Theory of retracts and infinite-dimensional manifolds

THEORY OF RETRACTS AND INFINITE-DIMENSIONAL MANIFOLDS S. A. Bogatyi and V. V. Fedorchuk

UDC 515.12

The survey is devoted to the theory of retracts and infinite-dimensional manifolds. The basic constructions and concepts are considered, and major attention is given to properties of mappings of softness type.

INTRODUCTION There exists a very broad class of topological problems known under the general name of extension (continuation) problems. The extension problem can be posed and formulated in a general categorical aspect. Here, depending on the category in which the extension problem is considered, the corresponding circle of questions and their solutions traditionally belong to different branches of mathematics. Thus, for example, the problem of extending a smooth [38] (Lipschitz [3]) mapping is traditionally studied within the framework of mathematical analysis. In order to indicate that part of the general extension problem to which the present survey is devoted, we make concrete the words "extension problem." Let ,7{' be some category, let A, X, Y be some of its objects, and let h: A + X, g: A § Y be morphisms in the category ~. In the extension problem a condition is sought under which there exists a morphism f: X § Y such that f 9 h = g. In other words, we seek conditions under which the diagram below can be closed (the dashed arrow) to a commutative diagram: g

A~Y

h-,,.47f X . Even in this form [diagrams of more complex form can be closed; for example in the theory of fiber spaces the dual diagram is closed (all arrows change direction)] the extension problem is too general, and the case of a nonmonomorphic h in the category of topological spaces pertains more to the theory of partitions, factor spaces, or the theory of multivalued mappings. In the case of an imbedding h the problem of extending g from h(A) to a mapping f on all of X decomposes into two qualitatively different steps. The first step consists in extending the mapping g from h(A) to [h(A)]. The possibility of this extension depends not so much on the topological properties of the spaces A, X, Y as on the uniform properties of the mapping g; therefore, questions of extending mappings from dense subsets traditionally belong to the theory of extensions (in particular, bicompact extensions) of topological spaces [44]. Assignment of an extension problem to some branch of mathematics is most often occasioned not so much by the form of this problem itself as by the methods invoked to solve it. Since any functor Y : j ~ - + ~ generates an extension problem in the category,, the lack of extension in the category ~ entails a lack of extension in the original category Jr. The problem of extending an individual mapping g is usually solved by construct ing a suitable functor F into some simpler category. Problems solved by means of functors into algebraic categories (the categories of groups, rings, modules) belong to algebraic topology and are usually considered within the framework of that science. Thus, properly the theory of retracts (extensors) studies the problem of extending a mapping g into a topological space Y from a closed subset A [we have already identified A and h(A)] to the entire space X. In such an "individual" form the extension problem in the category of topological spaces is complex and vast; therefore, its absolute fragments are usually considered. Let j~ be some class of spaces hereditary with respect to closed subsets. Definition i. It is said that a topological space Y is an absolute (neighborhood) extensor for the class j~ Y6AE(31p) (Y~ANE(~)) if for any space X ~ , any closed subset Translated from Itogi Nauki i Tekhniki, Seriya Algebra, 24, pp. 195-270, 1986.

372

0090-4104/89/4403-0372512.50

Topologiya,

Geometriya, Vol.

9 1989 Plenum Publishing Corporation

A ~ X of it, and any continuous mapping g: A § Y there exists a mapping f: X § Y (f: OA § Y, where OA is a neighborhood of the set A in X) such that flA = g" It is clear that in this definition it is possible to specify the class of subsets A considered, the class of extendable mappings g, and the class of mappings f among which an extension is sought, but we shall not consider such a general formulation, and results of this type will be specially formulated below. In addition to the category of topological spaces, the traditionally considered in the theory of retracts. In the going remark acquires major importance, and the two problems uous mapping g to a uniformly continuous mapping f and to a considered.

category of uniform spaces is case of uniform spaces the foreof extending a uniformly contincontinuous mapping f are usually

The shape category first constructed by Borsuk [500] has acquired great importance in the theory of retracts in the last fifteen years, and within the theory of retracts the theory of extension of a mapping of mappings [35, 759] has developed rapidly in the last five years. Since general categorical assertions in the theory of retracts are apparently still not sufficiently simple and clear, all definitions and assertions will be given specially for concrete catagories. The theory of retracts is connected with various areas of general topology, and itself uses their methods, while its ideas and results turn out to be necessary and useful in other areas of general topology. However, the recent solution by West [474, 476] of an old problem of Borsuk concerning the homotopy type of a compact absolute neighborhood retract and the large number of different functors naturally lead to absolute neighborhood retracts which are infinite-dimensional manifolds. In connection with the circumstances just mentioned, the authors have considered necessary a survey of the theory of retracts and infinite-dimensional manifolds within a unified framework. The survey contains results on absolute retracts, absolute extensors, and infinite-dimensional manifolds reviewed mainly in R Zh "Matematika" during the last 30 years, beginning in 1953. The limited size of the survey has not enabled the authors to illuminate all interesting problems connected with retracts and infinite-dimensional manifolds. Not even all important works are directly reflected in the survey. The authors have nonetheless tried to touch the key features of the theory. It should be mentioned that some questions concerning absolute retracts and infinite-dimensional manifolds found their way into the survey from different directions of topology [518, 524, 528]. For a systematic study of the questions considered in the survey the reader is referred to the monographs of Borsuk [499], HuSze-Tsen [263], Bessaga and Pelczynski [84], and Chapman [533]. The survey should be considered as a rather schematic guide to the direction of topology considered. It is possible to learn about special questions of the theory of retracts and infinite-dimensional manifolds by using the references to the bibliography in the corresponding section of the survey. i.

The Problem of Retraction and Extension

The problem of extending a mapping has an important special case which is almost always equivalent to the general problem and which explains the name "theory of retracts." By setting Y = A and assuming that g is the identity mapping Id of the space A, we obtain an extension problem which is called a retraction problem. If the mapping id admits an extension r: X § A, then the space A is called a retract of the space X, and the mapping r is called a retraction of the space X to the subspace A. The problem of extending the identity mapping Id is equivalent to the problem of extending a general mapping g in the following sense. Proposition i. A subspace A is a retract of a space X if and only if for any space Y each mapping g: A + Y admits extension to the space X. We shall be interested in the situation where the space X is floating. Definition i. It is said that a topological space Y is an absolute (neighborhood) tract in the class $~ F~A(N)N(JF) if Y62~ and for any homeomorphism h mapping Y onto a

re-

373

closed subset h(Y) of a space X of the class of the space X.

j~, the set h(Y) is a (neighborhood)

retract

The requirement that the set h(Y) be closed is imposed not so much from abstract considerations regarding uniform continuity of extendable mappings but because of the fact that any retract of a Hausdorff space is closed in this space [499]. However, it is sometimes useful to impose more stringent conditions on the way the set h(Y) is contained in the space X. In those cases where the class Jip is not written explicitly it is assumed to coincide with the class of spaces discussed in the corresponding assertion. The next result gives the obvious relation of absolute extensors and absolute retracts. THEOREM i.

The following inclusions hold:

a) A E (~) N O ' p A R (J~) ; b) A N E (a~) N J g _ _ A N R (a~). The two following standard techniques are presently used to prove the reverse inclusion. 1. We present the construction of an adjunction space for a given mapping g: A ~ Y defined on a closed subset A of the space X. In the disjoint union of the spaces XUY we identify points aEA and g(a)~Y. The factor space Z = XUgY is then the adjunction space. It is easy to see that the factor mapping p: XUY § Z generates an imbedding of the space Y in Z as a closed subset p(Y). The possibility of extending an individual mapping is described by Proposition 2. The subspace p(Y) of an adjunction space Z is a retract of Z if and only if the given mapping g: A + Y possesses an extension to all of X. Let us say that a class that also XU~FEYd.

3~

is invariant relative to pasting if from

X, F6Yf it follows

THEOREM 2. If a class Yd is invariant relative to pasting, then the following equalities hold: a) AE(Yd)NYd=AR(JiP); b) ANE(Jd~)fl~d=ANR(JF). THEOREM 3.

The following classes 3~ are invariant relative to pasting:

i) T1-spaces; 2) bicompact spaces; 3) Lindelof spaces; 4) normal spaces; 5) completely normal spaces; 6) collectively normal spaces. Invariance of many other classes of spaces is proved in the works 697, 698].

[263, 269, 271, 356,

From this it follows immediately that in classes of compacta, bicompacta, normal spaces, completely normal spaces, and many o t h e r s t h e concepts of an absolute (neighborhood) retract and an absolute (neighborhood) extensor coincide. 2. Since a retract of an absolute extensor for the class 3~ is itself an absolute extensor for the class Yd, the equalities A ( N ) R ( Y d ) = A ( N ) E ( Y d ) n Y d f o l l o w from theorems concerning the imbedding of spaces in A(N)E(Yd) -spaces. The most famous result in this direction is Urysohn's theorem. THEOREM 4. Any compact metric space can be imbedded in a closed manner in the Hilbert cube Q = Hi=z~Ii . THEOREM 5 (A. N. Tikhonov). Any bicompact space of weight T can be imbedded in a closed manner in a Tikhonov cube of weight T - I T . THEOREM 6. space.

Any complete metric space can be imbedded in a closed manner in some Banach

THEOREM 7 (Kuratowski-Wojdyslawski). Any metric space can be imbedded in a closed manner in a convex subset of a normed linear space. From this it follows immediately that although the class of metric spaces ~d'=~ is not invariant under pasting the equality A ( N ) R ( ~ ) = A ( N ) E ( ~ ) N ~ nevertheless holds as before. 374

2.

Operations

The property of being an AE(~)- and ANE(Jf)-space is not hereditary with respect to closed subsets, while the property of being an A E ( ~ ) -space is not hereditary with respect to open subsets. THEOREM i.

If 'YEANE(~I:) and U is an open subset of Y, then U ~ A N E ( ~ ) .

The theorem formulated shows that any ANE(Ji:) -space is also locally an ANE(J~ ~ -space. The question of the validity of the converse assertion is more complex and interesting. Sum theorems are now known under very different assumptions and go back to the works of Borsuk and Hanner. In Theorems 2 and 3 we assume that J : ~ - the class of normal spaces. THEOREM 2. If Y = YIUY~, where Yl and Y2 are open subsets of the space Y, and YI, ~ 6 A N E ( ~ ) then also Y~ANE(o~). THEOREM 3. I f Y = Y1UY2, w h e r e Yt and Y2 a r e open s u b s e t s YinY26AE(2:) then also Y ~AE(~). also

I f t h e c l a s s J{ c o n s i s t s for closed factors.

of hereditarily

normal spaces,

o f t h e s p a c e Y, and Y1, Y2,

then analogous theorems are true

THEOREM 4. I f Y = Y1UY2, where Y1 and Y2 a r e c l o s e d s u b s e t s A(N)E(~) t h e n Y 6 A ( N ) E ( ~ ) i f and o n l y i f YI, Y 2 6 A ( N ) E ( ~ ) .

o f t h e s p a c e Y, and Yln Y26

E q u i v a l e n c e o f t h e p r o p e r t y o f b e i n g a l o c a l and g l o b a l n e i g h b o r h o o d e x t e n s o r can be o b t a i n e d by means o f a n a l o g o u s sum t h e o r e m s w i t h r e s p e c t t o l o c a l l y f i n i t e c o v e r i n g s [ 2 4 3 , 244, 2 6 3 ] . G e n e r a l i z a t i o n s o f t h e u n i o n t h e o r e m s a r e p a s t i n g t h e o r e m s w h i c h go back t o B o r s u k [499] in t h e f i n i t e - d i m e n s i o n a l c a s e and t o ' W h i t e h e a d [499] i n t h e c a s e o f g e n e r a l c o m p a c t s p a c e s . THEOREM 5. I f X, A, Y a r e A ( N ) R - c o m p a c t a and ~ : j u n c t i o n s p a c e Z = XU~Y i s an A(N)R-compactum.

