Symposium on Infinite Dimensional Topology. (AM-69), Volume 69 9781400881406

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Annals of Mathematics Studies Number 69

SYMPOSIUM ON INFINITE DIMENSIONAL TOPOLOGY E D IT E D B Y

R. D. ANDERSON

P R IN C E T O N U N IV E R S IT Y PRESS AND U N IV E R S IT Y O F T O K Y O PRESS

P R IN C E T O N , N E W JE R SE Y 1972

Copyright ©

1972, by Princeton University Press A L L RIGHTS RESERVED

L C Card: 69-17445 IS B N : 0-691-08087-9 A M S 1968: 4160, 4601, 4610, 4615, 4625, 4630, 4635, 4690, 4785, 5340, 5374, 5720, 5731, 5755

Published in Japan exclusively by University o f T ok yo Press; in other parts o f the world by Princeton University Press

Printed in the United States o f Am erica

PREFACE The present volume constitutes the Proceedings of the Symposium on Infinite-Dimensional Topology held in Baton Rouge from March 27 through April 1, 1967.

The symposium was organized to bring together mathemati­

cians active in research in one or more of the following areas of infinite­ dimensional topology: the topology of linear spaces, fixed point theory, differential topology, and pointset topology.

In all of these areas, rather

striking new results had recently been obtained, and it was believed that a symposium which encouraged the interchange of ideas and information among those active in these various areas should be conducive to further and, hopefully, more broadly applicable research.

Recent results of vari­

ous of the participants have fully justified this belief. The idea of the symposium grew out of discussions by V. L. Klee, T. Ganea and the undersigned.

The organizing committee of the symposium

consisted of F. Browder, V. L. Klee, N. Kuiper, R. Palais and R. D. Ander­ son.

The symposium was sponsored by Louisiana State University under

funds made available from a National Science Foundation Science Develop­ ment Grant to Louisiana State University.

The sessions were held in the

Student Union on the Baton Rouge campus of L.S.U. cians from a dozen different countries participated.

About 70 mathemati­ There were about 30

invited hour and half-hour addresses with several problem sessions and many informal discussions.

Some of the talks were surveys, some were ex­

pository, and others represented current research. invited to submit papers for the Proceedings.

The participants were

The present volume is the

collection of those papers submitted at or shortly after the time of the sym­ posium, and represents contributions in all of the major areas of the sym­ posium. It is hoped that the publication of this volume will stimulate still further research.

vi

PR E FAC E

The editor sincerely apologizes to the contributors and to other inter­ ested mathematicians for the excessive delay in publication of this volume. The delay was his fault and his alone.

The editor wishes to thank the

Princeton University Press, and particularly John W. Hannon of the Press, for their great help and cooperation in the publication of these Proceedings and for their patience and forbearance with the editor. R. D. ANDERSON

C O N TEN TS P re fa c e ..................................................................................................

v

Topological Equivalence of Non-Separable Reflexive Banach Spaces by C. Bessaga

................................................................................

3

On Topological Classification of Non-Separable Banach Spaces by C. Bessaga and M. I.Kadec ................. ......................................

15

On Homotopy Properties of Compact Subsets of the Hilbert Space by Karel Borsuk ..............................................................................

25

Some Thin Sets in Frechet Spaces by H. H. Corson ..............................................................................

37

A Remark on Banach Analytic Spaces by A. Douady .................................................................................

41

Fibring Spaces of Maps by James Eells, Jr...........................................................................

43

An Approximate Morse-Sard Theorem by James Eells and JohnMcAlpin

.................................................

59

................................................................

63

Morse Theory for Closed Curves by Halldor I. Eliasson

Covering Properties of Convex Sets and Fixed Point Theorems in Topological Vector Spaces by Ky Fan .......................................................................................

79

On the Cohomology Theory in Linear Normed Spaces by Kazimierz Geba andAndrzej Granas .......................................... v ii

93

v iii

CONTENTS

Analyse de la Technique de Nash-Moser by G. Glaeser (R e n n e s).................................................................. 107 Generalizing the Hopf-Lefschetz Fixed Point Theorem for Non-Compact ANR-S by Andrzej Granas .......................................................................... 119 Some Questions in the Dimension Theory of Infinite Dimensional Spaces by David W. Henderson .................................................................. 131 On the Continuity of Best Approximation Operators by R. B. Holmes ............................................................................. 137 Some Self-Dual Properties of Normed Linear Spaces by Robert C. James ........................................................................ 159 Asymptotic Fixed Point Theory by G. Stephen Jones ............................................................... ...... 177 On the Relationship of Local to Global Fixed-Point Indexes by Ronald J. Knill .......................................................................... 185 C 1-Equivalence of Functions Near Isolated Critical Points by Nicolaas H. Kuiper

.................................................................. 199

Fixed Point Index and Lefschetz Number by Jean Leray ................................................................................. 219 Weakly Compact Sets — Their Topological Properties and the Banach Spaces They Generate by Joram Lindenstrauss

................................................................ 235

On Continuity and Approximation Questions Concerning Critical Morse Groups in Hilbert Space by E. H. Rothe ............................................................................... 275 On Homeomorphisms of Certain Infinite Dimensional Spaces by Raymond Y. T. Wong ................................................................ 297

SYMPOSIUM ON INFINITE DIMENSIONAL TOPOLOGY

T O P O L O G IC A L E Q U IV A LE N C E OF N O N -SE PA R A B LE R E F L E X IV E BAN AC H SPACES Ordinal Resolutions of Identity and Monotone B ases* C. BESSAGA

In 1965 Kadec [5] proved that all separable infinite-dimensional Banach spaces are homeomorphic. space of density character is still open.

The problem, if every non-separable Banach is homeomorphic to the Hilbert space l 2 (jtf),

Recently, Troyansky [13] proved that the space cQ( £< ), of

all continuous functions defined on the one-point compactification of a dis­ crete set of cardinality & vanishing at infinity, is homeomorphic to l 2(fc

Greek letters (except £ ) denote ordinal numbers.

means that

t

r 1


(r) = Sfx is continuous in r with respect to the order topology of the closed interval [1, llx ll for all x f span {e 7lf < a

then there exists a projection basis lSr! r< g such that e f e (S P ro o f:

in the space span

x - Sr)x for all r
k. Without loss of generality we may additionally assume that y 1 ^ 0 and yn + 1 ^ yn for all n. It is easy to check that the sequence xn = Yn + i “ Yn *s a Schauder

N O N-SEPARABLE R E F LE X IV E BANACH SPACES

basis in the space flexive space X.

Y = span{xnj.

7

Y is reflexive, as a subspace of a re­

Hence, by a theorem of James [5, Ch. IV, §3, Theorem 3]

the basis jx n! is boundedly complete, and therefore Inn y^ =

xn

exists, i.e., the original basis {Sf l is boundedly complete. P R O P O S I T I O N 4.

{ S^Jy.< ^5r and let x e X .

Let X be a Banach space with a projection basis Then the ordinal number a -

in fjr: x = Sr x! is

either isolated or is a limit of a countable sequence of ordinals. P ro o f:

If a is not isolated, then rn = infjr: ||Srx — x|| > 1/nJ * a.

P R O P O S I T I O N 5.

If X is a Banach space with a projection basis

{Srlr< £ then there exists in X an equivalent norm ||• || such that (v )

||* || is strictly convex, ||Sf + 1 x|| ;> ||Sr x||, ||Rr+1x|| ^ ||Sf x|| for x £ X and t < £.

P ro o f:

Let || ||x be the original norm in X.

Take a biorthogonal sys­

tem je r, frir < ^ associated with the basis, such that \\eT\\1 = 1 for all Then for every £ > 0, x € X we have card{r: |ff(x)| > £i ^

t .

Let

r(n, x) be the sequence of all ordinals r for which f r(x) ^ 0 and let

lfr(l,x)(x)l £ lfr(2,x)(x)l £ - • Define H 2 = Hxlli+

/% l2~nfr(n,x)W|2

v n= 1

•

|| • ||2 is an equivalent strictly convex norm, cf. Day [6]. Now it is easy to check that the norm l|X|1 =

« > P£ l|S« X~ S/SXll2

satisfies the condition (v ) and is equivalent to the original one.

C. BESSAGA

8

2.

T o p o lo g ica l equivalence of refle x iv e spaces. T H E O R E M 2.

L e t X be a re fle x iv e infinite-dim ensional Banach space.

Then X is homeomorphic to the space 1 ( onto Construction of h. Let ie f , i T\T < ^ be a biorthogonal system associated to the basis and let (2)

b (x ,y ) = ||x|H|y||+

%

(||Sr+1y||-||Sr y||)-||Rr+1x||,

T< € for (x, y) £ X x X ; b (x) = b (x, x) . Then (3)

hx = K sgnf (x ))-(b (S r+J x) - b(Sr x ))!r < ^

b(Srx0),

then Urn ||xk—xQ|| = 0.

for t < £ and b(xk) -► b(xQ) ,

NON-SEPARABLE R E FLE X IV E BANACH SPACES

P ro o f o f (1.1) and (1 .2 ):

By (v) we have ||Sr+1y|| —||Sr y|| ^ 0 and is

> 0 only for at most countably many values of r; 2

9

moreover

(||Sr+1y||-||Sr y||)= ||Sdry|| = ||y|| .

