Topology Seminar Wisconsin, 1965. (AM-60), Volume 60 9781400882076

Table of contents :
CONTENTS
Preface
CHAPTER I: 3-Manifolds
MONOTONE DECOMPOSITIONS OF E^3
EQUIVALENT DECOMPOSITIONS OF E^3
ANOTHER DECOMPOSITION OF E^3 INTO POINTS AND INTERVALS
CRUMPLED CUBES
SEWINGS OF CRUMPLED CUBES WHICH DO NOT YIELD S^3
CANONICAL NEIGHBORHOODS IN THREE-MANIFOLDS
BOUNDARY LINKS
SURFACES IN E^3
ADDITIONAL QUESTIONS ON 3-MANIFOLDS
CHAPTER II: The Poincaré Conjecture
NOW NOT TO PROVE THE POINCARÉ CONJECTURE
MAPPING A 3-SPHERE ONTO A HOMOTOPY 3-SPHERE
CONCERNING FAKE CUBES
CHAPTER III:
ON CERTAIN FIRST COUNTABLE SPACES
REMARKS ON THE NORMAL MOORE SPACE METRIZATION PROBLEM
TWO CONJECTURES IN POINT SET THEORY
ALMOST CONTINUOUS FUNCTIONS AND FUNCTIONS OF BAIRE CLASS 1
CHAINABLE CONTINUA
THE EXISTENCE OF A COMPLETE METRIC FOR A SPECIAL MAPPING SPACE AND SOME CONSEQUENCES
FINITE DIMENSIONAL SUBSETS OF INFINITE DIMENSIONAL SPACES
TYPES OF ULTRAFILTERS
ADDITIONAL QUESTIONS ON ABSTRACT SPACES
CHAPTER IV: N-manifolds
TAMING POLYHEDRA IN THE TRIVIAL RANGE
SOME NICE EMBEDDINGS IN THE TRIVIAL RANGE
APPROXIMATIONS AND ISOTOPIES IN THE TRIVIAL RANGE
ON ASPHERICAL EMBEDDINGS OF 2-SPHERES IN THE 4-SPHERE
GEOMETRIC CHARACTERIZATION OF DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN SPACE
WHITEHEAD TORSION AND h-COBORDISM
ADDITIONAL QUESTIONS ON N-MANIFOLDS
COMPLETELY REGULAR MAPPINGS, FIBER SPACES, THE WEAK BUNDLE PROPERTIES AND THE GENERALIZED SLICING STRUCTURE PROPERTIES
FIBER SPACES AND n-REGULARITY
FIBER SPACES WITH TOTALLY PATHWISE DISCONNECTED FIBERS
SOME QUESTIONS IN THE THEORY OF NORMAL FIBER SPACES FOR TOPOLOGICAL MANIFOLDS

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Annals of Mathematics Studies Number 60

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1.

Algebraic Theory of Numbers, by

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B ean

T ucker

TOPOLOGY SEMINAR WISCONSIN, 1965 S. ARMENTROUT

L. LININGER

R. H. BING

J. MARTIN

C. E. BURGESS

L. F. McAULEY D. R. McMILLAN, JR.

J. L. BRYANT A. C. CONNOR

D. V. MEYER

J. DANCIS

M. E. RUDIN N. SMYTHE

E. FADELL J. B. FUGATE

J. STALLINGS

C. H. GIFFEN

R. H. SZCZARBA

H. GLUCK

E. S. THOMAS, JR.

R. W. HEATH

P. A. TULLEY

D. W. HENDERSON

G. S. UNGAR

F. B. JONES

J. N. YOUNGLOVE

EDITED BY

R. H. Bing and R. J. Bean

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1966

Copyright © 1966, by P r in c e t o n U n iv e r s it y P r e ss All Rights Reserved L. C. Card 66-21829

Printed in the United States o f America by W e st v ie w P r e ss , Boulder, Colorado P rin c eto n U n iv e r sit y P r e ss O n D e m a n d E d it io n ,

1985

D EDICATION The contributors to this volume dedicate their w o r k to the memo r y of M. K. Port, Jr. whose w a r m t h and good w i l l have been felt b y the entire mathematical community.

CONTENTS Preface ...........................................................

v V

CHAPTER I: 3 -Manifolds MONOTONE DECOMPOSITIONS OF E3 ......................... Steve Armentrout

1 1

...

EQUIVALENT DECOMPOSITIONS OF E3 ............................. 27 Steve Armentrout, Lloyd L. Lininger and Donald V. Meyer ANOTHER DECOMPOSITION OF E3 INTO POINTS AND I N T E R V A L S ........33 33 Louis F . McAuley CRUMPLED C U B E S ...............................................53 53 L. Lininger SEWINGS OF CRUMPLED CUBES WHICH DO NOT YIELD S3 ............... 57 57 Joseph Martin CANONICAL NEIGHBORHOODS IN THREE-MANIFOLDS................... 6l 6i D. R. McMillan, Jr. BOUNDARY L I N K S ............................................... 69 69 N. Srnythe SURFACES IN E3 ............................................... 73 73 C. E. Burgess ADDITIONAL QUESTIONS ON 3- M A N I F O L D S ......................... 81 81

CHAPTER II: The Poincar^ Conjecture NOW NOT TO PROVE THE POINCAR^ CONJECTURE..................... 83 83 John Stallings MAPPING A 3-SPHERE ONTO A HOMOTOPY 3- S P H E R E ..................89 89 R. H. Bing CONJERNING FAKE C U B E S ....................................... 101 101 A. C. Connor CHAPTER III: ON CERTAIN FIRST COUNTABLE SPACES............................ 103 103 R. W. Heath REMARKS ON THE NORMAL MOORE SPACE METRIZATION P R O B L E M ........115 115 F. Burton Jones 121 TWO CONJECTURES IN POINT SET THEORY..................... . 121 J. N. Younglove ALMOST CONTINUOUS FUNCTIONS AND FUNCTIONS OF BAIRE CLASS 1 . . . 125 E. S. Thomas, Jr. CHAINABLE CONTINUA........................................... 12129 9 J. B. Fugate THE EXISTENCE OF A COMPLETE METRIC FOR A SPECIAL MAPPING SPACE AND SOME CONSEQUENCES..................................... 135 135 Louis F. McAuley

BING A N D BEAN FINITE DIMENSIONAL SUBSETS OF INFINITE DIMENSIONAL SPACES . . . . lifl Ikl David W. Henderson TYPES OF ULTRA F I L T E R S .................................................... 1 U11+7 7 Mary Ellen Rudin ADDITIONAL QUESTIONS ON ABSTRACT S P A C E S ..................... . . . . 15 1522 CHAPTER IV: N-manifolds TAMING POLYHEDRA IN THE TRIVIAL R A N G E .............................. 153 153 John L. Bryant SOME NICE EMBEDDINGS IN THE TRIVIAL R A N G E .......................... 159 159 Jerome Daneis APPROXIMATIONS A ND ISOTOPIES IN THE TRIVIAL R A N G E ................ 171 171 Jerome Daneis ON ASPHERICAL EMBEDDINGS OF 2 -SPHERES IN THE k -SPHERE............... 1 8198 9 Charles H. Giffen GEOMETRIC CHARACTERIZATION OF DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN S P A C E ................................................. 197 197 Herman Gluck 211 WHITEHEAD TORSION A ND h-COBORDISM ................................... 211 R. H. Szczarba ADDITIONAL QUESTIONS ON N-MANIFOLDS ................................. 217 COMPLETELY" REGULAR MAPPINGS, FIBER SPACES, THE W E A K BUNDLE PROPERTIES A N D THE GENERALIZED SLICING STRUCTURE PROPERTIES. 2 1 9 Louis F. McAuley FIBER SPACES A N D n - R E G U L A R I T Y .......................................... 2 2292 9 L. F. McAuley and P. A. Tulley FIBER SPACES W I T H TOTALLY PATHWISE DISCONNECTED F I B E R S ............235 235 Gerald S. Ungar SOME QUESTIONS IN THE THEORY OF NORMAL FIBER SPACES FOR TOPOLOGICAL M A N I F O L D S .............................................241 2^1 E. Fade11

PREFACE

T his volume co n ta in s the p ro ce ed in g s o f the to p o lo g y seminar h e ld un­ d der er the d iirr e c t iio o n o f R. H. B in g a t the U n iv e r s it y o of fW iscon W iscon sin in sin the sum­in the sum­ mer o f 1965. Most o f the a r t i c l e s were p re se n te d asas t a l k tsa ltkos the t o the toto p oplo o lo ­ ­ g i s t s assem bled, then expanded to make e x p o s ito r y a r t i c l e s w ith the d e t a i l s o f p ro o fs fr e q u e n t ly o m itted . No e f f o r t was made to c o n tr o l the s u b je c t m atter o f the t a l k s . On the c o n tr a r y , each p a r t ic ip a n t was in v it e d to d is c u s s the t o p ic o f h is cu rre n t in te r e s t. The ch a p te r headin gs are mine and I co n fe s s h a vin g to s t r e t c h a p o in t here and th e re to make the a r t i c l e s f i t . An attem pt was made to in c lu d e in the book a s many q u e s tio n s and con­ je c t u r e s as p o s s ib le . Most o f th e se f i t n i c e l y in to the a r t i c l e s and were in clu d e d by the a u th o rs. S t i l l o th e rs appear a t the end o f the c h a p te rs . Most o f the q u e s tio n s were asked a t s p e c ia l " c o n je c tu re s e s s io n s " o f the sem in ar. F o llo w in g i s a l i s t o f the grad u ate s t a f f p re se n t a t the sem inar. The l a s t th re e names are those o f s tu d e n ts who c o n tr ib u te d to t h i s volum e. In a d d itio n to th o se named, the s u cce s s o f the sem inar was due, in a la r g e idea­ s’ir e , to the stu d e n ts who were p r e s e n t. T h e ir p a r t ic ip a t io n , e s p e c i a l l y in the numerous d is c u s s io n s w hich fo llo w e d the t a l k s , was g r e a t l y a p p re c ia te d and, I am s u r e , m u tu a lly b e n e f i c i a l . A rm entrout, S teve Bean, Ralph J . B in g , R. H. B ry a n t, John Fade11, Edward F u g a te , Joseph G if f e n , C h a rles G lu ck, Herman Heath, R obert Henderson, D avid H u ss e in i, S u fie n J o n e s, Burton

U n iv e r s it y o f W isco n sin U n iv e r s it y o f G eo rgia

U n iv e r s it y o f G eo rgia

U n iv e r s it y U n iv e r s it y U n iv e r s it y U n iv e r s it y

of of of of

Iowa Tennessee W iscon sin M is s is s ip p i

U n iv e r s it y o f W iscon sin U n iv e r s it y o f Kentucky I n s t i t u t e f o r Advanced Study Harvard U n iv e r s it y .A r iz o n a S ta te U n iv e r s it y I n s t i t u t e f o r Advanced Study

U n iv e r s it y o f W iscon sin U n iv e r s it y o f C a l i f o r n i a , R iv e r s id e , C a lif o r n ia

BING A N D BEAN University of Michigan University of Maryland University of Missouri University of Wisconsin Rutgers University > University of Virginia

Kister, James Lehner, Guydo Lininger, Lloyd Martin, Joseph McAuley, Louis McMillan, Russell Meyer, Donald Roy, Prabir Rudin, Ma r y Ell e n Smy t h e , N.

Central College University of W i s consin University of W isconsin University of New South Wales Sydney, Australia Princeton University

Stallings, J. Szczarba, R.

Yale University of Michigan University of Maryland University of Houston University of Georgia University of Wisconsin Rutgers University

Thomas, E. S. Tulley, P. Younglove, James Connor, A. C. Dancis, Jerome Ungar, Gerald

We all express our gratitude to the secretarial staff at b o t h the University of W i s c onsin and the University of Tennessee, especially Mrs. Margaret Blumberg and Mrs. Sue Hart. R a l p h B ean University of Tennessee

CHAPTER I:

3 -MANIFOLDS

MONOTONE DECOMPOSITIONS OF

E3

by Steve Armentrout

*

§i . Introduction In 1921 +, R. L. Moore introduced the notion of an upper semi-continuous collection of point-sets [63] and in 1 9 2 5 , there appeared his proof [61*] of the following result: THEOREM (Moore). If G is an upper semi-continuous decomposition of 2 the E uclidean plane E into compact continua such that no element of G p separates E , then the decomposition space associated wit h G is homeo2 morphic to E . In the following ten years, Moore [65, 6 6 ], Alexandroff [ 1 ], Kuratowski [5 0 ], and others made significant contributions to the theory of upper semicontinuous decompositions. But none of these results dealt specifically w it h upper semi-continuous decompositions of E 3 into compact continua. G. T. Whyburn, in an address to the American Mathematical Society in 1 935 [8 0 ], raised the question of finding conditions on an upper semi-contin­ uous decomposition of E 3 sufficient to insure that the associated decompo­ sition space is hameomorphic to E 3 . He pointed out that even if the d e c o m ­ position has just one non-degenerate element and that element is an arc, some additional condition is necessary. Antoine had constructed [^] an arc O “3 -3 a in E J such that E - a is not hameomorphic to the complement, in E , of a one-point set. If G is the upper semi-continuous decomposition of •3 E whose only non-degenerate element is a, then the decomposition space associated w i t h

G

is not homeomorphic to

E 3.

W h y b u r n propored [80, p. 70] imposing the following "trial condition" on ea c h non-degenerate element g of an upper semi-continuous decomposition G of E into compact continua: E - g is homeomorphic to the complement, in E 3 , of a one-point set. Further, this is consistent w i t h the Moore theo2 rem on the plane since a compact continuum g in E fails to separate E 2 2 if and only if E - g is homeomorphic to the complement, in E , of a onepoint set. Re s e a r c h supported b y National Science Foundation Grant No. GP- ^ 5 0 8 .