A § Y is a c o n t i n u o u s mapping, the ad-

Since the class of metric spaces ~ is not invariant under pasting, a direct proof of an a n a l o g o u s t h e o r e m f o r m e t r i c s p a c e s i m p o s e s a d d i t i o n a l c o n d i t i o n s on t h e mapping 9. Hyman [641] i n t r o d u c e d a new c l a s s o f s p a c e s - M - s p a c e s - w h i c h i s i n v a r i a n t u n d e r p a s t i n g and c o n t a i n s a l l m e t r i c s p a c e s ; t h i s made i t p o s s i b l e t o g i v e an a n a l o g o f Theorem 5 f o r a c l a s s even b r o a d e r t h a n t h e c l a s s o f a l l m e t r i c s p a c e s . P a s t i n g theorems are of importance in p r i n ciple, since they make it possible to obtain directly the assertion that CW-complexes in the Whitehead topology are absolute neighborhood extensors for metric spaces and for CW-complexes themselves. One of the simplest operations on spaces is taking the product. THEOREM 6. ~ Y ~ 6 A E ( ~ )

if and only if Y=,6AE(2g) for each ~.

THEOREM 7. ~ ni=1 yiEANE($~o)

if and only if Yi6ANE(J~)

for each i.

Since a closed subset of a bicompact product has a cofinal system of cylindrical neighborhoods, for bicompacta we have THEOREM 8. A product of bicompacta HYa is an absolute neighborhood extensor if and only if almost all factors are absolute extensors, while the finite number of factors remaining are absolute neighborhood extensors. We note also that since the factors are retracts of the product, they cannot be "worse" than the entire product. Category theory makes it possible to view different particular operations on a space from the viewpoint of the theory of functors. In recent works many previous results of this direction were included in a general functorial formulation. We shall list results on preservation of A(N)R-compacta by concrete covariant functors. For this we indicate a list of authors, functors, and classes of spaces (if they are not ANR-spaces) transformable into ANR-compacta. The fiber version we call theorems on preserving properties of fibers of a mapping being ANR-spaces. i) Borsuk [499], Idk; 2) Wojdyslawski

[499], exp and exp c, Peano continua;

375

3) Ganea [212, 213], exPn finite-dimensional ANR-comapcta; 4) Floyd [212, 213], SPGn, finite-dimensional ANR-compacta; 5) Jaworowski [279] for exPn and [281] for SPGn; 6) V. V. Fedorchuk [48], locally convex subfunctors F c Pn; 7) van Mill [369] and A. V. Ivanov [509], A, metric continua; 8) V. V. Fedorchuk [48],exPn , SPGn, locally convex subfunctors F c Pn, fiber version; 9) Eberhart, Nadler, and Nowell [197], F and F c, Peano continua; i0) V. I. Golov [501], F and F c, Peano continua, fiber version; Ii) M. M. Zarichnyi [18], An and *An; 12) A. G. Savchenko [525], expn c, Peano continua, fiber version. It should also be noted that under minimal conditions on a functor it takes contractible spaces into contractible spaces. Therefore, the results listed are also theorems on preservation of absolute retracts. Basmanov [490] obtained very general results on the preservation of ANR-compact by functors of finite degree. They generalize all the results mentioned above pertaining to functors of finite degree. The main theorem of V. N. Basmanov can be formulated very simply. If F is a continuous monomorphic functor of finite degree n preserving intersections, and the spaces F(~) and F(n) are finite-dimensional ANR-compacta, then F preserves ANR-compacta. Cauty, Tashmetov, Curtis, Nguyen To Nku, Zarichnyi, Golov, Savchenko, et al., have investigated these questions in the class of (separable) metric spaces [19, 46, 502, 526, 570, 707] and obtained a number of interesting results regarding preservation of ANR-spaces by functors. In the nonmetrizable case the situation is considerably worse. M. M. Zarichnyi proved that if a normal functor preserves AR-bicompacta of we&ght e~2, then it is a power functor. Among nonnormal functors the functor of superextension A and its modifications preserve ARbicompacta (Ivanov [22, 23]). This circumstance is occasioned by the fact that, while preserving good properties of spaces, functors almost never preserve good properties (of the type of n-softness) of mappings even of metrizable compacta. Zero-softness of mappings of compacta, coinciding with openness, is preserved by many functors. As concerns one-softness, it is almost never preserved. M. V. Smurov obtained a strong result in this direction. He proved that if for a finite normal functor F there is a mapping f: X ~ Y of compacta for which the fiber f-1(y) over some nonisolated point y ~ Y is infinite, while F(f) is locally one-soft, then F = Id n. A. G. Savchenko augmented this theorem by proving that any normal functor taking soft mappings of compacta into one-soft mappings is a power functor. Normality of the functor F in the thoerem of M. V. Smurov is essential, since there is the following parametric version of the theorem of Vazhevskii, Vietoris, and Wojdyslawski. THEOREM 9 (Fedorchuk [49, 208]). following conditions are equivalent:

For a mapping f: X ~ Y between Peano continua the

i) f is one-soft and exp c f is open; 2) exp c f is one-soft; 3) exp c f is soft. It should hereby be mentioned that the mapping exp c f is very rarely open. Fedorchuk proved that if X and Y are nondegenerate Peano continua and Px: X • Y § X is a projection, then the mapping exp c PX is open if and only if the continuum X is a dendron. Questions of when functors preserve global geometric properlies of the type of m-connectivity have also been studied. Zarichnyi proved [508] that for any ANR-bicompactum X the spaces An(X) and *An(X) are simply connected for each n e 3. Basmanov, extending the basic geometric of idea of Zarichnyi, obtained a considerably more general result [5]. He proved that if F is a continuous, monomorphic functor of finite degree n preserving intersections and the empty set, F(1) is linearly connected, and F(n) is an ANR-compactum, then F(S I) simply connected implies F(X) simply connected for any connected ANR-bicompactum X. 376

Later V. N. Basmanov and A. N. Dranishnikov extended this assertion to connectivity in higher dimensions. In general and especially homotopic topology the space of continuous mappings yX = C(X, Y), which can be equipped with very different topologies, is of major importance. The extensor properties of the space C(X, Y) can be investigated by means of the following two methods. The "exponential" law yX• = (yX)Z makes it possible to replace mappings of the space Z into the space C(X, Y) by mappings of the space X • Z into the space Y, which in the case Z c T and Y 6 A {N)E({XXT}) makes it possible to extend mappings from Z into C(X, Y) to T. The second method consists in constructing on the space C(X, Y) various convexity structures on the basis of convexity structures on the space Y. In this way Borges and Cauty [563, 569] proved a theorem to the effect that if X is a compact space and Y is an absolute neighborhood extensor for the class of stratifiable spaces (which includes Hyman's class of M~spaces, and hence the classes of CW-spaces and metric spaces), then C(X, Y) in any admissible topology is an absolute neighborhood extensor for the class of stratifiable spaces. Since the space of mappings of CW-complexes is not a CW-complex, then, as shown in Milnor's work [695], just the theorem presently formulated (or weaker analogs of it) and the fact that any A N R ( ~ ) - s p a c e has the homotopy type of a CW-complex make it possible in homotopic topology to avoid many difficulties arising in the study, for example, of the loop space of a CW-complex. Many other results on operations in the theory of extensors are contained in [82, 88, ii0, 114, 129, 156, 159-164, 192, 193, 195-197, 212, 213, 219-221, 231-233, 265]. 3__~. Absolute Extensors in the Class of Metrizable Spaces The theory of extensors has been constructed most completely and deeply in the category of metric spaces ~ = ~ . Since within the framework of this survey it is not possible to present all important and interesting results on metric retracts, we consider only those criteria of membership in A R ( ~ ) which have a large number of applications in the solution of various problems and make it possible to illuminate the question of membership in AR('~) from maximally different viewpoints. Let K denote a simplicial complex in the Whitehead topology, and let L be a subcomplex of it which contains all vertices of K. For a covering a = {UI: ~ 6 A} of the space Y a partial a-realization is a mapping g: L § Y such that for any closed simplex o of K there exists an index ~ 6 A s u c h that g(oNL) ~ U~. In the case L = K we speak of a complete ~-realization Various characterizations of A N R ( ~ )-spaces in terms of "fine" extensions and "fine" homo ~ topics go back to Lefschetz, Dugundji, Hanner, and He Sze-Tsen [263, 499]. THEOREM i. A metric space Y is an ANR-space if and only if for any open covering a of the space Y there exists an open covering ~ inscribed in a such that any partial ~-realization extends to a complete a-realization. It is said that a space X a-dominates a space Y, where a is a covering of Y, if there exist mappings

~" X-~Y andS): Y-+X, such that the mappings Idy and ~ o ~ are a-homotopic, i.e., thereexists a h o m o t o p y H: Y x i * Y s u c h t h a t H ( y , 0) = y , H ( y , 1) = ~ ( ~ ( y ) ) a n d f o r e a c h p o i n t y ,6 Y t h e r e e x i s t s ~ ~ A s u c h t h a t H ( y , t ) 6 U% f o r a l l t E [ 0 , 1 ] . THEOREM 2. A m e t r i c s p a c e Y i s an A N R - s p a c e i f a n d o n l y i f f o r a n y o p e n c o v e r i n g ~ o f the space Y there exists a simplicial c o m p l e x i n t h e W h i t e h e a d t o p o l o g y w h i c h ~ - d o m i n a t e s Y.

There are many beautiful and powerful criteria of absolute extensors for finite-dimensional spaces. The theorem of Kuratowski-Dugundji is the kernel of a crystallization of such results [186, 263, 499]. A space Y is called an Lcn-space iffor anypoint y 6 Y and any neighborhood U of it there exists a neighborhood V such that a mapping of the k-dimensional sphere S k, where k ~ n, g: S k + V, admits extension to the entire ball B k+l into U: f: B k+1 § U. The space Y is called a cn-space if any mapping g: S k § Y, where k ~ n, admits extension to the entire ball B k+l f: B k+l § Y. 377

THEOREM

3.

For a metric

space Y the following

conditions

are equivalent:

i) Y 6 LC n. 2) If a mapping g: A § Y is defined on a closed subset A of a metric space X with dim (X\A) ~ n + I, then g can be extended to some open neighborhood OA of the set A in X. 3) If a mapping g: A + Y is defined on a closed subset A of a metric space X with dim (X\A) ~ n + i, then g can be extended to an open neighborhood OA of the set A in X such that dim (X\OA) ~ n. 4) For any point y 6 Y and any neighborhood U of it there exists a smaller neighborhood V ! U such that a mapping g: A ~ V defined on a closed subset A of a metric space X with dim (X\A) ~ n + i possesses an extension f: X + U. 5) For any point y 6 Y and any neighborhood U of it there exists a smaller neighborhood V ~ U such that any mapping g: X § V of a metric space X with dim X ~ n is homotopic in U to a constant mapping. 6) If Y is a closed subspace neighborhood retract of Z.

of a metric

space Z with dim (Z\Y) ~ n + i, then Y is a

7) For any open covering = of the space Y there exists an open covering B such that any partial B-realization admits extension to a full ~-realization under the condition that in the simplicial complex K all simplices have dimension ~n + i. 8) For any open covering ~ of the space Y there exists an open covering B inscribed in ~ such that two B-close mappings f, g: X § Y defined on a metric space X with dim X ~ n are ~-homotopic. 9) For any open covering ~ of the space Y there exist a simplicial complex K of dimension ~n and a mapping ~: K § Y such that for any mapping f: X + Y defined on a metric space X with dim X ~ n there exists a mapping ~: X § K such that the mappings f and ~ o ~ are ~homotopic. We present only the following two theorems a mapping to a finite-dimensional complement.

from global versions

of the extension

of

THEOREM 4. A metric space Y is a LC n- and Cn-space if and only if every mapping g: A + Y defined on a closed subset A of a metric space X with dim (X\A) ~ n + i has an extension to all of X. THEOREM 5. If a metric space Y is an LC p- and cq-space, then for any mapping g: A + Y defined on a closed subset A of a metric space X with dim (X\A) ~ n + i there exist two sets Fl, F 2 ~ X\A such that dim F I ~ n - p - i, dim F 2 ~ n - q - I, the set F I is closed in X\A, the set F 2 is closed in X, and an extension f: X\(FIUF 2) + Y is defined~ The result formulated goes back to Eilenberg's theorem on extension of mappings into the sphere and was subsequently strengthened by a number of authors [62, 493, 516, 540]. Having considered in metric space X an arbitrary closed set A, by Theorem 5 we obtain (Y = A, p = q = -i) a retraction onto a set A of the complement to some "meager" set. The presence of the corresponding retractions was first obtained by Levshenko and Smirnov [514, 515], and in [29] this served as the basis for an inductive definition of dimension. Engelking [606] proved that in a zero-dimensional space a full retraction of the space onto a closed subset can be taken in the class of closed mappings. The next theorem [499], which in many questions also makes it possible to simplify the verification of the existence of a global extension, gives a connection of the class of absolute extensors and the class of absolute neighborhood extensors. A space Y is called contractible (Y ~ C) if there exists a homotopy H: u • I ~ Y such that HIy• = Idy and HIy• ) is the trivial mapping. THEOREM 6.