T< £ Hence we conclude that for any fixed y ^ 0, b (x , y) is a well-defined norm in x, and ||x|| ||y|| < b (x , y) ^ ]|y|| (||x|| + sup j|R +1x||) S 2||x|| ||y||. T

Taking into account the continuity of the expressions Sf x and R^x with respect to

and the condition (v), it is easy to check by transfinite in­

t

duction that b (x , y) = ||x|| ||y|| +

2 ||say|| (||Ra x|| - ||Ra+1x||) . 2 ^a< £

This formula together with (v ) gives the conclusion, that for any x ^ 0, b (x, y) is a norm with respect to y. P ro o f of (1.3). obvious.

In the case where card £
Sr x0 for

T

and b(xk) -> b(xQ) .

< t

LEMMA 2. The conditions (v) and ( * * ) imply the following statements: (2.1)

||RxJ < lim

||Rxk|| for 1 < r
hxQ .

h is a homeomorphism into 1 (£ ).

P ro o f:

The statement (3.1) and implications:

(b) = > (a),

(b) = > (c)

follow from the definition of the map h and from the Lemma 1. From the continuity of Sf, ff and b follows that (a ) implies (b).

The equivalence

of the conditions (c ) and (d) is an obvious property of the space l ( ^ ) . It remains to prove that (c) implies (b). Assuming (c), from the statement (3.1) we obtain (4)

b(Sr xk) -> b(Sr x0) for all r ^ £ .

12

C. BESSAGA

Now suppose that (5)

f/ x k) -> f (x 0) for r < a ,

and denote fa(xk) = tk- By (4), b(Sa xk + tk ea)

b(Sa+1xQ), and using

(5), (1.2) and (3.1) we conclude

( 6) From the definition of h and the condition (c) we obtain (7)

sgn tk -» sgn fa(hx0) .

From the formula (2) defining the function b, the condition (v) of Proposi­ tion 5 and Lemma 1 it follows that the function g(t) = b(Sa xQ + t e ) is strictly increasing for t > 0, strictly decreasing for t < 0 and continuous. Hence, by (6) and (7) we obtain

Since, the inductive hypothesis (5) is obviously true for a = 1, by induc­ tion we establish the condition (b). The statement (3.3) is an obvious consequence of (3.2). P ro o f that h is onto. Forany z e 1 ( f ) If cr(z) =

1, then z

= 0=

denote

< t (z )

= infir: Sfz = z\ .

hO.To complete the inductive proofwe have to

show: (* * * )

For any z e 1 ( f ) the condition {y e 1 (f ):

cr(y) < g ( z )\ C hX im­

plies z e hX. The case

o { z ) = y + 1. Take a vector x in X with hx = S^z.

course £ = sgn f^(z) ^ 0.

Of

As we have observed, the function g(t) =

b(x + £te^) is continuous and increasing for t > 0. Hence there is a tQ, with g(tQ) = b(z).

It is easily seen that z = h(x+ S tQe^).

The case where cr(z) is a lim it ordinal.

By Proposition 4,

there are

* o (z) such that S z -> z. Take x e X with hx = S z. It is clear n n n that the sequence Jx } has the property stated in the Definition 2, and t

NON-SEPARABLE R E FLE X IV E BANACH SPACES

13

since the basis { S j is boundedly complete, we conclude that x = lim x n exists and that z = hx. R E M A R K 2.

In the case where the norm ||• || of the space X is uni­

formly convex, we may replace the functional b(x) by the norm ||x|| '

3. Problem s Problem 1. Is it true that every conjugate Banach space is homeo­ morphic to a space 1 ( f ) ? Re m

ark

3. For the spaces X = Y * for which wX = wY, the answer

is “ yes. ” Problem 2. Is it true that every Banach space with a monotone ba­ sis is homeomorphic to a space 1 ( f ) ? (Cf. Kadec [9 ].) Problem 3. Is it truethat every Banach space with an unconditional basis is homeomorphic to a space

1 (f) ?

The affirmative answer to this problem would follow from: C O N J E C T U R E 1. Assume that X is a Banach space with an uncondi­

tional basis of cardinality isomorphic to c Q(i i^0 and X does not contain any subspace

Then there is a subspace Y of X such that no sub­

space of Y is isomorphic to c Q, and wY = Problem 4. Is it true that every space C(Q), of all continuous funcr tions on a compact space Q, is homeomorphic with a space 1 ( f ) ?

U N IV E R S IT Y OF WARSAW, WARSAW, P O L A N D and LOUISIA NA S T A T E UNIVERSIT Y, B A T O N ROUGE, LA.

14

C. BESSAGA

REFERENCES [1] D. Amir and J. Lindenstrauss, “ On topological structure of weakly compact sets in Banach sp aces/ ’ to appear. [2] C. Bessaga, “ On topological classification of complete linear metric spaces,” Fund. Math., 56 (1965), 251-288. [3] ______ , “ Topological equivalence of non-separable reflexive Banach spaces.

Ordinal resolutions of identity and monotone b a se s,” Bull.

Acad. Polon. Sci., Ser. sci. math. astr. et phys., (1967), [4] _____ , and A. Pefczynski, “ On bases and unconditional convergence of series in Banach spaces,” Studia Math., 17 (1958), 151-164. [5] M. M. Day, Normed Lin ea r Spaces, Berlin-Gottingen-Heidelberg, 1962. [6] _____ , “ Strict convexity and smoothness of normed spaces,” Trans. Amer. Math. Soc., 78 (1955), 516-528. [7] M. I. Kadec, “ On homeomorphism of certain Banach spaces ” (Rus­ sian), Dokl. Akad. Nauk. SSSR, 163 (1953), 465-468. [8] _____ , “ On strong and weak convergence” (Russian), ibidem, 122 (1958), 23-25. [9] _____ , “ On topological equivalence of all separable Banach spaces” (Russian), ibidem, 163 (1966), 23-25. [10] _____ , “ On spaces isomorphic with locally uniformly convex spaces,” (Russian) Izvestia Vuzov, Matematika, 6 (13) (1959), 51-57. [11] _____ , Letter to the editor (Russian), ibidem 6 (1961), 139-141. [12] J. Lindenstrauss, “ On reflexive spaces having the metric approxima­ tion property,” Israel. Journ. Math., 3 (1965), 199-204. [13] S. Troyansky, “ On topological equivalence of cQ(fc a .

The subspaces L a and L a are now the same as in Example A. A projection basis lSa la

Must X be homeomorphic to the space l( i < )? THEOREM 8. E very abstract L -space is homeomorphic to a space 1M

([6], 9.3 xx). The above facts seem to suggest that the solution of the general clas­

sification problem of (non-separable) Banach spaces can be perhaps achiev­ ed by studying: 1)

geometrical properties of Banach spaces connected with the exis­

tence of “ nice” norms and Smodulars and “ nice” generalized [co-] Tsystems,

C. BESSAGA AND M. I. KADEC

22

2)

some isomorphic properties of Banach spaces, mainly the structure

of subspaces and linear images of a given space. We may expect also that the investigation of structural properties of Fr6chet spaces will allow to reduce the classification problem of Frechet spaces to that of Banach spaces.

(In the separable case this was possi­

ble thanks to Anderson’s theorem [1]:

1 ( ^ Q) — s, the countable product

of lines, Eidelheit’s result [12] stating that every non-normable Frechet space can be linearly mapped onto s, and Theorem 4.5.)

REFERENCES [1] R. D. Anderson, “ Hilbert space is homeomorphic to the countable in­ finite product of lines,” B ull. Amer. Math. Soc., 72 (1966), 515-519. [2] R. G. Bartle and L. M. Graves, “ Mappings between function spaces,” Trans. Amer. Math. Soc., 72 (1952), 400-413. [3] S. Bernstein, “ Sur le probleme inverse de la theorie de meilleure ap­ proximation des fonctions continues,” C. R. Acad. Sci. P aris, 206 (1938), 1520-1523. [4] C. Bessaga, “ Some remarks on homeomorphisms of itf Q-dimensional linear metric spaces,” Bull. Acad. P o l. Sci., set. sci., math., astr. et phys., 11 (1963), 159-163. [5] ______ , “ Topological equivalence of non-separable reflexive Banach spaces.

Ordinal resolutions of identity and monotone b a se s,” this

issue p. 3, [6] ______ , “ On topological classification of complete linear metric spaces,” Fund. Math., 56 (1965), 251-288. [7] ______ and A. Pefczynski, “ Some remarks on homeomorphisms of Fspaces,” B u ll. Acad. P o l. Sci., ser. sci., math., astr. et phys., 10 (1962), 265-270. [8]

______ , “ On extreme points in separable conjugate spaces,” Israel J. Math., 4 (1966), 262-264.

N O N-SEPARABLE BANACH SPACES

23

[9] H. Corson and V. Klee, “ Topological classification of convex sets,” P ro c . Symp. P u re Math., 7,

“ Convexity,” Amer. Math. Soc., 37-51.