1

2

STEVE ARMENTROUT

The "trial condition" proposed b y Whyburn came to be known as "point­ like" [ 1 9 ] and thus we can state Whyburn's question as follows: Is it true that if G is an upper semi-continuous decomposition of E3 into point like compact continua, then the decomposition space associat­ ed wit h G is homeomorphic to E 3 ? A n affirmative answer to this ques­ tion would yield a "natural" generalization to E 3 of the Moore theorem on p decompositions of E stated above. Wardwell, using conditions similar to these, established [78] a con­ dition sufficient for the topological equivalence of a compact metric space X and the space of a monotone decomposition of X. He obtained some theo­ rems concerning point-like decompositions of E 3 . In 1957 Bin g gave [2 0 ] a negative answer to Whyburn's question. One of the fundamental questions in the study of monotone decompositions of E 3 has accordingly been the following: For what monotone decompositions of E 3 is the associated decomposition space homeomorphic to E 3 ? The investigation of this fundamental question has led to the raising of a number of related questions concerning monotone decompositions of E 3, S 3 , and 3-manifolds. The purpose of this article is to give a comprehensive survey of known results dealing wi t h these various questions. In section 3, we consider results dealing w i t h the problem of finding conditions on monotone decompositions of E 3 under w hich the associated d e ­ composition space is homeomorphic to E 3 . In section k, we consider a con­ dition useful in showing that certain decomposition spaces are homeomorphic ■ 3 to E . Section 5 deals wit h the existence of pseudo-isotopies. In section 6 , we consider some conditions w hich imply that the elements of a d ecomposi­ tion are point-like. Section 7 deals wi t h the problem of characterizing spaces of monotone decompositions of E 3 . Section 8 concerns local properties of decomposition spaces. In section 9 , we consider homology and homotopy of decomposition spaces. Recent results by T. M. Price and others on the relationship b e ­ tween the homotopy of a space and that of an associated decomposition space have led to some interesting results. Dimension of decomposition spaces is the subject of section 1 0 . In section 1 1 , we consider the problem of e mbed­ ding decomposition spaces of euclidean spaces in euclidean spaces. Products of decomposition spaces w i t h E 1 and w i t h other spaces is the subject of section 1 2 . In section 13, we consider monotone decompositions of E that yield 3 -manifolds. Section 1 U mentions some closely r e ­ lated problems.

MONOTONE DECOMPOSITIONS OF E 3 §2.

3

Notation and Terminology

Suppose X is a topological space. The statement that G composition of X means that G is a collection of subsets of that e a c h point of X belongs to one and only one set of G. Suppose X is a topological space. upper semi-continuous decomposition of X of X such that if g is any element of containing g, then there is an open set and V is a union of elements of G.

is a d e ­ X such

The statement that G is an means that G is a decomposition G and U is any open set in X V in X such that g C V, V C U,

Suppose X is a topological space and G is an upper semi-continuous decomposition for X. Let B be the set of all subsets W of G such that the union of all the elements of W is an open subset of X. Then B is a base for a topology T for the set G, and (G, T) is the decomposi­ tion space assiciated w i t h G. The decomposition space associated x^ith G will be denoted by X/G. Throughout this paper, P denotes the projection map from X onto X / G (if x € X, P(x) is the set of G to which x b e l o n g s ) . P is a closed continuous function. Hq

denotes the union of all the non-degenerate elements of

G.

For a presentation of a number of fundamental results on upper semicontinuous decompositions of metric spaces into compact continua, see [67, Chap. V] or [81, Chap. 7]. We shall m e rely state a few of the most useful results concerning such decompositions. Suppose then that X is a separa­ ble metric space and G is an upper semi-continuous decomposition of X into compact continua. Then the following hold: 1. 2. 3. k. 5.

X/G If If If If of

is a separable metric space. X is locally compact, so is X/G. X is locally connected, so is X/G. M is a compact subset of X/G, then P - 1 [M] is compact. each set of G is connected and M is a connected subset X/G, then P ”1 [M] is connected.

There is a frequently used process for identifying a decomposition space. Suppose X is a topological space and G is an upper semi-continuous decomposition of X. Suppose further that f is a continuous function from X onto a space Y such that G = (f~1 [y]: y € Y) . There is a n a ­ tural m a p from X / G onto Y, namely f P “1 , and we shall let

1, we construct in Int A i>#>j six R^-cubes A^...^, k = 1 > 2 >• •• > as outlined in steps (2)-(5) for A but choose t, 0 < t < in step (5 ). These R^-cubes have the same linking properties w i t h respect to as the six R^-cubes A ^ . . . j _ 1 n , n=i , 2 ,..., 6 , have the respect to A ^ . . . ^ . The E x a m p l e ; In the description above, we have indicated a construc­ tion for what we call an R^.-cube A. In its interior, we have described other R^-cubes, and so forth. For each i, let A^ denote the collection of all R 7-cubes A where n. = 1,2,3,..., 6 . For i = 1 we have I 1 2 1 a collection A 1 v.'dch contains an element A 1 but no confusion should r e ­ sult. For i > i, the multiple subscripts should prevent any confusion in this regard. Consider the point set

^ m

. n i= 1

■

where A^* denotes the union of the elements of A^.. E a c h component of M is a straight line interval. Now, let H be the collection of these compon^ * ents. And, let G = H plus those points of E - H . Thus, G is an upper

LOUIS MCAULEY

l+o

semi-continuous (u.s.c.) decomposition of G, H is homeomorphic to a Cantor set.

E 3.

In the decomposition space

Properties of A and the collections A^. We shall use numerous fundamental theorems about the topology of E 3 from papers of R. H. Bing and E. E. Moise. Results from the paper [U] of Bing will be used extensive­ ly, and therefore, an attempt will be made to make definitions and use ter­ minology consistent wi t h his paper. Topologically, we ma y consider

A

as some s-ribhd of a linear graph

F w hich lies in a plane and consists of seven simple closed curves U^ and Lg, i= 1 ,2 , 3 , 1* and k=i, 2 , 3 such that each pair of different simple closed curves intersects in a point ter of

A,

c.

See Figure 6 .

and the simple closed curves in

Fig. 6

F

We shall call are the loops of

F

the cen­ F.

Fig. 7

Now let P 1 and P 2 denote two disjoint parallel planes such that P 1 n A = five planar discs w h i c h are pairwise disjoint and P 2 n A = four planar discs wh i c h are pairwise disjoint (i.e., P 1 cuts through the "top" holes of A and P 2 cuts through the "bottom” holes) . See Figure 7. A^, A.

We indicate in Figure 8 a topological picture of the six R^-cubes how they link around each other, and how they link around the holes in

If the center

B is the image of A under a homeomorphism h, we call h(F) of B (the center depends on the homeomorphism h ) . Suppose

ANOTHER DECOMPOSITION OF E 3 INTO POINTS A ND INTERVALS

hi

that F 1 is a topological graph homeomorphic to F, the center of A. We say that F 1 is homotopic to the center of A if there are continuous maps f(x, t ) , x € F and 0 < t < 1, of Fx[o, 1] into A such that f(x, o) = x and f(x, 1 )takes F home omorphi cally onto F 1 .Consider now, two properties of certain topological graphs in A. Property P(k, n ) . A topological graph F 1 homeomorphic to the center F of A has Property P(k, n) if and only if F 1 contains k + n points p^, qj, i = i,...,k and j = ,2, . , n where no two of these points lie in the same loop (simple closed curve) such that each arc from P-^ to q^ in F 1 intersects both P 1 and P 2 . Property Q(k, n) . A n image of A under a homeomorphism h has Property Q(k, n) if and only if each topological graph F 1 in h(A) which is homeomorphic to the center F of A and w hi ch is homotopic to the cen­ ter h(F) of h(A) also has Property P(k, n ) . The next objective is to show that if h is a homeomorphism of E 3 onto itself w h i c h is fixed outside A, then for each n, there is some e l e ­ ment of A n , say such that intersects b o t h P 1 and Pg. Our plan is to use arguments like those due to Bin g in [k]. However, the situation here is mu c h more complicated than in [U] although enough like [1+] to use many of the basic tools developed there. Bin g defined and used only Properties P(i, 1) and Q( i , l). Reasons for seven holes in the elements of A ^ . Consider a h omeo­ m orphism h of A onto itself fixed on Bd A. Then it is not difficult (although it will be proved) to see that for some i, i = 1, ..., 6, h(A^) intersects b o t h P 1 and Pg where A^ is an element of A 1 . H o w ­ e v e r , it m a y be that h(A^) has Property Q(k, n) for k + n < 7. If no n ( A i ) has Property Q(k, n) for k + n = 7, then at the next stage, we can find a homeomorphism g of h(A^) for each i onto itself fixed on B d h ( A i ) such that gh(A.^) fails to have even Property Q( 1, i) for any j. The homeomorphism h m ay shrink A^ for some i so that h(A^) misses P 1 U Pg altogether, while for some j, h ( A n.) has perhaps only one handle w h i c h intersects P., and two handles w h ich intersect Pg. It can be proved, however, that because of the seven holes and the special linking of the A^, there will exist m such that either three handles of h(Am ) on one end intersect P 1 while two handles on the other end of h(A^) i nter­ sect Pg or the other w ay around. That is, we can establish either P(3, 2) or P(2, 3). If we could prove P(i+, 3), then we could show that the decomposit i in space is not homeomorphic to E 3 . But, is either P(3, 2) or p ( 2, 3) sufficient to yield such a result? We prove the following: THEOREM 1 . Suppose that h is a homeomorphism of A onto itself w h i c h is fixed on the b o undary of A. Then h(A) has Property Q(4, 3).

1+2

LOUIS MCAULEY

P r o o f : If F is the center of A, then h(F) is homotopic in A to F since there is a graph F 1 on Bd A w h ich is homotopic to F in A. B y a result of Hamstrom [ 1 2 ], there is a n isotopy of E^ onto itself fixed outside A and pulls h onto the identity. Let U^ be one of the four "upper” loops of F, the center of A, and tU. be its image under a homotopy in A. It will follow that tU, intersects P 1 . Now, consider P 1 n A = U^ =1 D ^ , the union of five disjoint discs. Suppose that U^ links Bd D 1 where we use the same concept of linking (mod 2 ) as given in [1*.]. It wi l l follow b y an argument like the one for Theorem 1 1 of [1+] that tU^ intersects D.^ and therefore, P ]. Fur­ thermore, if tL, is the homotopic image of a "lower” loop of F, then tL, J k J intersects P g . Note also that P 2 n A = Ui = 1 D o i , four disjoint discs. It should now be clear that h(A) has Property Q(i+, 3). In the proof of the following theorem, we use Theorems 1-7 of [U] w h i c h are basic facts about E 5 . We do not mention these theorems specifi­ cally in the argument which follows because our proof parallels that given b y Bi n g in [U] for Theorem 1 0 . Nevertheless, they are essential. Admissible Sets of L o o p s . Since B has Property Q(U, 3), each t o ­ pological graph F 1 in it, homotopic to its center h ( F ) ,has Property P(U, 3). Thus, h(F) has Property P(i+, 3). Suppose that U 1 , U 2 , U^, and U^ are the four upper loops of h(F) while L 1 , L g , and L^ are the three lower loops of h ( F ) . Furthermore, there are four points p^ € U^ and three points q^ € Lj where O)

p ±,

qj 4 c = U ± n .Lj

(2 )

eac h arc p ^ j

in

and

h(F)

intersects b o t h

Clearly, h ” 1 (Ui ) is not homotopic to h ” 1 (Lj) sequently, U^ is not homotopic to L j .

in

A

P1

and

P2.

for

any

i, j.

Con­

For convenience, assume that h ” 1 (Ui ) and h ” 1 (Lj) for i = 1 , 2 , 3 , k and j = 1,2,3, are linked in the order around the four top holes a, b, c, d, of A and around the three bottom holes e, f, g of A, respectively, as shown in Figure 9 . Note that h ~ 1 ( U ^ intersects h ”1 ^ ) while h " 1 (Lj) intersects h ”1 (P2 ). Consider the six elements A^ of the collection A1 whi c h lie in Int A and w h i c h are linked as indicated in Figure 8. Let F^ be a topo­ logical graph in h(A^) w h i c h is b o t h homotopic in h(A^) to its center h( C i ) and homeomorphic to the center Ci of A^. There is n o loss of g en­ erality in supposing that no one of the the seven loops (simple closed curves) of F ± lies entirely in (P1 U P 2 ) n A. For each i, let u|, U 2 , U^, uj and L ^ , l |, L^, denote the four upper and the three lower loops, respectively, of F^. Since F^ is h o moto­ pic in h(A^) to its center h (Ci ), clearly, h ”1 (F^) is homotopic in to C^. A s a matter of convenience and without loss of generality, we shall

M O T H E R DECOMPOSITION OP E 3 INTO POINTS A N D INTERVALS

assume that

is

h(C^)

and that

h’1 (F^) = C^.

Prom each F^, we can pick a pair of loops the six pairs

[h_1 (Ujjj ), h”1 (1^ )]

(U* ,

)

such that

link as shown in Figure 10.

We shall

LOUIS MCAULEY say that a set of six pairs w h i c h link in this manner is an admissible set of loops. The number of such admissible sets of loops is (U ) • (3^) = 1*3 2 . THEOREM 2 . pair both

P1

that a n

If

), h_1 and 1 P 2 . Proof: e-nbhd

S )]

is an admissible set of loops, then at least one has the property that

1

U ^ U ij i i

intersects

Suppose that the theorem is false. There is an e > of each pair th”1 (U^ ), h " 1 (1^ )] lies in Int A^.

this nbhd b y D^. Thus D^ [1+] b y Bing. Furthermore, D^ tersect bot h P 1 and P2 .

0 so Denote

is a "dogbone" like those used in the paper can be selected so that h (Di ) fails to I n ­

Now, we claim that there exist six arcs xu, xv, xw, yu, yv, and yw such that ( 1 ) two arcs intersect if they have exactly one point in common which is an endpoint of each, (2 ) their union is a theta curve, ( 3 ) the image under h of eac h arc fails to intersect b o t h P 1 and P 2 , (4) e x ­ cept for "small" nbhds of their endpoints, these arcs lie, respectively, in A 1,..., Ag,

and

(5) these arcs are homotopic to those shown in Figure 1 1 .