1)

For a metric

space Y the following

conditions

are equivalent:

Y6AE(~),

2) YeANE(~)

and Y e, Cn f o r a l l

3) Y~ANE(~)

and Y o C.

n e 0,

A space Y is called locally contractible (Y 6 LC) is for every point y 6 Y and every neighborhood U of it there exists a smaller neighborhood V ~ U such that the inscribed im-

378

bedding is homotopically trivial. Since Y6ANE(~) implies the ihclusion Y6ANE({YXI}) every A N E ( ~ ) -space is locally contractible. The converse is true for finite-dimensional spaces [499]. THEOREM 7. i)

For a metric space Y with dim Y ~ n the following conditions are equivalent:

Y6A(N)E(~),

2) Y~LC~flC~(LC~), 3) Y6LCflC(LC). Already at the level of compacta Borsuk [499] constructed examples of LC ~- but not LCcompactaandof LC-, but not ANR-compacta. A detailed investigation of the question of extendability of mappings into LC ~- and LC-spaces is carried out in the works [32] and [487, 627], respectively. The concept of local contractibility can be strengthened to contractibility of a smaller neighborhood in itself and to "consistency" of contractions of different neighborhoods at different points [188, 637]. One of the simplest concepts is that of local equiconnectedness - ELC. Definition i. Y ~ ELC if there exists a mapping of the diagonal A in Y • Y, such that

H:O•

where O is a neighborhood

a) H(yl, Y2, 0) = Yl, b) H(yl, y~, i) = Y2 and

c) H(y, y, t) = y for all Yl, Y2, y 6 Y and t 6 I. Any A N E ( ~ ) - s p a c e is an ELC-space, but the question of the validity of the converse assertion remains open. The concept of a (locally) hyperconnected space is obtained by con ~ tinuous joining not of a pair of (nearby) points by segments but of collections of m (nearby) points by (m - l)-dimensional simplices. The presence of (local) hyperconnectedness is equivalent to A(N)E(~) and is useful, for example, in the study of a space of mappings (Sec~ 2). The Kuratowski-Wojdyslawski theorem (Sec. I) on the imbedding of a metric space in a convex subset of a normed vector space makes it possible by means of retraction to ~'transfer" convex structures to A(N)E(~)-spaces. On the other hand, the presence of many types of convex structures [569, 690] implies A ( N ) E ( ~ ) , which gives "topologized" versions of Dugundji's theorem on the extension of mappings into vector spaces (Sec. 6). Moreover, the imbedding noted shows that any metric space can be imbedded in a closed manner in a metric AE(~)-space. Many other results on imbedding into a space which is ~'good" from the viewpoint of extending mappings are contained in [726]. The problem not only of the extension of a mapping defined on a closed subset but also of the extension of a metric defined on a closed subset is important in the class of metrizable spaces [75, 708, 709, 737]. Various questions of t h e t h e o r y of extensors in the class of metric spaces are also considered in the works [238, 270, 295, 296, 298, 315, 398, 429]. 4.

Absolute Extensors in the Class of Bicompacta

A detailed investigation of absolute extensors in the class of nonmetrizable bicompacta began with the work of Haydon [629]. He proved that the class of Dugundji spaces coincides with the class of AE(0)-bicompacta (absolute extensors in dimension 0 for the class of bicompacta). Dugundji spaces were defined by Pelczynski [721] as bicompacta which under any imbedding into an enveloping bicompactum admit a regular operator of extension of functions. Here a linear operator u: C(A) + C(X) of extension of functions is called regular if it takes nonnegative functions into nonnegative functions and one into one. Thus, the problem of extending mappings from closed subsets of zero-dimensional bicompacta is closely related to the problem of simultaneous extension of functions. The implication "an AE(0)-bicompactum is a Dugundji space" is based on Milutin's lemma on covering a segment by a Cantor perfect set by means of a mapping admitting a regular op-

379

erator of averaging of functions. following theorem.

In verifying the reverse implication Haydon proved the

Any Dugundji space can be represented as the limit of the inverse, completely ordered spectrum of bicompacta

satisfying the conditions: a) the bicompactum X 0 is metrizable; b) for any limit ordinal 7 < 9 the natural projection from X7 to the limit of the spec ~ trum $~]~={X~, ~, ~ < ? } is a homeomorphism; c) all short projections ~ ~+i are open mappings with a metrizable kernel. Here a mapping f: X § Y is called metrizable or is said to have a metrizable kernel if there exist a metric space M and an imbedding X ! M • Y such that f = PYIX, where py: M • Y + Y is a projection. For mappings of bicompacta, instead of considering an arbitrary metric space M, it is possible to restrict attention to the Hilbert cube Q. Inverse spectra satisfying condition b) have come to be called continuous. Spectra satisfying all the conditions of this theorem are called Haydon spectra. Haydon's theorem served as the starting point for further investigations in this direc ~ tion. Shchepin defined [58] the concept of an n-soft mapping as a mapping which satisfies the condition of lifting mappings of n-dimensional paracompacta. Here a mapping f: X § Y (locally) satisfies the condition of lifting mappings of the pair (A, B) if any commutative diagram B

~X

A"

- u

can be completed by a diagonal mapping ~: A ~ X {~: OB + X, where OB is a neighborhood of the set B). In this definition for mappings f: X § Y of bicompacta it suffices to restrict attention to bicompact pairs (A, B). The concept of a (locally) n-soft mapping is equivalent to the concept of an absolute (neighborhood) extensor in dimension n for the category of mappings. A space Y iS an absolute (neighborhood) extensor in dimension n if and only if a constant mapping of it is (locally) n-soft. If a mapping f: X § Y is (locally) n-soft for any n ~ 0, then it is called infinitely (locally) soft. Mappings for which the softness condition is satisfied for all pairs without restriction on their dimension are called absolutely soft or simply soft. From Ferry's results [211] it follows that absolutely soft mapping, betweenANR-compacta are precisely Hurewicz fiberings all fibers of which are absolute retracts. Since any bicompactum is continuously zero-dimensional, it follows that any 0-soft mapping is surjective. From Michael's selection theorem it follows that a necessary and sufficient condition for local n-softness of an open mapping f: X + Y of bicompacta with a metrizable kernel is uniform local (n - l)-connectedness of the system of fibers f-(y). In particu lar, for a mapping of bicompacta with a metrizable kernel local 0-softness coincides with openness. Mappings whose systems of fibers are uniformly locally n-connected are known as homotopic n-regular mappings in the sense of Curtis [165]. Shchepin introduced [58] the concept of adequacy of a class of spaces G to a class of mappings JK. Adequacy consists of two conditions: completeness and representability. The condition of completeness is that the limit of the inverse spectrum, consisting of spaces which belong to the class ~ and of mappings of the class J/, belongs to the class ~. The condition of representability is that any space X of the class ~ can be represented as the limit of the inverse spectrum consisting of spaces Y of the class G of weight wY < ~X and mappings of the space Jff. Variants of this definition suppose that on the spectra it is possible to impose various conditions, for example, complete ordering and continuity. Haydon's theorem can be treated as a theorem on the adequacy of the class of AE(0)-bicompacta to the class of 0-soft mappings. Shchepin proved that the class of absolute retracts is adequate to the class of absolutely soft mappings and posed the question of whether

380

the class of AE(n)-bicompacta is adequate to the class of n-soft mappings. For n = 1 a positive answer to this question was obtained by Nepomnyashchii [36] and Fedorchuk [49]. In the general case the problem was solved by Dranishnikov [503]. The main step in the solution of this problem was the following fargoing generalization of a theorem of R. Anderson to the effect that the Hilbert cube is an open (and even monotone) image of a one-dimensional compactum. For any natural number n there exists an n-soft mapping fn: ~n+~ + Q of a Menger universal (n + l)-dimension compactum onto the Hilbert cube. Fedorchuk extended the theorem of Dranishnikov on adequacy to a category of mappings in the following form. Any n-soft mapping between bicompacta decomposes into a continuous, spectrum with n-soft metrizable short projections.

completely ordered

For 0-soft and absolute soft mappings this result was obtained earlier by Shchepin. It is worth mentioning that in all results mentioned here completely ordered spectra with metrizable short projections can be replaced by sigma-spectra, i.e., by inverse spectra with metrizable projections over a sigma-complete directed set (a set in which any countable subset has a supremum). The absence in general bicompact spaces of a distance complicates or even renders untrue many arguments of the theory of ANR-compacta. Shchepin [58] introduced the class of spaces in which a distance is defined from a point to a canonically closed set %-metrizable spaces. The class of ~-metrizable bicompacta turned out to be convenient for problems of the theory of extensors - m a n y results untrue in the case of general bicompacta carry over from compacta to ~-metrizable bicompacta [530]. It is necessary also to note specially the result of Shchepin [59] that any finite-dimensional ANR-bicompactum is metrizable. Interesting and important results on bicompact absolute (neighborhood) extensors are contained in [85, 99, 102, 106, 153, 190, 203, 227, 265, 499]. 5.

Extension of Mappings into a Metric Space

Although nothing close to the theory of A(N)E(J~) -spaces has been constructed when the class Yf goes beyond the framework of metric or bicomDact spaces, it turns out that for any sufficiently broad classes ~ the spaces of A(N)E(/f)nJf have a rather simple description. The theorems below go back to Harmer and Michael of considerable attention. THEOREM i.

[263] and were subsequently the object

For a metric space Y the following conditions are equivalent:

a) Y~A(N)E(~)

and the weight of Y ~ ~;

b) f 6 A ( N ) E ( ~ ) where ~ is the class of all perfectly normal spaces, and 9 is the the class of all collectively normal spaces. THEOREM 2.

For a metric space Y the following conditions are equivalent:

a) Y6A(N)E(~), b) F~A(N)E(~)

Y is topologically complete, and the weight of Y ~ T; where

~

is the class of all T-collectively normal spaces.

We formulate specially a result which does not fit into the "absolute" framework~ THEOREM 3. Any mapping from a bicompact subset A of a completely regular space X into a metric A(N)E(~) space Y can be extended to the entire space X (to a neighborhood of A). The theorems presented are proved by means of extension of coverings and pseudometrics defined on the closed subset A of the space X (in some class J s The problem formulated has major and independent interest. Definition i. A subset A (not necessarily closed) of a space X is called C-imbedded (respectively, C*-imbedded) if any continuous function F: A § R (respectively, f: A § I) extends continuously to all of X. Definition 2. A subset A of a space X is called P~-imbedded if any continuous pseudometric defined on A of weight ~z extends continuously to X.

381

The :connection of the manner a set is situated with the possibility of extending from it mappings into metric spaces is investigated in the works [75, 541, 726, 737]. We shall present only those assertions which clarify the foregoing theorems. THEOREM 4. are equivalent:

For a cardinal T and a subset A of the space X the following conditions

i) A is Pt-imbedded in X; 2) any continuous mapping g: A § Y, where Y is a complete metrizable weight ~t, possesses a continuous extension to all of X;

AE(~)

-space of

3) there exists a noncompact metrizable AE(~) -space of weight t, such that any contin uous mapping g: A § Y possesses a continuous extension to all of X; 4) A • Z is Pt-imbedded in X • Z, where Z is an arbitrary bicompactum of weight ~t; 5) thereexists a bicompact space Z of weight t such that A • Z is C*-imbedded in X • Z. THEOREM 5.