[10] R. 0. Davis, “ A norm satisfying the Bernstein condition,” Studia Math., 29 (1967), 219-220. [11] M. M. Day, “ Strict convexity and smoothness of normed spaces,” Trans. Amer. Math. Soc., 70 (1955), 516-528. [12] M. Eidelheit, “ Zur Theorie der Systeme linearen Gleichungen,” Studia Math., 6 (1936), 139-148. [13] M. I. Kadec, “ On homeomorphism of certain Banach spaces (R ussian),” D okl.

Akad.

Nauk SSSR (N. S.) 92 (1953), 465-468.

[14] ______, “ On topological equivalence of uniformly convex spaces (Rus­ sia n ),” Usp. Mat. Nauk, 10 (1955), 137-141. [15] ______, “ On connection between weak and strong convergence (Ukrain­ ian ),” D opovid i Akad. Nauk Ukrain., RSR 9 (1959), 465-468. [16] ______ , “ On strong and weak convergence (R ussian),” D o k l. Akad. Nauk SSSR (N. S) 122 (1958), 13-16. [17] ______ , “ On topological equivalence of cones in Banach spaces (Rus­ sian), ” ibidem, 162 (1965), 1241-1244. [18] ______, “ On topological equivalence of all separable Banach spaces (R ussian),” ibidem, 167 (1966), 23-25. [19] V. Klee, “ Mappings into normed spaces,’ ’Fund. Math., 49 (1960), 2534. [20] ______ and R. G. Long, “ On a method of mapping due to Kadec and Bernstein,” A rchiv. der Math., 8 (1957), 280-285. [21] D. Maharam, “ On homogeneous measure algebras,” P roc. Nat. Acad. Sci., USA, 28 ( 1942), 108-111. [22]

E. Michael, “ Convex structure and continuous selections,’ ’Canad. J. Math., 11 (1959), 556-576.

C. BESSAGA AND M. I. KADEC

24

[23] A. Pefczynski, ‘‘Linear extensions and averagings of continuous functions/’ Rozprawy Matematyczne, in print. [24] S. Troyanski, “ On topological equivalence of cQ(i H will be said to be a fundamental sequence from X to Y provided for every neighborhood V of Y (in the space H) there exists a neighborhood U of X (in H) such that the restrictions fn/U, fn 1 / U are homotopic in V for almost all n, that is, there exists a homotopy V such that cf>(x, 0) = fn(x),

: X -> Y is a map, then there exists a

map f: H -> H such that f(x) = 0 (x ) for every point x e X.

Setting f

= f

for every n = 1,2, ..., we get a sequence |fn! of maps which is a fundamen­ tal sequence from X to Y. One proves easily that if

X -> Y is another map homotopic to in

every neighborhood V of Y and if f ' is any map of the space H into it­ self satisfying the condition f ' ( x) = '(*) for every x e X, then setting f^ = f ' for every n = 1,2, ... one gets a fundamental sequence |f^i from X to Y homotopic to the fundamental sequence {fnj. Thus, if we denote, for every map cf> : X -» Y, by [0 lw the class of all maps of X into Y homotopic to in every neighborhood of Y (let us call such a class [ X is said to be the fundamental identity class for X. Manifestly, in general, there exist fundamental classes which are not

Y.

COMPACT SUBSETS OF THE H ILB E R T SPACE

generated by any map.

27

However, in the case when Y is an ANR-space,

one proves easily that the notion of the weak homotopy class is the same as the notion of the homotopy class and every fundamental class [f ]: X -* Y is generated by a map cf>: X -» Y.

In this case we have a one-to-one corre­

spondence between homotopy classes of maps of X into Y and the funda­ mental classes from X to Y.

Thus, in this special case the notion of the

fundamental class differs only formally from the notion of the homotopy class. For arbitrary compacta the situation is different and it is to some ex­ tent analogous to the situation which we have in the theory of real numbers, as given by Cantor.

The weak homotopy classes of maps play the role of

the rational numbers, the fundamental c la sse s—the role of real numbers.

§3. COMPOSITION OF FU ND AM ENTAL CLASSES Let X, Y, Z be three compacta lying in the Hilbert space H and let {fni be a fundamental sequence from X to Y, and fgn!, a fundamental se­ quence from Y to Z.

One sees at once that the maps gn^n: H

tute a fundamental sequence from X to Z.

H consti­

Moreover, if Jfn'} is another

fundamental sequence from X to Y homotopic to the sequence {f }, and if Ig^i is a fundamental sequence from Y to Z homotopic to the sequence {g nS, then the fundamental sequences SgnfnS and f ^ f ^ S are homotopic. It follows that the fundamental class from X to Z with the representative |gnfnl depends only on the fundamental classes [ f ] with the representa­ tive !fn! and [g ] with the representative lg n!. We will denote this funda­ mental class by [g ] [f ] and we will call it the com position of the funda­ mental classes [ f ]

and [ g ].

Evidently, if the fundamental class [ f ] is

generated by a map (f> : X -> Y and the fundamental class [g ] is generated by a map if/ : Y ^ Z, then the composition [g ][f]: X -» Z is generated by the map if/cf) : X -> Z. One sees at once that, for any three fundamental classes [f ], [g ], [h_], the composition [h ]([g ][f ]) is defined if and only if ([h ] [ g ] ) [ f ] is defined.

K AR E L BORSUK

28

In this case the associative law holds: [ h ]([g ][f ]) = ([h ] [ g ] ) [ f ]. notes this triple composition by [h ] [ g ] [f ].

One de­

It is defined if and only if both

compositions [g ][f ] and [h ][g ] are defined. Moreover, one sees easily that if [i y ] ‘ Y -> Y is the fundamental identity class for Y, then for each fundamental class [f ]: X -> Y the composition [i_ y ][fJ : X -> Y coincides with [ f j , and for every fundamental class [g ]: Y -> X the composition [g ][i y]: Y -> X coincides with [g ].

It follows that we obtain a category if

we consider the compacta lying in H as objects and the fundamental clas­ ses as mappings.

§4.

Let us call this category the fundamental category.

FU ND AM ENTAL DOMINATION AND FU ND AM ENTAL E Q U IVALEN C Y A fundamental class [g ]: Y -> X is said to be a right inverse of the

fundamental class [ f j : X -» Y if the composition [f ] [ g ]: X damental identity class for Y.

Y is the fun­

The fundamental classes [ f j : X

Y for

which there exists a right inverse will be said to be rightly inversible.

If

there exists a rightly inversible fundamental class [ f j : X -> Y, then we say that the compactum X fundamentally dominates the compactum Y. Let us notice that the relation of the fundamental domination does not depend on the position of compacta in the Hilbert space H. In fact, if X', Y', are two compacta in H homeomorphic to X and to Y respectively, then one proves easily that if X fundamentally dominates Y, then X ' fundamen­ tally dominates Y'. Moreover, let us observe that the relation of the fundamental domination is a generalization of the relation of the homotopy domination in the sense of J. H. C. Whitehead ([5], p. 1133).

In fact, if X homotopically dominates

Y, then there exists a map f: X -> Y and a map g: Y -> X such that fg: Y -* Y is homotopic to the identity.

It follows at once, that the fundamental class

[ g ]: Y -» X generated by the map g is a right inverse of the fundamental class [ f j : X -> Y generated by the map f. Hence X fundamentally domi­ nates Y.

29

COMPACT SUBSETS OF THE H ILB E R T SPACE

One proves easily that if [ f j : X -> Y and [g ]: Y -> Z are rightly in­ versible fundamental classes, then the composition [g ][f ]: X -> Z is also a rightly inversible fundamental class. fundamental domination is transitive.

Consequently the relation of the Manifestly it is also reflexive, but

in general it is not symmetric. It allows to introduce a classification of compacta based on their global topological properties. A fundamental class [f_]: X -> Y will be said to be inversib le if there exists a fundamental class [ g ]: Y -> X such that both compositions [f ][g ]: Y -> Y and [g ] [ f ]: X -> X are fundamental identity classes. Then [g ] will be called the inverse of [ f j .

Let us observe that, for every funda­

mental class [ f j : X -» Y, there exists at most one inverse fundamental class [ g ]: Y -> X. [ g '] = [ g ' K t L M

In fact, if [g '] : =

Y -» X is also an inverse of [ f j , then = [i L

Now we can generalize the classical notion of the homotopy type due to W. Hurewicz ([3], p. 525) as follows:

Two compacta X, Y C H are fun­

damentally equivalent if there exists an inversible fundamental class [ f j : X -» Y.

One sees at once that this relation is reflexive, symmetric

and transitive.

Hence the collection of all compacta lying in the Hilbert

space H decomposes uniquely into disjoint classes of fundamentally equiv­ alent compacta.

These classes will be said to be fundamental types.

It

is clear that two compacta of the same homotopy type belong to the same fundamental type.