The construction of these arcs may be carried out in almost exactly the same w a y as B i n g constructs the four arcs pq^r^s on pages U 9 3 -U 96 of [U], A l t h o u g h our example is more complicated, the techniques for producing these arcs are essentially the same. We feel that it is unnecessary to r e ­ produce his long but ingenious argument. that

B

The next objective is to show that the existence of such arcs implies does not have Property Q(i+, 3) .

h ”1 (P..) _1 W e have assumed that no one of these arcs intersects b o t h * and h (P2 ) . Let T denote the theta curve x u U xv U xw U yu U yv U yw. There are exactly three simple closed curves in T. E a c h of these consists of four arcs—two w i t h x an an endpoint and two w i t h y as a n endpoint. See Figure 1 1 . B y a theorem of Moise [ 1 6 ] (also, Bing), there is no loss of generality in assuming that h is piecewise linear and that ea c h of the simple closed curves Ju v , Ju w , and Jw in T is polygonal. The s ub­ scripts indicate the points other than x and y whi c h lie on the simple and h ” 1 (P2 ) closed curves. Furthermore, we m a y assume that ( 1 ) h - 1 ^ ) are polygonal planes (which, of course, do not intersect) and ( 2 ) each of the three simple closed curves in T is unknotted and bounds a disc. We shall denote these discs b y D u v , D u w , and D w , respectively. Suppose that Juv n h ”1 ^ ) = cp. Then we w i s h to show that D uv can be taken so that D uv n h ”1 ^ ) = cp. This m a y done b y considering h(Ju v ) , a polygonal simple closed curve, wh i c h does not intersect P 1 . Since Juv is unknotted and h is a homeomorphism of E 5 onto itself, h (J u v ) is u n “ knotted. Also, we have assumed that h Is piecewise linear. Hence, h(Ju v ) bounds a polygonal disc D, i.e., D has a (finite) triangulation T.

M O T H E R DECOMPOSITION OP E 3 INTO POINTS A N D INTERVALS

**5

Now, D can be adjusted so that no vertex or edge of a triangle lies in P 1 . Then each component D n P 1 is a simple closed curve. Consider the component C of D - P 1 w h i c h contains J u v . Thus, C is a disc w i t h a finite number of holes bounded by simple closed curves lying in P 1 . We now have a polygonal disc D bounded b y J uv whose intersection w i t h P 1 is a finite number of pairwise disjoint discs. Next, we adjust D slight­ ly to miss P 1 altogether. Consequently, we may assume that h[Du v ) and D uv

are P olyg°nal discs wh i c h miss

P1

and h - 1 ^ ) ,

respectively.

It is not difficult to see that there is a four leaf clover E 1 in Duv whi c h is homotopic in A to E, = h ' 1 (uj) U h ' 1 (u|) U h ' 1 (l|) U h ' 1 (L^). Now, recall that B = h(A) has Property Q(U, 3). Observe that E 2 is h o ­ motopic to a part of P, the center of A. We know that F consists of four upper loops U 1 , U 0 , U^, U^, and three lower loops L 1 , L 2 , L^. can label these so that E 0 is homotopic to E, = U 1 U Uj U L ] U L 2 . Figures 1 2 , 1 3 , and 1U.

We See

Since h(A) has Property Q(U, 5 ), it follows that h(F) has Property P(U, 3). Thus, there exist seven points, ? s € h(U^), i = 1,2,3,U; q^ € h(Lj), j = 1,2,3, such that ( 1 ) no two of these points b e l o n g to the same loop of h(F) and (2 ) each p ^ q 4 intersects bo t h P 1 and P 2 . Since E 1 is homotopic to E 2 and E 0 is homotcpic to E^, we have that h ( E 1) is homotopic to h ( E ^ ) . Consequently, htEj) intersects b o t h P 1 and P 2 since h(A) has Property Q(U, 3). But, h(Du v ) 0 P 1 = cp and D uv 3 E ^ Tliis involves a contradiction. Hence, J ^ h “1 (P1) ^ cp. In this way, it follows that J uv n h " 1 ( P ^ t ®, Juy n h ' 1 (Pi ) ? ... (t 1 • • •tR ) are all 0, and g n (t, t,..., t) is principal, generated by a knot polynomial. Al s o the longitudes of L lie in the second commutator subgroup of tt^ S 5- L) .

69

70

SMYTHE

III. W e a k linking can also be generalized to links of many components, a l ­ though the concept becomes somewhat unrecognizable. [ 3 ] L = U L U... -z * — U Hn in S (with solid torus regular neighborhoods V 1, V 2 ,..., V n ) will be called an homology boundary link if 1 ) There exist disjoint orientable surfaces ^ • S in S 5 -U Int V\ such that S^ n Vj = n ^ consists of a number of simple

closed curves x,ij2* * * •» ^ijm whi c h are longitudes of V j .

(w ith orientation induced from

S^)

2 >■ Uj,k Xijk 2 k Xijk ~ 0 k) (Shrinking

Z k Xijk ~ fi V\

ln in

V j if

^

1

V i-

down to its core

J^, we see that

L

is a boundary link

except that the surfaces are allowed to have singularities along the co m ­ ponents of L.) Then a link of 2 components is an homology boundary link if and only if it is not weakly linked. This follows from THEOREM: L is a hcmology boundary link if and only if there exists a homomorphism f : (S'- L) — F(n) onto a free group of r a n k n. Furthermore, L is a boundary link if and only if there exists meridians a l * * * *, an £n 3UCh that f(an) freely generate F(n) . To construct such a homomorphism given a homology boundary link, first thicken the surfaces S4 to open sets N. = S. x (0 , 1 ); then S - U V i can be mapped onto a wedge of n circles by mapp i n g points outside U to the vertex, and a point of x C t) to the point t of the i- t h circle. Since U Si cannot separate S 5- U V , , the induced mapping of fundamental groups is the required homomorphism. The idea of the proof in the other direction is to construct a retraction of S^- U onto a wedge of n circles. The surfaces will then appear as the inverse images of non-critical points (non-vertices). In the case of a boundary link, the n-leafed rose must be embedded in S^- L as a set of meridians. This algebraic characterization leads immediately to the fact that a homology boundary link has an Alexander matrix A ( t 1,..., t ) w i t h n columns of zeros (if the link is actually a boundary link, these columns correspond to meridians of 71^ (S^- L ) ) . Thus again = ••• = 6 n _-, » Also, the homomorphism f induces isomorphisms f : G / G a F /Fa of the quotient groups of the lower central series, where G = ir^S^- L) . Since F ^ = 1 , we have f^: G/G^ * F, and G ^ = ker f. The longitudes of L lie in G^, since they lie on the surfaces. The Milnor isotopy i n ­ variants ^ are therefore all 0. (In fact m u c h more can

BOUNDARY LINKS

71

be said: it turns out that any link which is isotopic to a homology b o u n d ­ ary link under a "nice" (e.g., polyhedral) isotopy, is an homology boundary link.) See [3], U L Eileriberg's question link w hich is not a boundary The answer is yes. (it is easy to shov that one

becomes: Does there exist a homology boundary link? The link of the figure is not a boundary link of its longitudes does not lie in G n) . But

it is a homology boundary link (either b y constructing the surfaces, or co n ­ structing a m a p of G onto a free group of ran k 2-but of course there is no such m ap under w hich a pair of meridians are free generators). The surface S 2 is shown in the diagram, to be thought of w i t h two boundaries lying parallel to ft] on a regular neighborhood V 1 of J^. in.ie surface S 1 is to have 3 boundaries, all longitudes of V 1, one having opposite orientation to the others (two lie "above" S 2 , and one "below"). Ques t i o n s : (1) (Fox) Suppose the longitudes of L lie in G". Is L a boundary link? (2) Similarly, suppose A ( t 1 «**t ) has n columns of zeros. Is L a homology boundary link (or a boundary link, if the columns correspond to m e r idians)? (3) Is every link isotopic to a boundary link also a boundary link (under a "nice" isotopy)? (h) Is there a corresponding theory for links in higher dimensions? (5) Generalize n-linking to arbitrary numbers of components and characterize a l g e b r aically. (6) There are a number of questions related to n-linking—see F ox [2]. (7) (Milnor) What does G^ look like? Can invariants (say isotopy i n ­ variants) of Gm be found to distinguish betwe en homology boundary links?

72

SMYTHE

REFERENCES [1]

S. Eilenberg, "Multicoherence p. 153; 29

I, II," Fundamenta Math., 27 (1936)

(1937) p. 107.

[2 ]

R. H. Fox, Some Problems in Knot T h e o r y , Prentice-Hall, 1 9 6 2 .

Topology of 3-Manifolds,

[3 ]

N. Smythe, Isotopy Invariants of Links, Ch. IV, Princeton Ph.D. Thesis (1965).

[U]

J. Milnor, "isotopy of Links, Algebraic Geometry and Topology" (Lefschetz Symposium) Princeton University Press, Princeton, 1957.

SURFACES IN E 5* by C. E. Burgess We present a brief summary, wit h references; of developments during the last fifteen years that are related to tame embeddings of 2 -manifolds in 3-manifolds. This is the basis of a twenty minute talk presented in the Special Session on Recent Developments in Topology at the meeting of the American Mathematical Society at Iowa City on November 2 7 , 1 9 6 5 . For sim­ plicity in the presentation, we restrict our discussion to 2 -spheres in E^, but most of the theorems can be extended to 2 -manifolds in a 5 -manifold. We have included an extensive list of references on this topic, but we presume that there are some serious omissions and some lack of current i n ­ formation in this list. We find it necessary to place some restriction on the number of papers listed here, and we suggest that other wor k essential to this development can be found in the references cited in the papers that are included in this list. For example, see Harrold's expository paper [37]. 1 .Fundamental wo r k related to

A 2 -sphere

S

tame imbeddings of 2- spheres in E? . is defined to be tame in E 5 if there is a homeomorphism h

of E^ onto itself that carries S onto the surface of a tetrahedron. Alexander [ 1 ] showed in 1 9 2 U that every polyhedral 2-sphere in E 5 is tame under this definition. Graeub [ 3 0 ] and Moise [Ui, II], working independently, showed in 1 9 5 0 that every polyhedral sphere in E^ can be carried onto the surface of a tetrahedron with a piecewise linear homeomorphism of E^ onto itself. A fundamental characterization of tame spheres in E 3 was given b y Bi n g in 1959 wi t h the following theorem. (In the various theorems -on characterizations of tame surfaces in E 3 , we do not include the ob­ vious statement that every tame sphere satisfies the requirements of the characterization.) i.i A 2-sphere is tame if it can be homeomorphically approximated in each of its complementary domains. (Bing [ 1 0 ].) The proof of this theorem depended upon Bing's Approxlmation Theorem for Spheres [8], whi c h was later generalized to the following Side A p p r o x i ­ mation T h e o r e m . *

This paper was not presented at the seminar, but it is included by i n v i ­ tation since it treats recent developments and questions discussed at the seminar. This w o r k was supported b y NSF Grant GP-3882. 73

BURGESS

7^

1.2 If S is a 2-sphere in E^ and e > o, then there exist ( 1 ) a polyhedral 2-sphere S', (2) a finite collection D 1, D 2 ,..., D n of disjoint e-discs on S', and (3) a finite collection of disjoint e-discs E 1 , E 2 , ..., E n on S such that (1) S' is homeomorphically w i t h e of S, (2) S f - TD± C Int S, and (3) S - EE± C Ext S ' . F u r t h e r m o r e , Int S and E x t S ’ can be replaced wi t h Ext S and Int respectively, in conclusions (2) and (3) above. (Bing [iU]# [17].)

S 1,

Essential to some of the recent developments on tame surfaces are papers on locally tame sets [7], [^1, VIII], triangulation of 3-manifolds [9 ], [^ 1 , V], and Dehn's Lemma and the Sphere Theorem [U2], [U 3 ]. Brown's w o r k on the generalized Shoenflies Theorem [ 1 8 ] and on locally flat imbeddings [ 1 9 ] is applicable, but for surfaces in E 5 , or in 3-manifolds c om­ parable results can be obtained from the w o r k of Bing and Moise cited above. 2 . Examples of wild 2 -spheres in E 5 . A 2-sphere in E 5 is d e ­ fined to be wild if it is not tame. Ear l y examples of wild spheres in E5 were given b y Antoine [^] in 1 9 2 1 and b y Alexander [2 ] in 1 9 2 U. The set of

wild points in Antoine's example is a wild Cantor set, and in Alexander's example this set is a tame Cantor set. A third type of wild sphere, which has only one wild point, was described b y Fox and A r t i n [2 7 ] in 1 9 ^ 8 . In 1 9 6 1 , B i n g [ 1 2 ] described a wild 2 -sphere in w h i c h every arc is tame. E x ­ amples of other types of wild spheres can be found in papers b y Alford [ 3 ], Ball [5 ], B i n g [6], [ 1 6 ], Cannon [2 3 ], Casler [2k]f Fort [2 6 ), Gillman [28], and Martin [Uo]. Of course other wild spheres can be described b y combining some of the methods used in describing these examples. Relatively little has been done on the equivalence of imbeddings of wild spheres in E 5 .For horned spheres, wo r k b y Ball [5 ] and Casler [2U-] shows that the horned sphere described b y Ball [5 ] is imbedded differently from the one described b y Alexander [2 ], and Cannon [2 3 ] has describd a 3horned sphere that is imbedded differently from either of these. These three examples are free of any knotting in the horns. 3.Other characterizations of tame spheres in E ^ . Let S be a sphere in E^. Then S is tame if the requirements of any one of the f o l ­ lowing statements are fulfilled. 3.1 3.2

2-

E v e r y disc in S has b o t h the strong enclosure property and the hereditary disc property. (Griffith [ 3 1 ].) S is locally peripherally unknotted; i.e., for eac h p € S and e ac h e > 0 , there exists a tame 2 -sphere S 1 such that p € Int S ’ , diam S 1 < e, and S n S' is a simple closed curve. (Harrold, [ 3 2 ], [33].)

SURFACES IN E 3 *

75

3.3 3.^

The complement of S is i-ULC. (Bing [ 1 1 ].) S can be deformed into each complementary domain; ±.e., for e ach component U of E 3-S, there is k homotopy h: S x I cl(U) such that h(x, o) = x and h(x, t) € U for 0 < t < i . (Hempel [35].) 3.5 S is the image of a tame sphere S ’ under a map. f of E 5 onto itself such that f |S1 is a homeomorphism and f(E3- S ’ ) = E 3 - S. (Hempel [35].) 3.6 S can be locally spanned from each complementary domain; i.e., for ea c h p € S, for each e> 0, and for eac h component U of E 3- S, there exist discs D and R such that p € Int R C S, Int D C U, D n R = Bd D = Bd R, and diam (D + R) < e. (Burgess [2 1 ].) 3.7 S can be uniformly locally spanned in each complementary d o ­ main; i.e., for each component U of E 3 - S and for every e > 0 there exists a 5 > osuch that for any 5 -disc R on S and for any cr > 0, there is an e-disc D in U such that Bd R can be shrunk to a point in D + (some o-neighborhood of Bd R ) . (Burgess [21].) 3.8 (Let K be a Sierplnski Curve; i.e., K is a universal one­ dimensional plane curve.) For each component U of E 3- S and ea c h p € S, there is a homeomorphism h: K x I -*• E 3 such that h ( K x o) C S, p is an inaccessible point (relative to S) of h ( K x 0), and h ( K x t) C U for 0 < t < 1 . (Burgess [21].) 3.9 S can be locally spanned in each complementary domain on tame simple closed curves; i.e., for each component U of E 3 - S, for each p € S, and for eac h e > 0, there exists an e-disc R in S such that (1) p € Int R ’ C S, (2) B d R is tame, and (3) for each ce > o there is a n e-disc D in U such that Bd R can be shrunk to a point in D + (an a-neighborhood of Bd R) . (Loveland [38].) 3.10 For each p € S and each e > o there is a 2-sphere S ’ sat­ isfying the following requirement: (i) p € Int S', (2) diam S 1 < e, (3) S n S' is a continuum, and (l*) S can be sideapproximated missing S n S'. (Loveland [38].) (See the next section for a definition of the requirement in ( k).)

if. Tame subsets of 2 -spheres in E 3 . A subset H of a 2-sphere S in E 3 is defined to be tame if H is a subset of a tame sphere. We say that S can be side-approximated missing H i f , in the Side Approximation Theorem stated above, it is further required that H C S - Z E^. Bing's methods of u s ing the Side A pproximation Theorem to prove the following theorem have been essential in subsequent w o r k on tame subsets of spheres.

