equivalent to the existence of a mapping cp : X ~ R such that a = po <p, where the universal covering projection p: lr ~ S is defined by p(t) = exp(27l'it) for t E lr ([5, Theorem 1, p. 162]; compare [12, Chapter 11, 6, Corollary 6.22, p. 226]). We say that a mapping I : X ~ Y from a space X onto a space Y has the Eilenberg property provided that for every mapping 9 : Y ~ S. the implication holds (1) if go I '" 1 on X, then g'" 1 on Y. Note that the condition 9 '" 1 on Y obviously implies 9 0 I rv 1 on X for every surjection I: X ~ Y (namely one can put 1fJ = cpo/: X ~ lr to have gol = po1fj). Thus the implication (1) can be replaced by the equivalence of the two conditions 9 0 I '" 1 on X and 9 '" 1 on Y Mathematics Subject Classification. Primary 54CI0, 54E40, 54F15, 54F55. Key words and phrases. acyclic, confluent, continuum, Eilenberg property, inessential, mapping, monotone, open, unicoherent. -95-

2 JANUSZ J. CHARATONIK AND WLODZIMIERZ J. CHARATONIK Since a locally connected continuum X is unicoherent if and only if the space SX is connected ([1, Theorem 38, p. 195]; [5, Theorem 7, p. 167]; compare property (b) of [5, p. 168]; cf. [12, Chapter 11, 7, Theorem 7.4, p. 228]), the following assertion holds. 2. Unicoherence of locally connected continua is an invariant under mappings having the Eilenberg property. For other important consequences of the implication (1), in particular for the groups S-Y, P(X), B(X) and so on, the reader is referred to Chapter 8 of Kuratowski's monograph [8], as well as to Whyburn's book [12, Chapter 11, 8, p ]; in particular see [12, Theorems 8.3 and 8.4, p ]. A space X is said to be contractible with respect to if every mapping Q : X --t S is homotopic to the constant mapping Qo : X --t {1} C. This is equivalent to the condition that the space SX is arcwise connected (or connected, if X is compact), which in turn is equivalent to Q '" 1 on X for each Q : X --t (see e.g. [8, 57, I, Theorem 1, p. 434]). Thus we have the following result. 3. Contractibility of space with respect to S is an invariant under mappings having the Eilenberg property. A compact Hausdorff space X is said to be acyclic provided that all of its Cech cohomology groups are trivial (we take here the Cech cohomology groups based on an arbitrary open covering and with integer coefficients). For one-dimensional (in the sense of the covering dimension) compact spaces X acyclicity is equivalent to the condition that the first cohomology group H,1 (X) is trivial, which in turn is equivalent to the fact that each mapping Q : X --t S is inessential (see [4, Theorem 8.1 (Bruschlinsky's Theorem), p. 226], where a relation between the elements of one-dimensional Cech cohomology group and the mappings into S, for paracompact normal spaces is established; compare also [7, Corollary, p. 150]). Therefore an acyclic curve means a curve (i.e., a one-dimensional continuum) X for which Hl(X) is trivial. This concept should not be confused with another one, of the same name, used in a different sense in [12, p. 88]. Accordingly, the following statement is a consequence of the above mentioned results. Thus it is both interesting and important to know what mappings have the property. The following results, due to S. Eilenberg, are shown in [5, Theorem 5, p. 165, and Theorem 14, p. 174] (compare [8, 56, XI, Theorem 4, p. 433]). 5. Open mappings as well as monotone ones (of compact metric spaces) have the Eilenberg property. 4. Acyclicity of curves is an invariant under mappings having the Eilenberg property. -96-