The converse is not true for arbitrary compacta, but it

is true for ANR-spaces.

Two ANR-spaces of the same fundamental type

have necessarily the same homotopy type. The following example illustrates to some extent the sense of these notions.

Let X and Y be two continua lying in the Euclidean plane E 2,

r\

which we consider as embedded in the Hilbert space H. If E — X and E 2 — Y have the same number of components, then X and Y are of the same fundamental type.

It follows that the collection of all fundamental

types of plane continua is only countable.

However one shows easily

K A R E L BORSUK

30

that the cardinality of the collection of all homotopy types of plane compacta is 2 ^ ° .

§4a.

EXTENSION OF FUNDAM ENTAL SEQUENCES FUNDAM ENTAL RETRACTIONS Let J[ = jfnl be a fundamental sequence from X to Y and i_' = jf^j

a fundamental sequence from X" to Y.

If X C X" and

fn(x) = f^(x) for every point x e X then the fundamental sequence f_ is said to be a restriction (to X) of the fundamental sequence f_', and the fundamental sequence f/ is said to be an extension (onto X ') of the fundamental sequence f_ . One can prove that the existence of an extension of a fundamental se­ quence _f from X to Y onto a compactum X / D X depends only on the fundamental class of _f . The notion of the extension for fundamental sequences allows to ex­ tend the notion of the retraction.

A fundamental sequence r_ - jrn} from

X to Y C X is said to be a fundamental retraction o f X to Y, if rfl(y) = y for every point y e Y. Manifestly, if

Then Y is said to be a fundamental retract of X .

is a fundamental retraction of X to Y and i_ is a funda­

mental sequence from Y to Z, then the composition f_r_ is an extension of the fundamental sequence f_ onto X. Now we can extend also the notions of the absolute retract and of the absolute neighborhood retract.

A compactum Y is said to be a fundamen­

tal absolute retract (F A R ) provided it is a fundamental retract of every compactur X D Y, and Y is said to be a fundamental absolute neighborhood retract (F A N R ) provided for every compactum X 3 Y there exists a compact neigh­ borhood Z of Y in X such that Y is a fundamental retract of Z.

Among

plane sets, FAR-sets are the same as continua which do not decompose the plane and the FANR-sets are the same as compacta with finite number of components which decompose the plane into a finite number of regions.

COMPACT SUBSETS OF THE H ILB E R T SPACE

31

Many theorems on AR-sets and ANR-sets can be extended onto FAR-sets and FANR-sets.

§5.

FU N D A M E N TA L CLASSES AND HOMOLOGY GROUPS The classical definition of the homology groups of a compactum X, as

given by L. Vietoris ([4 ]), is based on the notions of the 8 -simplex and of the true cycle.

Let us recall shortly these definitions.

By an n-

dimensional 8 - sim plex in X (where 8 is a positive number) one under­ stands a system aQ, a^ ,..., an of points of X with diameter less than 8. The linear forms

k

=

a x + a2 a 2 + •*' +

where a. are oriented

n-dimensional simplexes in X and a^ are elements of an Abelian group 21, are said to be n-dimensional chains in X over St. By a true n-dimensional c y c le in X over 31 one understands a sequence y = {y^}, where y^ are 8 ^-cycles in X over St with 8^ converging to zero and if for every k = 1,2, ... there exists an 8^-chain in X over St having

as its

boundary. If one defines, in the standard way, the addition of n-dimensional true cycles in X over St and the relation of the homology for them, then one gets the group Z n(X, St) of n-dimensional true cycles in X over St and its subgroup B n(X, St) consisting of all true cycles homologous to zero in X. The factor group Z n(X, St)/Bn(X, St) is the n-dimensional homolo­ gy group Hn(X, St) of the space X over

St .

If X is a subset of the Hilbert space H, then let us modify the basic definition of the 8-simplex (a Q, a^, ..., an) in X so that, instead of the hypothesis a. e X, let us assume that a- e H and that p (a., X) < 8. One easily sees that this modification is rather unessential and it leads to the same homology groups of X as the classical definition of Vietoris. Now, if we have two compacta X and Y lying in the Hilbert space H and if £ = {f^} is a fundamental sequence from X to Y, then it is easy to prove that, for every n-dimensional true cycle (in the modified sense) y = |y^|, there exists an increasing sequence ji^l of indices such that

32

K A R E L BORSUK

the sequence

)S is an n-dimensional true cycle in Y over 91. More­

over, one proves that the homology class of this true cycle doe not depend on the choice of the sequence li^i and it does not change if one replaces the true cycle y by another true cycle y ' homologous to y in X, or if one replaces the fundamental sequence Jf^j* by another representative of its fundamental class.

It follows that if [y] is an element of the homology

group Hn(X, St) with the representative y = jy^S, and if the fundamental sequence jf^S is a representative of the fundamental class [f_]: X -> Y, then assigning to [y] the element of the homology group H (Y, SI) with the representative ping the group Hfl(X,

)| one obtains an uniquely defined function map­ 31) into the group H (Y, 91). It is evident that this

function is additive, hence it is a homomorphism, which we call the homo­ morphism induced by the fundamental class [f ]. Let us denote it by [f ]* . Hence [f > :

Hn(X, 31) -» Hn(Y , * ) ;

One sees easily that 1) if [ f j :

X -» Y and [ g ]: Y -» Z are fundamental classes, then their

composition [g ] [f ]: X -> Z induces the homomorphism ([g ][f])*

= [g ]* [f_ ]*

;

2) if 0 : X -» Y is a map, then the fundamental class [ f j : X -> Y gen­ erated by cf> induces the same homomorphism [ f j * : Hn(X, 31) -> H (Y, 91) as the map cf> . In particular, the fundamental identity classes induce the identity homomorphisms. It follows from 1) and 2) that 3) if X fundamentally dominates Y, then the group H (Y, 31) is iso­ morphic to a direct factor of the group Hn(X, 9l)> i-e-» there exists a group © (depending on n and 91) such that Hn(X, 91) is isomorphic to H (Y, 91)

x S. 4) if X and Y are fundamentally equivalent, then the group Hn(X, 91)

COMPACT SUBSETS OF THE H ILB E R T SPACE

33

is isomorphic to the group Hn(Y, 91). Thus we infer that, if one assigns to every fundamental class [ f j from X to Y the induced homomorphism [fj* :

Hn(X, 91) - Hn(Y, 91) ,

then one gets a covariant functor Hn from the fundamental category to the category of Abelian groups.

§6.

R ELA TIVIZA T IO N As it was already remarked, the notion of the fundamental class may

be considered as a generalization of the notion of the weak homotopy class. For the sake of brevity, I limited myself to the case of fundamental clas­ ses from one compactum X to another compactum Y.

But a systematic de­

velopment of the theory requires the extension of the basic notions onto the case where instead of two compacta lying in the Hilbert space H, one considers two pairs of compacta (X, XQ) and (Y, YQ) lying in H. A pair (U, U Q) of subsets of H will be said to be a neighborhood o f the pair (X, XQ) provided U is a neighborhood of X and U Q is a neigh­ borhood of XQ. By a fundamental sequence from (X, XQ) to (Y, YQ) one understands a sequence of maps ffl: H -> H such that for every neighbor­ hood (V, V Q) of the pair (Y, YQ) there exists a neighborhood (U, U Q) of the pair (X, XQ) such that for almost all n the formula 0 n(x) = fn(x) for every point x e U defines a map c/>n: (U, U Q) -> (V, V Q) and that (Y, Y Q) induces a homomorphism [ f j * : Hn(X, XQ, 91) -> Hn(Y, Y q, 91) of each homology group of the pair

34

K A R E L BORSUK

(X, XQ) into the corresponding group of the pair (Y, YQ).

The homomor­

phism induced by the composition of two fundamental classes is the com­ position of the homomorphisms induced by these classes.

§7. HOMOTOPY GROUPS Let us add some remarks concerning the relations between the homoto­ py groups induced by the fundamental classes.

It is evident, that the clas­

sical notion of the homotopy groups is not suitable for this aim. In fact, there exists a plane continuum X decomposing the plane into exactly two regions and such that n j(X ) is trivial.

As we have already remarked, X

is fundamentally equivalent to a circle Y, for which the group 77^ (Y ) is trivial.

Thus we have two compacta of the same fundamental type with dif­

ferent first homotopy groups. However, there exists another notion of the homotopy groups, which is related to the classical one so as the notion of the homology groups in the sense of Vietoris or of Cech is related to the notion of the singular homolo­ gy groups.

An exposition of the theory of so modified homotopy groups,

based on the ideas of Cech, was given in 1944 by D. E. Christie [2].

We

proceed here by another way, not using the notion of the net homotopies considered by Christie, but using the operation of the homotopy join (see [1], p 46) applied, instead of to maps (as in the classical theory of homotopy groups) to the so-called approximative maps.