76

BURGESS 4. 1

If S is a 2-sphere in E^ and e > 0, then there exists a tame Sierpinski curve K in S such that each component of S - K has a diameter less than e. (Bing [ 1 3 ].)

The following four theorems have been announced in the chronological order in w hich they are listed, and each of the last three is a generaliza­ tion of the preceding one. 4.2

4.3

4.4

4.5

If the 2-sphere S in E 5 can be side-approximated missing the arc H in S, then H is tame. (Gillman [2 9 ].) If S is a 2-sphere in E^, H is a closed subset of S such that the set of all diameters of the components of H has a positive lower bound, and S can be side-approximated missing H, then H is tame. (Hosay [37].) If the 2-sphere S in E 5 can be side-approximated missing the closed subset H of S and e > 0, then there exists a tame subcontinuum M of S and a null sequence of ediscs in S such that ( 1 ) M = S - £ Int and (2 ) H C M - Z Di = S - Z (Loveland [3 9 ] •) If each of H 1 , H 2 ,..., is a closed subset of the 2-sphere S in E^ and, for each i, S can be side-approximated missing Ep then Z is tame and S may be side-approximated missing £ H^. (Loveland [39]-)

Proofs of the following two theorems depend upon the methods used in proving the above four theorems. 4.6

4.7

If S is a 2-sphere in E^, H is a closed subset of S such that the set of all diameters of the components of H has a positive lower bound, and for each e > 0 there is a 5 > 0 such that each 5 -simple closed curve in E^- S can be shrunk to a point in an e-subset of E^- H, then H is tame. (Hosay [37] and Loveland [ 39] -) If S is a 2-sphere in E 5 , H is a closed subset of S such that the set of all diameters of the components of H has a positive lower bound, and S can be locally spanned at each point of H from each complementary domain of S, then H is tame. (Loveland [38].)

5. Spheres in E^ which are tame except on a tame s e t . Let a 2 -sphere in E^ and let H be a tame subset of S such that S is locally tame at each point of S - H. We present here some additional requirements on H which imply that S is tame. The foundation for the first result of this type can be found in M o i s e 's wo r k [41, VIII], and this led to the follow­ ing generalization. 5.1 S is tame if H is a tame finite graph. (Doyle and Hocking [2 5 ].)

S

be

77

SURFACES IN E 3 *

U sing consequences of Dehn's Lemma, the Side Approximation Theorem, and his earlier theorem characterizing tame spheres, Bing proved the follow­ ing more general 5.2 S 5.3 S of

theorems. Is tame if H is a tame Sierpinski c u r v e . (Bing [ 1 5 ].) is tame if H is the union of a finite number of sets each w h ich is either a tame finite graph or a tame Sierpinski

curve.

(Bing C 1 53 -)

Other results have been obtained where

H

is the intersection of a

tame sphere w i t h S. 5 .ij- If S is a 2 -sphere in E^ and S' is a tame 2 -sphere such that ( 1 ) S is locally tame at each point of S - S ’, (2 ) S n S ’does not separate S, and (3) the set of all diameters of components of S n S' has a positive lower bound, then S is tame. (Hosay [ 3 6 ].) 5 . 5 If C is a 3 -cell in E 5 and S is a 2 -sphere in C such that S is locally tame from Int S at each point of S-Bd C, then S + Int S is a 3 -cell. (Burgess [2 0 ], [2 2 ].) The interior of a disc one side if for each p € Int D that p € Int D* C D and E n D

D in E^ is defined to be locally tame from there exists a disc D* and a 3 -cell E such = D 1 CBdE.

If C is a 3 -cell in E 5 and D is a disc in C such that D is locally tame at eac h point of D - Bd C, then Int D is locally tame from one side. (Burgess [2 2 ].) 5.7 If p is an isolated wild point of a 2 -sphere S in E^, then S is locally tame at p from one of the components of E^- S. (Harrold and Moise [3^].) 5.6

6. Questions on tame spheres and on tame subsets of spheres in E ^ . The questions raised here have arisen in papers, seminars, and informal d i s ­ cussion in various places. We include a reference where we know a question

has appeared in the l i terature . 1 W h i c h of the following conditions imply that a 2-sphere

S

in

E5

is tame? 6.1 6.2

6.3 6.k

S can be pierced b y a tame disc at S can be pierced b y a tame annulus

each arc in S. at each simple closed curve

in S. S can be approximated w i t h a ma p of S in each complementary domain of S [35, p. 280]. E a c h Sierpinski curve in S ean be pushed wit h a homotopy (an isotopy) into eac h component of E 5- S.

1 Some of these questions were included in an address b y R. H. Bing at the International Congress of Mathematicians in 1 9 6 2 . (See pp. 1+57-5^8 of the Proceedings of that Congress.)

78

BURGESS 6.5

For each p € S, for each component U of E^- S, and for each 6 > 0 , there is an 6 -disc R such that p € Int R C S and the identity m ap on Bd R can be shrunk to a point in an e-subset of Bd R + U [21, p. 8 9 ]. 6.6 S Is locally tame at each point of S - H, where H is a n o n ­ degenerate tame subcontinuum of S. 6.7 S is locally tame at each point of S - H, where H is a tame closed subset of S having no degenerate component. 6.8 S can be locally spanned in each component of E^- S. 6 . 9 For each p € S and each e > 0 , there exists a tame 2-sphere S' such that p € Int S 1, diam S f < e, and S n S 1 is a continuum. 6 . 1 0 F or each p, q e S, there is a homeomorphism of E^ onto itself that carries S onto itself and p onto q [2 8 , p. 2 5 3 ]. 6 . 1 1 E a c h horizontal plane that intersects S either intersects S in a point or a simple closed curve [1 ], [ 1 6 ].

REFERENCES [l]

J. W. Alexander, " O n the subdivisions of 3-spaces by a polyhedron,11

[2 ]

_______ , "An example of a simply connected surface bounding a region

Proc. Nat. Acad. Sci., U.S.A.,

10 (1924) pp. 6-8.

w h i c h is not simply connected," Proc. Nat. Acad. Sci. U.S.A., 1 0 ( 1 9 2 4 ) pp. 8 - 1 0 . [3]

W. R. Alford, "Some ’ nice' wild 2-spheres in E^," Topology of 3-M a n i ­ folds and Related Topics, Prentice Hall, 1 9 6 2 , pp. 2 9 - 3 3 .

[43

L. Antoine, "Sur l ’ homeomorphie de deux figures et leurs voisinages," J. Math. Pures Appl., 4 ( 1 9 2 1 ) pp. 2 2 1 - 3 2 5 .

[5]

B. J. B a l l , *"The sum of two solid horned spheres," Ann. of Math.,

(2 )

69 (1959) PP. 253-257. [6 ]

R. H. Bing, "A homeomorphism between the 3-sphere and the sum of two solid horned spheres," Ann. of Math. (2) 56 ( 1 9 5 2 ) pp. 354-362.

[7]

_______ , "Locally tame sets are tame," Ann. of Math. 145-158.

[8 ]

_______ ,

(2 ) 59 (1 9 5 4 ) pp.

"Approximating surfaces w i t h polyhedral ones," Ann. of Math. (2)

65 (1957) pp. 456-483. [9]

_______ , Ann. of

[ 1 0 ] _______ ,

"An alternative proof that 3-manifolds can be triangulated," Math. (2 ) 69 (1959) pp. 37-65. "Conditions under wh i c h a surface in E 5is

47 (1959) PP. 105-139.

tame, Fund.

Math.,

REFERENCES [n ]

_______ , "A surface is tame if' its complement is Math. Soc., 1 01

79 Trans. Amer.

( 1 9 6 1 ) pp. 29^-305.

[1 2 ]

_______, "A wild surface e a c h of whose arcs is tame/* Duke Math. J., 28 ( 1 9 6 1 ) pp- 1 - 1 6 .

[13)

_______ , "Each disc in E 3 contains a tame arc," Amer. J. Math., 81+ ( 1 9 6 2 ) pp. 583-590.

[U]

_______, "Approximating surfaces from the side," Ann. of Math. (2 ) 77

[15]

_______ , "Pushing a 2-sphere into its complement," Mich. Math. J., 1 1

(1963) pp. 11 +5 - 1 9 2 . (1961+) pp. 33-14-5. [1 6 ]

_______ , "Spheres in E 3 ," Amer. Math. Monthly, 71 ( 1 9 6 U) pp. 353-361+.

[ 1 7]

_______ , "improving the side approximating theorem," Trans. Amer. Math.

[18]

Morton Brown, "A proof of the generalized Schoenflies Theorem," Bull. Amer. Math. Soc., 66 ( 1 9 6 0 ) pp. 714--7 6 .

[191

_______ , "Locally flat imbeddings of topological manifolds," Ann. of Math. (2) 75 (1 962) pp. 331- 3 M .

[20]

C. E. Burgess, "Properties of certain types of wild surfaces in E 3," Amer. J. Math., 86 (1961 +) pp. 325-338.

[ 21]

_______ , "Characterizations of tame surfaces in E 3 ," Trans. Amer. Math. Soc., 1 1 1 + ( 1 9 6 5 ) pp. 80-97.

[2 2 ]

_______ , "Pairs of 3-cells w i t h intersecting boundaries in E 3 ," Amer. J. M a t h . , 88 ( 1 9 6 6 ).

h23]

L. 0. Cannon, "Sums of solid horned spheres," Trans. Amer. Math. Soc., (to a p p e a r ) .

[ 21+]

B. G. Casler, "On the sum of two solid horned spheres,” Trans. Amer.

Soc., 1 1 6

(1965) pp. 511-525.

Math. Soc., 1 1 6

(1965) pp. 135-150.

[25]

P. H. Doyle and J. G. Hocking, "Some results on tame discs and spheres i n E 3 ," Proc. Amer. Math. Soc., 1 1 (i 96 0 ) pp. 832-836.

[ 26]

M. K. Fort, Jr., "A wild sphere w h i c h can be pierced at e a c h point by a straight li_ie segment," Proc. Amer. Math. Soc., 11 + ( 1 9 6 3 ) pp. 991 +995.

[27]

R. Fox and E. Artin, "Some wild cells and spheres i n three-dimensional space,"Ann. of Math. (2 ) 1+9 ( 191 +8 ) pp. 979-990.

[28]

David S. Gillman, "Note concerning a wild sphere of Bing," Duke Math. J., 31 ( 1 961 +) pp. 21 +7 - 2 5 U.

[29]

_______ , "Side approximation, miss i n g an arc," Amer. J. Math., 85 ( 1 9 6 3 ) pp. 1+59-1*76.

6o

BURGESS

[30 ]

W. Graeub, Semilinear Abbildungen. Heidelberg, 19 5 0 , pp. 20 5 - 2 7 2 .

Sitz.-Ber.d. Akad. d. Wissensch

[31 J

H. C. Griffith, "A characterization of tame surfaces in three space," Ann. of Math. (2 ) 69 (1959) pp. 2 9 1 - 308 .

[32 ]

0. G. Harrold, Jr., "Locally tame curves and surfaces in three-dimensional manifolds," Bull. Amer. Math. Soc., 63 (1957) pp. 293 - 3 05 .

[ 33 J ______ , "Locally peripherally unknotted surfaces in E 5," Ann. of Math. ( 2 ) 69 (1959) pp. 276 -2 9 0 . i 34 ]

______ , and E. E. Moise, "Almost Math. (2) 57 (1953) pp. 575-578.

locally polyhedral spheres, " Ann. of

[35]

John Hempel, "A surface in S^ is tame if it can be deformed into each complementary domain," Trans. Amer. Math. Soc., ill (1964) pp. 273-287.

[36]

Norman Hosay, "Conditions for tameness of a 2-sphere which is locally tame modulo a tame set," Amer. Math. Soc., Notices, 9 (19 6 2 ) p. 1 1 7 .

[37]

______ , "Some sufficient conditions for a continuum on a 2-sphere to lie on a tame 2-sphere," Amer. Math. Soc., Notices, 11 (1964 ) pp. 370371 .

[38]

L. D. Loveland, "Tame surfaces and tame subsets of spheres in Trans. Amer. Math. Soc. (to appear).

[39]

______ , "Tame subsets of spheres

[ 1+0 ]

Joseph Martin, "A rigid sphere," Fund. Math, (to appear).

[1+1 ] [41]

E. E. Moise, "Affine structures In in 3-manifolds, I-VIII," I-VTII," Ann. of Math. 51+0951) pp. 506-553; 506-533; 55 ('952) (1 9 5 2 ) pp. 1 7 2 - 1 7 66;; 55(1952) 5 5 (1 9 5 2 ) pp. 203203 (2 ): 54(1951) 211 +; 55 1 9 5 2 ) pp. 1 5 - 2 2 2 ; 56 (19 5 2 ) pp. 96 - 1 1 4 ; 58 (1 9 5(1953) 2 ) pp. pp. 96 - 111 214; 55 ((1952) pp. 2215-222; 107+; ; 58 58 (1953) (1953) pp. pp. i+03-1+08; 403-408; 59 (1954) pp. 159-170; 0954) pp. 159-1 70;

E^,"

in E^," Pacific J. Math, (to appear).

