Certain geometric properties of covariant functors

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Certain geometric properties of covariant functors

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1984 Russ. Math. Surv. 39 199 (http://iopscience.iop.org/0036-0279/39/5/R10) View the table of contents for this issue, or go to the journal homepage for more

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Uspekhi Mat. Nauk 39:5 (1984), 169-208

Russian Math. Surveys 39:5 (1984), 199-249

Certain geometric properties of covariant functors V.V. Fedorchuk

CONTENTS Introduction §1. The Wazewski-Vietoris and Wojdyslawski theorems §2. The Curtis-Schori-West theorem §3. The relation between functors and absolute extensors §4. The preservation of ANR-compacta and β-manifolds §5. Homology and homotopy properties §6. The non-compact case §7. The non-metrizable case References

199 201 213 219 229 234 237 242 245

Introduction 1. One of the basic problems of topology, and not only of topology, is that of extending continuous maps. Particular cases of this are the problem of extending linear functionals, of extending analytic functions, of the existence of solutions of differential equations, of computing homotopy and homology groups, and many other problems. Therefore, a study of those spaces in which partial maps (maps from subspaces) admit continuous extensions is an urgent task. Such spaces are called absolute extensors (AE-spaces). For many classes of spaces the concept of an absolute extensor is the same as that of an absolute retract or an AR-space. Such is the case, for example, for the class of metric or bicompact spaces. The study of absolute retracts and absolute neighbourhood retracts (ANR-spaces) is interesting not only from the view point of the possible extension of maps, but also because the property of being an A(N)R-space is basic for certain other properties, for example, the property of being a manifold, both finite- or infinite-dimensional (see §2). The important role of the class of A(N)R-spaces in topology stems not only from its good properties, but also from the fact that this class is rather wide. Many topological operations preserve the property of being an A(N)R-space: summing, glueing, taking cones, or products. Some of these operations are of a functorial character. This makes the following problem urgent:

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V. V. Fedorchuk

(A) What covariant functors in the category of topological spaces preserve the property of being an A(N)R-space? The property of being an A(N)R-space often goes with other geometric properties such as: connectedness and local connectedness in particular dimensions, contractibility and local contractibility, the triviality of homology groups, and so on. Therefore, attempts to solve Problem (A) lead to the necessity of solving the following more general problem. (B) How does one or other of these geometric properties behave under the action of various covariant functors'? Since the properties of a topological space can be regarded as those of its constant map, Problem (B) naturally extends to the problem: (C) How does one or other of these properties of maps behave under the action of various covariant functors'! Disconnected results of past years bearing on this topic (the first of which is perhaps the WaSewski-Vietoris theorem of 1923 (see §1)) has in recent years merged into one broad stream forming a new geometrical direction in topology. It should be noted that the question of the behaviour of properties of spaces and maps under the action of functors includes the following question: to what extent do functors improve one or other of these properties of spaces and maps. This naturally leads to the problem: (D) How do the geometric properties of spaces and maps behave on transition from a space F(X) and a map F(f) to a space X and a map f respectively (here F is a covariant functor)'! In particular let iP be a certain geometric property. For what functors F does the fact that F(X) {or F(f)) has the property &> imply that X (or / ) also has this property"! Or, conversely, what are the geometric properties of spaces and maps which, for a given functor F, go over from F(X) and F(f) to X and f respectively"! 2. In this paper we give a survey of results relating to the solution of the Problems (А), (В), (С), and (D). In spite of the fact that in the survey the results touched upon are basically those of the author and his students in the last three or four years, it reflects perfectly adequately the present state of affairs in this branch of topology. The division into sections is fairly artificial. For example, a separate seventh section is devoted to non-metrizable bicompact spaces, but the third section is, in essence, also about non-metrizable bicompact spaces, although the results are new and interesting even in the metrizable case. All spaces in the paper are Tikhonov spaces and, as a rule, metrizable and compact, all maps are continuous and, as a rule, epimorphisms, that is, maps 1 "onto". The fibres of a m a p / : X -*· Υ are the complete inverse images/" ^ of points у Ε Υ, and also the images f(x) of points χ G X for many-valued maps, that is, for m a p s / : X -> exp Y. In the discussion of hyperspaces the closed sets of a space X are also called points of the space exp X.

201


By C(X) we denote the space of all continuous functions on a bicompact space X with the topology of uniform convergence, and by Comp the category of all bicompact spaces and all continuous maps. A (2-manifold is a separable, metric and, as a rule, compact space that is locally homeomorphic to the Hubert cube Q. The definition of normal functors in the category Comp, their properties, the concept of support, of inverse-image, and so on, can be found in [40]. Unless otherwise stated, all functors are assumed to be normal, although some properties of normality, such as the preservation of epimorphisms and of inverse images, are very rarely used. A description of specific functors, such as the superextension functor λ, the probability measure functor Ρ and its subfunctors, the functor SP^ of the G-symmetric power and so on, can be found in [32] and [40]. For a normal functor F we denote by Fn the functor that associates with a space X the set of all those elements a Ε F(X) whose supports consist of at most η points. We use the modification of this definition due to Basmanov for an arbitrary continuous functor F: Comp -* Comp, in particular, for one that does not preserve a point. We denote by η the discrete space consisting of η points, and by Yx the space of all continuous maps from X to Y, furnished with the compact-open topology. The map nFXll: Xn X F(n) -v F(X) defined by nFXn(t,

a) = F(l)a

where £ G Χ", α Ε F(n), is continuous if F is continuous. The subfunctor Fn of a continuous functor F is defined as follows: the space Fn(X) is the image of Χ" χ F(n) under nFXn, and the map Fn(f) is the restriction of F(f) to the space Fn(X). §1. The Waêwski-Vietoris and Wojdyslawski theorems Let X be a topological space and exp X its exponential, that is, the space of all non-empty bicompact subsets of X, furnished with the Vietoris topology. A base for this topology is formed by all sets of the form (Uu

. . ., Ut)

= {A e exp X:Ad

\J{Ut: i = 1, . . -, s} and A f] Ut φ

0

for all i = 1, . . ., s}, where Ult ..., Us is an arbitrary finite collection of open sets in X (taken from any base). For a metric space X the Vietoris topology is generated by the Hausdorff metric pH: pH(A,

B) = inf {ε > 0: A cz Ot(B), В cz Οε(Α)}-

We denote by exp c the so-called continual exponential of a space X, which is the subspace of exp X consisting of all continua (connected compact c subsets) of X. The operations exp and exp are covariant functors in the category of all topological spaces and continuous maps and in some of its

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full sub categories, such as the categories of bicompact spaces, of metric spaces, and of compact metric spaces. In 1923 Wazewski [93] and Vietoris [91] independently obtained the following result. Theorem 1.1. If X is a metric continuum, equivalent: 1) X is locally connected, 2) exp X is locally connected, 3) exp c X is locally connected.

then the following conditions are

The Wazewski-Vietoris theorem was strengthened by Wojdyslawski in 1939 [95]. Theorem 1.2. If X is a metric continuum, equivalent: 1) X is locally connected, 2) exp X is an absolute retract, 3) exp c X is an absolute retract.

then the following conditions are

These theorems imply that the functors exp and exp c not only remain in the class of locally connected compacta, but take this class to the class of ANR-compacta, or more precisely, the class of finite discrete sums of ANR-compacta (see [32]). Turning from Problem (B) to Problem (C) we are led to the following problem: what are properties such that if the fibres of a map / : X -* Υ has one or other of them, then so do the fibres of exp / and exp c /? This problem was solved by the author in [33], where the basic results were announced. Here we introduce some stronger results and also answer some questions in [33]. For what follows we require some preliminary remarks. Let X be compact. An arc а С exp X is said to be ordered if for any А, В € a either A d В от В С A.

Lemma 1.1 (Mazurkiewicz [73]). / / П а е ехрс X, then the whole ordered arc α lies in exp Xе. Lemma 1.2 (Kelley [70]). Let А, В G exp X and А С В. Then there is an ordered arc in exp X from A to В if and only if each connected component of В intersects A. Lemma 1.3. Let X be a continuum, Υ a compactum and f: X -> exp У а continuous map. Then every connected component of the set 2 = U {/(s): x € X} exp X joining χ to X. We put h = roexp fog,


203

where r: exp(exp Y) -> exp Υ is the union retraction. Then the map h:I -*• exp Υ is an ordered arc from fix) to Z. For if ίγ < t2, then g(h) С g(t2) and Α(ί,) = U {/(j·'): jr' 6 £(';)}· Hence, Л(^) С /j(f2)· Moreover, *(0) = № ' ) : i ' 6 g(0)} - f(x) and A(l) = U {/ 0 there is η > 0 such that: for every finite polyhedron W, every triangulation j ~ of it, and each of its subpolyhedra Wo (with respect to its triangulation) containing all the vertices of y~ any map f0: Wo-+ Ζ for which diam / 0 (σ Π Wo) < η for every simplex σ £ 3~ has an extension f: W -*• Ζ such that diam /(σ) < ε for every σ 6 J. Thus, we have to check that the Lefschetz condition is satisfied for Ζ = G. Since G is locally a connected compactum, for any ε > 0 there is an η > 0 such that whenever pH{A, B) < η, the points A and В are joined by an arc Г С G of diameter Υ is a map onto a compactum Y, then (exp Z)"1^*} denotes the inverse image of the "point" У Ε exp Υ, that is, the set of all A E exp X for which fA = Y. There is a similar definition for (expc /)~1{^'}. Theorem 1.4 [33]. If f: X -+ Υ is a map from a compactum X onto a continuum Υ with at least one point y0 Ε Υ such that the inverse image /~гу0 is locally connected, then (exp 0 /)- 1 ^} is an absolute neighbourhood retract. Proof. If the space Ζ = (expc f)~l{Y} is empty, then by definition it is an absolute neighbourhood retract. If, however, Ζ is not empty, then it is a growth hyperspace and according to Theorem 1.3 it is sufficient to verify that it is locally connected. Let Α Ε Ζ and let О A be an arbitrary neighbourhood of A. There is a basic neighbourhood U of the form ( Vx, ..., Vs) Π Ζ in this neighbourhood, where the intersection У; П j~ly Υ is a map from a compactum X to a continuum Y, then all fibres of expc / are absolute neighbourhood retracts if and only if all fibres of f are locally connected.