The notion of the approxima­

tive maps, being in fact only a slight modification of the notion of “ map­ pings towards a space” considered by Christie, may be defined as follows: Let X, Y be two compacta lying in the Hilbert space H. mative map o f X towards Y we understand a sequence

By an approxi­ of maps

5?n(Y, yQ) and that the compo­ sition of two fundamental classes induces the homomorphism being the com­ position of the homomorphisms induced by these fundamental classes. Since the fundamental identity classes induce the identity homomorphisms, we infer that the corresponding homotopy groups of two fundamentally equivequivalent compacta are isomorphic. Finally, let us remark, that if one assigns to each fundamental class [ f j from (X, xQ) to (Y, yQ) the induced homomorphism [ f j * : nn(X , xQ) -» i n(Y, yQ) , then one gets a covariant functor IIn from the fundamental category to the category of groups (Abelian, if n > 1). If xQ = yQ e Y C X and _f is a fundamental retraction of X to Y, then one proves easily that the homomorphisms of the homology and homotopy groups induced by f_ are so-called r-homomorphisms (see [1], p. 32).

It

follows in particular that all homology and homotopy groups of a fundamen­ tal absolute retract are trivial, and the homology and homotopy groups of a fundamental absolute neighborhood retract are r-images of the correspond­ ing groups of a polyhedron.

K A R E L BORSUK

36

REFERENCES [1] K. Borsuk, Theory of Retracts, Monografie Matematyezne 44, Warszawa 1967. [2]

D. E. Christie, Net homotopy for compacta, Trans. Amer. Math Soc., 56 (1944), pp. 275-308.

[3] W. Hurewicz, Beitrage zur Topologie der Deformationen III, Proc. Ak. Amsterdam, 39 (1936), pp. 117-125. [4]

L. Vietoris, Uber den hoheren Zusammenhang kompakter Raume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann., 97 (1927), pp. 454-472.

[5] J. H. C. Whitehead,

On the homotopy type of AN R ’s, Bull. Amer. Math.

Soc., 54 (1948), pp. 1133-1145.

Warsaw

SOME THIN SETS IN FR ECH ET SPACES H. H. CORSON

Let s be the countable product of real lines, and let l 2 denote all square summable sequences.

It has been known for some time that the

complement of a point in l 2 is homeomorphic to l 2, with a similar result holding for s . This follows from results of 0. H. Keller, V. Klee, and R. D. Anderson.

In order to prove that the complements of other subsets

of these spaces are homeomorphic to the original space, one approach has been to straighten out these subsets with respect to a topologically com­ plemented subspace and then remove one point at a time.

We will show

how this straightening can be accomplished in a linear way which depends only on the classical theorems of functional analysis, in contrast to the other methods which are ad hoc topological techniques. This follows a suggestion of C. Bessaga at the L. S. U. conference that it might be inter­ esting to prove results of this type by the methods of functional analysis. (See [1] for the necessary theorems used below.) Our results hold in the general context of Frechet spaces, that is, complete metric locally convex vector spaces. THEOREM.

L e t X be a Frech et space with in fin ite dimensional

c lose d subspaces E p E2, ... . L e t K be a o-co m p a ct subspace o f X. Then there are linearly independent points x- € E- such that the clo s e d linear span o f {x-i has only 0 in common with K. P ro o f:

Since X is locally convex and K \ {0} has the Lindelof prop­

erty, we can find a countable collection \K^\ of compact convex subsets of X \ {0! which cover K \ |0}.

Hence we may assume that K- C K, all i.

37

H. H. CORSON

38

By induction, we will show that one may find closed subspaces X- of finite co-dimension in X and points x. e X- H

with these properties:

(a) X ^ C X - X U j } , (b) Y- HR = {Oi where Y- is the linear span of ix 1,x 2, ...,x-S, and (c) Ki+1 + Y- does not meet Xi+1, where Y Q = \0\. In fact, X j

can be chosen asthe null space of a continuous linear

functional separating 0 from K j, by the separation theorem.

Then X 1 H E x

is closed and infinite dimensional, hence not locally compact; so a cate­ gory argument shows that there is a point x 1 in ( X 1 D E 1) \ K . Now suppose that x 1, . . . , x m and X 1, . . . , X m are chosen.

It is easy to

see that Y _ + . 1 is closed and convex, and by (b) 0 / Y + . m m+ i m m+ i Again by the separation theorem one may pick Xm+ 1 of finite co-dimension such that Xm+ 1 does not meet Ym + Km+ 1 , and we may obviously suppose that Xm+1 C Xm \ i x j . contains some point

+

As above, (X m+1 0 Em+ 1) \ (K + Ym)

Clearly (a ) and (c ) are valid.

For (b) suppose

that axm+ 1 + S'*1 a-x- e K \ iO}. Then a ^ 0 for otherwise Ym f! K 4 {0}. If a ^ 0, then

m+1 1 would be inK + Y _ which is m not the case. To complete the proof, suppose that y e K \ iOi and y =

lim % a ^ x ; m -* oo .

where each of the sums is finite.

,

By the Hahn-Banach theorem there is a

continuous linear map of X onto Y- such that X^+1 goes to 0, since (a) implies that X-+1 fl Y- = {0}.

Since (a) also implies that {x p ...,x -i is a

set of linearly independent points in X, it follows that a- =

lim

a?1

m-> oo

exists for each i . Suppose that y e

. Then y = a ^

lim

X

+ ••• + a ^ ^ x ^ j + z where

a? xi e Xi r

i - 10 It follows that z e K-q + Y j0—1, contradicting (c) and completing the proof.

SOME THIN SETS IN FRECH ET SPACES

COR OLLAR Y 1.

39

Given a a-com p a ct subset K o f £2 there is an infi­

nite dimensional close d subspace H o f 12 and a complementary subspace H / of f?2, each isom orphic to t 2, such that each translate o f H meets K in at most one point. P ro o f:

We may suppose that K is a linear subspace.

Pick any infinite

dimensional closed subspace E of t 2 such that E is of infinite co-dimension.

Let E- = E for all i.

Let H be the closed linear span of }x-S of

the Theorem, and let H / be its orthogonal complement.

Then it is well-

known that H and H ' are isomorphic to f?2 . Clearly H has the desired property. COR OLLAR Y 2. R eplace l 2 by s in C orollary 1. P ro o f: Partition the integers into infinite subsets N 1, N 2, ... . Let Ebe the subspace of s composed of all sequences whose support is con­ tained in N-. Let Ej

Let H be the closed linear span of the {x^S of the theorem.

be a closed proper subspace of E- such that E ^ is isomorphic to

s and E-' U }x-! generates E^. U fE / :

Let H ' be the closed linear span of

i > 1}.

To prove that. H and H ' are isomorphic to s it is only necessary to ob­ serve that Ej is isomorphic to E ^ x fax-: a real! and s is isomorphic to n ~ =1 E|. Clearly H has the desired property. COMMENT.

For the topological applications of the straightening theo­

rems it is sometimes enough to know that X is homeomorphic to X j x X2 where X- are infinite dimensional Frechet spaces and the image of K in X j x X2 meets each translate of X2 in at most one point. This can al­ ways be accomplished by applying the above Theorem together with the Bartle-Graves theorem [2].

It does not seem likely, however, that one can

assert that there is an isomorphism of X into some X 1 x X 2 and also spe­ cify the linear properties of X 1 and X2 in terms of those of X, as in Corollaries 1 and 2.

H. H. CORSON

40 Question.

Can one always find an isomorphism as in the Comment

above for some X 1 and X2 ? The answer is affirmative if every infinite dimensional closed subspace contains an infinite dimensional closed sub­ space which is complemented in X . Note.

If E- = X for all i and X is not a reflexive Banach space, then

the conclusion of the theorem holds under the assumption that K is crcompact in the weak topology.

REFERENCES [1] M. M. Day, Normed Linear Spaces, Berlin, Springer-Verlag, 1958. [2] E. Michael, Continuous selections, J. Ann. of Math. (2), 63 (19 56), 361-382.

UNIV E RSITY OF WASHINGTON, S E A T T L E

A REM ARK ON BAN AC H A N A L Y T IC SPACES

A. Douady (Faculte des Sciences, Nice)

L ’ Universite mene a tout, a condition d’en sortir. (French proverb.)

In [1], Banach analytic spaces were used as a tool for a moduli problem in complex analytic geometry.

However, the only interesting aspect of

these spaces is their use as intermediates in constructing an analytic space

whicheventually turns out to be (locally) finite dimensional.

I be­

lieve that there isno point in studying Banach analytic spaces for their own sake.

For instance, the fact that a topological space is homeomorphic

to a Banach analytic space gives no information about its topological type. It is likely that any complete metric space is homeomorphic to a Banach analytic space. Let us prove: PR O PO SITIO N .

Any com pact m etric space is homeomorphic to a Banach

analytic space. Proof.

Let X be a compact metric space.

Denote by d its distance and

by A the Banach algebra of Lipschitz functions from X to C provided with the norm ||f|| = sup |f(x)| + sup 1f(x) - f(y)| xfX xeX d(x, y) yeX x^y

41

.

JAMES EELLS, JR.