[ 1+ 2 ]

C. D. Papakyriakopoulos, "On Dehn*s Lemma and the asphericity of knots," Ann. of Math., 66 0957)pp. 1 - 2 6 .

[1+3]

A. Shapiro and J. H. C. Whitehead, "A proof and extension of Dehn's Lemma," Bull. Amer. Math. Soc., 64 (1958) pp. 174-178.

ruu]

John R. Stallings, "Uncountably many wild disks," Annals of (2 ) 71 (i960 ) pp. 1 85-186.

Math.

58 (1953

ADDITIONAL QUESTIONS ON 3 -MANIFOLDS 1.

(R. H. Bing)

Can Dehn's Lemma be generalized in either of the following

two directions? a) Does a simple closed curve bound a disc in a 3-manifold if it can be shrunk to a point in its own complement? 2 3 b) Suppose f is a piecewise linear map of E into E wh i c h 2 takes unbounded sets onto unbounded sets, D is a disc in E such that f " 1 is a homeomorphism on f (D), and N is a n e i g h ­ borhood of the singularities of f. Is there a homeomorphism 2 2 h of E onto a closed set in f(E ) U N wh i c h agrees with f on D ? 2.

(David Henderson) Let D be a disc-with-holes and f a map of (D, Bd D) into (M5 , Bd M 5) , where M 5 is a 3-manifold. We say that f is n o n pullable if there does not exist a map F: D -►Bd M 5 such that F|Bd D = f|Bd D. Conjecture: If M is an orientable 3-manifold, D a disc-withholes, and f: (D, Bd D) -*• (M, Bd M) is non-pullable, then there exists a disc-with-holes D T and an embedding f 1 such that f 1: ( D 1, Bd D')— (M, Bd D) is non-pullable. Note that if D is a disc, then this conjecture reduces to the loop theorem and Dehn*s Lemma. Usi n g results of Shapiro and Whitehead [i], it is enough to gr, g a map f 1 ■such that is non-pullable and is an embedding w hen r e strict­ ed to a neighborhood of B d D 1. A n affirmative answer to this conjecture would give an affirmative answer to Question 13 of [2] and would be a useful tool in "scissors and paste" topology. References: 1. 2.

Shapiro and Whitehead, "A proof and extension of Dehn's Lemma," Bull. A m e r . Math. Soc., vol. 6k (1958) pp. 174-178. Sumner Institute on Set-theoretic Topology, Madison, Wisconsin, 1955.

3.

(R. H. Bing) W h i c h of the following conditions is enough to imply that a 2-sphere S in E 5 is tame? (a)

S can be approximated from either side b y a singular 2-spbere

(b) (c)

S can be touched from either side b y a pencil [2], At each point • S lies locally between two tangent 2-spheres

[ 1 ].

[ 2 ]. 81

82

BURGESS (d) (e) (f)

(g) (h) (i)

References: 1 . 2.

E a c h h orizontal cross-section of S is connected. E a c h horizontal cross-section of S is either a point or a simple closed curve [ 2 ]. For each e > 0 there is a 5 > 0 such that each simple closed curve of S of diameter less than 5 can be shrunk to a point on e-sets on either side of S. S can be pierced a long e a c h arc in it b y a tame disc [l ]. S is homogeneous under a space homeomorphism Cl]. S is tame mo d a closed set X such that X lies in a plane and has no degenerate components. (Hosay*s thesis gives some partial solutions.) Bing, R. H . ,"Embedding surfaces in 3-manifolds," Proc. Inter national Congress of Mathematicians, 1 9 6 2 , pp. 1+57-1+58. _______ , "Spheres In E 3 ," Amer. Math. Monthly, vol. 71 ( 1 9 6 4 ) pp. 353-361+.

1+.

(R. McMillan) If M is a compact 3 -manifold wit h boundary (non-empty) and h is a homeomorphism of M onto itself such that h|Bd M is the identity, what conditions on h are sufficient to insure that h is isotopic to the identity?

5.

(F. Burton Jones) Suppose that M is a non-separating subcontinuum of S 2 . Then if a and b are points of M w h i c h are not cut from each other in M by any zero dimensional closed subset of M, then there is a closed disc in M w h i c h contains a and b. [Jones, about 1 9 6 0 .] Wh a t are possible true extensions to S 3 ? For example: If M is a point-like subcontinuum of S 3 and no 1 -dimensional closed subset of M cuts a from b in M, does M contain a locally connected subcontinuum K containing b o t h a and b such that no 1 -dimen­ sional closed subset of K separates a from b in K ?

6 . (R. H. Bing)

Suppose h: E 3 -►E 3 is a periodic map whose fixed point set is a straight line. Is h equivalent to a rotation about E 1?

7.

(Robert Craggs) Can ea c h periodic transformation of S 3 onto S 3 be approximated b y a piecewise linear periodic transformation? If the a n ­ swer is yes, must the fixed point set for the approximating periodic transformation be the same type of sphere as the fixed point set of the original periodic transformation?

8.

(R. H. Bing) If h: E 3 -*• E 3 is a periodic m a p w i t h a point as the fixed point set, is h the composition of two periodic maps hj and h 2 where h 1 is a rotation about a line (possibly wild) and h 2 is a reflection through a plane (possibly wild)?

CHAPTER II: THE POINCARE CONJECTURE H OW NOT TO PROVE THE POINCARE CONJECTURE by J o h n Stallings* Introduction I have committed—the sin of falsely proving Poincare's Conjecture. But that was another country; and besides, until now no one has known about it. Now, in hope of deterring others from m a king similar mistakes, I shall describe m y mist a k e n proof. W h o knows but that somehow a small change, a new interpretation, and this line of proof m a y be rectified! In the b a c k of m y m i n d wh e n I conceived my proof was this theorem. THEOREM 0: (For n ^ 2) . Let f: M -*■K be a m ap of a connected orientable n-manif old into a n n-complex, and let C 1 ,. •., C ^ be some of the n-simplexes of K such that the degree of f on each C^ is zero (that is, the homology map induced b y f , HJl(M) H ^ K , K-int C ^ ) , is zero) . Suppose f induces a homomorphism of tr1 (M) onto tr1 (K) . Then f is homotopic to a map into K - (int C 1 U...U int C^) . A special argument establishes this theorem for n = 1 . For n > 3 we m a y argue as follows. W e shall make a number of changes on f w h i c h will be independent of each other, since n > 3; hence we need only consider the case that k = 1 and suppose that C 1 is covered twice w i t h opposite orien­ tations by f (M) . The inverse image of a small cell in C 1 is the union of two cells A and B in M. Let P be a pa t h in M from A to B; fP represents an element of tt1 (K) . Since iry (M) -*• i ( K ) is onto, we can m o d i ­ fy P by adding on i loop whose image represents the inverse of fP; thus we can suppose fP is a null-homotopic loop in K. Since the dimension of M is at least 3, we can choose P to be a non-singular p a t h and change f b y a homotopy in the neighborhood of P so that f(P) C C ] . If T is a tube around P,. then A U T U B wi l l be an ncell mapped into C 1 w i t h degree zero. A further homotopy within A U T U B

Sloan Foundation Fellow 83

STALLINGS

84

will uncover a point of of

C 1;

by pushing away from that point, we uncover al.

int C 1 .

But, in my proof of Poincare's Conjecture, I need this theorem for n = 2 . The argument above fails in this case for several reasons. We canno uncover 2-cells independently of ea c h other; we cannot make the pa t h P nonsingular; if P were non-singular, the homotopy bringing f(P) into C 1 might cause us to cover up cells which we want to uncover. The reader may be able to patch up some of these points. If he patches up all these points, he will have proved the Poincare Conjecture (for we shall show how Theorem o for n = 2 implies the Poincare Conjecture incorrectly. For, Theorem 0 is false for n = 2: Consider a torus w i t h two 2 -cells C 1 and C 2 attached to kill the fundamental group; there is a map of the 2-sphere into this complex; b y a homotopy we can uncover either C 1 o: C~, but not bo t h simultaneously. 1.

A conjecture about the 3-sphere

In the 3-sphere S^ let T be a tame 2-manifold such that b o t h coi ponents of S^- T have free fundamental groups. Let U and V denote the closures of the components of S^- T. According to theorems of Papakyriakopoulos, b o t h U and V are handlebodies. The only conceivable such embedding of T is shown in Fig. 1

Fig. 1 However, if the genus of T is greater than 1 , this standard e m ­ bedding has a property whi c h does not seem to follow immediately from the fact that U and V are handlebodies. Namely, there is a simple closed curve C on T, not contractible on T, yet bounding 2-cells b o t h in U and in V. * CONJECTURE A: The existence from the hypothesis that b o t h U and tempt

of such a curve C can be proved only V are handlebodies.

If we hope that Conjecture A is true, a reasonable direction to a proof of the Poincare Conjecture can be made as follows.

at­

H O W NOT TO PROVE THE POINCAKg CONJECTURE

85

Poincare's Conjecture is that any simply-connected 3-manifold M is a 3-sphere. It is known that any orientable 3-manifold such as M, has a Heegaard representation as U U V, where U and V are handlebodies and U n V is their common boundary, a 2-manifold T. If the genus of T should happen to be one, then space, and so, if simply-connected, is a 3-sphere.

M

is a lens

Assume that we could prove Conjecture A for M, rather than for the 3 -sphere: That is, if the genus of T is greater than one, then on T there is a simple closed curve C, not contractible on T, yet bounding 2-cells in bo t h U and V. Then we could write M as the connected sum M 1 # M 2 of two manifolds whose Heegaard representations would have less genus. A nd so b y induction on the genus, we would know that M is indeed a 3-sphere.

2.

R eduction to group theory

Let M = U U V, T = U n V be a Heegaard representation of a 3manifold. We obtain a diagram of fundamental groups, w i t h homomorphisms in­ duced from inclusions:

Since

x is onto, since t t 1 (U) x (V) is the product of kernels of the projections onto its factors, it follows that tr1 (T) is product of ker q> and ker >|r, and hence M is simply connected.

the the

Conversely, if M is simply connected, then tt^T) = (ker •••> n from Lemma 2 that T^ lies in a cube w i t h handles 7? in M 5 such that T 3 can be shrunk to a point in Int X and some open subset U of f(a ) misses X^ v5 Let be the lateral side of P, and S 4 be a disk on intersecting each end of P^ in an arc. Let be a disk on Bd T ; U Int Aj such that R ^ n (P^5 U R ^ 2) = o (i £ j) and R ±2 n Si 2 - Int A i . See Fig. 5. If Ri + Si the lateral edge of a pill b ox 3 would be a double pill gox as mentioned in Lemma 1.

A,

in There is no

MAPPING A 3 -SPHERE ONTO A HOMOTOPY 3 -SPHERE assurance that this is the case. However, since w hich in turn can be shrunk to a point in Int

3

95

R ^ 2 U S ^ 2 lies in there is a map

3

T^, of

3

the double pill box P U Q of Lemma i such that g, takes P homeo3 3 2 morphically onto P., , the later sides of P^ onto A., , the lateral 3 2 2 3 3 side of Q homeomorphically onto S^ U R ^ and Q into Int X' (per­ haps singularly) . Let B. 2 = f ”1 (R j 2 U ends of P.,5). It is a disk on B d B 5 resem3 3 3 b l i n g two disks joined b y a narrow band. Let Cj , C 2 , 0^ be m u t u ­ ally exclusive 3-cells in B 5 such that C ^ 5 n B d B 5 = B ^ 2 . Let h^ be a h^ieomorphism of 0 ^ onto P 5 U Q 5 of Lemma 1 such that * f/Bj_2 Note that f(Bd C .5 - B, 2) C X 5 . We define f on the pits C.,5 b y f / o ^ - gl V 3

'

3

Part 3. Let B denote the 3-ball wh i c h is the closure of B U C±5 . Note that f(Bd B 5 ') C X 5 . Since X 5 is a cube w i t h handles, 7rg (X5) = o. Hence f/Bd B^ can be extended to m a p of B 3 * into Int X 5 . 3 3 —1 It m a y be shown that f : S' -► K ' is of degree 1 since f is 3 3 a homeomorphism on U . The only points of M' that have more than one inverse lie on a compact subset of Int X^. Hence the image of the singu­ larities of f lie in Int X^. 4.

Repl a c i n g cubes w i t h holes b y cubes w i t h handles

Suppose C 5 is a polyhedral 3-cell and C ^ , C 2^,..., is a collection of mutually exclusive polyhedral 3-cells i n such that Bd C 5 n Bd is the union of two mutually exclusive disks A i 2, A ^ 2 . We regard as A ^ 2 x [0 , 1 ] where A ^ 2 x o * A ^ 2 and

Ai2 x 1 * A^

96 Then

BING C5 - U ( I n t

x [0, 1 ])

i s a cube w ith h o le s and each

I n t A^ x [ 0, 1 ]

i s a h o le . L e t p^ be a p o in t o f A ^ . The h o le I n t Ai 2 x [ 0, 1 ] i s k n o tted or un kn otted a cc o rd in g as p^ x [ 0, 1 ] i s a k n o tte d or unknotted spanning a rc o f oK A tame spanning a rc o f i s un kn otted or k n o tte d a c c o rd in g as i t , t o g e th e r w ith an a rc on Bd (P , does or does n ot bound a d is k i n C5 . I t i s known th a t e ach p olyh edron in bounded by a con nected 2m an ifo ld i s t o p o lo g ic a l ly a cube w ith h o le s . Hence f - 1 (X^) o f Theorem 1 i s a cube w ith h o le s and was o b tain ed from by rem oving a cube w ith h o le s and r e p la c in g i t w ith a cube w ith h a n d le s. Theorem 1 a d v is e s us t h a t any homotopy 3-sphere can be o b tain ed a s such a s im p l if i c a t io n o f S^. Theo­ rem 1 t e l l s us a way o f g e t t i n g a counterexam ple to the P o in care c o n je c tu r e i f th e re i s one. The fo llo w in g r e s u l t i s r a t h e r e a s y to v e r i f y . THEOREM 2. Suppose Y 5 ' i s a tame cube w ith h o le s in S5, q i s a 3 1 3 . 3 1 7i map Y onto a cube w ith han d les Y such t h a t g/Bd Y ta k e s Bd Y 3 3 1 3 3 hom eom orphically onto Bd Y . I f one r e p la c e s Y by Y in S (th e 3 • sew ing b e in g done by a g a lo n g Bd Y ) , one o b ta in s a homotopy 3 -sp h ere. Theorem 1 t e l l s us t h a t any homotopy 3-sp h ere can be o b tain ed from by rem oving some cube w ith h o le s and r e p la c in g i t w ith a cube w ith h an d les w h ile Theorem 2 t e l l s us th a t anytim e t h i s rep lacem en t i s done in a c e r t a i n way, one o b ta in s a homotopy 3 -sp h ere. Hempel showed th a t i f X map f o f X onto a cube w ith hom eom orphically onto Bd f ( X ) . can be r e p la c e d i n the " c e r t a i n

i s a cube w ith one h o le , then th e re is a a handle such th a t f/Bd X ta k e s Bd X Whether or n o t each cube w ith h o le s in way" depends on the answers to the f o llo w in g

q u e s tio n s . Q uestion k. I f X i s a cube w ith two h o le s , i s th e re a map f of X onto a cube w ith two h an d les such th a t f/Bd X ta k e s Bd X homeomorphic­ a l l y onto the boundary o f th e cube w ith handles? Q uestion 5 . I f X i s a cube w ith h o le s , under what c o n d itio n s i s th e re a map f o f X onto a cube w ith han dles such th a t f ta k e s Bd X hom eom orphically onto the boundary o f the the cube cube wwith ith handles? handles? A cube w ith a k n o tte d h ohle o lei si s t tooppool ol oggi cicaal ly lly d iffe d r ei fnf te r from e n t from a cube a cube w ith an un kn otted h o le s in c e one has a n on -A b elian fundam ental group and the o th er has a fundam ental group w hich i s i n f i n i t e c y c l i c . However, a cube w ith two h o le s one o f which i s k n o tte d and the o th er o f w hich i s un kn otted may be t o p o l o g ic a l l y e q u iv a le n t to a cube w ith two s t r a ig h t h o le s . F ig u re 6 shows how to s t a r t w ith a cube w ith two s t r a ig h t h o le s , g r a d u a lly move the bottom b ase o f the r i g h t h o le a lo n g