3 ON MAPPINGS WITH THE ElLENBERG PROPEIITY A mapping f : X -t Y between spaces X and Y is said to be confluent provided that for each sub continuum Q of Y and for every component C of f-1(q) the equality f(c) = Q holds. On compact spaces monotone mappings are obviously confluent, as well as open ones are ([12, Chapter 8, Theorem 7.5, p. 148]). Answering a question of mine ([2, p. 219]), A. Lelek extended Eilenberg's results quoted above to con.fl.uent mappings ([9, Theorem, p. 229]) as follows. 6. Each confluent mapping of a countable compact space X onto a metrizable space Y has the Eilenberg property. He has also shown that that both assumptions (countable compactness of X and metrizability of Y) are indispensable in the result ([9, Remarks III and IV, p. 231]). In spite of this, B. A. Pasynkov asked ([9, P 558, p. 233]) whether the result can be extended to confluent mappings f between Hausdorff compact spaces X and Y. An affirmative answer was given by J. Grispolakis and E. D. Tymchatyn in [6, Theorem 5.2, p. 354]. 7. Each confluent mapping f : X -t Y of a Hau.sdorfJ continuum X onto a Hau.sdorfJ space Y has the Eilenberg property. The next significant progress has been made, also in [6], for semi-con.fl.uent mappings. Recall that a mapping f : X -t Y is said to be semi-confluent provided that for each subcontinuum Q of Y and for every two components C 1 and C 2 of f-1(q) either f(cd c f(c 2 ) or f(c 2 ) c f(c 1 ). The following result is a consequence of [6, Theorem 4.2, p. 350] (where the authors consider mappings g from continua into an arbitrary graph G in place of 8). 8. Each semi-conflu.ent mapping f : X -t Y of a continuu.m X onto a hereditarily unicoherent continuum Y has the Eilenberg property. According to [6, Problem 1, p. 353] it is not known whether hereditary unicoherence of Y is an essential assumption in the above quoted result. Thus we have the following question which is a particular case of Problem 1 of [6]. 9. Question. Is it true that each semi-confluent mapping between continua has the Eilenberg property? In connection with semi-confluent mappings recall the following result, due to T. Mackowiak [10, Lemma 4.3, p. 257J (which was used to prove that a semi-confluent image of a 'x-dendroid is a 'x-dendroid, see [10, Theorem 5.2, p. 262]). 10. There is no semi-conflu.ent mapping from an arc onto a circle. The original proof given in [10, p J is long, complicated and rather cwobersome. Recently A. Illanes suggested the following short and elegant argument -97-

4 JANUSZ J. CHARATONIK AND WLODZIMIERZ J. CHARATONIK to get a direct proof of the above statement. To present it, recall that a class M of mappings between spaces X and Y is said to have the composition factor property provided that for each mapping f : X -+ Y if f E M and f = go h, then gem. For various classes M in connection with the composition factor property see [11, Chapter 5, Part B, p ]. In particular, the following result is known [10, Theorem 3.5, p. 254J. 11. The class of semi-confluent mappings has the composition factor property. Proof of 10. Suppose on the contrary that there is a semi-confluent mapping 9 : [0, 1J -+. Thus g is inessential ([12, Theorem 6.1, p. 225]), and therefore 9 = po<p, where <p: [0,1]-+ lr and p is the universal covering projection. Then <p([0, 1]) is a closed interval, say [a, b] C R By the composition factor property for semiconfluent mappings (quoted above), the partial mapping p\[a, b] : [a, b] -+ is a semi-confluent s:urjection. If p( a) =F p( b), consider the two arcs in S whose union is having pea) and pcb) as its end points. Exactly one of them, denote it by Q, has the property that if C a and Cb are the components of (pl[a, b])-1(q) with a E Ca and b E Cb, then Ca = {a} and Cb = {b}. Therefore p(c a ) = {pea)} and p(cb) = {pcb)}, contrary to the definition of semi-confluence of p. If pea) = pcb), we denote by Q c an arc containing pea) in its interior. If C a and C b are the components of (pl[a,b])-1(q) with a E C a and b E C b as previously, then again neither p(c a ) C p(c b ) nor p(cb) C p(c a ), a contradiction. 