205

The necessity follows from Wojdyslawski's Theorem 1.2, and the sufficiency from Theorem 1.4. Theorem 1.5. // G С ехр X is a growth hyperspace of an arbitrary compactum X, the following conditions are equivalent: a) G is an absolute connected neighbourhood retract; b) G is an absolute retract. Proof. We only have to show that a) implies b). For this it is sufficient to verify that G G C" (that is, it is connected in dimension n) for every η > 0. When η = 0, this follows from the fact that G is connected. Suppose that η > 0 and t h a t / : S " -»• G exp X is continuous. We take the standard retraction r n + 1 : Bn+1 -*• exp c Sn (see the proof of Theorem 1.3) and define /: 5" + 1 ->-ехр X as the composition r°exp c /r,,+1, where, as usual, r: exp(exp X) -> exp X is the union retraction. It remains to verify that ]{Bn+1) c= G. Let b£Bn+\ Then /(6) = U{/(o): я 6 rn+1(b)}. By Lemma 1.3, every component of f(b) intersects/(a) G G for any α ζ rn+}(b). Hence f(b) Ε G, and Theorem 1.5 is proved. Theorems 1.3 and 1.5 have the following corollaries. Corollary 1.2. For a growth hyperspace G of any compactum to be an absolute retract it is necessary and sufficient that G is connected and locally connected. Corollary 1.3. Every absolute neighbourhood retract that is a growth hyperspace is the sum of finitely many absolute retracts. Theorem 1.5 answers a question in [33]. The next theorem supplements Theorem 1.4. Theorem 1.6. Let f:X~* Υ be a map from a compactum X to a continuum Υ such that the inverse image 1~гу0 of at least one point y0 G Υ is locally connected. Then the non-empty set (expr j)~x{Y) is an absolute retract in each of the following cases: a) X is connected; b) the inverse image j~xx of at least one point χ G X is connected. Proof By Theorems 1.4 and 1.5, it is sufficient to verify only that (expc /)"Χ{Γ} is connected. For any С^Сг 6 (expc ί)~χ{Υ) there is a continuum С containing them in each of the cases a) or b). Consequently, by Lemma 1.2, there is an ordered arc from C,· to С and hence from Cx to C2. This proves Theorem 1.6. The constant map of a continuum that is not locally connected shows that the existence of a locally connected fibre is essential for Theorem 1.4. However, as Example 1 in [33] shows, it is not necessary. But the following question remains unsolved.

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Question 1.1. Suppose that for a monotonic (and open) map f:X-*Y c г between continua the space (exp })~ {У} is locally connected. Does it follow from this that some fibre o f / i s locally connected? Example 1.1. Theorem 1.4 becomes false if we replace the functor of continuum exponentiation exp c by exp. For let Υ = {(t, 0): 0 ^ t ^ 1} be the unit interval on the χ-axis in the plane X = Υ \J {(Vn, 1 / n ): η = 1, 2,. .} and / the orthogonal projection onto the x-axis. Then it is easy to see that (exp /)-'{l~} is homeomorphic to the perfect Cantor set. We recall certain other facts that will be needed later. If С is a Peano continuum (locally connected compactum), then, as Bing has shown [43], there is a (locally) convex metric ρ on С This is a metric such that any two points xu x 2 S C (with the condition p(xlt x2) < δ, where δ > 0 is fixed for О can be joined by an arc in С that is isometric to [0, p(xb x2)]. All ε-balls (where ε < δ) in a (locally) convex metric are connected, and the closure of an open ε-ball is the closed ε-ball. Lemma 1.4. Let С be a Peano continuum. Then exp С contracts to {C} 6 exp С in such a way that each path from A to С is an ordered arc. This is still true for the continuum exponentiation exp c С Proof. Let ρ be a convex metric on С and diam С = δ. We define ht{A) as the closed ί-neighbourhood of A. Then the family ht: exp С -* exp C, 0 < t < δ, is easily seen to be the required contraction. In the case of the continuum exponentiation we have to use Mazurkiewicz's Lemma 1.1. The next theorem, which follows from Theorem 1.6, supplements Corollary 1.1. Theorem 1.7 [ 3 3 ] . / / / : X -*• Υ is a map from a continuum X onto Y, all the fibres of exp c f are absolute retracts if and only if all fibres of f are Peano continua. Theorem 1.8. If f: X -+ Υ is a map from a compactum X onto Y, all fibres of exp / are contractible if and only if all fibres of f are Peano continua. Proof. The necessity follows from Wojdyslawski's Theorem 1.2. To prove the sufficiency we have to show that the space Ζ = (exp/)- 1 {Y} is contractible. In the compact space Υ there is an everywhere dense set D = {yu . . ., yn, . . .) such that j'W is everywhere dense in X. By Lemma 1.4 there is a contraction gn:

ехр(/-Ч,) X

from the hyperspace exp^-1;/,,) to /-χι/η such that gn(A,i g,,(exp(/-Vn) X {l — ^ j } ) = /~Vn> 1

a n d t n e

~~) = A,

at

P h of each point

A 6 exp(Z" !/.,) is an ordered arc. We now define a map h : Ζ χ [0, 1 ] ->· Ζ in


207

the following way: h (A, t)=\

if if

1—i-s η

i = l.

Clearly, h(A, 0) = A. It only remains to verify that h is continuous. As is easy to see, it is continuous for t < 1, and we have to check that it is continuous when t = 1. Let U = ( Vv ..., Vs) be any basic neighbourhood of the "point" Χ Ε Z. We have to find a neighbourhood OA С Ζ and an ε > Osuch that h(OA X (1 — ε, 1]) cr U. We choose ОA to be (Vh, . . .. Vih) where {F,,, . . ., F ifc } is a subfamily of the family x {Fj F J consisting of all sets that intersect A. Since j~ D is everywhere dense in X, there is some η such that

l%- П (Γ1*/, LI- · · U/-1*'») ^ 0 for all / = 1, ..., s. As is easy to see, then ε = l/и is the required number. This proves Theorem 1.8. Theorem 1.9 [33]. If f: X -+ Υ is an epimorphism between locally connected compacta (or Peano continua), then both (e.xp f)~*{Y} and (expr /)~1{5*} are absolute neighbourhood retracts (or absolute retracts). Proof. We only give the proof for the functor exp, since for exp c it is simpler. The compactum X has a locally convex metric p. Consequently, there is a δ > 0 such that for ε < δ all closed ε-balls are connected. We claim that in the metric pH for ε < δ the closed ε-ball ΒεΑ with centre at an arbitrary point Α ζ Ζ = (exp/)" 1 {Υ} is connected. Let В — {χ £ λ': p(x. A) ^ ε}. The set В is the union of the closed ε-balls Bey around points у Е A. As remarked above, these balls are connected. Hence, В splits into a finite sum of connected components Bu ..., Bk. We note also that pH(A. Β) ; ζ F. Since each connected component of В intersects A, by Lemma 1.2 A and В can be joined by an ordered arc a, which obviously lies in BrA. Now let С ξ BtA be an arbitrary point. Clearly, С С В. Let us suppose that there is a certain component Bt that is not intersected by C. By definition of В there is a point у Е A such that Bt ZD /JEy. This implies that p(y, C) > ε , consequently, pH{C, Λ) > ε. This contradiction shows that С intersects each connected component of B. Consequently, by Lemma 1.2, there is an ordered arc joining С and B. Then β a BtA, and the union of the arcs α and j3 joins A to С and lies in BtA, which thus turns out to be connected. Hence, Ζ is locally connected and, by Theorem 1.3, is an absolute neighbourhood retract. If X is a Peano continuum, then Ζ is connected and, by Theorem 1.5, is an absolute retract. This completes the proof of Theorem 1.9. Theorems 1.8 and 1.9 have the following corollary.