A subset A of a space X is C-imbedded if and only if it is

p~0

-imbedded.

THEOREM 6. A Tl-space X is t-collectively normal if and only if any closed subset of it is Pt-imbedded. THEOREM 7. A completely regular space X is C*-imbedded in a bicompact extension bX if and only if the extension bX is the Stone-~zech extension SX. Since any (complete) metrix space can be imbedded in a closed manner in a (complete) metric AE(~) -space of the same weight (Sec. i), in the hypotheses of Theorem 4 it is possible to take an A N E ( ~ ) -space Y, but the extension of the mapping is obtained correspondingly on a neighborhood. Theorem 4 shows that in the definition of an A E ( ~ ) -space for a class Jip beyond the framework of metric and bicompact spaces it is natural to require extendability of mappings not from arbitrary closed subsets but from subsets better situated (for example, P-imbedded)

[536]. In homotopy theory (for example, in the tube lemma of Sec. 7) it is important to be able to extend mappings defined on a finite sum of rectangular subsets of a product. Such generalizations of parts 4) and 5) of Theorem 4 are contained in the works [743, 744]. THEOREM 8. If R is a closed subset of a t-collectively normal space X and B is a closed subset of a bicompactum Z of weight ~t, then A • ZUX • B is C*-imbedded in X • Z. The class ~p "combining" metric and bicompact spaces [31] also turned out to be natural for the theory of extensors. THEOREM 9. If Y is a metric space and the class of paracompact p-spaces.

Y 6A~N)E(~)

then

Y~A(N)E(~p)

where

~p is

Actually, many assertions for metric A(N)E(~)-spaces have presently been proved also for

A(N)R(~p )=A(N)E(~p)n~p-spaces.

We have already noted (Sec. 2) that a certain "theory" of A ( N ) E ( ~ ) -spaces has also been constructed for the class ~ of all M-spaces in the sense of Hyman [641]. In the works [560, 561, 569] a theory of A(N)E(~) oped for the class ~ of all stratifiable spaces.

-spaces has been quite deeply devel-

Since according to Tikhonov's theorem any bicompactum of weight t can be closedly imbedded in I t and the space I t, according to Theorem 2 of Sec. 6, is an absolute extensor for the class ~ of all normal spaces, it follows that the equalities A ( N ) R ( ~ ) = A ( N ) E ( ~ ) ~ = A(N)E(~)n~ hold; therefore, the question of the possibility of extending a mapping into a bicompact spaces is solved "at the level" of bicompact spaces. 6.

Concrete Spaces, Dimension

The most important and famous example of an absolute extensor is given by the TietzeUrysohn theorem which can be formulated in the following form. THEOREM i. The unit segment I = [0, i] is an absolute extensor for the space X (.~= {X}) if and only if the space X is normal.

382

The necessity of the inclusion

$s

is clarified by

proposition i. If the class ~ contains at least one nonnormal dorff A N E ( ~ ) -space is a one point space.

space, then any Haus-

From the Teitze-Urysohn theorem and Theorem 6 of Sec. 2 we automatically obtain THEOREM 2.

The Tikhonov cube I t is a AE(~) -space.

In order to present at least something from extensors outside the class of normal spaces, we note that a connected two-point space (Y = {0, i}, the closed subsets are the entire space Y, the point {0}, and the empty set) is an absolute extensor for the class of all topological spaces. The next example of an absolute extensor, which is of major importance in functional analysis, if given by Dugundji's thoerem [598]. THEOREM 3. then Y~AE (~).

If Y is a convex subset of some locally convex, linear topological space,

The majority of constructions of an extension of mappings from a closed subset of a metric space are based on the adjunction covering [499]. Borges [560] introduced a broader class of spaces ~ - stratifiable spaces - in which it is also possible to construct adjunction coverings and by means of them determine extensions of mappings. Since the adjunction covering is constructed not on the basis of the quadruplet (X, A, Y, g) but on the basis of the pair (X, A), extension of mappings can be determined by a canonical procedure for the consistency of structures in corresponding spaces of functions. To clarify our words we present such a formulation of Dugundji's theorem (but immediately for stratifiable spaces [560]). THEOREM 4. Let A be a closed subset of a stratifiable space X, let Y be a locally convex, linear topological space, and let C(X, Y) denote the linear space of continuous mappings from X into Y and similarly for C(A, Y). Then there exists a mapping u: C(A, Y) + C(X, Y) such that i) u(g) is an estension of g(u(g)[A = g) for any g 6 C(A, Y); 2) the image of u(g) is contained in the convex hull of the image of g; 3) u is a linear transformation. In Sec. 4 we have already discussed extension operators which preserve additional structure (regular operators). The presence of extension operators preserving some other structures, for example, the ordering in the spaces C(X) = C(X, R) and C*(X) [630-635] or the product in C+(X) [749], is connected with characteristics of various extensor properties. In Sec. 2 we already noted that pasting theorems make it possible to draw conclusions regarding extensor properties of CW-complexes. Moreover, the corresponding assertions are also true for metric topologies. THEOREM 5.

Any simplicial complex in a metric topology is an A N E ( ~ ) -space.

The next result is important in connection with Theorem 2 of Sec. 5. THEOREM 6. A simplicial complex in a metric topology is topologically complete if and only if it does not contain an infinite complete subcomplex. Theorem 1 has already been formulated in a form so that in it not so much the class ~=~ determines the space Y = I as the opposite. The next theorem of this type is due to Aleksandrov [524] and has basic character in the framework of general topology. THEOREM 7. The n-dimensional sphere S n is an absolute extensor for the space X (J~= {X}) if and 0nly if the space X is normal and dim X in. There is the individual version [52]: THEOREM 8.

n+l

A continuous mapping g: A + S n = FrI~[--l, 1]~

of a closed subset A of a

normal space X can be extended to all of X if and only if between the (n + l)-st pair of n+l

closed sets g-l(Bi-), g-1(Bi+) there are partitions C i closed in X such that

n C~=O

where

i=I

Bi g is the set of all points of the cube I n+l whose i-th coordinate is equal to ~.I.

383

The possibility of extending mappings into the sphere from a "factor" of the space X is investigated in the works [13, 42, 53]; this makes it possible to construct interesting examples of spaces having thickenings to bicompacta. The possibility of extending an individual mapping into the sphere S n from a subset of (n + l)-dimensional space is described by the famous theorem of Hopf in terms of properties of homomorphisms of the homology or cohomology groups; therefore, in correspondence with what was said in the introduction, usually Hopf's theorem and various strengthenings of it relative to the classification and extension of mappings of CW-complexes solvable in terms of homology or cohomology groups are considered in courses in algebraic topology. In the works [48g, 513, 640] the question of characterizing cohomological dimension by means of extension of mappings is investigated. An Eilenberg-MacLane complex K(~, n), where n e 1 and ~ is an Abelian group, is a CW-complex having trivial homotopy groups in all dimensions different from n and in dimension n having the group ~. According to Sec. 2, we have the inclusion K(~,n)~ANE(~). Up to homotopy type, and it will become evident in Sec. 7 that just the homotopy type is important, there is exactly one Eilenberg-MacLane complex K(~, n) which can moreover be constructed with better extensors and algebraic properties. THEOREM 9. An Eilenberg-MacLane tum X if and only if c dim~ X ~ n.

complex K(~, n) is an absolute extensor for a bicompac-

The assertion formulated is valid for a broader class of spaces, for example, for paracompacta, but in connection with Theorem 7 of Sec. 5 by definition it is possible to set c dim~ X = c dim~ ~X for a nonbicompact, completely regular space X. It is just this definition which is meant in the next theorem. Theorems 7 and 9 show that any theory of Lebesgue dimension dim and cohomological dimension c dim~ are "problems" of the extension of mappings. Usually, however, only those results and methods of dimension theory which are based on the extension of mappings and not on other characteristics of dimension are assigned to the proper theory of extensors. In the series of results of this type [506, 513] one of the first and fundamental results is the following theorem of Zarelua on factorization which in special cases has been proved by combinatorial-topological methods. THEOREM i0. Suppose there is given a family {X%} of power ~ of closed subsets of a normal space X, a family of mappings {G~} of power ~ of the space X into bicompacta Y~ of weight ~ , a family of transformations T~: X + X of power ~ , and numbers n~, o - the dimensions of the set X% with respect to countable or finite groups no; the power of the family {~o} is also assumed ~ . Then there exists a bicompactum B of weight ~ , a mapping G: X § B, mappings F~: B + Y~, and transformations UB: B + B such that I) F~ o G = G~; 2) US o G = G o T~; 3) dim B[GX~] = dim X~; 4) c dim~o B[GX~] = n~,~. The question of the possibility of replacing the sphere S n in Theorem 7 by a broader class of spaces Y is investigated in the works [348-350]. 7.

The Homotopy Category

Absolute (neighborhood) extensors in the homotopy category are considered in a number of works [467]. Since very often (J~=~, ~, ~ n ~ ) any space of the class Jip can be closedly imbedded in an AR(~)space, any homotopy absolute (neighborhood) retract for the class J~ is homotopically dominated (or even homotopically equivalent to [695]) by a A(N)R(J~)-space. This circumstance "solves" the problem of describing HA(N)R($~) -spaces, and we shall therefore not consider it in more detail. Considerably more important and interesting is the connection of problems of the homotopy and exact extension of a mapping, for example, into an A N E ( ~ ) space. The following tube lemma of Borsuk [499] is fundamental here. LEMMA i. If A is a closed subset of a normal space X, then for any neighborhood O of the set B = A x IUX x {0} in X x I there is a mapping ro: X x I + O such that ro(b) = b for all points b 6 B. From the lemma formulated we automically obtain THEOREM i. If g0, g1: A + Y are two homotopic mappings of a closed subset A of a normal space X into a ANE({%Xf}) -space Y, then the mappings go and gl either simultaneously extend to the entire space X or simultaneously do not extend to the entire space X. The theorem formulated shows that the possibility of extending a mapping g: A § Y into an ANE(J~) -space Y depends not on the topological properties of this mapping but on its 384

homotopy type. The problem that mappings go, g1: A + Y be homotopic is also the problem of the possibility of extending the mapping g = go U gz: B = A • {0} O A • {i} § Y to the en ~ tire space A • I. The investigation of homotopic dependence of mappings [89] is based on the facts noted. Since according to the works already cited regarding the space K(~, n) [489, 640] the set (moreover, the group) of homotopy classes of mappings X into K(~, n) is isomorphic to the cohomology group Hn(x, ~), the theory of cohomology groups is also a "part" of the theory of extension of mappings. The work [621] is based on such a homotopy approach. We note also that deformation and strong deformation retracts [387, 499] are also of major importance in various topological problems. A ~ X is called a deformation retract of X if there exists a retract r: X § A E X such that the mapping r and !dX are homotopic. In conclusion we note that various homotopy identities can be understood as extensor properties of some pairs of the spaces. 8.

The . Shape Category

The shape category was first constructed by Borsuk [500] for compacta by means of an imbedding into an AR-compact~m and consideration of a "consistent" sequence of mappings. Subsequently this construction was extended to a broader class of spaces. Mardesic and Segal [680] constructed a shape category by means of mappings of inverse spectra from ANR-bicompacta. Mardesic and Porter [678, 723] proposed constructions of shape categories by means of mappings into ANR-spaces. All that has been said shows that shape theory can be considered a theory of systems of ANE($~) -spaces and can thus be considered a part of the theory of extensors. However, within the framework of the shape category itself it is possible to consider the problem of extending morphisms and arrive at the concepts of fundamental absolute (neighborhood) retracts FA(N)R - in various classes of spaces. Here different geometric representations of morphisms in a shape category pose the new important problem of comparing the concept of "extension" in different models [104, 700, 701, 720].