42

Consider the Banach space A ' dual to A with the norm topology, and the map 8 : X -* A ' defined by = f(x),

f €A .

We have the inequalities

I +do / y ) -

< d(-X’ y^’

X’ y e X ’

the second one is obvious and the first one is obtained by considering the function on X z

inf (d(x, z), d(x, y)) .

These inequalities show that 8 is a homeomorphism of X onto its image in A'. It is well-known that 8 (X ) is the spectrum of A, gebra homomorphism from A to C.

i.e., the set of al­

In other words, let B denote the

Banach space of bilinear maps from A x A to C, and the map from A ' to B © C defined by 0 ( f ) = (0, 0 (1 ) — 1), where e a , g) = f ( f g ) - f ( f ) f ( g ) ,

f,g 6 a .

Then cf> is analytic and §(X) = B q is the restriction, then we have the natural map C q (E ,E ')->

C q (E q,E /), which under mild conditions on (B ,B Q) is a Hurewicz fibra­ tion. See I. M. James, The space of bundle maps. Topology 2 (1963), 4559, where applications are made.

FIBRING SPACES OF MAPS

(D )

EXAM PLE.

closed subgroup.

53

Let G be a metrizable topological group and K a

A theorem of Michael [22] asserts that if K is com plete

and lo c a lly convex, there is a lo c a l section of the cos et map G-»G/K; therefore we have a homogeneous K-bundle. ing when G = G L(V ),

That is particularly interest­

the group of linear automorphisms of an infinite

dimensional Hilbert space, topologized as an open subset of the Banach algebra L (V ) of endomorphisms of V. is an absolute retract, and G ^G /K

For then by a theorem of Kuiper G

is a universal K-bundle.

Similarly, if V is a separable infinite dimensional Hilbert space and U L S(V ) denotes the group of unitary operators with its strong operator topology, then U L g(V ) is metrizable and contractible [5, §10]. For any locally compact group K we take its Lebesgue space L 2(V ) using Haar measure; then K can be imbedded as a closed subgroup of U L g( L 2(K )) — and whenever it has a local section we have a universal K-bundle. REMARK.

It would be interesting to know whether every compact Lie

group can operate principally on every C°°-manifold X modeled on the infinite dimensional separable Hilbert space E. principally and smoothly on E its e lf.

It is true that G operates

For G can be represented faith­

fully as a closed subgroup of an orthogonal group 0 n, which operates principally and analytically on the Stiefel manifold V n(E ) of orthonormal n-frames in E.

But by a theorem of Bessaga, coupled with a remark made

to me by Husemoller,

V n(E ) is C°°-diffeomorphic to E.

5. Certain fibrations o f mapping spaces (A )

That certain evaluation maps define a sort of fibration (with suffi­

cient structure to insure the covering homotopy theorem for cells; such a map is called a Serre fibration) follows from Borsuk’s extension theorem [13, 24].

Such fibrations of path spaces have been used extensively in the

celebrated thesis of Serre. T H E O R E M [24].

L e t S be a com pact space and A a close d subspace.

Then for any absolute neighborhood retract M the evaluation map f: C(S,M )-*C(A,M ) defined by f(x) = x|A is a Serre fibration over the image.

JAMES EELLS, JR.

54

Here C(S,M) denotes the mapping space of all continuous maps of S into M, topologized by the compact-open topology. A similar theorem with somewhat different hypotheses is the following [17]: L e t S be a loca lly compact absolute neighborhood retract and A a closed subset which is also an absolute neighborhood retract. any space M the evaluation map f:C(S,M) -> C(A,M)

Then for

is a Hurewicz fibra­

tion over the image. Suppose that M is a C r+2-manifold (r > 0) modeled on a Cr+2-smooth Banach space, and that S is a compact space.

Then C(S,M) and C(A,M)

are C r-manifolds; furthermore, then f is a C l -map, and is loca lly C °trivial.

If S is metrizable and A has separable frontier, then f is a

foliation map (see [14, §11]).

In that case the sequence (1) becomes

0 -» Ker f* -» C(S,T(M)) -> f"- 1C(A,T(M )) -> 0 ; and we can construct a locally Lipschitz splitting to apply Theorem 3B, giving a differentiable interpretation of that result. (B )

If in the above theorem S and A are compact C°°-manifolds and

f: A h>S a C^-embedding, then for any C°°-manifold M the induced C°°-map f: Cr(S,M )-*Cr(A,M) is a foliation map over its image (0 < r < oo). We then obtain [25] the THEOREM,

f: Cr(S,M )-*Cr(A,M) is a loca lly C ° -triv ia l fibration over its

image. The space Emr(S,M) of C r-embeddings of S in M is an open sub­ manifold of Cr(S,M).

The following result is due to Thom [29], Cerf [3],

and Palais [25]; see also [20]: THEOREM,

f: Emr(S,M)->Emr(A,M) is a loca lly C ° -triv ia l fibration over

its image (2 < r < °o). We can suppose that S and A have boundaries; and S need not be com­ pact. Analogously [4], le t ®

be an open subgroup of ®(S).

ates principally on Emr(S,M).

Then 5) oper­

FIBRING SPACES OF MAPS

EX A M PLE .

Taken together with Example 4B, we find the following result,

of interest in the calculus of variations: (S) > 2,

55

If S is a closed surface of genus

then the orbit map Emr(S,M) ->Emr(S,M)/®0 (S)

is a homotopy equi­

va len ce. (C ) We fix an embedding h:S->M and thereby view S as a submani­ fold of M. Denote by Emr(S,M;A) the totality of C r-embeddings of S into M which induce the identity on A.

Let J^Emr(S,M;A) denote the space

of r-jets of these embeddings which are tangent through order r at every point of A(1 < r < oo). The following result — and its many variants — play a fundamental role in the theory of Cerf [3,4]: THEOREM.

The canonical map Emr(S,M;A) -> J^Emr(S,M;A)

is a lo ca lly triv ia l fibration. E X A M PLE .

If M is a compact m-manifold without boundary and Dp the

closed p-dimensional Euclidean disc centered at 0 (p < m), then ?: Em°°(DP,M) - jjEm^CDP.MjO) is a locally trivial fibration.

Now we have a canonical identification of

jQEm°°(DP,M;0) with the Stiefel manifold V m p(M) of p-frames of M. Furthermore, f has aspherical fibres, whence we have a homotopy equi­ valence T. Em°°(DP,M) -> V m (M). (D ) A corresponding theory of fibrations for immersions is more deli­ cate and difficult. The space Imr(S,M) of C r-immersions of S in M is also an open submanifold of C r(S,M).

The following result was first established (in a

slightly different form) by Smale and Thom ([29]; see [26] for further biblK ography), with refinements made by Hirsch-Palais. THEOREM .

I f dim S < dim M,

then the res triction map f: Imr(S,M) ->

Imr(A,M) is a Hurewicz fibration over its image (2


Y is a fim -Sard map, provided that

r > max (n —m, 0), where n = dimX, m = dim Y, and fi

is a Lebesgue

m -measure on Y. We now consider the possibility of obtaining a theorem of that sort for manifolds modelled on Banach spaces.

There have been some results in

this area—notably that of Smale [13] to the effect that Fredholm maps be­ tween smooth manifolds, modelled on separable Banach spaces, are Sard maps.

Such results have been highly restrictive; on the other hand, ex­

amples, such as that of Kupka [8 ], show that strong restrictions are neces­ sary. For many applications, however, it will suffice to know that a given map can be approximated (in a suitable sense) by a Sard map. In that di1 Research partially supported by NSF Grant GP-4216.

2

Research sponsored by the Air Force Office of Scientific Research, Office of

Aerospace Research, United States Air Force, under AFOSR Grant Nr. 1243-67.

59

JAMES E ELLS AND JOHN M cALPIN

60

rection we have established the THEOREM:

L e t X and Y be smooth manifolds m odelled on a H ilbert

space E and a Banach space F, respectively. ble.

Suppose that X is separa­

Then the Sard maps are dense in the fine topology on C °(X , Y). The proof is reduced to a local situation by standard techniques.

The

approximation itself is achieved by using a smooth partition of unity, whose summands are specially constructed functions (using the method of scal­ loping [9]) with critical points that can be kept under control—so that we can use the Morse-Sard theorem. The above theorem permits us to extend a standard transversality theorem as follows: THEOREM:

theorem.

L e t X and Y be smooth manifolds as in the preceding

L e t B be a closed direct submanifold.

Then for any smooth map

(ft: X -> Y and any smooth function 8 : X -> R (> 0) there is an E -approxi­ mation ift: X -» Y of eft which is transversal over B. A smooth submanifold B of Y is direct if every tangent subspace B(y) is a direct summand of Y(y). As an application (of a special case) we can show that every c lose d subset of a smooth H ilb ert manifold has a fundamental system o f smooth neighborhoods.

(An open neighborhood is called smooth if its boundary is

a smooth 1-codimensional submanifold.)

This should be viewed in conjunc­

tion with the basic separation theorem [5, p. 412] for closed convex subsets of a Hilbert space. We are also able to establish a form of the Atiyah-Thom duality Theorem for Hilbert manifolds.