MAPPING A 3 -SPHERE ONTO A HOMOTOPY 3 -SPHERE

97

parallel to the dotted line until the right hole becomes knotted. Question 6. Suppose C 5 - (Int A 1 x [0,1]) - (Int A g x [0,1]) is a cube wi t h two holes tamely embedded in E 5 and topologically equivalent to a cube with handles such that Int A 1 x [o,i] is unknotted but Int A g x [0,1] is knotted. In Fig. 6 the knotted hole is a trefoil knot. Other toroidal knots could have been similarly obtained. What kind of knots could have been obtained?

One cannot tell at a glance whether or not a cube w i t h holes is topologically equivalent to a cube with handles. Even if b o t h holes are knotted, one cannot be sure it is topologically inequivalent to a cube with nandles. For example if one winds the straight hole of Fig. 6 about he knotted hole, one obtains two holes as shown in the bottom part of

Pig. 7.

98

BING

5. f

Simplifying handles One might try to prove the Poincare conjecture by simplifying either X 5 of Theorem 1.

or

THEOREM 3. In order for M 3 of Theorem 1 to be a counterexample to the Poincare conjecture it is necessary that X 5 have more than one handle.

3-cell.

Proof: If X 5 has no handles, each of X 5 and M 3 - Int Hence, in this case, M 3 is topologically equivalent to

X3 S3.

is a

If X 3 has one handle, one of f " 1(X3), S 3 - f ’^ I n t X 3) is a solid torus. If S 3 - f “1 (Int X') is a solid torus, each of X 3 , M 3 Int X 3 is a solid torus and no counterexample results. If f ' ^ X 3) is a solid torus, there is a homeomorphism h of f ~ 1(X3) onto X 3 that agrees with

f

on

Bd f ~ 1(X3)

and

M3

is homeomorphic to

S3.

One might try proving the Poincare conjecture b y reducing the n u m ­ ber of handles in X 3 . If one can find a handle in f - 1 (X3) one can cut off this handle and redefine f so that the images of the singularities of the modified f lie in a cube wit h fewer handles than X 3 has. Question 7. What special can be said about a cube w i t h handles tamely embedded in S 3 so that some simple closed curve in Bd X bounds a disk in S 3 - Int X but none on Bd X? It might be of help i n reducing the handles on X 3 of Theorem 1 if one knew, for example, that such a B d X contained two transverse simple closed curves one of w hich bounded a d i s k in X and the other of whijch. bounded a disk in E 3 - X. This m a y be d i f f i ­ cult to show since I do not believe it is even known to be true in case S3 - Int X is a cube w i t h handles. As noted in the following paragraph, if we start modifying f and X 3 so as to reduce handles we arrive at a situation where there is a simple closed curve on the modified Bd X 3 w h i c h bounds a d i s k in M 3 - Int X 3 but none on Bd X 3 . If Bd X 3 contained two transverse simple closed curves one of wh i c h bounded a disk in X 3 and the other a di s k D in M 3 - Int X 3 , we could remove another handle from X 3 b y thickening D and adding it to X 3 . It follows from the loop theorem that there is a dis k D in S3 such that D n f ~ 1(X3) = Bd D n Bd f ”1(X3) and Bd D does not bound a di s k on Bd f " 1 (X3) . If D C S 3 - f - 1 (Int X 3), we have the situation m e n ­ tioned in the preceding question. If D C f _ 1 (X3) and Bd D fails to se­ parate Bd f " 1 (X3), then D shows us where to cut off a handle from X3 and redefine f as suggested in the paragraph following the proof of T h e o ­ rem 3. Finally, we suppose D separates f _ 1 (X3). We suppose f is a homeomorphism near D so that we can remove a thickened f(D) from X 3 and be left w i t h two pieces X ^ , X 2 3 so that' X 13 U X 23 contains the images

MAPPING A 3 -SPHERE ONTO A HOMOTOPY 3 -SPHERE

99

of the singularities of f . B y continuing splitting we finally arrive at 3 a finite set of X. *s whose u nion contains the images of the singularities of f and the sum of the handles in the X ^ ' s is the same as the number of handles in X 3 . Then we find a di s k D such that D n f ’1(U X ^ ) = B d D n Bd f “1 (U X , 5) and Bd D does not separate the Bd f ’1(X.5) on whi c h it lies. We join the f (X, )'s b y tubes which miss D to.obtain a —1 5 5 new f (X ) such that the new X has the same number of handles as the old but also, either f(D) enables us to move a handle from 7!? or there is a simple closed curve on Bd X^ w h ich does not bound a disk on Bd X^ but does bound one in M 3 - Int X^. Instead of reducing the number of handles in X 5, one instead might try to simplify f ”1 on the handles. If D, x [o,i] is a handle of X^, _i it is possible to define f so that each f (D^x t) is a disk with handles such that the set of singularities of f on f ' U D ^ x t) is a finite number of figure eights. By drilling holes in each f - 1 (D,x [0,1]) so as to sepa^ X rate the figure eights in the middle level one can change X (by increas­ ing handles) so that the inverse of each handle is the Cartesian product of an arc and a di sk-with-a-handle. Hence, we obtain the following result: THEOREM 4. For ea c h homotopy 3-sphere M 5 there is a map f of S 5 onto M 3 such that the image of the singular set of f is a bouquet of simple closed curves, the inverse of the vertex point of the bouquet is a cv.be w i t h holes and the inverse of eac h other point of the bouquet is a f ig­ ure eight.

REFERENCES [1]

R. H. Bing, "Some aspects of the topology of 3-manifolds related to the Poincare c o n j e c t u r e Lectures on M odern M a t h e m a t i c s , John W iley and

[2 ]

Wolf g a n g Haken, "On homotopy 3-spheres," Illinois Journal of Mathematics to appear.

[3 ]

E. E. Moise, "Simply connected 3-manifolds,M Topology of 3-Manifolds and Related Topics. Prentice Hall ( 1 9 6 2 ), pp. 1 9 6 - 1 9 7 .

[4]

Russel McMillan, these notes.

Sons (1964), vol. 2, pp. 9 3 - 1 2 8 .

CONCERNING FAKE CUBES by A. C. Connor A fake cube is a compact, simply connected 3-manifold wi t h boundary whose boundary is a 2-sphere. It has been shown by R. H. Bing [i] that a fake cube K is a cube if it has a triangulation T such that (1) each vertex of T is cn Bd K and (2) each 3-simplex of T has at least two edges on Bd K. The principal object of the present paper is to show that condition (2)can be replaced by ( 2 1) each 3-simplex of T has at least one edge on Bd K. A point of a 2-complex K which has no open 2-cell neighborhood in K is called a singular point or a singularity of K; the set of all singular points of K will be denoted by S ( K ) . Two kinds of singulari­ ties will be of interest here. A singular point is of the first kind if it has a neighborhood homeomorphic to the product of an arc and a triod and is of the second kind if it is not of the first kind and has a n e i g h ­ borhood homeomorphic to the cone over a set consisting of a circle to­ gether w i t h three of its radii. A compact 2-ccmplex is said to be of Type 1 if each of its singularities is of the first kind and to be of Type 2 if eac h of its singularities is of the first or second kind. The following theorem, wh i c h is used in the proof of the main r e ­ sult, m a y be of independent interest. THEOREM.

E v e r y simply connected 2-complex of Type 1 has a n o n ­

trivial second homology group. This theorem is proved b y showing that the set of singular points of a 2-complex K of Type 1 is the union of a finite number of disjoint simple closed curves eac h of w h i c h separates K, and that if K is any 2-complex of Type 1 w h i c h is separated by every simple closed curve in S ( K ) , then (a) K contains a closed 2-manifold without boundary and (b) H 2 (K, Z 2 ) ^ o. A corollary is that no 2-complex of Type 1 is contractible. Now suppose that K is a fake cube having a triangulation satis­ fying (1) and (2'). It follows as in [l] that K has a triangulation T 10 1

CONNOR

102

w h ich satisfies (1) and ( 2 1). It follows as in [1] that K has a trian­ gulation T whi c h satisfies (1) and (2») as well as (3) each 2-simplex of T which has two edges on Bd K is a subset of Bd K and (b) no 3-simplex of T has exactly two of its 2-dimensional faces on Bd K. Suppose T contains more barcycentric subdivision of T simplexes of T*1 w hich have no that N is a 2-complex of Type

than one 3-simplex. Let T 1 be the first and let N denote the collection of all vertex on Bd K. It is easily verified i.

Let R denote the second derived neighborhood with respect to T of Bd K and let K 1 = !( - R. Since K 1 is a regular neighborhood of N, K 1 collapses to N and hence, since K is homeomorphic to K 1, K collapses to N. It follows that N is contractible, which is impossible, since N is a 2 -complex of Type 1. Hence T contains just one 3-simplex, so K is a cube. A further result is that if the Poincare conjecture is false, then there exists a fake cube K which is not a cube and which has a spine N of Type 2 such that S(N) is connected and each component of N - S(N) is an open disk.

REFERENCE [1]

R. H. Bing, 'Some Aspects of the Topology of a 3-Manifold Related to the Poincare Conjecture" Lectures on Modern Math. Vol. 11 edited by T. L. Saaty, Joh n Wiley & Sons, Inc., 1 9 6 U.

CHAPTER III O N CERTAIN FIRST COUNTABLE SPACES by R. W. Heath Some first countable spaces wit h interesting variety of structures are semi-metric spaces, developable spaces and (first countable) M 1-spaces. E a c h seems to be characterized by having a basis wit h a certain kind of uniformity, and in fact these seem to be the three basic types of u niform­ ity in the spectrum from first countable to metric spaces. Definition 1.1 A T 2-space S is semi-metrie if there is a d i s ­ tance function (or "semi-metrie") d for S such that (i) for each x and y in S, d(x, y) = d(y, x) > o and d(x, y) = 0 only if x = y, (ii' for x € S and M C S, inf(d(x, y) : y e M) = 0 if and only if x is in the closure of M. Definition 1.2 A development for a space Sis a sequence G 2 ,,.. of open coverings of S such that, for each p € S and each open set R containing p, there is an n such that every element of Gn containing p lies in R. A developable space is a T 2-space having a development. REMARKS (l) For any metric space S, the sequence G 1, G 2 ,... in which, for each i, G i is the set of all open sets of diameter less than 1/i would be a development for S. (2) A space X wit h development G 1, G 2 ,... has the natural semi­ metric d: d(x, y) = i n f { i /n: x € G and y € g for some g € G ) (where we have let X e G 1 and G^ D G i+1 for all i -without loss of ge n e rality). Note that spherical neighborhoods under but d is not continuous—hence cannot be a metric.

d

will be open,

(3) Regular developable spaces are Moore space, i.e., satisfy the first three parts of A x i o m 1 in [ 3 2 ]. (k) Russian mathematicians call semi-metric spaces "symmetrizable" and they call a development a "countable complete family of open coverings',' also what they call a space wit h a "uniform base" is simply a pointwise paracompact developable space and a "symmetrizable Cauchy space" is eq u i v ­ alent to a developable space. This difference In terminology led to the 103

10k

HEATH

unfortunate situation that they did not know until 1 962 that there was a non-developable semi-metric space (for such a space see [28], [ 1 7 ] or Example 2 . 1 below). In fact the only "example" they now have, by A. Lunts in 1 962 Doklady, is incorrect (the set of all countable ordinals wi t h the order topology!). This communications problem has also led to the recent rediscovery of a number of theorems. Definition 1.3

A collection

A

is called closure-preser v i n g t if,

for every B C A, the union of the closures of members of B is closed. A collection is a-closure-preserving if it is the union of the countably many closure-preserving collections. A n M 1-space is a T^-space w i t h a a-closure-preserving basis. R E M A R K (5) A n M 1-space is paracompact b y Professor E. Michael's theorem [ 3 0 ]: A T^-space in w h i c h every open cover has a a-closure-preserving open refinement is paracompact. A semi-metrie for a first count­ able M^-spacS can be defined in somewhat the same way as for a d evelop­ able s p a c e . In Summary we have first countable

M. = >

paracompact semi-me trie

metric = >

= >

developable

semi-metrie

first countable

and examples show that none of the arrows can be reversed— except that developable M^-spaces are metric. 2.

Examples

Example 2 . 1 [28]. A paracompact semi-metric apace w hich is an space but not de v e l o p a b l e . 2

M 1-

The points of S are the points of E and the semi-me trie d is defined as follows. If either x or y is on the x-axis, let d(x, y) = Ix-y| + a(x, y ) , otherwise let d(x, y)= |x-y|, where |x-y| is the ordinary E uclidean distance and a(x, y) is the (radian) measure of the smallest non-negative angle formed b y a line through x and y with a horizontal line. Thus a "spherical" neighborhood of a point on the x-axis with respect to d is "bow-tie" shaped. \

y \ \

x

ON CERTAIN FIRST COUNTABLE SPACES

105

These bow-tie neighborhoods of points on the x-axis and discs elsewhere are used as a basis for the topology. REMARKS C O . If one uses as basis elements bow-tie regions at every point of E 2 (and let d(x, y) = |x-y| + a(x, y) for all x and y ) , the space is then a semi-metric space which is not paracompact or an M 1space. Also, though it is connected, locally connected and complete, it is not arc-wise connected [173. 2 (2 ) The semi-metric space S wit h points those of E and a basis consisting of all bow-tie regions centered on irrational points of the xaxis and open discs centered on all other points is paracompact but has the following property: there is no semi-metrie for S with respect to which all spherical neighborhoods are open sets [ i5 JExamples of developable spaces are given in [2 5 ] snd [ 3 9 ] of these pr o c e e d i n g s . 3.