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V.V. Fedorchuk

Corollary 1.4. If f:X -+ Υ is a map between locally connected compacta and every fibre of it is a Peano continuum, then (exp Ζ)" 1 χ {Υ} 6 AR.

Simple examples show that the condition on X to be locally connected is essential for Theorem 1.9 but, as the following Theorem 1.10 shows, in many cases it is not necessary. Definition. W esay t h a t χ ε Χ i s a p o i n t o f reducibility o f a m a p f : X - * Y if there is a closed set Л С X that does not contain χ and maps onto F. Theorem 1.10. If χ & X is a point of reducibility of a map f: X -*• Υ between compacta, then local connectedness of (exp /)~Х{У} implies local connectedness of the compactum X at x. Proof. Let A be closed in Χ, χ Ε Α, and fA = Y. Suppose that the assertion of Theorem 1.10 is not true. Then there is a neighbourhood Ox of χ containing no connected neighbourhood whose closure lies in X\A. We put 4 , = i [ J {x} and О = (Ox, Χ \ [Οχ]) |~| (exp fl"1^}· T h e n Λ ε Ο. We claim that О contains no connected neighbourhood of Ao £ (exp /)" 1 {F}. There are points x0 Ε Ox arbitrarily near to χ such that Ox = t/j1 U £/», where £/{· Π Щ = 0, the t/" are open, and χ 6 tfj1, *„ 6 i 7 "· We can find points An = .4 U {л:,,} arbitrarily near to Ao- They cannot be joined by an arc to O. For О splits into the disjoint sum of open subsets:

where Л о belongs to the first term and An to the second. Hence, О cannot contain a connected neighbourhood of Ao. This proves Theorem 1.10. Corollary 1.5 [33]. If an epimorphism f: X -*• Υ is open, then the local connectedness of (exp /)~Х{У} implies that of X at every χ that is not a oneto-one point of f. Returning to Problem (C) it is worth while remarking that among the geometric properties of a map / : X -*• Υ we can distinguish global, local, and point properties. The latter—properties of the fibres of the map—is what we have just discussed. More difficult is the problem of transferring properties of a space to maps in the form of global properties. This is the problem of finding the so-called parametric versions of theorems about spaces. As far as the WaSewski-Vietoris and Wojdyslawski theorems are concerned, this problem was solved by the author in [34] and [58]. We recall that a map / : X -> Υ is called и-so ft [38] if every commutative diagram 7 IX Y V

t


209

where Ζ is an arbitrary «-dimensional paracompact space and Z o is a closed subspace, can be complemented to a commutative diagram by a map h : Ζ -*• X. If X and У are bicompact, it is sufficient to consider bicompact Ζ and compact Ζ if X and Υ are compact. If Υ is restricted to a one-point space, then the property of the constant map / being «-soft is the same as that of X being an absolute extensor in dimension η (an AE(«)-space). Thus, η-softness of a map is a global property of it, corresponding to the property of a space being an absolute extensor in dimension n. In particular, soft (or °°-soft) maps correspond to AE(°°)-spaces or, what is the same, absolute retracts. As regards the continual exponential, the Wazewski-Vietoris and Wojdyslawski theorems can be combined in the following way: Theorem 1.11. The following properties are equivalent for a compactum 1) X is a Peano continuum, 2) exp c X is a Peano continuum, 3) exp c X is an absolute retract.

X:

Since by the Hahn-Mazurkiewicz theorem the Peano continua are linearly connected, locally linearly connected, and compact, by the KuratowskiDugundji theorem on map extensions (see [3]) the Peano continua are precisely the AE(l)-compacta. Therefore, the following assertion is a parametric version of Theorem 1.11. Theorem 1.12 ([34, [38]). The following conditions are equivalent for a map f: X ->• Υ between Peano continua: 1) / is 1 -soft and exp c / is open; 2) exp c / is \-soft; 3) exp c / is soft. It is worth mentioning that the condition on exp c / to be open is automatically satisfied in Theorem 1.11 if it is restated by replacing a space by a constant map. A detailed proof of Theorem 1.12 will be published in the volume of the "Lecture Notes in Math." devoted to the Proceedings of the Leningrad International Topology Conference 1982, and we only sketch it here. The main difficulty is in proving the implication 1) => 2). We denote by Μ the set of all points у G Υ such that | / - 1 y | > 2. In the first place one proves that: (1) Υ contains no simple closed curve intersecting M. The idea of the proof of (1) is close to the proof that if p1: S1 χ / -> S1 is the projection, then the map exp c px is not open. Then, on the basis of (1) one can prove the next assertion. (2) Every subcontinuum К С М is a dendron, that is, a Peano continuum that does not contain a topological circle.

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V. V. Fedorchuk

The proof of this uses the fact that every sub continuum of a dendron is a dendron (see [79]). Now to prove that e x p c / i s 1-soft, we have to verify in accordance with Michael's selection theorem [45], that a) e x p c / is open, b) the fibres of e x p c / are Peano continua, c) the family {(expc f)~lA: A 6 exp e Y) is uniformly locally linearly connected, that is, for any ε > 0 there is some δ > 0 such that, whenever By, B2 € (cxpc i)~lA and pu(B1, B2) < 6. then there exists an arc σ с: (expc f)~xA joining Bx and B2 whose diameter is less than ε. Now a) is contained in 1) of the theorem; b) follows from Theorem 1.7, since the fibres of a 1-soft m a p / are Peano continua. The verification of c) is technically the most difficult. Since / is 1-soft, for a given e > 0 there is a δ < e/G such that, whenever д-1% хг £ j~xy and p{x\, x2) < δ, there is an arc τ cr /-1i/joining хг to x2 whose diameter is г \ Л 0 φ- 0, then В = В ι []Β2ζ (ехр 0 /)" 1 /!, andB1 is joined to B2 by an arc of diameter 0 there is a point a G A and a map к : X -> exp X (or exp c X) such that α $Ξ k(x) for any χ G X and к is г-close to the identity embedding of X in exp X (or exp c X). This assertion obviously extends to the case when Λ' is a locally connected compactum with no isolated points. We now apply this to the case when X is our fibre F and homeomorphic to {y} X F = /г-^/-1!/). Since by Wojdyslawski's theorem the hyperspace exp F (or exp c F) is an ANRcompactum, the map к = k0 can be connected by an ε-small homotopy kt to the identity embedding ki'.F-*- exp F (or exp c F). Now let V be a neighbourhood of у whose closure lies in U, and ψ : Υ -> [0, 1 ] a map such c that ψ(γ) = 0 and φ(Υ\ V) = 1. We now define a map g : Τ (or T ) -* 7 (or с Г ) as follows: * (B) = U {/* (/ (x) X А-5(!-а,й/г» (г): x 6 β П /-»£/} U (5 Π Λ 1

(YW)),

where ρ2 '• U χ F -*• F is the projection. The map g associates with a closed set 5 С I a closed set g(B). For g(J5) = Βγ U 5 2 , where JSj = 5 Π /-' and

= U{ № ) X


215

Let us check that B2 is closed. Identifying /-'[V] with the product [V] χ F, we can consider }ιφ: [V] χ F-* exp([ F] χ F) to be defined as the composition

mOl [V] X exp F [F] χ [ F] is the diagonal embedding. P u t t i n g ^ = r exp hlfi, where r: ехр(ехр([Г) X F)) -*• exp([F] X / ) is the union retraction, we find that 5 2 coincides with g^B Π/'HV!) and consequently is closed. This implies that g : exp X -* exp X is continuous. By the definition of g, g(B) Ε Τ, when 5 ε 71. If В & Tc, then the map к in the Assertion 2.1 and the homotopy kt take points into continua. Therefore, g(B) is also a continuum, as the continuous image of the continuum В under a continuous many-valued map whose fibres are continua. Thus, the restriction of g to Tc takes Tc into itself. For small ε the map g is near to the identity. Finally, the point a £ A dj'ly (see Assertion 2.1) does not belong to g(B) for any В & Τ (or Tc). This means that the map g "drags" our space Τ (or Tc) away from the set CA, which therefore turns out to be a Z-set. This proves Lemma 2.1. We now complete the proof of the theorem. We fix a locally convex metric ρ on the fibre F » f-ly and cover f-*y by finitely many closed ε-balls Ax, ..., An. (Here ε is chosen so small that every 2e-ball is connected.) By Lemma 2.1, С = (J CAi is a Z-set. We now consider a continuous function φ: Υ —ν [0, 2ε] such that qi(i/) = 2ε and (г(У'\Г) = 0, where V is as in Lemma 2.1. For any χ ζ /~ ι ί* and δ > 0 we denote by В 6(.r) a closed δ-ball with centre at χ in f~lfx relative to the metric p. We define a map gx:T-* Τ by putting = U {#Ф(./Л) (*): x 6 δ

Π Ζ-1^}

U Β.