C O R N E L L UNIV. C H U R C H ILL C O L L E G E , UNIV. OF C O LORADO

61

AN A PPRO XIM ATE MORSE-SARD THEOREM

BIBLIO G RAPHY [1] R. Abraham and J. Robbin, Transversal Mappings and F low s, Benjamin (1967). [2] M. Atiyah “ Bordism and Cobrodism,” P roc. Camb. P h il. Soc., 57 (1961), 200-208. [3] R. Bonic and J. Frampton, “ Smooth functions on Banach Manifolds.” J. Math. Mech. 15 (1966), 877-898. [4] P. Conner and E. Floyd,

“ Differentiable periodic maps,”

Erget.

Math. Bd., 33 (1964). [5] J. Dunford and J. Schwartz,

Linear Operators, Interscience, 1958.

[6 ] J. Eells, “ A setting for global analysis, Bull. Am. Math. Soc. 72 (1966), 751-807. [7] E. Feldman, “ The Geometry of Immersions I, TAMS, 120 (1966), 185224. [8]

I. Kupka, “ Counterexample to the Morse-Sard theorem in the case of Infinite Dimensional Manifolds,” P roc. A. M. S., 16 (1965), 954-F.

[9] S. Lang, Introduction to D ifferentiable Manifolds, Interscience, 1962. [10]

A. Sard, “ The measure of the critical values of differentiable maps,” Bull. A. M. S., 48 (1942), 883-890.

[11] ______ , “ Images of critical sets,” Annals o f Math., 68 (1958), 247-259. [12] ______ , “ Hausdorff measure of critical images on Banach manifolds,” Am. J. Math., 87 (1965), 158-174. [13]

S. Smale, “ An infinite dimensional version of Sard’s theorem,” Am. J. Math., 87 (1965), 861-866.

[14] R. Thom, “ Quelques proprietes globales des varites differentiables,” Comm. Math. H elv., 28 ( 1954), 17-86. [15] H. Whitney, “ A function not constant on a connected set of critical points,” Duke Math. J., 1 (1935), 514-517.

MORSE THEORY FOR CLOSED CURVES H A LL D O R I. ELIASSON

Introduction. Let X be a complete Riemannian manifold of class C°°, modeled on separable Hilbert spaces and let F be a C°° function on X. In order to provide Morse theory for F, we must prove:

F satisfies condition (C ) of

Palais and Smale [9] and F has only nondegenerate critical manifolds Bott [1], Meyer [10], Wasserman [12]. We will here discuss the energy function defined on a Hilbert manifold of closed curves and present some intrinsic methods, which are both prac­ tical and extendable to more general variational problems.

The main dif­

ference between the case of closed curves and curves connecting two fixed points is, that we have necessarily degenerate critical points in the first case (critical manifolds of dim > 1).

Several people have observed,

that the proof of P alais [8 ] for condition (C ) in the second case can also be used for closed curves; we will, however, give an independent com­ pletely intrinsic proof here (§3).

We will first outline the theory and then

present the proofs and the tools from differential geometry used.

A more

general and complete introduction to the differential geometric aspect can be found in [5]. §1. Statement o f results. In what follows M is a compact Riemannian manifold of class C°°, connected and without boundary, bundle and exp: (or circle).

r: TM -» M w ill denote its tangent

TM*-* M the exponential map. S = R/Z is the 1-torus

We will use H ° and H 1 to denote the class of maps, which

63

HALLD O R I. ELIASSON

64

are square integrable and have square integrable derivatives respectively. H 1 (S, M) the set of maps x: S -> M of class H 1 is a Hilbert manifold of class C°°.

The tangent space at x can be identified with the Hilbert

space of H 1-fields

S -» TM along x, t o £ = x, or equivalently with

H (x t), sections of class H t

of M by x.

in the pull-back x r of the tangent bundle

£ -> exp ° £ can be interpreted as the exponential map for

H 1(S, M) given by a certain connection for this manifold [5] and provides it in particular with charts.

A Riemannian metric can be defined in

H ^S , M) by:

< £ 77 >j

=

f 0

< £ (t ), 77(0 >dt +

< V f ( t ) , V 77(1 ) >dt

f 0

= V>0 + < v €>V v>0 ’ where < , > denotes the Riemannian metric on M and A the correspond­ ing covariant differentiation. T H E O R E M 1.

H *(S , M) is a com plete Riemannian manifold of class

C°°, modeled on separable H ilbert spaces. The energy function

E (x) = 1

f

||||2 dt = i

2 J0

H^xHg

2

is now a C°° function E : H1 (S, M) -> R and its derivative, as a section in the cotangent bundle, is

d E (x ) • rj = Qf

rj € HX(x *T M ) .

Suppose x is a geodesic, i.e., x is of class

and V dx = 0, then

dE(x)- rj = - < Vdx, 7j>0 = 0, thus x is a critical point of E, dE(x) = 0. Conversely, if x e H 1(S, M) and dE(x) = 0, then Q = 0 for all rj e H 1(x *T M ) is easily seen to imply, that x is of class C°° (regularity) and thus V TM be the curvature tensor on M, put V 2 =

A (Laplacian) and define Kx: H 1(x*r) -» H 1(x*r)

Kx • f =

THEOREM 4.

R ° ( H 1 (x*r) ax

is the self-ad joint Fredholm operator:

= i +

.

COROLLARY 1. I f x is a c r itic a l point o f E, then wehave an orthogo­ nal decom position: H 1(x *r) = T ° + T “ + T + is the sum of eigenspaces o f Ax corresponding to zero, negative and p os i­ tiv e eigenvalues A. Moreover T°

= k e r (- A + Kx) C C °°(x *r)

T “ = 2 A < o k e r ((A - l )A + Kx ~ A) C C°°(x*r) and Nullity (x) = dim T x < oo Index (x) = dim T “ < oo .

.

H ALLD O R I. ELIASSON

66

P roo f: ( A - l ) A + Kx —A is an elliptic differential operator and there are only finitely many negative eigenvalues A of A x-

D efinition .

Given a C°° curve c in

M, a C°° field a along c

is

called a Jacobi field, iff it satisfies the differential equation: V 2a + R ° (a, dc, dc) = (A - Kc) • a = 0 . We will say, that M has property (J), iff given a closed

geodesic c : S -> M

and a closed Jacobi field f : S -* TM along c, then there is an infinitesi­ mal isometry Y on M, such that £ = Y ° c . COROLLARY 2. M imbedded in H 1 (S, M) as a submanifold (point curves) is a nondegenerate c ritic a l manifold of index zero of the energy function. P ro o f:

The imbedding i: M -> H 1 (S, M) is given by i (p )(t ) = p,

t e S.

Then T i (v )(t ) = v for all v e TM and obviously the Jacobi fields along x = i (p ) are exactly the constant maps into TpM, i.e.,

T x is the tangent

space to i(M ) at i(p ) = x. Moreover (observe that dx = 0 implies

Kx =

0 ): [ ( A - l ) A + Kx - A] *

0

has no periodic solutions if A < 0, as it reduces to the ordinary differen­ tial equation: =

£ 1 - A

COROLLARY 3.

in

TM P

.

Z e ro is an isola ted c r itic a l value of E and there is

a sm allest p os itiv e c ritic a l value, establishing the existen ce o f a closed geod esic o f minimal length on M. P roo f:

If there are no positive critical values, then M is contained as

a deformation retract of HX(S, M), which is absurd, as then the loop space of M is homotopically trivial, but not M. Using Theorem 3, the set

MORSE THEORY FOR CLOSED CURVES

67

of critical points with bounded energy is compact, thus the lower bound for positive critical values is taken and must be positive by Corollary 2 and Morse lemma (nondegenerate critical levels are isolated).

The existence

of closed nontrivial geodesics on even a simply connected manifold is a result obtained by Fet [ 6 ] and others, obviously we have one in any non­ trivial homotopy class of 771 (M) as a minimum point of E in the correspond­ ing component of H 1 (S, M). Let I(M ) be the group of isometries of M. I(M ) is a compact L ie transformation group of M and acts on H 1 (S, M) by composition (g, x) ^ g o x .

This map I(M )x H 1(S, M) -* H*(S, M) is of

class C°° and its tangent at (g, x) is (Y, £ ) -> Y ° g ° x + (dg o x) • £ , where Y is an infinitesimal isometry and

e H 1(x*r).

is an effective Lie transformation group of H 1 (S, M).

In particular I(M )

The orbits are com­

pact submanifolds and the tangent bundle of any orbit is obtained by re­ stricting the infinitesimal isometries to the curves in the orbit.

The energy

function is obviously invariant under this action, so every orbit I(M )* c, c a closed geodesic, is a critical manifold of E. THEOREM 5.

Suppose M has property (J), then E has only nondegen­

erate c ritic a l manifolds. P ro o f: Property (J) implies that T^ is exactly the tangent space at c to the orbit I (M) • c and as A Q is a linear homeomorphism of T “ + T * , the Hessian is nondegenerate on T ~ + T * . My conjecture is, that every (irreducible) globally Riemannian symmet­ ric space has property (J).