-

Just as in the homotopy category (Sec. 7), FANR-compacta are precisely compacta which are shape-dominated by ANR-compacta (finite simplicial complexes), while FAR-compacta are precisely compacta which are shape-dominated by AR-compacta [500]. Therefore, we shall present only one theorem in which there are formulations new in principle [267, 496, 500, 573, 574, 677]. THEOREM i.

For a compactum Y the following conditions are equivalent:

i) Y ~ FAR; 2) Y is shape-dominated by some FAR-compactum; 3) Y has shape points; 4) any mapping of Y into an arbitrary ANR-compactum is homotopic to zero; 5) Y is a fundamental retract of the Hiibert cube Q; 6) Y is a movable compactum which is approximately connected in all dimensions; 7) Y is approximately connected with respect to the class of all polyhedra; 8) Y has a finite fundamental dimension and is approximately connected in all dimensions; 9) Y is the limit of the inverse spectrum of FAR-compacta; i0) if Y is the limit of an inverse sequence {Zn, pn n+1} of ANR-compacta, then there is a cofinal subsequence in which projections are homotopically trivial; !i) if Y is a Z-subset of the Hilbert cube Q, then Y is cellular in Q, i.e., there exists a sequence of Hilbert cubes Qn ~ Q.such that Qn+z ~ Int Qn and nn=1~Qn = Y; 12) Y is cellularly similar to (CE), i.e., Y can be cellularly imbedded in Q; 13) if Y is a Z-subset of Q, then Q/Y ~ Q; 14) if Y is a Z-subset of Q, then the Q-manifolds Q\Y and Q\{p} are homeomorphic where p is some point of Q;

385

15) if Y is a Z-subset of Q, then the Q-manifolds Q\Y and Q\{p} are properly homotopically equivalent; 16) if Y is a subset of an arbitrary ANR-compactum Z, then the factor space A/Y is also an ANR-compactum, and the projection p: Z + Z/Y is a homotopy equivalence; 17) if Y is a subset of an arbitrary ANR-compactum Z, then the factor space Z/Y is also an ANR-compactum, and the projection p: Z § Z/Y has a left homotopy inverse mapping. There are finite-dimensional versions of the theorem (under the condition dim Y ~ m the space Q is replaced by I#m+5). The equivalence of parts 3), 14), and 15) is a special case of the following theorem of Chapman [574]. THEOREM 2 . If compacta X and Y are Z-subsets of a Hilbert cube Q, then their shapes coincide (Sh X = Sh Y) if and only if their complements are homeomorphic (Q\X ~ Q\Y). The analogous fact for subsets of Hilbert space is untrue, since according to Anderson's theorem [545] any compactum Y is negligible in ~2, i.e., ~2\Y is homeomorphic to ~2- In the work [498] it is shown that the shapes of compacta in ~2 are characterized by their posi ~ tion (Pos). We have already noted that the theory of shapes can be "imbedded" in the theory of ex ~ tensors. Indeed, Chapman proved a theorem stronger than Theorem 2 which establishes an isomorphism of the shape category and a certain category of Q-manifolds, i.e., which "imbeds" shape theory in the theory of infinite-dimensional manifolds. Regarding the equivalence of parts 3), ii), 12), 13), and 16) it is necessary to observe that cellular and cell-like subsets have special importance in the investigation of factor spaces of finite-dimensional manifolds. So many works [550, 552, 660-662, 725] have been devoted to the partitioning of manifolds (even three-dimensional) into cellular and cell-like sets that within the framework of the present survey it is not possible for us to list all these works. We shall consider the basic questions. Rather interesting criteria for cellularity in terms of properties of the complement were obtained by McMillan already in 1964 (before the creation of shape theory). We shall present a stabilization theorem [688, 689]. THEOREM 3. If X and Y are compact, absolute retracts lying in the interiors of manifolds M m and N n, respectively, where m, n e 1 and m + n e 5, then X x y is cellular in M m x N n. back case Rn\~ ized

Investigation of topological properties of the space Rn/~, where ~ is some arc, goes to Bing-Andrews~urtis-Kwun-Bryant [550, 661]. This space is a manifold (and in this it is homeomorphic to R n) if and only if the arc ~ is cellular in R n if and only if is homeomorphic to S n-1 • R. The space Rn/~ is always an absolute retract and a generalmanifold and Rn/~ x Rm/~ ~ R n+m. There is an analogous version of the Q-theorem.

Cell-like mappings are of major importance for the theory of retracts and cohomological dimension. Definition i. A mapping f: X + Y of a compactum X onto a compactum Y is called a CEmapping if the full preimage f-Z(y) of any point y 6 Y is an FAR-compactum (cell-like). The majority of results on preservation the following theorem of Smale [342].

of properties under CE-mappings go back to

THEOREM 4. If f: X § Y is a CE-mapping of an ANR-compactum X onto Y, then Y E LC = and fn*: ~n( X, x0) § ~n [Y, f(x00] is an isomorphism for all n ~ i. Theorem 7 of Sec. 3 shows that if the compactum Y in the hypotheses of Smale's theorem is finite-dimensional, then it is an ANR-compactum and even an AR-compactum provided that X is such. Taylor [753] constructed an example of a CE-mapping of the Hilbert cube Q into a non-ANR-compactum Y. A construction of improving such mappings so that the preimages of points of the new mapping are even AR-compacta is proposed in the works [308, 371]. There is a homological "version" of Smale's theorem (proved in earlier works of Vietoris and Begel) to the effect that under the hypotheses of Theorem 4 the homomorphism of cohomologies fn#: Hn(y, ~) ~ Hn(x, ~) generated is an isomorphism. Dyer [599] proved that a CE-mapping cannot increase (on the image side) the cohomological dimension for any group of coefficients. However, the question of the existence of a CE-mapping f: X + Y, which is finite-dimensional in the sense of dim, of a compactum X onto a compactum Y of higher dimension (and

386

then, according to the theorems of Dyer and Aleksandrov the compactum Y will be infinitedimensional) remains open. Moreover, as Walsh showed [769], the existence of a dimensionraising CE-mapping is equivalent to the existence of an infinite-dimensional compactum with finite cohomological dimension with respect to the group of integers. As Aleksandrov showed [513], for finite-dimensional compacta in the sense of dim there is the equality dim = c" dim Z. The question of whether the requirement of finite-dimensionality is essential has been open already 50 years. Recently Dranishnikov solved another problem of Aleksandrov by constructing a compactum which is finite-dimensional with respect to one group and infinite-dimensional with respect to another. In the case of manifolds we shall present criteria of cellularity of a mapping due to Lacher, McMillen, and Seibenmann [321, 664-668, 738]. THEOREM 5. If f: M § N is a mapping of closed, compact m-manifolds and 2k + i e m e 5, then the following conditions are equivalent: i) The preimages f-Z(n) of all points n ~ N are approximately connected in all dimensions ~k~ 2) f is a CE-mapping. 3) The preimages f-Z(n) of all points n 6 N are cellular in M. 4) For any open subset U of the space N the mapping flf-Z(U): f-Z(U) + U is a proper homotopy equivalence. 5) The manifolds M and N are homeomorphic,

and f is the uniform limit of homeomorphisms.

A more detailed survey of results on CE-mappings can be found in the survey [667]; the infinite-dimensional case is discussed in Sec. 15. A theory of retracts of the category of morphisms of a shape category, i.e., shape analogs of the results of Sec. ii, has developed recently. Thus, I. E. Prokhorov showed that a mapping of continua is monotone if and only if its continuum exponent is a CE-mapping. On the whole, regarding many questions of FAR- and FANR-spaces, 567] can be recommended. 9.

the monographs

[500,

Uniform Spaces

The first sufficiently complete and systematic investigation of absolute (neighborhood) uniform retracts was carried out by Isbell [642] and goes back to the works of Katetov where a uniform analog of the Tietze-Urysohn theorem (Theorem 1 of Sec. 6) was given. We note at once that in questions of global extension of uniform mappings boundedness of the space is of major importance. We illustrate this with the uniform analog of Dugundji's theorem (Theorem 3 of Sec. 6 and Theorem 2 of Sec. 5). THEOREM i. The unit ball of the Banach space of all bounded, real functions on an arbitrary set is an absolute uniform extensor (in the category sense) for the class of all uniform spaces. Since any uniform space can be imbedded in a product of metric spaces, any metric is uniformly equivalent to a bounded metric, and a metric space M with a bounded metric can be imbedded isometrically in the Banach space of all bounded, real functions on the set M (Theorem 5 of Sec i), it follows that any uniform space can be imbedded in an absolute uniform extensor, whence it follows, in particular, that the extensor and retract concepts coincide in the uniform category. THEOREM 2. For any uniformly continuous, mapping g: A e U*(M. R) of a subset A (not necessarily closed) of a uniform space X into the Banach space U*(M, R) of all bounded, uniformly continuous, real-valued functions on an arbitrary metric space M there exist a uniform neighborhood OA and a uniformly continuous mapping f: OA + U*(M, R) such that fIA = g" In the theorem presented the space U*(M, R) can be replaced by an arbitrary uniformly convex Banach space or by an injective Banach space, i.e., by an absolute extensor in the category of Banach spaces and bounded linear operators. However, the question of the validity of Theorem 2 for an arbitrary Banach space remains open. In the works [10-12, 520] a uniform analog of the Kuratowski-Dugundji theorem (Theorems 3 and 4 of Sec. 3) is obtained from which it follows, in particular, that under the

387

additional condition of finite-dimensionality of X in the sense of the large uniform dimension A (AX < ~) in place of U*(M, R) it is possible to take a closed, convex subset Y of an arbitrary Banach space L, while under the condition that Y be bounded it is possible to assume that OA = X. As we have noted in the introduction, the problem of extension of a mapping in uniform spaces may have different (noncategorical) formulations. Thus, Michael [363] investigates extensions of f which are uniformly continuous only on the set A. The equivalence of the extensor and retract properties is proved, and criteria in terms of extending partial realizations are given (analogs of Theorem i of Sec. 3). i0.

G-Spaces

In this section properties of extensors in the category of spaces with continuous action of a (fixed) group G and equivariant continuous mappings are discussed. The development of the theory of G-extensors goes through basically in parallel to the theory of topological extensors and goes back in all probability to Gleason's work where an equivariant analog of the Tietze-Urysohn theorem was obtained (Theorem i of Sec. 6). The basic idea of obtaining an equivariant extension into a linear space consists in taking an arbitrary topological extension with subsequent averaging (with respect to Haar measure) over the group G. We shall present the equivariant analog of Theorem 4 of Sec. 6 [i, 2]. THEOREM i. If a bicompact group G acts on a complete, locally convex, linear space L as a group of linear transformations, then for any closed invariant subset A of a metric G-space X there exists a linear operator

u : Co(A, L)-+Co(X, L), satisfying the following conditions: i) u(g) is an extension of g for any g ~ CG(A, L); 2) the image of the mapping u(g) lies in the closed, convex hull of the image of the mapping g. Many functors defined on the category of topological spaces "withstand" the action of groups which makes it possible to carry over important constructions of general topology to the theory of G-spaces. Thus, for example, in the works [488, 527] a natural action of a group G on the space C(G, L) is considered which makes it possible by means of topological theorems on imbeddings (Sec. i) to obtain equivariant imbedding theorems (to "linearize" the action of the group G) and on the basis of analogs of Theorem 1 to construct a closed imbedding into a G-extensor and thus prove the equivalence of the G-retract and G-extensor properties, for example, for the classes j~=~, ~, ~p, ~n~Major attention is devoted to results which go back to the works of Jaworowski [280, 517, 646-650] on the reduction of the description of G-extensors to the description of topological properties of the set of all points which are fixed for a subgroup H ~ G. For a G-space Y and a subgroup H ~ G we set Y[H] = {y E y: hy = y for all elements h G H}. Then for an equivariant mapping f: X + Y and any subgroup H ~ G there is the inclusion f(X[H]) ~ Y[H] which "proves" one part of the following equivariant analog of the Kuratowski-Dugundji theorem (Theorems 3 and 4 of Sec. 3). Theorem 2. For a metric space Y with the acs ing conditions are equivalent:

of a finite Abelian group G the follow-

i) Y[H] E LC n (LC n n C n) for all subgroups H of the group G. 2) For any closed, equivariant subset A of a metric G-space X with dim (X\A) ~ n + 1 and any equivariant mapping g: A + Y there exist an equivariant neighborhood OA of the set A(OA = X) and an equivariant extension to this neighborhood f: OA + Y. In the case of more general groups positive results have been obtained under the assumption of orbit-type finiteness of the action. The theorems of imbedding spaces into uniform extensors and into G-extensors have recently made it possible to construct the theory of uniform and G-shapes, respectively.