Just as for g in Lemma 2.1, one can verify that gy is well-defined, continuous, and for any continuum В the set gy(B) is also a continuum. Finally, с х g(T) С С , since an arbitrary 2e-ball Bit(x) in /- г/ contains one of the ε-balls Ay, ..., An, namely the one that contains x. It is also clear that for small ε the map gy is near to the identity. Thus, the identity map on Τ (or Tc) can be approximated by Z-maps. This proves Theorems 2.1 -2.3. Corollary 2.1 [33]. If f:X -> Υ is a locally trivial fibration whose fibres are locally connected compacta (or Peano continua) without free arcs or isolated e points, then all fibres of exp / are homeomorphic to a Q-manifold {or the Hilbert cube).

216

V.V. Fedorchuk

Corollary 2.2 [33]. / / / : X-> Υ is a locally trivial fibration over a compactum Υ whose fibres are locally connected compacta (or Peano continua) without isolated points, then all fibres of exp f over locally connected compacta (or Peano continua) are homeomorphic to a Q-manifold (or the Hilbert cube). Remark 2.1. In many cases the absence of free arcs is superfluous for the validity of Theorems 2.1 and 2.2 and Corollary 2.1. This is shown by the example of the projection/ of a square onto an interval. The fact that in this case (expc Z)-1 {/} is homeomorphic to the Hilbert cube is a consequence of the following theorem, of Curtis, on growth hyperspaces (for the definition, see § 1). Curtis' theorem [51]. Let G be a non-trivial growth hyperspace of a Peano continuum X such that either X contains no free arcs or G is an inclusion space. Then the following conditions are equivalent: 1) G is homeomorphic to Q, 2) G\{A'} is contractible, 3) {X} is a Z-set in G. With the help of this theorem one can give an alternative proof of Theorem 2.3 in the connected case, by showing that {λ'} is a Z-set in (e.xp f)-x{Y) (or (expc /)-'{)'}). But this is not much simpler, all the more since Theorem 2.3 was proved simultaneously with Theorem 2.1 and 2.2, where Curtis' theorem does not apply. Nevertheless, the application of Curtis' theorem gives a simple proof of Theorem 2.3 without requiring that the fibre has no free arcs, provided that it contracts onto a proper subspace. In this connection there arises the following question. Question 2.1. Let f: X -• Υ be a map between Peano continua having a trivial patch whose fibres are infinite, locally connected, and compact. Is it true that (expc f)"l{Y} is homeomorphic to Ql Several variants of the parametrical version of the Curtis-Schori-West theorem were obtained in [34] and [58]. Here we give stronger and in a certain sense definitive results. For detailed proofs, see [59]. Theorem 2.4. Let X and Υ be non-degenerate Peano continua and c Px: Χ χ Υ -> X the projection. Then the map f = exp px is a trivial fibration whose fibres are Hilbert cubes Q if and only if X is a dendron and Υ contains no free arcs. Proof. The necessity follows from Theorem 1.13. To prove the sufficiency we note first of all that by Michael's selection theorem (see the proof of Theorem 1.12) the projection ρχ is 1-soft and by Theorem 1.13 the map exp c pi is open. Hence, by Theorem 1.12, exp c ργ is soft and consequently, is a Hurewicz fibration with contractible fibres. Now we require the following parametric version of Torunczyk's theorem on a characterization of the Hilbert cube.


217

The Torunczyk-West theorem [ 8 9 ] . Let f:X -> Υ be a Hurewicz fibration of an ANR-compactum with contractible fibres J'1}/. Then f is a trivial fibration whose fibres are Hilbert cubes Q if and only if for any ε > 0 there are maps g{: X -> X (i = 1 or 2) such that

Ui = /.

p(gh id) < ε,

Im g1 Π I m ? 2 = 0 .

Taking this theorem into account, to complete the proof of Theorem 2.4 we have to construct maps gt: expe(X X Y) -*- expc(X X Y) (i = 1 or 2) with non-intersecting images that are ε-close to the identity. The idea of the construction is the same as that applied to obtain Z-maps approximating the identity map in the proof of Theorems 2.1-2.3 (applying Lemma 5.4 in [48] and an ε-inflation). This is simpler in the present situation, since px is a trivial fibration. Remark 2.2. Theorem 2.4 can be regarded as a theorem on necessary and sufficient conditions for trivial fibrations of non-degenerate Peano continua to be preserved under the functor exp c . For if Χ χ Υ is a Peano continuum, then X and Υ are also Peano continua. Since exp c px is soft, we find according to Theorem 1.13 that X is a dendron. Finally, if Υ contains a free arc, then any fibre of the form (expc Pi)'1^} contains an open twodimensional set, whereas for any non-trivial continuum А С X the fibre (exp0/?])"1/! is locally infinite-dimensional. Thus, for non-degenerate Peano continua a trivial fibration of the form exp^p! can only be a Q-fibration. Question 2.2. Let f: X -*• Υ be a \-soft map of a continuum X onto a dendron Υ such that the fibre j-xy is non-degenerate and contains no free arcs. Is it true that exp e / is a trivial fibration'] The Wojdyslawski and the Curtis-Schori-West theorems, as well as the theorems on the maps of Peano continua into absolute retracts and Hilbert cubes have in recent years been extended to other functors of exponential type. Thus, in [57] two new functors Γ and Г с were introduced. We denote by Г(Х) (or ГС(Х)) the set of all ordered arcs in the exponential exp X (or the continual exponential exp e X) of a compactum X. The set Г(Х) is closed in the second exponential exp(exp X) and consequently is compact, and YC{X) is closed in Г(Х). The operations Г and Г с are sub functors of the functor expo exp. Tlieorem 2.5 (Eberhart, Nadler, and Nowell [57]). Let X be a metric continuum. Then the following assertions are equivalent: 1) X is locally connected, 2) ГС(Х) is an absolute retract, 3) Г(Х) is homeomorphic to the Hilbert cube. Golov [4] has obtained a fibre version of this theorem of the type of Theorems 1.4, 1.9, 2.1, and 2.3. We quote his results.

218

V.V. Fedorchuk

Theorem 2.6 (Golov [4]). Let X and Υ be metric continua, у £ V{Y), UV = У, and let f:X -*• Υ be an epimorphism, where the inverse image of at least one point у Ε Υ is a locally connected compactum (or continuum). Then (Γ°/)-ι(ν) is an ANR or (AR)-co трас turn. Corollary 2.3 (Golov [4]). If all fibres of a map fare locally connected compacta (or continua), then all fibres of Tcf are ANR or (AR)-compacta. Theorem 2.7 (Golov [4]). Let f:X~* Υ be an epimorphism between metric continua, 7 €= Г(У), Uv = Y, and let Πγ and i'l({\y) be locally connected compacta (or continua). Then (Γ/)-χ(γ) is an ANR (or AR)-compactum. Theorem 2.8 (Golov [4]). Let f: X -> Υ be an epimorphism of metric continua having a trivial patch whose fibres are non-degenerate locally connected compacta (or continua) without free arcs; let у £ Γ°(Υ), \jy = Υ, and suppose that the inverse image of at least one point у ζ f]y is a locally connected compactum (or continuum). Then (Γ°/)"1(γ) is homeomorphic to a Q-manifold (or the Hilbert cube). Theorem 2.9 (Golov [4]). Let f:X-+ Υ be an epimorphism between metric continua, having a trivial patch whose fibres are non-degenerate, locally connected compacta (or continua); let у € Г(У), Uv = Υ, and let Π Υ and f~1(f\y)be locally connected compacta (or continua). Then (Γ/)" χ (γ) is homeomorphic to a Q-manifold (or the Hilbert cube). Problem 2.1. To give examples of Q-manifolds in the results of §2 that do not reduce to finite sums of Hilbert cubes. We mention that in Theorems 2.1 and 2.3 and in the Corollaries 2.1 and 2.2 the g-manifolds are finite sums of Hilbert cubes, by Corollary 1.3. Savchenko has studied the functor exp£. The space exp£ X consists of all non-empty closed subsets of a compactum X having at most η connected components. This functor is a natural generalization of the functor of continual exponentiation. Savchenko [24] extended Theorems 1.9, 1.11, and 2.3 and Corollary 2.2 to the functor exp£. This implies, in particular, the following result the space exp£ X is homeomorphic to Q if and only if X is a nondegenerate Peano continuum without free arcs. He has constructed an example to show that Theorems 1.4, 1.7, and 2.1 and Corollary 2.1 do not extend to the functor exp£. In his example the map / is a trivial fibration with fibre Q. Let us give here an assertion, which is a parametric version of a theorem that follows from results of Keller [69] and Klee [71], namely the theorem that the space P(X) of probability measures on an infinite compact space X is homeomorphic to the Hilbert cube. Theorem 2.10 [34]. Let X and Υ be compact spaces and p , : I x Υ -> X the projection. Then the map f = P(pi) is a trivial fibration with fibre Q if and only if Υ is infinite.