Metrization of Developable Spaces Professor R. B. Jones proved the following in [2 2 ]:

is

THEOREM 3.1. If 2^° < 25*1 then every separable normal space -compact (i.e., every uncountable subset of it has a limit point).

COROLLARY 3.2. If 2^° < 2**l then every normal separable Moore space (regular developable space) is metrizable. M u c h effort has been made b y man y people to rid the corollary, in one w ay or another, of the assumption 2^° < 2 1 . There has bee n very little success (see [53, [2 0 ], [353, [393), although the converse to Theorem 3.1 was proved in [2 1 ] so that: THEOREM 3.2. E v e r y separable normal and only If 2*° < 2S l .

T--space is

« 1-compact if

There is also the following result in which the hypothesis 2^0 < is replaced b y another, possibly weaker, hypothesis about the reals (and equivalence to the metrization proposition is s h o w n ) .

2 1

THEOREM 3.3. [203 E v e r y separable normal Moore space is metrizable if and only if every uncountable subset M of E 1 contains a subset w hich is not a G_. set (relative to M ) . 5 It was also conjectured by Professor Jones in [22] that every normal Moore space is metrizable. The only major results in this direction have been the following due to Professor R. H. B i n g in [6]. THEOREM 3.it-.

A paracompact Moore space is metrizable.

106

HEATH

THEOREM 3.5.

A collectionwise normal Moore space is metrizable.

R E M A R K . It is Theorems 3.4 and 3.5 in particular w h i c h have bee n rediscovered in several forms b y the Russians (for example see [1, p. 44], [2, p. 40]). (Refer b a c k to Rema r k 4 in Section 1 for equivalent termi­ nology) . F or further details on the normal Moore space metrization problem, see [2 5 ] and [ 3 9 ] or [2 3 ]. Finally, a proposition w hich I think might be true (but will probably require the assumption 2 0 < 2 1 -because of Theorem 3 in [2 0 ]) is: Eve r y normal pointwise paracompact Moore space is metrizable.

4.

Metrizability and Developability of Semi-metric Spaces.

There are the following characterizations of developable and m e t r i z ­ able spaces in terms of special semi-metrics possessed b y the space. THEOREM 4.1 [ 3 ]. A T^-space S is developable if and only if there is a semi-metric d for S such that whenever n

lim d(x , p) = lim d(y , p) = 0 00 n 00

then n THEOREM 4.2 [37]. there is a semi-metrie d n

Ujn d(y , x^) «

=

0

.

A T^-space S is metrizable if and only if for S such that, whenever lim d(x , p) - 11m d(x_, y_) = 0 « n * 11 11

then lim d(y n —►00

p) = 0

.

A question posed b y Morton B r own [8] remains open, however: "What ’ t o p ological1 property can be added to a semi-metrie space to get a Moore space?" The following property (suggested b y Professor Jones) is a rather natural candidate: for every compact subset M of the space, there is a sequence XJ1, U 2 ,... of open sets such that, for every open set R c on­ taining M, there is an n such that M C U n C R. Unfortunately, I can give an example of a non-developable semi-metric space (which is p aracom­ pact) wh i c h does have that property. Example 4.1. Let M be an uncountable subset of the real numbers all of whose compact subsets are countable ([ 2 7 , p. 4 2 2 ]). Let S be the

ON CERTAIN FIRST COUNTABLE SPACES

107

o space whose points are the points of E not on the x-axis or belonging to a subset M 1 of the x-axis congruent to M, and let a basis consist of all open disks centered on points of S - M 1 and of all bow-tie regions (see Example 2 . 1 ) centered on points of M 1. Then S is semi-metric but not developable, and it can be shown that S has the above property, that there is a "countable neighborhood system" for each compact subset of S. Some theorems w h i c h m ight be helpful in answering Morton Brown's question are to be found in [173 and [363- The following, for example, is a reformulation of Theorem 3.6 of [173. THEOREM 4.3. A regular semi-metrie space S is a Moore space if there is a decreasing sequence G 1 3 G 2 3 ••• of open coverings of S such that, if p € S, then any decreasing sequence, g 1 D g 2 D ,*•* w i t h p € g^ e G^ (i = 1, 2,...) is a local base for p, and for each p € S there Is such a decreasing sequence. The following two theorems, certain special cases.

and 4.5, answer Brown's question in

D e f i n i t i o n . A semi-metrie space S is strongly complete if there is a semi-metric d for S w i t h respect to w hich every nested sequence M 1 D Mg 3 • •• of closed sets such that for eac h n there is some point Pn for w h i c h M n C (y: d(y, pn ) < i/n) has non-empty intersection. THEOREM 4.4 [173. is a Moore space.

A strongly complete, regular semi-metric space

THEOREM 4.5 [193.

A semi-metric space w i t h a point-countable base

i j dev e l o p a b l e . REMARK.

A base

B

is point-countable if ea c h point of the space

belongs to only countably man y members of B. A pointwise paracompact developable space (same as a T 2-space wit h a "uniform base" in Russian terminology—see [23) lias a point-countable base. There does exist a Moore space w i t h a point-countable base w h i c h is not pointwise paracompact [143. See [123 for some theorems that can be extended from metric spaces to pointwise paracompact Moore spaces (but not to Moore spaces in ge n e r a l ) . The following metrization theorem shows how strong "strong complete­ ness" is. THEOREM 4.6 [213. is metrizable.

A strongly complete separable semi-metric space

Finally, I have found Theorem 4.7 extremely useful in constructing examples of semi-metrie spaces.

1C8

HEATH

THEOREM k.l [ 1 7 ]. A T 2-space S is semi-metric if and only if there is a collection ( g ^ x ) : x € S, n = 1 , 2 ,...} of open sets such that ( 1 ) for each x € S, ( g ^ x ) : n = 1, 2,...} is a local base for x and (2) if y € S and, for each n, x n € S such that y €g ^ x ^ , then x 1, x 2 ,... converges to y. R E M A R K . If in addition (gn (x): x € S, n = 1, 2 ,...} satisfies (3), below, then S is developable, and if it satisfies (3) and (1+), S is metrizable [ 1 7 ]: (3) If x € S and y and z are point sequences in S such that, for each n, x € gn (yn ) and zn € gn (yn ) , then z ]f z 2 ,... converges to x. that

5.

(10 For each x and y in S and eac h n, x € gn (y) implies y € gn ( x). Compare these conditions w i t h Theorem 5.1 below.

Semi-metrie, M 1- and Related Spaces

In [9 ], Jac k Ceder defines Mj-, M 2- and M^-spaces. M 1 -spaces are defined as above in Section 1, as a T^-space wit h a o-closure-preserv­ ing basis, and an M 2-space is similarly defined but in terms of a "basis" whose members are not necessarily open. A characterization of (first countable) M^-spaces (which are defined b y Ceder in terms of cushioned collections) is given b y Theorem 5 . 1 . B y Ceder's definition " M 1 implies M 2 " and "M 2 implies M^" but it is not known whether either converse is true (I believe that it is, at least in the first countable c a s e ) . Note that M^-spaces are called stratlflable spaces b y Borges in [7 ] and first countable M^-spaces are the same as Nagata spaces [9 ]. THEOREM 5.1 [18]. A T 1-space Y is a Nagata (first countable M^space if and only if there is a collection (gn (x): x € Y , n = 1 , 2 ,...] of open sets such that for e a c h x € Y ( 1 ) (gn (x): n = 1 , 2 ,...) is a local base for x and (2 ) for every neighborhood R of x, there is an n such that gn (x) n gn (y) ^ $ implies that y € R. Theorem 5 . 1 shows the uniformity w hich characterizes M^-space (and hence is at inherent in M 1 -spaces). Also, a comparison of Theorem 5.1 w ith the Moore metrization theorem (or the Alexandroff-Uryshon theorem — see [2 3 ] and [ 1 , p. U 5 ]) shows that the relationship of semi-metric spaces to first countable M^-spaces is exactly analogous to that of developable metric spaces. Moore's Metrization T h e o r e m : Let S be a T 1 -space. Then S is metrizable if there is a sequence G 1, G 2 , ... of open coverings of S such that p € S, and R is an open set containing p, then there is a n n such that p e g , g € G , h €.Gn and g n h ^ 0 imply h C R. That analogy between the semi-metric-M^-space and the developablemetric relations suggests that a paracompact semi-metric space should be

109

ON CERTAIN FIRST COUNTABLE SPACES

an M^-space. Such a conjecture was made b y J a c k Ceder in [9 ] (on the basis of other evidence). In [ 1 6 ], however, an example is given of a paracompact semi-metric space w h i c h is not M^. It remains unknown, then, what condition on a semi-metric space gives an M^- (or M^-) space.

6.

Continuous Images of Metric Spaces

It was shown by V. I. Ponomarev [34] that a T^-space is first count­ able if and only if it is the continuous open image of a metric space. In [4] A. Arhangelskii showed that a T 1-space has a uniform basis (i.e., is a pointwise paracompact developable space) if and only if it is the image of a metric space under a bicompact open mapping ( f : x -► S is bicompact if f ~ 1 (p)

is bicompact for every

p € S).

This suggested to me the follow­

ing characterization of developable spaces. THEOREM 6 . 1 [18]. A T 2-space S is developable if and only if there is a metric space X and an open m apping f from Y onto S such that for every point p e S and every open set R containing p there is an s > o such that f ( N ( f ' 1 (p), s) C R (where N ( f ”1 (p), e) denotes the spherical (x: |x-y| < 7

neighborhood of radius e for some y € f “1 (p)}).

about f _ 1 (p),N ( f ~ 1 (p),

s) =

R E M A R K S . ( 1 ) If in place of "every point p € S" (In the hypothesis of Theorem 6 . 1 ) we put "every compact subset p C S," the conclusion would be that S is metrizable. Similar characterization of semi-metric and first countable M^-spaces are also given in [ 1 8 ]. (2) Another w ay of stating Theorem 6 . 1 would be: "Every de v e l o p ­ able space is a decomposition space of a metric space corresponding to a lower-semi-continuous decomposition M wi t h the following open sets N £ (g) sufficing for a basis: for each g € M and each e > o, let Ng (g) = [h: for some x e g and y € h |x-y| < e } . M Q u e s t i o n : Under what condition on the function f is the space S of Theorem 6 . 1 normal? A n answer to this question might make it possible to use that characterization of developable spaces to make some progress on the normal Moore space metrization problem. One might restrict one's attention to continuous images of "nice" mappings of general metric spaces. A T^-space wh i c h is the continuous image of a separable metric space is called a cosmic sp a c e . Such spaces are clearly LindeTof, separable and perfectly normal. For some other p r o p ­ erties see [ 3 1 ], from w h i c h comes the following characterization of cosmic spaces (see Lemma 4.1 of [31]). THEOREM 6 .2 . A T^-space S is a cosmic space if and only if there is a countable collection { B ^ B 2 ,...J of subsets of S such that for

HEATH

110

each

p e S and each open

that

p €BR

set R containing

p

there is an n

such

C R.

A cosmic space then has what one might call a countable pseudobase (except that Professor Michael has already called something else a ps e u d o ­ base in [31]). The following questions were raised b y C. J. R. Borges in [7]: (1) Is every separable M^-space a cosmic space? (2 ) Is every-cosmic space an M^-space? and (5) Is every countable T^-space (which would of course be a cosmic space) an M^-space? The examples in [ 1 6 ] and [ 1 3 ] give negative answers to ( 2 ) and ( 3 ) respectively, and it has been shown in [ 1 3 ] that a first countable cosmic space is semi-metrie at least, but question ( 1 ) remains open and one can ask also: Question:

7.

Is every Lindelof semi-metric space a cosmic space?

Summary

In conclusion some of the known relations among first countable spaces are: ( 1 ) A paracompact Moore space is metrizable. (2 ) A separable normal Moore space is metrizable (assuming the continuum hypothesis or some related hypo t h e s i s ) . (3) There is a characterization of developable spaces as lowersemi-continuous decompositions of metric spaces—i.e., open continuous images of metric spaces—of a certain type. (4) A semi-metric space can be far from be i n g developable—it m a y eve n fail to have a semi-metric under w h i c h all spherical neighborhoods are open. (5) A semi-metric space w h i c h is strongly complete or w h i c h has a point countable base is developable, but not conversely (even for complete Moore spaces in the first c a s e ) . There is still lacking a satisfactory "topological" characterization of those semi-metrie spaces w h i c h are d ev e l o p a b l e . (6 )

Developable spaces can be characterized as those having a semi­

metric d such that x^ -* p and y^ -*• p implies d(x^, y^_) -* o. Metric spaces are those having a semi-me trie d such that x^ —■p and d(y ±, x.) — 0 implies y ±-* p. (7) A first countable M 1-space is a paracompact semi-metric space, but the converse is not true; nor is it true that a cosmic space (continuous imate of a separable metric space) is necessarily an M 1 -space. (8 )

A first countable cosmic space is a Lindelof semi-metrie space.

The following are some questions that remain open. 1. Is every normal Moore space metrizable? Is every pointwise p a r a ­ compact normal Moore space metrizable (assuming the continuous hypothesis if necessary).

ON CERTAIN FIRST COUNTABLE SPACES

111

2. Is every normal semi-metric space paracompact? (There m a y be a counter example to this w h i c h is not a counter example to question 1.) 3. What is a necessary and sufficient "topological" condition for a semi-metric space to be developable? Wh a t is a necessary and sufficient condition for a semi-metric space to be an M 1-space? 5. Is every M^-space an M 1-space? 6. Is every separable M 1-space a cosmic space? Is every Lindelof semi-metrie space a cosmic space? REFERENCES [1]

P. S. Alexandrov, "On some results concerning topological spaces and their continuous mappins," Proc. Sympos. Prague ( 1 9 6 1 ), pp. 1+1 - 51+.