388

ii. Iniective Objects with Respect to a Functor. Characterizations of Absolute Extensors and Soft Mappings by Means of Functors The concept of an injective object can be extended by introducing covariant functors into the category Top of topological spaces and some of its subcategories. We shall consider the compact case which has already been rather well studied. A bicompactum Y is called injective with respect to a covariant functor f :~-+~ acting in the category of bicompacta (briefly, F-injective) if for any mapping g: A + Y and any imbedding i: A § X there exists a mapping f: F(X) + F(Y) for which F(g) = f o F(i). Here the spaces A and X are also assumed to be bicompacta. It is clear that absolute extensors are precisely the injective objects with respect to the identity functor Id. Haydon's theorem on the coincidence of AE(0)-bicompacta and Dugundji spaces (Sec. 4) can be reformulated as follows: AE(0)-bicompacta are precisely the P-injective objects where P is the functor of probability measures. Wojdyslawski's theorem on hyperspaces of Peano continua (Sec. 2) augmented by a special case of the Kuratowski-Dugundji theorem [the class of Peano continua coincides with the class of AE(1)-compacta] can be reformulated in the following manner. For a compactum Y the following conditions are equivalent: i) Y is expC-injective; 2) Y is exp-injective; 3) Y 6 AE(1). G. M. Nepomnyashchii

and V. V. Fedorchuk extended this theorem to bicompacta

[49, 395,

531]. If we require of the bicompactum Y in the definition of F-injectiveness that the mapping g: A + Y extend only to a mapping f: X + F(Y), then we obtain the definition of a weakly F-injective space. For the monad functor, i.e., the functor F for which there is the natural transformation F o F § F which is a retraction, the concepts of injectiveness and weak injectiveness coincide. F-injective bicompacta are naturally called absolute extensors with respect to the functor F or AE(F)-bicompacta. AE(F)-bicompacta are precisely the absolute G-valued retracts, i.e., bicompacta Y for which for any imbedding Y c X there exists an Fvalued retraction X + F(Y). For the exponential functor F = exp an exp-valued retraction is called multivalued. If in the definition of an F-injective bicompactum we restrict ourselves to spaces X of dimension dim X ~ n, then we obtain the definition of an AE(n - F)-bicompactum. These concepts extend to the category of mappings in an ambiguous way even in the case of monad functors. it is said that a mapping f: X § Y is n-F-soft (respectively, strongly n-F-soft) if for any bicompactum Z of dimension dim Z ~ n, any closed subset A of it, and mappings g: A + X and h: Z ~ Y [respectively, h: Z + F(Y)] for which the composition f o g coincides with h on A there exists an extension k: (F)(Z) + F(X) of the mapping g such that F(f) o klz = h. If no restrictions on the dimension of the space Z are imposed, then we obtain the concept of an absolutely F-soft (respectively, absolutely strongly F-soft) mapping. If F ~ exp and of the mapping h: Z + F(Y) we assume only upper semicontinuity, then we obtain the concept of an n-F+-soft mapping. The analog of an absolute F-valued retract in the category of mappings is called an A(F)R-mapping. For the monad functor F and a mapping f the following conditions are equivalent: i) f ~ A(F)R; 2) f is absolutely F-soft; 3) f is absolutely strongly F-soft. There are the following characterizations THEOREM i [531].

of zero-soft and one-soft mappings.

For a mapping f the following conditions are equivalent:

a) f is zero-soft; b) f 6 A(P)R; 389

c) f is n-P-soft, n ~ 0; d) f is strongly n-P-soft, n e 0; e) f E A(expti)R; f) f is 0-expCsoft; g) f is 0-exp-soft; h) f is n-exp+-soft, n ~ 0; i) f is strongly n-exp+-soft, THEOREM 2 [531].

n e 0.

For a mapping f the following conditions are equivalent:

a) f is one-soft; b) f 6 A(exp)R; c) f ~ A(expC)R; d) f E A(expC+)R; e) fis n-exp-soft, n e i; f) f is strongly n-exp-soft, n e i; g) f is n-expC-soft, n e i; h) f is strongly n-expC-soft, n ~ i; i) f is n-expCi-soft,

n e i;

j) f is strongly n-expC+-soft,

n ~ I.

For arbitrary n Dranishnikov characterized [503] AE(n)-bicompacta by means of multivalued retractions of enveloping spaces. Fedorchuk extended these characterizations to n-soft mappings. lliadis [24a] considered the general concept of an absolute retract in a category. 12.

Selections and Factorization Theorems in the Theory of Extensors

Application of theorems on selections in the extension of mappings is based on the following simple argument. A given mapping g: A + Y is extended to some multivalued mapping Sm: X + Y from which a selection is then made which is the extension of the mapping g. At the present time theorems on selections (not necessarily continuous, for example, measurable) form a vast area of mathematical investigation [157, 202, 217, 245, 318, 319, 332, 345, 346, 364-366, 377, 414, 717]. The fundamental results on the existence of continuous selections were obtained by Michael [358-362] of whose results we present only one which is needed in augmentative definitions. THEOREM i.

For an arbitrary T1-space X the following conditions are equivalent:

I) X is paracompact; 2) for any Banach space L and any lower semicontinuous multivalued mapping ~-: X + L with nonempty closed and convex images of points there is a single-valued selection. Now for a continuous mapping g: A + L of a closed subset of a paracompact space X into a Banach space L the multivalued mapping@': X ~ L given by the formula

for

x~A,

satisfies the conditions of Theorem I (lower semicontinuity),and therefore has a single-valued selection f, which completes the proof of a somewhat weakened version of a combination of Theorem 2 of Sec. 5 and Theorem 3 of Sec. 6. The Kuratowski-Dugundji theorem (Theorems 3 and 4 of Sec. 3) can be obtained in a similar way by means of selection theorems, and, moreover, an analog of the Kuratowski-Dugundji theorem in the category of mappings (the description of n-soft mappings in Sec. 4) can be obtained precisely on the basis of Michael's selection theorems.

390

On the basis of selection theorems it is also possible to obtain multivalued (but rather "meager") extensions. Thus, for example, on the basis of a multivalued selection Nikiforov recently obtained the following theorem which, by the way, can be obtained in the special case dim X ~ n by combining Engleking's theorem (Sec. 3) and the theorem on the existence of a closed "economic" mapping of a zero-dimensional space onto X [524]. THEOREM 2. For any closed subset A of a metric space X with dim (X\A) ~ n there exists an upper semicontinuous mapping R: X § A, such that RIA = IdA and dim {x E X: IR(x) I ~ i} n - i + 1 for all i ~ i. Factorization theorems in essence pertain to dimension theory [524]; therefore, without presenting new formulations of factorization theorems, we only demonstrate their use (for example, Theorem i0 of Sec. 6) in extension of mappings. THEOREM 3. If a mapping g: A § Y is defined on a closed G6-subset A of a normal space X with dim (X\A) ~ n + I, Y 6 LC n N Cn and Y is a separable metric space, then g can be extended to all of X. Indeed, we imbed Y in the Hilbert cube Q. Then from the condition Q~AE(~) it follows that there exists an extension f: X + Q. On the other hand, since A is a G6-set, there exists a function~: X § [0, i] such that A = ~-i(0). We consider the diagonal mapping G = fA~: X + Q • I given by the formula G(x) = [f(x), (x)]. Then A = G-I(Q • {0}). We apply Theorem i0 of Sec. 6 to the mapping G and a c o u n t able family of closed sets whose union is X\A. Then there exists a metric compactum B and a mapping Go: X + B and F: B § Q • I such that F'G 0 = G. The space Z = G0(X) ~ B is a metric space, the set G0(A) = G-I(Q • {0}) N Z is closed in Z, and dim [Z\G0(A)] ~ n + I. Here F[G0(A)] = G(A) ~ Y • {0}; therefore, according to Theorem 4 of Sec. 3, there exists an extension of the mapping FIG0(A) to all of Z,which gives the extension of g to all of X. We note that in the proof of Theorem 3 we first proved a new factorization theorem (we constructed Z) and only then easily and simply used concepts of the theory of extensors (LC n N cn). 13.

Fixed Points

The class of ANR-compacta is one of the most reasonable classes of spaces for the development of a substantial theory of fixed points of mappings in no way coordinated with the metric of the space being mapped. Monographs and surveys have been devoted to the theory of fixed points, and we therefore consider only the basic features. Since absolute neighborhood extensors have a simple local structure, for them the singular and Cech homology and cohomology coincide [336], while since any ANR-compactum Y is homotopically dominated by a finite polyhedron (a version of Theorem 2 of Sec. 3), its homologies and cohomologies are finitely generated, and hence for any continuous mapping f: Y § Y in the homologies (over the field of rational numbers Q ) there is defined the alternating trace - the Lefschetz number:

C--,

A(/)= 2~ (-- 1)'tr f~. i=0

The famous Hopf-Lefschetz theorem then goes as follows. THEOREM i. If for a continuous mapping f: Y + Y of an ANR-compactum Y into itself the Lefschetz number A(f) is nonzero, then there exists a point y E Y such that f(y) = y. There are versions of Theorem I in which the condition of compactness of Y is weakened to various forms of compactness of the mapping f [724]; Theorem i is true for ANR-bicompacta and many other classes of spaces carrying weaker forms of simpliciality [653]. Theorem i has a strong form in which the "contribution" of the fixed points to A(f) is considered. Indices of fixed points and open sets can be defined from several different viewpoints. Let ~ denote the set of all pairs (f, V) such that V is an open subset of Rome ANR-compact~m Y and f: Y + Y is a mapping of it such that f has no fixed points on the boundary of V. THEOREM 2. There exists precisely one function I:J$-+Q which satisfies the following five conditions. This function actually takes integral values and is called the index. 391

(I) Localization.

If (f, V) and (g, V) 6J~ are such that f = g on [V], then l[(f, V)]=

i[(g, v)]. (2) Homotopy. V)] = I [(fl, V)]. (3) Additivity.

If ft is a homotopy such that ([t,V ) E ~ If (/, ~ / ) ~

for all 0 <_ t ~ i, then I[(f0,

and U contains pairwise disjoint sets Ui, i = i, .... k, k

such that f has no fixed points in U\Ui=ikui, then I((/, U ) ) = ~ I ( ( / , Ui)). In particular, if f i=l has no fixed points in U, then I[(f, V)] = O. (4) Normalization.

l[(f, Y)] = A(f).

(5) Commutativity.

If f: X + Y, g: Y + X, and (gof, V ) 6 ~

then l[(g o f, V)] = l[(f o

g, g-~(v)]. An estimate of the Nielsen number of a mapping [607] is a global method of obtaining a lower bound of the number of fixed points. A homotopically converse version of Theorem 1 has been obtained by means of Nielsen numbers for a rather broad class of spaces~ To the direct side Theorem 1 has been proved for upper semicontinuous, multivalued mappings with acyclic images of points and finite compositions of mappings of the type indicated for which, in particular, the Lefschetz number is defined [620]. There are also substantial results on the more general problem of finding where mappings f, g: X + Y coincide. Dold investigated the question of finding all relations on indices of an open set under iterations of a mapping. Some forms of his results are valid also for multivalued mappings. THEOREM 3. If ~-: Y § Y is a multivalued mapping of an ANR-compactum Y into itself which is a finite composition of upper semicontinuous mappings with acyclic images of points, then for all m _> i there are the congruences ~(d)A(~)

----0m~

where p( ) is the Mobius function. The proof is carried out on the basis of the equality'(~'k)n*=(~-n~) ~ and of a proposition of the work [497]. It follows from Theorem i that any AR-compactum possesses the fixed-point property, but there are examples of noncontractible ANR-compacta with the fixed-point property and also examples of contractible compacta without the fixed-point property. The majority of operations destroy the fixed-point property already in the class of (noncontractible) polyhedra [607]. 14.