Certain geometric properties of со variant functors

219

For a detailed proof, see [59]. We give a sketch of it. The necessity follows from considering the fibres of/ over the Dirac measures δχ G P{X). To prove the sufficiency we must note first of all that according to Michael's selection theorem [74] the m a p / i s soft. Consequently, as in Theorem 2.4, we have to construct maps g1 and g2 satisfying the Torunczyk-West criteria. For a non-isolated point y0 e Y, let σ: λ' -»- Χ X {г/„} с-* X Χ Υ be the embedding. For e > 0 we put #,(ц) = (1 — ε)ιι 4- ε/>(σ)(/μ). For a small ε the map gx is near to the identity and (2-1)

\

dgiM>0

-Yx Υ be the Milyutin map of a zero-dimensional compactum Ζ onto У and Μ : C(Z) -»• C( K) the corresponding regular averaging operator (see [23]). Then the dual map u*: M(Y) -»• Λί(Ζ) takes />(F) to />(Z). Let V be a small open-and-closed neighbourhood of h~1yoand z0 £ ν'χΛ"1^,,. We consider the map Α·: Ζ -* (2\V) U {г0}, which is the identity on Z \ F and takes V to zQ.

The map π = i d m - , X P{h)oP{k)°u* from P(Z) χ Д У ) into itself preserves the fibres of / and is near to the identity if V is small. It extends linearly to P(X) 2), then dim Ρ(ρ1)'1δχ = η —1 for any χ e X. On the other hand, if ^j Φ χ2, then dim Рф^-ЧЧЛ*, -τ ν,δ*,) = 2?г - 2 > η - 1. Question 2.3. Is it true that if / : X -> F is an open map with infinite fibres, then P(f) is a trivial fibration? A partial answer to this question (when X and Υ are polyhedra) was given recently by Zarichnyi. §3. The relation between functors and absolute extensors The relation between absolute retracts or absolute extensors in some dimension or another and covariant functors, which was outUned in Question (A) of the Introduction is not far-fetched. There is a procedure enabling us to link the classes of absolute extensors in certain dimensions with welldetermined functors.

220

V. V. Fedorchuk

Definition 3.1 [40]. A bicompact space X is said to be injective with respect to a certain covariant functor F: Comp -> Comp (briefly, F-injective) if for every map / : Υ -*• X and every (closed) embedding i : Υ -> Ζ there is a map g : F(Z) -> F(X) for which F(f) = g ° F(i). Theorem 3.1. A bicompact space X G AE(0) if and only if X is injective with respect to the functor of probability measures P. This theorem is in fact due to Haydon [65], who proved that the class of AE(0)-bicompacta coincides with the class of Dugundji spaces. We recall that a bicompact space X is a Dugundji space [23] if for any embedding /: A" ^- Υ in another bicompact space X there is a regular extension operator и : C(X) -> C(Y), that is, a linear operator such that 1) (г* о u)
0 for every non-negative function φ G C(X); 3)u(lx)=

\Y.

Thus let Ζ G AE(0), let / : У -* Z be an arbitrary map and i : У -• Z an embedding. There is an embedding /: X c~ Ιτ for which, by Haydon's theorem there is a regular extension operator и : С(Л0 ->· C(/ T ). By Uryson's theorem, / can be extended to a map g : Ζ -*• Γ', that is, / = g о /. The extension operator и determines a dual map и*: Д / т ) ->• ДЛЗ by the formula (Μ*(μ))φ = μ(Μ(φ))· Then и* о P{g) :P(Z) -> P(X) satisfies the definition of /Mnjectivity. Conversely, if a bicompact space X is P-injective and г: Χ ^> Υ is an embedding, then by Definition 3.1 (with У = X, f — id x ) there is a map g :P(Y) -> P(X) such that P(.idx) = goP(i). This means that g is a retraction. Therefore, the formula u{ C(Y). Consequently, Л' is a Dugundji space and X G AE(0) by Haydon's theorem. Theorem 3.2. If X is a bicompact space, the following conditions are equivalent: 1) X is expc-injective, 2) X is exp-injective, The implication 1) =>2) is obvious; 2) =*-3) was proved by Nepomnyashchii (see [80]). Let us prove 3) =M), which was verified in a somewhat weaker form in [34]. Let / : У -> X be a continuous map and г : У ->· Ζ an T embedding. As in Theorem 3.1, we take an embedding / : X -> / . By Theorem 2 in [34] there is a one-dimensional bicompact space Τ and a monotinic open map g from this Τ onto / r . Thus, we have the following


221

commutative diagram:

,3.1,

1

where To = 8~гХ. g» ~ g \ To, and h is an extension of/. Since X G AE(1), the diagram (3.1) can be completed to a commutative diagram by a map k:T -+ X. Then the map r» expc/, where I = expc bg-^h: Ζ -*• exp c Λ' and r : exp e (exp c X) -> exp c X is the natural retraction, satisfies the condition for expe-injectivity. Theorems 3.1 and 3.2 are supplemented by the following obvious theorem. Theorem 3.3. If X is a bicompact space, then X G AE(°°) if and only if X is Ы-injective, where Id is the identity functor. Thus, in dimensions 0, 1, and °° we have functors P, exp and exp c , Id that indicate whether bicompact spaces belong to the class of absolute extensors in these dimensions. This one-to-one correspondence between functors and absolute extensors enables us to solve the problem of the adequacy (see [38]) of the class of AE(«)-bicompacta for the class of n-soft maps in the given dimensions (for η = 0, see [65], for η = 1, see [34] and [22], and for η = °°, see [38]). It appears that for further progress in this direction one has to find functors Fn for which the class of Fn-injective bicompacta is the same as that of AE(«)-bicompacta. The latter problem is very difficult and even contrary to intuition. It remains unsolved to the present day, whereas the problem of adequacy was solved by Dranishnikov. One of the basic features in the solution of this problem is his proof of the following theorem: Theorem. For every η > 0 there is an η-dimensional compactum Xn and an (n - \)-soft map fn : Xn -> Q onto the Hilbert cube Q. Let us turn to the concept of F-injectivity. It admits certain modifications. Definition 3.2. We say that a bicompactum X is an F-valued absolute retract (or a strong F-valued absolute retract) (written X G AR(F) (or X G AR S (F)) if for every embedding of X in another bicompactum Υ there is a continuous map / : Υ -> F{X) (or f:F(Y)-> F(X)) such that f\X = id*. (The definition makes sense if X has a natural inclusion in F{X).) Proposition 3.1. Every strongly F-valued absolute retract is an F-valued absolute retract. Proposition 3.2. A bicompactum X is F-injective if and only if it is a strongly F-valued absolute retract.

222

V.V. Fedorchuk

The necessity follows immediately from the definition. The sufficiency is proved by embedding X in the Tikhonov cube Γ and extending / : Υ -* X to a map g : Ζ -*• Γ.

Remark 3.1. We do not change the Definition 3.2 if instead of embedding X in an arbitrary bicompactum Υ we only consider embeddings in the Tikhonov cube IT of weight r = wX. For let X c - Υ be an embedding. We consider the embedding Υ (Z-* Iх'. There is a set Л С т' of cardinality \A I = wX such that pA \X is an embedding, where pA: Iх' -*• IA is the projection. There is a map g :IA -> F(X) (or g : F(/^ ) -* F(X)) such that g I P A W = PA · Then the map / = g ° ( P y l I У) (or / - g о F{pA i 7)) is the required F-valued retraction of Υ onto X. Proposition 3.3. For the functors F = exp, expc, ant/ Ρ f/ге concept of an F-valued absolute retract and a strongly F-valued absolute retract coincide. For this is ensured for the functors exp and exp c by the union retraction r : exp(exp X) -*• exp X. For the functor Ρ such a retraction is given by the regular extension operator и : C(X) ->· C(P(X)) defined by (κ(^))μ = μ(φ). Therefore, Theorems 3.1 and 3.2 can be restated as follows. Theorem 3.1 χ. If X is a bicompactum, then Χ ε ΑΕ(0) if and only if X € AR(F). Theorem 3.2X. / / X is a bicompactum, equivalent: 1 ) I € AR(expc), 2) Χ Ε AR(exp), 3)leAE(l).

the following conditions are

Problem 3.1. To give an example of a normal functor F for which the classes AR(F) and AR^(F) do not coincide. Theorems 3.1 χ and 3.2i were improved by Dranishnikov. We recall that a many-valued map / : X -> exp Υ is continuous if it is both upper and lower semicontinuous. Here, / is upper semicontinuous (or lower semicontinuous) if for every open set U С Υ the set {i f X: f(x) a U) (or [χ £ X: f(x) Π U Φ 0}) is open in X. If F is a subfunctor of exp, we say that a bicompactum X is an F-valued upper semicontinuous absolute retract (Χ ζ АЩГ\)) if for every embedding of it in an arbitrary bicompactum Υ there is an upper semicontinuous retraction/: Υ -*• F(X). Theorem 3.4 (Dranishnikov [7]). If X is a bicompactum, then the conditions Χ Ε AE(0) and X f AR(exp f ) are equivalent. Theorem 3.5 (Dranishnikov [7]). If X is a bicompactum, then the conditions Χ Ε AE(1) and X 6 AR(expc f ) are equivalent. Just as we passed from the concept of an absolute retract to that of an absolute extensor in a given dimension и, so we can pass from the concept