This is a similar condition as ‘ Variational com­

pleteness” required by Bott and Samelson [2] in their study of Morse theory for loop spaces.

Their proof of variational completeness in the symmetric

case, implies (J) for those Jacobi fields vanishing at some point of the curve, however more is needed if the rank is larger than 1 . Example.

M = Sn the unit sphere in Rn+1.

68

H ALLDOR I. ELIASSON

For every nonnegative integer j we have a nondegenerate critical mani­ fold Wj of the energy function. of length 2tt].

Wj is the submanifold of closed geodesics

We can identify WQ with Sn and Wj with the Stiefel mani­

fold of 2 -frames in Rn + 1: Wj = Vn + 1,2

= SOn + 1/ S O „_r

j > 1 .

We can easily solve the differential equation involved (Corollary 1) to com­ pute the index kj = Index (c) for c e Wj. We obtain kQ = 0 and k. = ( 2j - l)(n — 1)

for j > 1 .

Let aj be a sequence of numbers with E (Wj) < aj < E(W j+ 1 ), and put Xj = E- 1([0, aj]), j > 0.

j > 0

If H* denotes the singular homology

functor, we obtain from Morse theory ([10] and [12]) using any group G of coefficients: Hk(x r x j _ i ; G) = Hk _ k.(W j;G)

j > 0 ,

for the homology group in dimension k. THEOREM 6. F o r n > 2 we have for a ll k > 0 : oo Hk(H 1(S ,S n) ; G ) = Hk(Sn; G) +

P ro o f:

We prove H^(Xj) =

2

Hk _ (2j _ l ) (n _ l ) (V n + 1 2 ; G) .

o ^ k ^ i >^ i - 1 ^ usinS induction on j.

As this formula is trivial for j = 0, suppose it is true for j and consider the exact sequence -

- Hk + 1 (Xj+ 1 , X j) -

Hk(x j + r x j ) -

Hk(X j) - Hk(Xj+ 1 ) Hk _ i ( X j ) —

•

Now the first non-vanishing homotopy group of Vn + 1 2 in positive dimen-

MORSE THEORY FOR CLOSED CURVES

69

sion is 77’n_ i ( V n + 1 2) = Z if n is odd and = Z 2 if n is even (see [13]), therefore V n + 1 2 has nonzero homology at most in the dimensions 0 , n — 1 , n, 2n — 1. Then using

= kj + 2n —2 and n > 2 implies 2n —2 > n,

2n - 2 + n — 1 > 2n — 1 we see that if H ^(X j+1, Xj) ^ 0 then H^(Xj) = Hk - i ( x j> = 0 and if Hk(x j) ^ 0 then Hk( x j + i ' V = Hk + i (x j + r x j ) = 0 and so for all k : H ^(X j+1) = H^(Xj + 1, X j) + H^(Xj), which proves the theorem. Using Z 2 coefficients, we obtain for the Poincare polynomial (or series):

P (H 1(S, Sn); z) = P (S n; z ) + P ( V n + l j 2 ;z )z n- 1( l - z 2( n- 1) ) - 1 . This formula and a comparison with the polynomial having the circular connectivities or sensed circular connectivities of Sn as coefficients (see Bott [ l ] ) , suggests that those are exactly the mod 2 Betti numbers of the spaces H 1 (S, Sn)/ 0 2 and H 1 (S, Sn)/S02 respectively (where the ac­ tion of

0 2 is the unusal

rotation of the closed curves).

Using the information Hk(Vn + 1 2; Z) = Z if k = 0 , 2n —1 or n — 1, n and n odd and = Z 2 in case k = n — 1 and n even and = 0 elsewhere (as follows from our knowledge of

using Poincare duality and the

universal coefficient theorem), we obtain for the integer homology: If n is odd, then . ( Z for k = j(n — 1) and j(n — 1) + n for all j > 0 Hk(H (S, S )) = ( 0 otherwise. If n is even, then [ Z for k = 0 , ( 2j + l)(n - 1) + 1 j-j (H 1(S Sn)) — k

'

and (2j + l)(n — 1) for all j >

0

j Z 2 for k = 2j(n — 1) all j > 1 V 0 otherwise.

v This result agrees with Svare [11] and Eells [4], except in [4] the torsion

70

HALLD O R I. ELIASSON

(n even) is missing.

This however, is due to an incorrect statement in

Lemma (b) page 121 [4], the homomorphism j n: Hn(X ) -> Hn(X, Y *) is not an isomorphism for n even, but rather a multiplication by 2 : z -> Z, thus Hn(Y * ) = Z 2 and not 0.

Eells uses the Banach manifold C°(S, Sn) of

continuous maps and we know from a general theorem of Palais [7], that the continuous inclusion:

§2.

H1(S, M) C C°(S, M) is a homotopy equivalence.

The manifold H1(S, M). As before r : TM -* M is the tangent bundle and e x p : TM -> M the

exponential map.

Let D2 exp :

TM -> L(TM, e x p *T M ) denote the fibre

derivative of exp: exp (v + tu) 11 = 0 .

D2 exp (v) • u = Let

0 be an open neighborhood of the zero section in TM, such that

(r, exp) maps G diffeomorphic onto an open neighborhood of the diagonal in M x M.

Then for v eGp D2 e x p (v ):

TPM - T e x p v M

is a linear isomorphism. Given a C °° map c : S -> M, Gc = c * G is an open neighborhood of the zero section in the pull-back c*TM, and the correspondence C ° (0 c) j £ « - x = exp ° £ e C°(S, M) is one-to-one.

Moreover the transition exp ° ^

tions in GCl and GQ2 for two close F °

= exp o ^

between sec­

maps c ^, c2 is given by £ 2 =

where F is a fibre-preserving map of class C°°.

Now this holds

for any compact C°° manifold S and the main idea in the construction of manifolds of maps from S to M, Eells [3], is to use this 1: 1 correspond­ ence as a chart, that is by restricting it to the Banach space of sections in pull-backs by C°° maps c : S -> M used as model.

A class of Banach

spaces of sections serving this purpose has been axiomatically described

71

MORSE THEORY FOR CLOSED CURVES

by P alais [7] and also in [5] by the author. 0 < k < oo, spaces and the Sobolev

This class includes the C^,

spaces if k > Vi dim S.

We will

here only be interested in H°, H 1 and the case, where S is the circle. Let E and F be vector bundles over S, say pull-backs by C°° maps from S into M, and give them the Riemannian metric and connection in­ duced from the tangent bundle of M. We then have inner products in H °(E ) and H 1 (E ) by

0 =

f

! =

< £ » ? > o + < V £ V i7>0

< £ (t ), 7/(t)>dt

we denote the

corresponding norms by || ||Q and ||and

usual norm ||

in C ° (E ) by

= sup)|£(t)||

, define the

(t £ S).

We then have the following properties: We have continuous linear inclusions

Property 1.

H 1(E ) C C °(E ) C H °(E ) , where the first inclusion is completely continuous. In fact:

||£||0 < I lf I L We have a continuous linear inclusion

Property 2.

H 1(L (E , F » C L (HV(E), H " ( F » In fact:

||A-£||„ < 2

^

• H\\v

= 0, 1 .

.

Let 0 be an open subset of E projected onto S and let

Property 3.

e H 1(F )

f : 0 -» F

be a fibre map of class C°°, then f (£ ) = f ° £

whenever

£ € H 1(0 ) and the map f : H 1(0 ) -» H 1(F ) is continuous.

From these three properties one obtains easily the following fundamen­ tal lemma (see [5], §4). Lemma. map:

f~: H 1(0 ) -» H X(F ) is of cla ss C°° and its derivative is the

72

HALLD O R I. ELIASSON

D f” = (D 2£)': H*(G) -> H 1(L (E , F )) C L ( H 1(E), H 1(F )) , where D2f: 0

L (E, F ) is the fibre derivative of f, i.e ., D2f |0^ =

D (f | 0 t) for t e S. Then using the results described in §5 and 6 in [5] we obtain the following: (a) H 1(S, M) is a manifold of class C°°, modeled on the separable Hilbert spaces H1 (c *T M ) with c e C°°(S, M) and the map exp~:

H 1(0 C) -> H ^ M )

provides a chart, the natural chart centered at c . (b) We have C°° vector bundles H^CH^S, M)* TM) -> H ^ S .M ) with H ^ x ^ T M ) as the fibre over x e H ^S , M).

v = 0, 1 A local trivialization over

the natural chart at c is given by: (D 2 exp)’ : H X(Gc) x H ^ c + T M ) -» H ^ H ^ S , M )*TM ) . For v = 1 this bundle is naturally equivalent with the tangent bundle of H 1 (S, M) and the local trivialization above corresponds to the tangent trivialization (exp . T exp ) of the tangent bundle under the equivalence (if we put a (s , t) = e x p (£ (s ) + tr ](s )) with £,rj e H X(x*r), then