Miscellaneous

i. "Retractions" which carry only part of the properties of an ordinary retraction are considered in many works [16, 17, 153, 222, 227, 287-290, 652, 730, 731]. This may be either a weakening of the continuity property [495, 670] or admissibility of a "small" perturbation on the range of the "retraction" [155, 399, 582]. Such concepts are needed both with the purpose of carrying over certain results of the theory of extensors (for example, results on fixed points [237, 620]) to broader classes of spaces and with the purpose of comparing concepts of extensoriality in different categories. Retracts in the category of pairs of spaces have been defined [386], and for sufficiently good pairs, in particular, exactness of the homology sequence of a pair has been proved. 2. The famous Hahn-Banach theorem on extension of a bounded linear functional also fits naturally into the framework of the extension problem. Injective Banach spaces were mentioned in Sec. 9. Extensions into semifields are investigated in the work [643]. The theory of extension of a partial homeomorphism to a global homeomorphism is interesting and has been greatly advanced. In the case of infinite-dimensional manifolds the problem is solved by the concept of a Z-subset [543]. Important results on extension of homeomorphisms have been obtained for "meager" subsets [84, 729]. The extension of commuting functions defined on a segment is studied in the works [542, 699, 713]. There are substantial results on open retractions [33, 761] and retractions of homogeneous spaces [523, 761]. The possibility of extending a mapping with low-dimensional preimages to a mapping with l o w 392

dimensional preimages is investigated in the work [215]. Since a cellular mapping of the sphere can be uniformly approximated by homeomorphisms (Theorem 5 of Sec. 8), a cellular mapping of the sphere extends to a mapping of the ball which is a homeomorphism on the interior [214]. 3. The extension of a mapping onto a CW-complex can be constructed inductively on the dimension of skeletons which makes it possible to develop a powerful method of extending an individual mapping - the theory of obstructions [40]. Many examples of ANR-compacta possess subsets which are naturally called the m-dimensional skeleton - a subset into which it is possible to deform a mapping of any m-dimensional compactum. 4. Major attention has been devoted to deformation and strict deformation retracts [263, 499]. Thus, for example, it has been proved [700, 701] that compacta have the same (shape) homotopy type if and only if they are (fundamental) deformation retracts of some common enveloping compactum. 5. The majority of special questions of the theory of retracts are substantial and nontrivial already at the level of compacta; to the end of this section all spaces are assumed to be compact. In the work [499] an example was constructed of a two-dimensional ANRcompactum which is not a finite (or even countable) union of AR-compacta; an example of an AR-compactumwhich does not decompose into a finite (or even countable) number of AR-compacta of arbitrarily small diameter is also given there. Spaces having decompositions into elementary block-briquests are considered in analogy with finite simplicial decompositions [532, 639]. An ANR-compactum with a briquet decomposition has the homotopy type of the nerve of this decomposition. Recently West proved (Sec. 15) that any ANR-compactum has the homotopy type of some finite polyhedron. 6. The construction of examples of ANR-compacta with bad subsets to the work of Bing and Borsuk [558].

[466, 485]goes back

THEOREM I. For any m e 3 there exists an m-dimensional AR-compactumYm, contain a proper ANR-subcompactum of dimension greater than one.

which does not

We note that the space of continuous functions on Ym in the topology of pointwise conver ~ gence cannot be linearly homeomorphic to the corresponding space over a polyhedron. 7. There are examples [499] of AR-compacta Yk,m in which some set of dimension k ~ 1 can be contracted to a point only along a set of dimension ~m which is impossible in polyhedra. Borsuk [499] introduced the following important class of spaces. Definition i. A compactum Y satisfies the condition A if for any e > 0 there is a 6 > 0 such that any closed set A ~ Y with diameter less than 6 can be contracted to a point along a subset of the space Y having dimension ~dim A + 1 and diameter
If X and Y are compacta satisfying condition A, then dim X x y = dim X +

The problem of the validity of Theorem 2 for arbitrary ANR-compacta remains open. 8. In 1953 Lubanski [327] constructed an example of an ANR-compactum which is the common boundary of three regions in R 3. However, as shown already in 1924 by Kuratowski [499], a continuum which is the common boundary of three regions in R 2 is indecomposable or is the sum of two indecomposable subcontinua. Such a continuum on the plane was constructed by B~auer in 1920 [499]~ Plane AR-compacta were described by Borsuk [499] as precisely the locally connected continua not decomposing the plane, while plane ANR-continua are precisely the locally connected continua with a finite one-dimensional Betti number, i.e., locally connected continua decomposing the plane into a finite number of connected components. 9. One-dimensional A R - c o m p a c t a a r e precisely the dendrites. However, dendrites can be described in an intrinsic manner by means of the existence of monotone retractions onto their subcontinua [619, 674, 675]. 10. Uniqueness of the decomposition of ANR-compacta into a product of one-dimensional compacta (if it exists) was proved in the works [404, 405, 719]. At the present time results of this type have been proved under rather general assumptions [601, 609]. 393

ii.

There are many special results on imbeddings in the class of ANR-compacta.

THEOREM 3. If in an n-dimensional ANR-compactum Y there is given an uncountable family {X~} of n-dimensional subcompacta, then there exist two distinct indices ~ ~ ~ such that dim (X~ n X$) = n. Bing and Borsuk [499] proved that it is possible to imbed very few umbrellas in an n-dimensional ANR-compactum. Bothe [109] proved that any n-dimensional compactum can be imbedded in an (n + l)-dimensional AR-compactum. There are deep results on imbeddings in Euclidean space [406, 471]. Borsuk [499] constructed already in R ~ a continuum family of two-dimensional "strongly different" AR-compacta. Questions of compactification have been studied [185, 433, 705]. 331].

12. The connection of the properties ANR and homogeneity are studied in the works [204, Thus, for example, no AR-compactum can be strongly homogeneous.

13. spaces.

In the monograph

[499] an entire chapter is devoted to the r-classification of

14. In the works [339-343, 682] fixed points are studied not "inside" but "outside" which, by the way, can be treated also as a relaxation of retraction properties. THEOREM 4. A compactum Y is an absolute retract if and only if it is an absolute neighborhood retract and for any imbedding of it in a compactum Z there exists a mapping f: Z § Z such that Y = {z 6 Z: f(z) = z}. For a finite-dimensional compactum Y the ANR condition can be omitted. 15. The study of the space of subsets in various topologies in the metric of continuity, began from investigations of Borsuk. special properties of ANR-spaces are studied in [499]. 15.

[88, 315], for example, Many other interesting

A General Definition and Properties of Y-Manifolds

A topological space X is called a manifold modeled on a space Y or a Y-manifold if any point of the space X has a neighborhood homeomorphic to an open subset of the space Y. If necessary, additional conditions are imposed on the space X: connectedness, the presence of a countable base, etc. Such a general definition can be advantageous only for model spaces Y possessing sufficient good (local) properties. Examples of Y-manifolds are: i) a topological m-dimensional manifold (model space

Rm);

2) Q-manifolds; 3) s 4) Tikhonov manifolds (model space IT). One of the impetuses to the construction of a theory of manifolds modeled on infinitedimensional linear spaces was the Anderson-Kadets theorem that any infinite-dimensional separable Frechet space (a complete, locally convex, linear metric space) is homeomorphic to a countable power of the line R" [84]. After this at the end of the 1960s Anderson, Shori, Henderson, West, and Chapman obtained theorems on classification and representations of infinite-dimensional manifolds. The basic theorems for s Classification Theorem. the same homotopy type.

are the following. Two s

are homeomorphic if and only if they have

Theorem on Triangulability and Stability. A separable metrizable space X is an s fold if and only if X is homeomorphic to the product K • s where K is a countable, locally finite simplicial complex. Imbedding Theorem. as an open subset. of s

A space X is an s

At the beginning of the 1970s Torunczyk of s

if and only if it can be imbedded in s [756, 757] gave the following characterization

If X is a complete, separable, metrizable space, then the product X • s fold if and only if X is an ANR-space. 394

is an s

The basic theorems of the general theory of manifolds modeled on infinite-dimensional linear spaces were augmented by results from infinite-dimensional differential topology [118, 119, 226, 347, 472, 702, 715]. The theory of Q-manifolds was created in parallel with the theory of ~2-manifolds but with some lag. Although Q-manifolds are strongly infinite-dimensional just as ~2-manifolds their local compactness brings them close to finite-dimensional spaces. This remark, which is light at first glance, is not so superficial. To obtain the most complex results regarding Q-manifolds (triangulation and classification theorems) it is necessary essentially to apply ideas and methods of the theory of finite-dimensional manifolds. Generally, as Chapman writes in the preface to his "Lectures on Q-Manifolds," "The theory of Q-manifolds is in some sense the theory of 'stable' piecewise linear m-manifolds." This can partially explain the lag of the theory of Q-manifolds as compared with the theory of ~2-manifolds, since problems of finite-dimensional topology are, as a rule, more complex than the analogous problems of in ~ finite-dimensional topology. The theory of Q-manifolds owes its origin to Anderson [67-72, 543-549]. He introduced the concept of and investigated the properties of Z-sets of the Hilbert cube, he proved its homogeneity, and together with Shori obtained a theorem on the stability of Q-manifolds. For Q-manifolds there are basically the same results as for ~2-manifolds: triangulability, stability, and imbedding theorems and a classification theorem. It is clear that a) a compact Q-manifold not homeomorphic to Q cannot be imbedded in Q as an open subset; b) a compact Q-manifold has the homotopy type of a noncompact Q-manifold. Therefore, the imbedding and classification theorems [Chapman, 135, 533] acquire the following specific form. i) If X is a Q-manifold, then the product X • [0, i) can be imbedded in Q as an open subset. 2) Q-manifolds X I and X 2 have the same homotopy type if and only if their products with the interval [0, I) are homeomorphic. Within the framework of the theory of Q-manifolds Chapman [533] obtained the following deep results. Classification Theorem for Polyhedra. A perfect mapping f: X + Y between po!yhedra is an infinite, simple homotopy equivalence if and only if the mapping

[xId

: XxQ-+YXQ

is a perfect homotopy homeomorphism. The topological invariance of Whithead torsion follows from this theorem in the compact case.

The theorem on CE-mappings proved by West [476] was an essential step in the construction of the theory of Q-manifolds. If X is an ANR-compactum, then X is the CE-image of a compact Q-manifold. This theorem was preceded by Miller's theorem [373, 374] that the product of a finitedimensional ANR-compactum with the half interval is the CE-image of a finite-dimensional manifold. Moreover, Miller's proof made it possible to replace finite-dimensional manifolds by Q-manifolds. From the theorem on CE-mappings, the triangulation theorem, and Haver's theorem [246] that CE-mappings between ANR-spaces coincide with finite homotopy equivalences it follows that any ANR-compactum has the homotopy type of a compact polyhedron. Thus, West sol~ed the old conjecture of Borsuk. Further advancement of the theory of Q-manifolds is due to Edwards~ He proved [200, 201] that I) a CE-mapping between Q-manifolds is an almost homeomorphism and 2) the product of a locally compact ANR-space with Q is a Q-manifold. Torunczyk's characterization theorems are the crown both of the theory of Q-manifolds and of the theory of ~2-manifolds. ~orunczyk's First Characterization Theorem [460]. A locally compact, metrizable space X is a Q-manifold if and only if the following conditions are simultaneously satisfied: i) X is an ANR-space;

395

2) the identity mapping IdX can be approximated by Z-mappings. In particular, a characterization of the Hilbert cube follows from this theorem and Chapman's theorem [135] that a compact, contractible Q-manifold is homeomorphic to Q. T0runczyk's Second Characterization Theorem [461]. A separable, complete ANR-space X is an s if and only if the following condition is satisfied. For any mapping

/:I~9=IQ~-+Xof

a countable free sum of Hilbert cubes and any open cover-

ing ~ of the space X there exists a mapping a-close

g:~iQi-+X

such that the mappings f and g are

and the family {g(Qi)} is discrete.