223


of an F-valued absolute retract to that of an F-valued absolute extensor in dimension n. Definition 3.3. We say that a bicompactum X is an F-valued absolute extensor (or a strongly F-valued absolute extensor) in dimension η (written X G АЕ(и - F) (or X G AEs(n - F)) if for every bicompactum Υ of dimension dim Υ < η and every map f:A-*X from a closed subset А С Υ to X there is a map g : У -»• F(Jf) (or £ : F ( / ) -»• F(X)) such that g\A = f. Proposition 3.4. £Very strongly F-valued absolute extensor in dimension η is an F-valued absolute extensor in dimension n. Theorem 3.6. AE(co-F) = AR(F). AEs(oo-F) = AR'(F). Proof. The inclusion С follows immediately from Definition 3.3 with / = i d x . Let us check the reverse inclusion for AKS(F). Let A be closed in Υ and / : A -+ X. Since Χ G ARS(F), there exists a retraction r : F(/ T ) -* F(X) for X С / T . The map / admits a continuous extension h : Υ ^ IT. Then g = r°F(h) is the required map satisfying Definition 3.3. The case for AR(.F) is examined similarly. Proposition 3.5. // F is a functor for which there is a natural transformation F о F ->· F that is a retraction, then АЕ(и — F) = AEs{n - F) for every n. Corollary 3.1. AE(n-P) =- AES(n-P), AE(«-oxpc) = AE5(/?-exp• Υ is 1) weakly η - F-soft, 2) left и-F-soft, 3) upper η — F-soft, 4) strongly ή - F-soft, if for every bicompactum Ζ of dimension dim Ζ < η, every closed subset A in Z, every map g : A -+ X, and 1) 2) 3) 4)

h: Z-+ Г, h: Z-+F(Y), h: Ζ ->· Υ, h: Z-+F(Y),

for which fog = h on A, there exists an extension 1), 2) к: Z-+F{X), 3), 4) k: F(Z) -Η. F(X) of g such that F(f) ° k\Z = h. If η = °°, then an η - F-soft map is said to be F-soft. Remark 3.2. The conditions 1), 2), 4) and 1), 3), 4) are in increasing order. A natural problem is to sort out the comparability of 2) and 3). Remark 3.3. If we take for/, the constant map then 1) and 2) give for X the concept of an AE(« — F)-bicompactum and 3) and 4) that of an AEs(n — F)-bicompactum.


225

Proposition 3.7. // there is a natural transformation F'° F -*• F that is a retraction, then 1) = 3) is weaker than 2) = 4). If / : X -> У and F is a functor, we put F0(f) = F(J) \ F(f)~lX, where F{f)-xX = U{F{j-ly): у 6 У). Theorem 3.10 (Nepomnyashchii [22]). For a map F: X -> У between bicompacta having a metrizable kernel the following conditions are equivalent: 1) e x p 0 / й да/ί; 2) / is strongly txp-soft; 3)//s 1-so//. We note that the softness of e x p 0 / follows from the weak exp-softness of /. For since / has a metrizable kernel, there is an embedding X С Υ χ Q such that / = ρλ\Χ. The map exp 0 px is soft, as a trivial fibration with fibre exp Q * Q. In Definition 3.4 we put Ζ = Υ χ Q, A - X, g = id^, h = pi: Υ χ Q -> Y. Then there is a retraction к : Υ χ Q -> exp X such that £?! = (exp/)о &, that is, the image of к actually belongs to (exp0 /)"ХУ. By means of the union retraction we can extend A: to a fibre retraction r: (expoPi)-1}'-»- (exp0 f)~lY. Then expo / is soft, as a retract of a soft map. The same arguments work for the functor exp c . Thus, we have proved that (3.4) if f is weakly exp- (or expc)-soft, then expo/ (or ехр§/) is soft. Now we show that (3.5) if f is soft, then f is strongly expc-soft. As before, let X be embedded in У χ β so that / = ρχ\Χ. Suppose that A is closed in Ζ and that g : A -*• X and h : Ζ -*• exp c У are maps such that fog = h on A. By Michael's theorem on the selection of a convex-valued map [74], the constant many-valued map Φ : Ζ -*• Q with image Q has a selection φ that is an extension of the partial selection pi°g A -*• Q where Pi '• Υ x Q -*" Q is t n e projection. Let r : F χ β ->• I be a p^fibre retraction, which exists because / is soft. Then к = exp c r ο (/ζΔιρ), where /ζΔιρ : Ζ -> c -> (exp У) χ β is the diagonal product of h and ψ, is also a map for which h = (exp c /)o/c. An application of Proposition 3.7 completes the proof of (3.5). An immediate consequence of (3.5) by an application of the union retraction is the following result: (3.6) if expo / is strongly expc-soft, then f is strongly expc-soft. Finally we verify that c

(3.7) if f is \-soft, then f is weakly exp -soft. The proof uses the standard device already applied, which relies on the theorem that every bicompactum is the open monotonic image of a onedimensional bicompactum.

si dvui

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J/OS-J-U

joj

р э л р эцЗшге в о; saonpaj joojd эщ JOJ jUBjJoduii si ipnjM uoijisodojd J9ij;jnj e иоциэш OS|B эду uv si / (£ ЧЭШ1ЭЛ рэпрл-ахз ajnjosqO uv si / (3 'ί/os-I я / ( 1 suoijipuoo Зщмоцо/ эщ \эшщ d\qOzuidiu ν Зщлщ щэюашоэщ иээлиэц А · P{ZX) and u*2:P(IT) -> P(Z 2 ) be the dual maps and Ψ = ut®u%. Then Ψ maps P(Y) ® Ρ(/ τ ) = Ρ (Υ Χ / τ ) into Ρ(Ζ,) ® Ρ(Ζ,) = />(Ζ, Χ Ζ 2 ). Finally, we show that к = P{h) οψ\Υ χ Ιτ is the required P-valued retraction of py onto /. We note first of all that Im к С Р{Х) by the definition of k. Next, £(δ*) — 8X for χ = (у, t) Ε Χ. This follows from the fact that the support of the measure k(8x) is concentrated at x. Finally, we check that ργ = P(f)k. Let (y, t) P(X). Then roP(k), where r : P(P(X)) -> P(X) is the natural retraction, retracts the soft map P 0 (pi) onto P o ( / ) , which is therefore soft.

рив

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'(Лвл°}[)

АЭ - X \ / determines an inclusion F(i,): F(X χ {t})-+ F(X X /). But F(Xx{t}) is naturally homeomorphic to F(X) χ {?}. Therefore, there is a natural embedding F(X) χ / -> F(X χ /). Then the map F(H)\F(X) χ / is a continuous homotopy joining F(h0) and

Fih,).

If we disregard Borsuk's theorem on products of ANR-spaces and Wojdyslawski's theorem, perhaps the first person to work on functors preserving ANR-compacta was Ganea. In 1954 he proved in [62] a theorem on the preservation of finite-dimensional ANR-compacta. We cite a list of authors, functors, and the classes of spaces (if they are not ANR's) which they transform into ANR-compacta. As fibre version we single out the

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lit


233

Next, Ρ is divided by R 4 into two symmetric contractible polyhedra /\ and P2. Multiplying Ρ by the Hubert cube Q we obtain a representation of Q as the sum of two copies Q1 and Q2 of the Hilbert cube such that £?ι Π Q2 is not a contractible Q-manifold. Multiplying X by Q we obtain a representation of the Hilbert cube as the sum of Ql and Q2 such that Qx Π Q2 is not locally contractible. Question 4.1. Can a finite-dimensional cell B" be represented as the sum of η-cells B\ and Bn2 such that В^СЛ Bn2 0 some sphere S" С Q\h{Q) is homologically linked with h(Q). Example 4.2. There exists an embedding h : Q -*• Q such that Q\h{Q) has trivial homology groups and trivial homotopy groups for η > 2, but does not have the homotopy type of a finite polyhedron. The following question is concerned with the disposition of one (^-manifold in another. Question 4.2. Let Μ D N be a pair of Q-manifolds. When is this pair triangularizable, that is, when is there a pair of polyhedra S D Τ such that the pairs (M, N) and (S χ Q, Τ χ Q) are homeomorphic ? This question is interesting even when Μ and N are homeomorphic to Q. We remark that a necessary condition for this is the existence of a pseudoisotopy in the sense of Keldysh [20] taking a certain neighbourhood ON С Μ to N. Is this condition sufficient for the pair (Μ, Ν) to be triangularizable? Returning to the main theme of this paper we ought to note that the problem (D) (see the Introduction) contains the following question. Question 4.3. For what functors F does the property of being an A(N)R-space carry over from F(X) to ΧΊ The above results imply that the basic standard functors of infinite degree exp, expe, P, and λ are no use for this. However, Zarichnyi has proved in [8] that if Xn(X) or *Xn(X) are ANR-compacta, then so is X. For other functors of finite degree the question remains open. In particular, there is no answer yet to the following question. Question 4.4 [37]. Let e x p 2 Z be an AR-compactum. retract•?

Is X then an absolute

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§5. Homology and homotopy properties 1. Homotopy and homology properties of symmetric powers were among the first to be studied by Richardson and Smith. In 1935 Richardson [82] computed the Betti numbers of the symmetric powers SP2ATn and SP 3 £ n in terms of the Betti numbers of the и-dimensional simplicial complex Kn. In 1936 Smith [86] proved that for any CW-complex Ε 71

E) =s HÊ)

when η > 1.