12. Remark. Note that Statement 10 is a particular case of Theorem 4.1 of [6, p. 349J saying that if the first cohomology group of a continuwn X is trivial, and if f : X -+ Y is a semi-confluent surjection onto a Hausdorff space Y, then Y is unicoherent. It is also a consequence of Corollary 4.6 of [6, p. 353] saying that tree-likeness of Hausdorff continua is an invariant under semi-confluent mappings. Coming back to the Eilenberg property for various classes of mappings, let us recall that ones of open and of monotone mappings are subclasses of the class of confluent mappings of continua, which is in turn contained in the class of semiconfluent mappings. Thus one can ask if Statement 8 can be generalized not only by deleting hereditary unicoherence of Y (Question 9) but by considering a wider class of mappings. In the hierarchy of mappings presented in Table II of [11, p. 28] three classes of mappings are immediate successors of the class of semi-confluent mappings: weakly confluent, joining and locally semi-confluent ones. A mapping f : X -t Y is said to be - weakly confluent provided that for each subcontinuwn Q of Y there is a component C of f-l(q) such that the equality f(c) = Q holds; - joining provided that for each sub continuum Q of Y and for every two components C 1 and C2 of f-1(q) we have f(ct) n f(c 2 ) =1= 0; -98-

5 ON MAPPINGS WITH THE ElLENBERG PROPERTY - locally semi-confluent provided that for each point x E X there is a closed neighborhood V of the point x such that f(v) is a closed neighborhood of f(x) and the partial mapping fly is semi-confluent. 13. Remark. It is evident that the mapping a : [0,1] -t S defined by a(t) = exp( 41Tit) for t E [0,1] is both weakly confluent and joining. It is also locally semiconfluent (see [11, 7.18, p. 64]). Namely as the needed closed neighborhood V of in [0,1] one can take V = [0, eo] U [1/2 - eo, 1/2 + eo] for sufficiently small eo> 0, and similarly, V = [1 - eo, 1] U [1/2 - eo, 1/2 + eo] is a suitable closed neighborhood of 1. For points x E (0,1) one can take V = [x - eo,x + c]. Remark 13 and Statement 2 lead to the following assertion. 14. Neither weakly confluent, nor joining, nor locally semi-confluent mappings have the Eilenberg property. Therefore in the light of Statements 8 and 14 it is natural to ask if the Eilenberg property holds for the considered three classes of mappings under the additional assumption that the range space Y is a hereditarily unicoherent continuum. Below an example is described showing that none of such generalizations of Grispolakis and Tymchatyn result (i.e., Statement 8) is possible. 15. Example. There exist an arc-like continuum X, a hereditarily unicoherent continuum. Y, and a weakly confluent, joining and locally semi-confluent mapping f : X -t Y and a surjection g : Y -t S such that go f '" 1 on X and 9 non '" 1 on Y. Proof. Let P denote the pseudo-arc. Choose two points a and b belonging to two distinct composants of P, and put X = (P x {O, 1})/ {(a, 1), (b, O)} and Y = P/{a,b}. Thus X and Y are hereditarily unicoherent continua, and X is arc-like. Define f: X -t Y by f((8, t)) = s for s E P and t E {0,1}. Denoting we have p = {(a, 1), (b, O)} E X and q = {a,b} E Y f-l(q) = {(a, O),p, (b, I)} and f-l(y) = {(y, 0), (y, I)} for y E Y \ {q}. The reader can verify that f is weakly confluent and joining. We will show that it is locally semi-confluent. To this aim let x E X. If x f-l(q), then there is a closed -99-

6 JANUSZ J. CHARATONIK AND WLODZIMIERZ J. CHARATONIK neighborhood V of x such that either V c Px {OJ \f-l(q) or V C P x {I} \f-l(q). Then flv is a homeomorphism. Let U a and Ub stand for closed e-balls about a and b (in P) respectively, where e> 0 is small enough such that U a n Ub = 0. If x = (a,o) or x = p, then we put If x = (b, 1), then V = (U a X {OJ) U (U b X {OJ) U (U a x {I}). V = (Ub X {OJ) U (U a x {I}) U (Ub x {I}). One can verify that f(v) is a neighborhood of f(x) = q and that flv is semiconfluent. By the construction of Y there is an essential mapping 9 : Y ~. Note that the mapping 9 0 f : X ~ S is not essential because X is arc-like. Thus the proof is complete. 