In the case where X is an AR-space we obtain a characterization of the Hilbert space Zz. These deep theorems of Torunczyk immediately obtained many applications in the proof that some space is a Q-manifold or an s (see, for example, [23. 50, 161, 492, 501, 526, 531]). The application of Torunczyk's theorem in the recognition of Q- and s among spaces of the form F(X), where F is some covariant functor in the category Tor, turned out to be especially fruitful. Here not only many new results were obtained, but simple proofs were given of an entire series of theorems already known. Thus, for example, the well known theorem of West and Shori [433, 734] that the exponential of the segment exp I is homeomorphic to Q can be proved in the following manner. By Wojdyslawski's theorem exp I is an absolute retract. On the other hand, the mappings f, g: exp I + exp I defined by the equalities

f(A) =[O~(A)],

g(A) =

(l--e)AU{sup A}

are e-close to the identity and have disjoint images, since no set F(A) has isolated points. This completes the proof, since the first characterization theorem of Torunczyk in the compact case can be formulated as follows. An ANR-compactum X is a Q-manifold if and only if for any e > 0 there exist mappings f, g: X + X e-close to Idx and having nonintersecting images. 16.

Examples of the Occurrence of Y-Manifolds

In the preceding section it was shown that a Y-manifold can be obtained by multiplying the model space Y by a "good" (ANR) space. From theorems of West [473] and Edwards [201] it follows that the product of a countable number of nonsingleton AR-compacta is homeomorphic to the Hilbert cube. Many examples of Y-manifolds arise in applying covariant functors to spaces. The problem of whether the exponential (hyperspace) of the segment exp I was homeomorphic to the Hilbert cubeQ remainedunsoived for a long time. This problem arose already in the 1920s in works of Polish topologists. It was formulated in full generality by Wojdyslawski for the exponential of arbitrary nondegenerate Peano continua. An analogous question arose also for the continuum exponential exp c (the hyperspace of all subcontinua of the space X), although it should be mentioned that a necessary condition for the equality exp c X = Q is the absence in X of arcs with nonempty interior, since otherwise exp c X contains open, two-dimensional sets. At the beginning of the 1970s this problem was solved in the works of the American topologists West, Curtis, and Shori (see [160, 162, 163, 428, 592, 733, 734, 773]) first for the segment, then for a one-dimensional polyhedron, then for an arbitrary polyhedron, and finally for any Peano continuum. Torunczyk's criterion made it possible to considerably simplify the original proofs, but the technique developed in solving the problem of hyperspace turned out to be one of the main working tools in constructing the theory of Q-manifolds. We therefore present here some basic facts which lie at the foundation of the original proof. The key concepts here were Q-factors (compacta X with the property that X • Q is homeomorphic to Q), almost homeomorphisms, and "interior approxlmation" by means of inverse limits. As indicated in Sec. 15, Edwards later established [200] that any AR-compactum is a Q-factor. The basis of the solution of the problem of hyperspace was formed by the geometric technique of West applied in the proof of the fact that certain finite-dimensional subspaces of exp I are Q-factors and

396

his theorem that the product of an infinite-number of Q-factors is homeomorphic to Q [473]. Moreover, the following lemma, explicitly formulated in [160], was applied together with Brown's lemma [568] on the limit of an inverse sequence of almost homeomorphisms. Lemma on Interi0r Approximation. Let X be a compactum, and let (Xi, fi) be an inverse sequence of subcompacta of it satisfying the conditions: i) X i converges to X in exp X; 2) d(f i, !d) < 2 -i for each i; 3) {fi . . . . .

4:

j ~ i} is an equiuniformly continuous family for each i.

Then X is homeomorphic to lim (X i, fi)It should also be noted that in obtaining the general result on the continuum exponent of a Peano continuum Curtis and Shori used the following theorem of West [773]. If X is a finite, connected graph, then Q • exp c X is homeomorphic to Q; if X is a dendron with a dense set of branch points (or, equivalently, without free arcs), then exp c X is homeomorphic to Q. We recallthat fora compactum X a nonempty compactum G ~ exp X is called a hyperspace of growth if the fact that A ~ G, B ~ exp X, A ~ B and each connected component of the set B intersects A implies B'6 G. If the conditions on the connected components are omitted, then the corresponding space is called a hyperspace of inclusion. Curtis proved [590] the following theorem. Suppose G is a nontrivial hyperspace of growth of a Peano continuum X such that either X does not contain free arcs or G is a space of inclusion. Then the following conditions are equivalent: i) G is homeomorphic to Q; 2) G\{X} is contractible; 3) {X} is a Z-set in G. Very often Q- and Z=-manifolds arise under the action on spaces by other functors of exponential type, for example, such functors as F and F c of order and continuum order of arcs, the functor exPn c of closed subsets having no more than n connected components (see [197, 501, 502, 525, 526]). From results of Keller and Klee [651] it follows that the space P(X) of probability measures on an infinite compactum is homeomorphic to the Hilbert cube. Ditor and Haydon [594] proved that the (infinite) bicompactum P(X) is an absolute retract if and only if X is a Dugundji space of weight ~ l o On the basis of this and on the basis of a characterization he obtained of theTikhonov cube as an absolute retract of homogeneous character, Shchepin proved [61] that for a nonmetrizable bicompactum X the space P(X) is homeomorphic to the Tikhonov cube if and only if X is a Dugundji space of weight ~ i of homogeneous character. Van Mill [369] and independently Ivanov [23] proved that application of the functor of superextension ~ to a metrizable nonsingleton continuum gives a Hilbert cube. Ivanov extended this result [24] to the nonmetrizable case. He proved that a superextension of a bicompactum X is homeomorphic to I ~ if and only if X is a connected, openly generated bicompactum of weight T of homogeneous character. Here a bicompactumis openly generated if it is the limit of the sigma-spectrum of compacta with open projections. A number of interesting results on the occurrence of infinite-dimensional manifolds and on the position of one manifold in another were obtained by applying covariant functors to noncompact spaces, infinite iterations of functors, and functors going beyond the limits of (bi) compact spaces (see, for example, [19, 161, 507, 508, 758]). A large store of Y-manifolds arises in considering spaces of mappings, in particular homeomorphisms. Let M be a compact m-manifold. We denote by H(M) the space of homeomorphisms h: M + M and by Hs(M) the subspace of H(M) consisting of homeomorphisms which are the identity on the boundary ~M, One of the most interesting problems at the present time concerning s the following "problem of a group of homeomorphisms."

is

Is the space Hs(M) for a compact m-manifold an s 397

Geoghegan [224, 612] proved that the space Hs(M) contains Z2 as a factor. Therefore, by Torunczyk's theorem to solve the problem of a group of homeomorphisms it suffices to show that Hs(M) is an ANR-space. This problem has so far been solved for n = i (Anderson) and n = 2 (Luke and Mason). As concerns the infinite-dimensional case, i.e., compact Q-manifolds M, the problem of a group of homeomorphisms for them has been solved. First Geoghegan proved that, as in the finite-dimensional case, H(M) contains s as a factor. Then Ferry [209] and Torunczyk [457] independently proved that H(M) is an ANR-space. The space PLH(M) of piecewise linear homeomorphisms of a compact, piecewise linear manifold M has also been studied. Completing previous work of several authors, Wilson and Keesling proved [285] that PLH(M) is a s where s f is the linear subspace of the Hiibert space s consisting of all points with only a finite number of nonzero coordinates. In analogy to the problem of a group of homeomorphisms of a compact, m-dimensional manifold the following question remains unsolved. Is the space R(I m) of retractions of an m-dimensional cube into itself homeomorphic to the Hilbert space s ~ For m = 1 a positive answer was given by Basmanov and Savchenko [492]. For compact Q-manifolds M this question has been solved. First Chapman proved that R(M) e ANR, and Sakai completed this theorem by proving [732] that R(M) is an s The examples presented suggest that spaces of mappings naturally arising in nature of sufficiently "movable" objects are never locally compact. Geoghegan's result [611] is thus the more surprising: he found a class of mappings of smooth manifolds such that all mappings of one manifold into another belonging to this class form a Q-manifold. Fiber versions of theorems on the occurrence of manifolds modeled on some space under the action on the mappings by a functor have also been obtained (see [50, 501, 502, 525, 526, 531]). These are theorems to the effect that for some mappings f: X + Z and functors F (particular) fibers of the mapping F(f): F(X) + F(Z) are Y-manifolds. However, parametric versions are the most interesting, i.e., theorems of when a mapping of the form F(f) is a Y-fibering, i.e., is a locally trivial fibering with fiber Y. Fedorchuk proved [see [49, 208, 531]) that for nondegenerate Peano continua X and Y the mapping exp c PX, where PX: X x y + X~ is a Q-fibering if and only if X is a dendron and Y does not contain free arcs. The question remains openof whether the projectionPX here can be replaced by a 1-soft mapping F: X + Y of a continuum onto a dendron whose fibers are not degenerate and do not contain free arcs. A necessary of compacta, and and infiniteness in the following

condition for a Q-fibering of the mapping P(f), where f: X + Y is a mapping P is the functor of probability measures, is the openness of the mapping f of its fibers. Fedorchuk proved that this necessary condition is sufficient cases:

i) the compacta X and Y are finite-dimensional; 2) the compactum Y is finite-dimensional, and the fibers f-1(y) have no isolated points; 3) the compactum Y is zero-dimensional. Zarichnyi augmented these results by proving that if an open mapping f: X + Y of compacta admits local sections passing through nonisolated points of the fibers f-l(y), then P(f) is a Q-fibering. It was also proved (Fedorchuk) that for an open mapping f: X + Y of a nonmetrizable bicompactum X onto a diadic bicompactum Y the mapping P(f) is a Tikhonov fibering (a fibering with fiber I ~) if and only if X is a Dugundji space of weight ~i of homogeneous character and Y is a compactum. For the functor ~ Ivanov obtained the following definitive result in the class of continua. Let f: X + Y be an epimorphism of connected, openly generated bicompacta of homogeneous character. Then %f is a Tikhonov fibering if and only if the mapping f is open and has no points of multiplicity one. Ivanov obtained a similar result with connected bicompacta replaced by zero-dimensional ones and the Tikhonov cube I ~ replaced by the Cantor discontinuum.

398

All the theorems on Y-fiberings mentioned above were obtained on the basis of the following transfer of Torunczyk's criterion to mappings (West-Torunczyk). Let f: X + Y be a fibering of Hurewicz ANR-compacta with contractible fibers f-Z(y). Then f is a Q-fibering if and only if for any E > 0 there exist mappings gl, g2: X + X such that f o gi = f,

p(g~, Id) <e, g1(X)Ng2(X) =~. On the basis of this criterion West constructed an example of a Hurewicz fibering f: Q Q with fiber Q which is not a Q-fibering, thus solving a well known problem; see the Supplement to Chapman's monograph "Lectures on Q-Manifolds" [533]. In conclusion it is appropriate to note that the theory of manifolds modeled on the m-dimensional universal Menger compactum Dm has developed rapidly. Bestvina gave a fine topological characterization of the compactum pm which made it possible for him and Dranishnikov to basically construct at the present time a theory of Dm-manifolds. ~m-manifoids turned out to be interesting and important objects both in the theory of cohomological dimension and in the theory of transformation groups in connection with Hilbert's problem of free actions on m-manifolds. ~m-manifolds and many classes of infinite-dimensional manifolds admit free and some other special actions of many classes of groups, which gives interesting properties of the corresponding factor spaces. LITERATURE CITED i. 2. 3. 4. 5o 6. 7. 8. 9o I0.

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