In 1938 Richardson and Smith [83] computed some homology groups of the cyclic product SPJ К of the complex K, where Zn С Sn is a subgroup of cyclic permutations of order η in the symmetric group. These results were generalized by Dold and Puppe. In 1958 Dold [55] indicated a procedure for computing the homology groups of an arbitrary symmetric power SVQX of a polyhedron X, provided that the coefficient domain is a hereditary ring. In 1961 Dold and Puppe proved in [56] under broad assumptions that the homotopy groups of the symmetric powers SP"£ are the same as the homology groups of the connected CW-complex E. The cohomology operations on the symmetric powers were studied by Bott [45]. Some questions on the preservation of finite-dimensional manifolds by symmetric and hypersymmetric powers of functors were touched upon in [30]. A complete survey on this topic was given by Wagner in [92]. 2. Bott's theorem and Kozlovskii's example. Bott's theorem [44] that ехрз^ 1 is homeomorphic to S3 turned out to be of supreme importance for general topology. Here it is worth mentioning that the simplest part of this theorem turned out to be interesting for applications, namely, that the fundamental group of the hyperspace exp 3 S' is trivial. This fact is the basis for Szankowski's theorem [87], which proves that for a compactum X the following are equivalent: 1) X is a Peano continuum; 2) for some embedding X is a countable-valued retract of the Hubert cube Q; 3) for some embedding of X in Q there is a multiplicative operator extending C+(X) to C+(Q), where C+(Y) denotes the space of non-negative functions on Y. The fact that the fundamental group of the hyperspace ехрз^ 1 is trivial implies that there is a three-valued retraction of the two-dimensional disc B2 onto its boundary circle Sl. Applying this fact Szankowski constructed a countable-valued retraction of the Hilbert cube onto an arbitrary Peano continuum. In explicit form such a three-dimensional retraction r of B2 to S1 was described by Kozlovskii. The graph Φι of this retraction is a two-dimensional polyhedron, which is projected onto B2 by an open, three-valued, and inessential map g. Taking


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some triangulation of «^ and replacing each two-dimensional simplex of this triangulation by a copy of 3. Basmanov by applying the basic idea in Zarichnyi's argument, has obtained a considerably more general result.

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Theorem 5.1 (Basmanov [2] ). Let F be a continuous monomorphic functor of finite degree h preserving intersections and the empty set, where F(n) is an ANR-compactum and F ( l ) is linearly connected. Then the following conditions are equivalent: 1) for every bicompactum X having the homotopy type of a connected compact polyhedron the space F{X) is simply-connected; 2) for every connected ANR-bicompactum X the space F(X) is simplyconnected; 3) FiS1) is simply-connected; 4) any map a: S1 -*• FOS1) whose image belongs to F2(S1) is homotopic to the constant map. Consequences of this theorem are not only Zarichnyi's result, but also the fact that for η > 3 the functors exp n and Pn take connected ANR-bicompacta into simply-connected spaces. We recall that a space X is said to be λ?-connected (written Χ Ε С") if for every к = 0, 1, ..., η any map f:S"~*X is homotopic to the constant map. The class LC" is defined similarly (maps of spheres whose images have small diameter contract to the constant map by small homotopies). By the Kuratowski-Dugundji theorem on map extensions (see [3]), C" Π LC" = = AE(« + 1). Thus, the assertions just stated concern functors that take 0-connected spaces to 1-connected spaces. Quite recently Basmanov and Dranishnikov have obtained results on functors that preserve and improve the connectedness of spaces in higher dimensions. Theorem 5.2 (Basmanov and Dranishnikov). Let F be a continuous monomorphic functor of finite degree η preserving finite intersections, where F{n) and F(0) are finite-dimensional ANR-compacta. If X G ANR and F(\), X e Ck, then F(X) G Ck for each k. Dranishnikov has strengthened this result by showing that under the assumptions of Theorem 5.2 the functor F preserves LCfc-compacta, and when the condition F(\) G Ck is added, then F preserves AE(A:+ l)-compacta. Theorem 5.3 (Basmanov). Under the assumptions of Theorem 5.2, if Χ Ε ANR, X G Ck, and F(Sk+1) ζ Chi\ then F{X) 6 Ch+1. This theorem generalizes Theorem 5.1. From Theorem 5.2 it follows that the functors />„, ехр„, SPg, λ η , and *λη take ANR-bicompacta that are connected in dimension к into spaces that are connected in dimension к. Theorem 5.3 implies that the functors Pn, Х„, and *λη, where η > 3, raise the connectivity of ANR-bicompacta by 1. A recurrent problem is, on the one hand, the search for conditions on abstract functors that would be sufficient for the functor to raise the connectivity of a space by more than 1. On the other hand, not less interesting for future applications is the problem of computing the connectivity of spaces of the form F(Sk) for specific functors F.


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Question 5.1. The condition F(Sh+1) 6 C' i + 1 in Theorem 5.3 is necessary for the functor F to take the class of all Ck-compacta into that of all C'l+1-compacta. But is it necessary for the functor F to take any compactum Χ ξ Ch \ Ch+1 into a Ch+1-compactuml Question 5.2. Can a functor of finite degree take spaces that are not connected in some dimension into absolute retracts (or into C°°-compacta)r>. §6. The non-compact case 1. To begin with we recall that the Wazewski-Vietoris and Wojdyslawski theorems were carried over to complete metric spaces by Tashmetov in [29]. Curtis proved that if X is a metric space, then the hyperspace exp X is homeomorphic to Q \ {a point} if and only if X is connected, locally connected, and locally compact, but not compact [53]. He even proved that exp X is homeomorphic to the Hilbert space l2 if and only if X is connected, locally connected, separable, topologically complete and nowhere locally compact [52]. His proof relies on the following remarkable characterization of Hilbert space. Torunczyk's theorem [90]. Let X be a separable, topologically complete AR-space. Then X is homeomorphic to l2 if and only if it satisfies the following condition: ОС

For each map f: ^

A"; -»- X from a countable disjoint sum of finite

polyhedra and for each open cover 41 of X there is a map g: V, A",- -*• X such that f and g are ^l-near and the family {g(Kj)} is discrete. This criteria of Torunczyk was used by Golov [5] to study the hyperspace of ordered arcs of complete metric spaces (for the compact case, see §2). Theorem 6.1 (Golov [5]). The hyperspace F(X) is homeomorphic to the separable Hilbert space l2 if and only if X is a metric space that is connected, locally connected, separable, topologically complete, and nowhere locally compact. Apart from Torunczyk's criterion the proof uses Curtis' theorem (already mentioned in this section) about exp X being homeomorphic to the Hilbert space /2 and the following analogue of Tashmetov's theorem for ordered arcs. Theorem 6.2 (Golov [5]). If X is a complete metric space, the following conditions are equivalent: \) X is connected and locally connected; 2) T(X) is connected and locally connected; 3) FC(X) is connected and locally connected; 4) Г(Х) is an absolute retract; 5) ГС(Х) is an absolute retract.

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Question 6.1. Is Theorem 6.1 true for the hyperspace Г с ? Ал interesting question is also to what extent the connectedness of X is essential in Theorem 6.1. In particular, we ask the following question. Question 6.2. Is it true that if X is a metric space that is locally connected, separable, topologically complete, and nowhere locally compact, then the hyperspace T(X) (and ГС(Х)) is homeomorphic to an l2-manifold1 Note that a positive answer to this question automatically gives a positive answer to the same question with separability of X replaced by local separability. A substantial contribution to the study of the interrelationship of functors and absolute retracts was made by Savchenko. In the compact case his results on the functor exp£ were already mentioned in §2. Theorem 6.3 (Savchenko [25]). If f:X -»• Υ is a map from a local, linearly connected (or connected and locally linearly connected) metric space X to a compactum Y, then the space (exp^/)" 1 {Y} is an absolute neighbourhood retract (or absolute retract)^. Corollary 6.1 (Savchenko [25]). Let f:X -+ Υ be a locally trivial fibration of metric spaces with fibres that are a locally linearly connected (or connected and locally connected) space. Then all fibres of exp£ f over locally connected compacta are absolute neighbourhood retracts (or absolute retracts). Theorem 6.4 (Savchenko [25]). / / / : X -*· Υ is a map from a metric space X onto a compactum Υ for which the inverse image of at least one point у Ε Υ is locally linearly connected (or connected and locally linearly connected), then the space (expc j)'x{Y) is an absolute neighbourhood retract (or absolute retract). Corollary 6.2 (Savchenko [25]). The functor exp c preserves the property that the fibres of maps between metric spaces are absolute (neighbourhood) retracts. Remark 6.1. Theorem 6.4 cannot be extended to the functor exp£, η > 2. As Savchenko has shown, even in the compact case the condition in Theorem 6.3 on X to be locally connected is essential. Thus, although the functor exp£ preserves according to Theorem 6.3 the property of being an A(N)R-space, it does not preserve the property that the fibres of a map are A(N)R-spaces. From Theorem 6.3 for the constant map / we obtain the following generalization of Tashmetov's theorem and consequently also of the Wazewski-Vietoris and Wojdyslawski theorems. and further on we assume that this space is not empty.