16. Remark. Let us recall that if a continuum Y is hereditarily unicoherent and hereditarily decomposable, then there is no essential surjection 9 : Y ~ (see [2, XI, p. 217]). Thus if Y is hereditarily unicoherent and if there is an essential surjection 9 : Y ~, then Y has to contain an indecomposable continuum. Hence the range space Y in the example above cannot be hereditarily decomposable. A further step forward concerning the Eilenberg property was made by Davis and Marsh in [3J for hereditarily weakly confluent mappings. Recall that a mapping f : X ~ Y is said to be hereditarily weakly confluent provided that for each sub continuum K of X the partial mapping flk is weakly confluent. Note that this class of mappings neither includes nor is included in the classes of monotone, open, confluent and semi-confluent mappings (see e.g. [11, p. 17]). The following result is shown as Theorem 4.7 of [3, p. 854J. 17. Each hereditarily weakly confluent mapping f : X ~ Y of a continuum X onto a hereditarily unicoherent continuum Y has the Eilenberg property. Again, as for semi-confluent mappings, it is not known whether hereditary unicoherence of Y is an essential assumption in the above quoted result. So, one can ask a question similar to Question 9 above. 18. Question. Is it true that each hereditarily weakly confluent mapping between continua has the Eilenberg property? 19. Remark. One can also ask whether Statement 17 can be generalized to some' classes of mappings that properly contain hereditarily weakly confluent ones. In the above mentioned hierarchy of mappings presented in Table II of [11, p. 28] immediate successors of the class of hereditarily weakly confluent mappings are -100-

7 ON MAPPINGS WITH THE ElLENBERG PROPERTY weakly confluent and hereditarily atriodic ones. Recall that a mapping 1 : X -t Y is said to be atriodic provided that for each subcontinuum Q of Y there are two components C 1 and C2 of l-l(q) such that I(Ct} U I(C 2 ) = Q and for each component C of l-l(q) either I(C) = Q or I(C) C I(Ct} or I(C) C I(C 2 ). A mapping 1 : X -t Y is said to be hereditarily atriodic provided that for each subcontinuum K of X the partial mapping 11K is atriodic. The reader can verify that the mapping 1 of Example 15 is hereditarily atriodic. Since it is weakly confluent, too, we again see that none of the two generalizations of Statement 17 considered above is possible. REFERENCES 1. K. Borsuk, Quelques theoreme sur les ensembles unicoherents, Fund. Math. 17 (1931), J. J. Charatonik, Confluent mappings and unicoherence of continua, Fund. Math. 56 (1964), J. F. Davis and M. M. Marsh, Hereditarily weakly confluent mappings on acyclic continua, Houston J. Math. 21 (1995), C. H. Dowker, Mapping theorems for non-compact spaces, Amer. J. Math. 69 (1947), S. Eilenberg, Sur les transformations d'espaces metriques en circon/erence, Fund. Math. 24 (1935), J. Grispolakis and E. D. Tymchatyn, Semi-confluent mappings and acyclicity, Houston J. Math. 4 (1978), W. Hurewicz and H. Wallman, Dimension theory, Princeton University Press, Princeton, K. Kuratowski, Topology, vol. 2, Academic Press and PWN, New York, London and Warszawa, A. Lelek, On confluent mappings, Colloq. Math. 15 (1966), T. Ma.tkowiak, Semi-confluent mappings and their invariants, Fund. Math. 79 (1973), T. Mackowiak, Continuous mappings on continua, Dissertationes Math. (Rozprawy Mat.) 158 (1979), G. T. Whyburn, Analytic topology, reprinted with corrections 1971, Amer. Math. Soc. Colloq. Pub!. 28, Providence, MATHEMATICAL INSTITUTE, UNIVERSITY OF WROCLAW, PL. GRUNWALDZKI 2/4, WROCLAW, POLAND INSTITUTO DE MATEMATICAS, UNAM, CIRCUITO EXTERIOR, CIUDAD UNIVERSITARIA, MEXICO, D. F., MEXICO DEPARTAMENTO DE MATEMATICAS, FACULTAD DE CIENCIAS, UNAM, CIRCUITO EXTERIOR, CIUDAD UNIVERSITARIA, MEXICO, D. F., MEXICO address:jjc@math.uni.wroc.pljjc@gauss.matem.unam.mx wjcharat@math.uni.wroc.pl wjcharat@lya.fciencias.unam.mx Received Nlay 7,