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Theorem 6.5 (Savchenko [25]). If X is a complete metric space, the following conditions are equivalent: 1) X is locally connected (or connected and locally connected); 2) exp£ X is locally connected (or connected and locally connected); 3) exp£ X is an absolute neighbourhood retract (or an absolute retract). Applying Torunczyk's criterion Savchenko obtained the following partial strengthening of Theorems 6.3 and 6.4. Theorem 6.6. If f:X -> Υ is a map from a separable topologically complete metric space X onto a compactum Υ such that the inverse image of at least one point у е Υ is a (connected) locally connected and nowhere locally compact space, then (e.\pc f)'1^} 's a n h-manifold (homeomorphic to the Hilbert space). Theorem 6.7. If f: X ->• Υ is a map from a separable topologically complete (connected) locally connected metric space X onto a compactum Υ such that the inverse image of at least one point у Ε Υ is nowhere locally compact, then (expc/)- 1 ^'} is an l2-manifold (homeomorphic to the Hilbert space). The reader can draw his own consequences of these theorems for the constant map. The example noted by Savchenko of the projection p2: h x Μ -+ Μ of the product of the Hilbert space by the graph of sin — with the limiting interval shows that the functor exp£ for η > 2 does not preserve the property that the fibre of a map is a separable Hilbert space (up to homeomorphism). Michael considered the hyperspace ехр°°ЛГ consisting of all A Ε exp X that have infinitely many connected components. He proved in [76] that if a Peano continuum X either is a finite graph or does not contain free arcs, then e x p T is homeomorphic to l2. 2. Questions about the mutual disposition of sets, in particular, the location of one set in another, is important for topology. Therefore, the behaviour of the mutual disposition of sets is a very urgent question under the action of functors on the space. Strong results in this direction were obtained by Zarichnyi. He introduced in [10] the concept of a quasinormal functor in the category of metric spaces. This concept is weaker than that of normality, in particular, it does not assume that functors preserve inverse images. The functors ехр„, SPQ, λ η and *λΛ are quasinormal. Zarichnyi studied the problem of whether the pairs (Q, s) and (Q, B(Q)) are preserved by functors, where s is the pseudo-interior of the Hilbert cube, ω ω that is, 5 = (0, 1) if Q = [0, 1 ] and B(Q) = Q\s is its pseudo-boundary. Note that one problem does not reduce to the other, since

F(Q) \ F(s) Φ F(Q 4 s).

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Theorem 6.8 (Zarichnyi [10]). Let F be a quasinormal functor such that F(Q) is homeomorphic to Q and one of the following two conditions is satisfied: a) F is a functor with finite supports; b) F is a functor with continuous supports. Then (F(Q), F(s)) % (Q, s), (F(Q), F(B(Q)) « (Q, B(Q)). Under the conditions of Theorem 6.8 the functor F preserves complete separable metric spaces, complete separable AR(i)ft)-spaces, and σ-compact AR(i)31)-spaces. Zarichnyi also mentions that the pair (Q, B{Q)) is preserved by any quasinormal functor F that preserves the Hubert cube. We recall that a Σ-manifold is a metric space that is locally homeomorphic to B{Q). Theorem 6.9 (Zarichnyi [10]). If a quasinormal functor with continuous compact supports preserves compact Q-manifolds, then it preserves l2-manifolds and Σ-maniJbids. Question 6.3. Under the conditions of Theorem 6.9 the functor F preserves complete separable ANR-spaces and σ-compact ANR-spaces. Is this still true under the conditions of Theorem 6.8? Theorem 6.10 (Zarichnyi [10]). If a quasinormal functor with finite supports preserves the Hilbert cube and local connectedness, then it preserves complete separable ANR-spaces. Remark 6.2. Since a functor with finite supports is not necessarily of finite degree, Theorem 6.10 in the compact case does not reduce to Theorem 4.3 of Basmanov, but is a nice complement to it. But in contrast to Basmanov's theorem, where only intrinsic conditions are imposed on the functor, here it is assumed that the functor preserves local contractibility of spaces, which in practice is very difficult to verify. Remark 6.4. Theorem 6.10 implies that the functors ехр„, SPg, Х„, and *λη preserve complete separable ANR-spaces. But whereas the first two functors preserve non-compact Q-manifolds, Zarichnyi has shown that the functors λη and *Х„ always take non-compact spaces with non-isolated points into non-locally compact spaces. 3. A whole series of covariant functors carry compact spaces into noncompact spaces. Among the most important of these are functors FM, AM, FG, and AG of the free and free Abelian topological groups in the sense of Markov and Graev (see [6]) and the functor SP of Dold-Thom of the infinite symmetric power (see [54]). We recall that R°° (or Q°°) denotes the direct limit of the finite powers R' (or Q') under the natural embedding in one another. The spaces R°° and Q°° do not satisfy the first axiom of countability. They are homeomorphic to the σ-products of R and Q in their countable power, taken with the box topology. A space that is locally homeomorphic to R°° (or Q°°) is an R°°-manifold (or a Q°°-manifold).

Certain geometric pmperties of covariant functors

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Theorem 6.11 (Zarichnyi [96]). Let X be an ANR-compactum, 0 < dim X < °o and let e be a non-isolated point of X. Then the spaces FG(X), AG(X), and S?(X, e) are R-manifolds. Theorem 6.12 (Zarichnyi [96]). Let X be an ANR-compactum containing the Hilbert cube Q topologically and let e G Q С X. Then the spaces FG(X), AG(X), and SP(X, e) are Q°°-manifolds. With the assumptions of Theorems 6.11 and 6.12 the functors FG and SP carry homotopy equivalences into maps that are homotopic to homeomorphisms. Analogues of Theorems 6.11 and 6.12 and the subsequent remark also hold for the functors FM and AM of the free (Abelian) topological group in the sense of Markov. If the compactum X in Theorems 6.11 and 6.12 is an absolute retract, then FG{X) and SPX) the direct limit of the countable spectrum formed by the spaces εχρ,-Ζ (or λ,·(Χ)) and their natural embeddings in one another. We note that for an infinite space X the spaces exp^X and \„Х do not satisfy the first axiom of countability and consequently do not coincide topologically with the subspaces U {ex Pi X: i = 1, 2, . . .} - exp X and U {λ,(Α'): ί = 1, 2, . . .. }c= λ(λ'). Question 6.5 (Zarichnyi). Is it true that if X is a finite-dimensional nondiscrete ANR-compactum, then the spaces exp m X and λ^Χ are R"-manifolds? Question 6.6 (Zarichnyi). Is it true that if X is a compact Q-manifold, then the space X^X is a Q°°-manifold1 Koval, has considered yet another functor exp p deriving from the limit of p compact spaces. Let Qxp X denote the subspace of the continual exponential expc X consisting of all Peano subcontinua of X. Theorem 6.13 (Koval). // X is a Peano continuum, absolute retract.

then exp p X is an

4. Chigogidze considered the question [36] of the extension of normal functors from the category Comp of bicompact spaces to the category Tikh of all Tikhonov spaces. In the definition of normality the preservation of surjections must be replaced by the transfer of A:-covering maps into surjections. For every normal functor F: Comp -> Comp one defines a

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functor F$: Tikh -*• Tikh in the following way: = {x 6 FftX):

supp a r c X) and Ffi(f) = /-(β/)

The functor F0 also turns out to be normal. The definition implies that F$ is a functor with bicompact supports and its restriction Fp I Comp to Comp coincides with F. Furthermore, any normal function L : Tikh -> Tikh with bicompact supports can be obtained by the method indicated above provided that it takes the category Comp into itself. The countable power functor I d " : Tikh -»· Tikh is a functor without bicompact supports. Hence, I d " is not isomorphic to the functor (Id® | Comp) p == I dp. Chigogidze has proved that I d " is multiplicative. Thus, in the category Tikh there is a multiplicative but no power functor. In the category Comp this is an unsolved question. §7. The non-metrizable case In this section we touch upon that part of results on the theme of the paper concerning non-metrizable bicompacta. The majority of the results in §3 are also concerned with non-metrizable bicompacta. Schepin's spectral theorem on homeomorphisms [38] and the theorems on the adequacy of AE(«)-bicompacta and «-soft spaces (see §3) reduce the problem of the preservation of the class of AE(«)-bicompacta by a functor F to the following problem. Problem 7.1. Let / : X -*• Υ be an «-soft map between bicompacta. Is the map F(f) «-soft? In particular, to solve the question about the structure of bicompacta of the form F(DT) and F(IT) one must solve the question of the structure of the map F(pd, where px is the projection D° χ D° -> D°, or Ισ χ 1° -> 1° for σ < r. As a rule, the properties of these maps turn out to be bad. For example, all the known functors of finite degree, with the exception of the symmetric power functor SPg, do not preserve the openness of maps. As regards the functor SPg, while preserving the openness of maps, when G Φ {