[2 ]

_____ , "Some results in the theory of topological spaces, obtained within the last twenty-five years," Russian Math. Surveys, 15 ( 1 9 6 0 ), pp. 23-83.

[3 ]

P. S. Alexandrov and V. V. Nemitskii, "Der allgemaine Metrisatienssatz und das Symmetricaxiom " (in Russian; German Summary), Mat. Sbornik

[k]

A. Arhangel skii, "On mappings of metric spaces," Soviet Math. Dokl. 3 (I962)pp. 953-956.

[5 ]

R. H. Bing, "A translation of the normal Moore space conjecture,"

[6 ]

_____ , "Metrization of topological space," Canad. J. Math., 3(1951)

3

(1+5) 1938 pp. 663-672.

Proc. Amer. Math. Soc., 1 6 ( 1 9 6 5 ) pp. 6 1 2 - 6 1 9 . pp. 175-186. [7 ]

C. J. R. Borges, "On stratifiable spaces," to appear.

[83

Morton Brown, "Semi-metric spaces," Summer Institute on Set-Theoretic Topology, Madison, Wisconsin, Amer. Math. Soc., (1955) pp. 62-64.

[9 ]

J a c k Ceder, "Some generalizations of metric spaces," Pacific J. Math., n

(1961) pp. 1 0 5 - 1 2 5 .

[ 1 0 ] D. R. Taylor, "Metrizability in normal Moore spaces," Pacific J. Math., to appear. [ 1 1 ] B. Fitzpatrick and D. R. Traylor, "Two theorems on metrizability of Moore spaces," Pacific J. Math., to appear. [1 2 ]

E. E. Grace and R. W. Heath, "Separability and metrizability in pointwise paracompact Moore spaces," Duke Math. J., 31 (l964) pp. 603-610.

[133

R. W. Heath, "A note on cosmic spaces," to appear.

[11+3

______, A non-pointwise paracompact Moore space w i t h a point-countable base," to appear.

HEATH

112

[15]

______, "A regular semi-metrie space for wh i c h there is no semi-metrie under w hich all spheres are open," Proc. Amer. Math. Soc., 12 ( 1 9 6 1 ) p p . 81 0 -811.

[16]

______, nA paracompact semi-metric space whi c h is not an M^-space," to appear.

[17]

______, "Arcwise connectedness in semi-metric spaces,” Pacific J. Math.,

[16]

______, M0n open mappings and certain spaces satisfying the first countability axiom,1' Fundamenta Mathem a t i c a e , to appear.

[19]

______ , "On spaces w i t h point-countable bases," Bull. Pol. Akad. Nauk., to appear.

[2 0 ]

______, "Screenability, pointwise paracompactness and metrization of Mocre spaces," Canad. J. Math., 16 ( 1 9 6 4 ) pp. 763-770.

[2 1 ]

______, "Separability and pp. 11-14.

1 2 ( 1 9 6 3 ) pp.

[2 2 ]

1301-1319.

^-compactness,M

Coll. Math., 12 ( 1 9 6 4 )

P. B. Jones, "Concerning normal and completely normal spaces," Bull. Amer. Math. Soc., 43 (1937) pp. 671-677.

[23]

______, "Metrization," Amer i c a n Math. Monthly, to appear.

[24]

______, "Moore spaces and uniform spaces," Proc. Amer. Math. Soc., 9 (1958) pp. 483-486.

[25]

______ , "Remarks on the normal Moore space metrization problem," these

[26]

______, R. L. Moore's A x i o m 1 1 and metrization," Proc. Amer. Math. Soc., 9 (1958) p. 487.

[27]

C. Kuratowski, "Topologie I," 4th ed. Monografie Mathmatyczne, vol. 2 0 , Panstwowe Wydawnictwe Naukowe, Warsaw, 1 9 5 8 .

[28]

L. F. McAuley, "A relation between perfect separability, completeness and normality in semi-metric spaces," Pacific J. Math., 6 (1956)

[2 9 ]

______, "A note on complete coliectionwise normality and paracompact­ ness," Proc. Amer. Math. Soc., 9 (1958) pp. 769-799.

[30]

E. A. Michael, "Another note on paracompact spaces," Proc. Amer. Math. Soc., 8 (1957) pp. 822-828.

[31]

______,

[32]

R. L. Moore, "Foundation of Point Set Theory," Am. Math. Soc. Coll. Publ. 13, Revised Edition, (Providence, 1 9 6 2 ).

[33]

V. W. Niemytzki, "On the ‘ third a xiom of metric s p a c e s ’ ," Transactions Amer. Math. Soc., 29 ( 1 9 2 7 ) pp. 507-513.

proceedings.

pp. 315-326.

N q -spaces,

to appear.

REFERENCES

113

[34]

V. I. Ponomarev, "Axioms of countability and continuous mappings," Bull. Pol. Akad. N a u k . , 8 (1 96 0 ) pp. 127-134.

[3 5 ]

D. R. Traylor, "Normal separable Moore spaces and normal Moore spaces," Duke Math. J., 30 ( 1 9 6 3 ) pp. 4 85-493.

[ 3 6]

J. M. Worrell, Jr., and H. H. Wicke, "Characterizations of develop­ able topological spaces," Canad. J. Math., 17 ( 1 9 6 5 ) pp. 820-830.

[37]

W. A. Wilson, "On semi-metric spaces," Amer. J. Math., 53 (1931) pp. 361-373.

[38]

J. N.Younglove, "Two conjectures in point set topology," these p r o ­

[3 9 ]

______, "Concerning metric subspaces of non-metric spaces," Fund.

ceedings . Math., 48 (1959) pp. 15-25.

REMARKS ON THE NORMAL MOORE SPACE METRIZATION PROBLEM by P. B u rton Jones As far as I know the first example of a non-metric Moore space was discovered (probably b y Moore himself) in the late 1 9 2 0 's. It is a dendron D in the plane w i t h uncountably man y endpoints w h i c h has the r e l a ­ tive topology of the plane at all of its points except at the endpoints and m a y be roughly described as follows: axis.

Let C be the standard (ternary) Cantor set on [0 , l] of the xDesignate the components of [0 , 1 ] - C (in order of length and

from left to right for those of the same length) b y

I1 5

I n , I 12; I 11l#

I|i2 , ^ 1 2 2 ; ••• • Selsct a point p 1 one unit below the midpoint of I 1 and from p 1 draw two straight line (closed) intervals one half-way tow­ ard the center of 1 ^ and another half-way toward the center of I 12. Let p 11 and p ]2 denote the n o n - p 1 endpoints of each of these arcs r e ­ spectively. Repeat the process b y drawing two intervals from p 11 ha l f ­ w a y toward the centers of I 111 and I 1l2 and two intervals from p 12 ha3f-way toward the centers of I 121 and I 12 2 # Continue this process indefinitely and the u nion of C with all of the arcs so constructed is the dendron D. P r o m p 1 to a point c of C there is only one arc T(c). A ny half-open subarc (z, c] of T(c) plus appropriate half-open arcs at e ach b r a n c h point (these are not to contain any other br a n c h point) is to be an open set containing c. Under this change of topology, n o point of C is a limit point of C in the space D. Moore had another example derived from this w h i c h he called the automobile road space w h i c h went something like this: Start two roads out from the origin in opposite directions. After e ac h has gone a mile let each b r a n c h into two roads. Continue eac h of the now four roads for a mile and let e a c h bra n c h into two roads. Continue this a countable infinity of times, so that none of the new branches ever intersect and so that all roads (regardless of w h i c h turns are taken) p r o ­ ceed indefinitely far from the origin. Now, wh e n eac h road (there are c of them) comes to the edge of the plane, continue it straight on away from the plane another countable infinity of miles. The boundaries of the roads 115

116

JONES

do not belong to the space.

Neighborhoods (or regions) are now open disks

and quite a number of Moore's other axioms hold true in the autom bile road space.

In particular, Axioms 0 ,

1, 2, 3, U, 5, 5 1

and

52 M ] hold

true. The usual way to see that these spaces were not metric was to ob­ serve that each contained a closed separable subspace (the whole space in the case of

D)

which was not perfectly separable ( i . e . ,

had no countable topological b a s i s ).

the subspace

This kind of observation left me

mildly restless and it was only after I discovered that neither was normal that I felt that I was closer to the ’’real reason” for non-metrizability. The argument I used totoprove prove the the non-normality non-normality of of DD mobile road space contains

(and (and the the auto­ auto­

D D as a subspace) subspace) has has aa good good deal deal in in common common

with Professor Younglove's and goes like this: Let and

1

M M denote denote the the set set of of aall ll points points

such that

In the space

D

m

both

M

and

C -M

normal, there would exist for each T(m)

from

has no limit point in

in p1

of points 1]

xx

belongs M; belongs to to M;

of of discontinuity discontinuity of off

[0,

I f the space

0

1] - C. D

were

M, a half-open arc (z , m] of to

C - M.

m] such that in the space Now let

[o,[o, 1] 1]to tothe thenon-negative non-negative real real numbers numbers defined defined

| |where IIiJtng iJtng (z)(z) where [0,

are closed.

m

[T(m) was that unique arc from

U (z , m] m € M

m m of of of C CC different different different from from0

is not a limit point of a component of

otherwise, otherwise,

f(x)) f(x

f

as as follows: follows: = 0. 0. =

fis is exactly exactly thethe set

D,

be a function

Since Since

f(x f(x)) = = the set set the

M, set MM,(as Ma (as subset asubset of of

with its usual topology) must be be an an FFo. o. This This is is obviously obviously impossi­ impossi­

ble ble since it would imply that

[o,[o,1] 1]is is of of the first category.

In fact, a little expgnsion^of this argument was the argument I first got to prove that i f space is metric.

2 0 0 < < 2 2 1 1then then every separable normal MooreMoore

For suppose that that S S Is Is aa separable normal Moore spacespace

with with a monotone descending development

G 1, G2 , G ^ ,, . . .

.

I^t Let

p 1, p2 , p ^ , . . .

be a countable dense subset ofofSS and uncountable be and suppose suppose that that MM is is an an uncountable subset of set

M

S - (p 1 + p2 + p^ + +• • • )• ) which which has has no no limit limit point point (i f (i nof such no such

exists,

For each point

S m

is perfectly separable and hence metric and normal). of

such that for each property that ly

pm -*■m.

pn

i,

M

let

m1, m2, m ^ , . . .

be the sequence of integers

m^ is the smallest natural number

belongs to an element of

Using these sequences,

G^

n having the

which contains

{m^}, the points of

M

m.

EvidentEvident

may be given

the lexicographic order and embedded in an order preserving manner in [o, of the real numbers. M

into

Let

T

denote a 1-1 transformation that thus embeds

[0, 1 ]. Without loss of generality we may assume that every point of T(M)

is a point condensation of T (M ).

Since

M

is uncountable (but of cardi­

nality < c) and the number of countable subsets of that of

2 0 < 2 1) there are more than c subsets T(H) and T(M-H) is dense in the other.

H

M of

is M

c

(assuming

such that each

i]

REMARKS ON THE NORM A L MOORE SPACE METRIZATION PROBLEM Now for eac h such set function

f

from

T(M)

H, let

f

117

be a real valued (non-negative)

defined as follows:

Since S is normal there exists an open set U such that M - H C U C U C H . If k is a point of M - H, let n be the smallest natural number such that every element of Gn containing k is a subset of U and let f[T(k)] = 1/n; otherwise, let f be sero (i.e.,v f (T(H)) = 0). O n T ( M ) , f is continuous exactly at points of T(H) (and discon­ tinuous at points of T(M - H ) ); hence T(H) is G 5 -set relative to T ( M ) . So there exists a sequence Q 1, Q 2 , Q^, of open subsets of [ 0 , 1 ] such that (Q1 - Q 2 • Q 3 . ...) • T(M) = T(H) . Now let Q(H) - Q, • Q 2 • Q 5 • ••• Then Q(H) is a Gg-set in [0 , 1 ]. But suppose that H ’ is another subset of M of the type that H is and that Q ( H ’ ) is the corresponding G &-set in [ 0 , 1 ] . If H and H' are different then so are Q(H) and Q(H') different. Since there are only c distinct G & -sets in [o, l ] we have a contradiction. So no such set M exists and S is metric and normal. Amo n g several examples of non-separable non-metric Moore space there is one w h i c h I had high hopes of proving to be normal. (I don't believe that I ever proved it was not normal.) It is a tree branching upward as did space D but continuing through uncountably many branching levels (in a well-ordered fashion) such that each b r anch at each branching level con­ tinues upward along a countable infinity of branches instead of two. It can be started in the plane but must be continued uncountably man y times out beyond the plane producing an uncountable monotone increasing wellordered sequence of connected domains whose union is the space. Start w i t h a point p Q and from p Q draw closed straight line i n ­ ternals of length 1 / 2 , 1 /4 , 1 / 8 ,... wh i c h have only p Q in common. Call the set of n on-pQ endpoints of these intervals M 1 . Repeat this process at ^ ach point of M 1 producing a new level of end points wh i c h we shall call Mg. While M 2 is a countable infinity of infinities it is still countable. Now continue this process to produce an uncountable sequence of branching levels M Q , M 1, M 2 , M^,..., M z , ... (z < (d }) so that each level is count­ able and if p belongs to M z , z* is an ordinal larger than z, and n is a positive integer, there is a point p* belonging to M , such that there is a pat h of intervals: p p 1 from p of M z to p 1 of M z+1 , p^g from p 1 to p 2 of M z + 2 ,... w hich starts at p and runs consecu­ tively through the intervening branching levels and terminates at p' of M z , and whose length is l/2n . To see how the construction can be made to carry on this property inductively consider the following two cases: Case 1 . Let z ' be a non-limit ordinal such that the branching levels have already been defined so as to possess the property and let us see how to define M z i . We simply go from M z ,_1 to M , exactly as we did from M 1 to M p . F r o m a point p at any lower level M z (z < Z f - 1 ) there would be a pat h to q in M ,_ 1 of length i/2n + 1 . Join this path

JONES

118 the one from q to a point the required pat h of length

p 1 of M , of length i/2n+1 and one gets l/2n from p of M z to p 1 of M ,.

Case 2. Let z 1 be a limit ordinal less that cd^ such that the inductive property is possessed for B.11 smaller ordinals. Let z be an ordinal less than z and let z < z 1 < z 2 < ... be a simple countable monotone increasing sequence of ordinals such that z^ -*z'. Fr o m • p in M z to a point p 1 in M z there is a path of length l/2n + 1 , from p 1 to a point p 0 in M th^re is a pa t h of length 1/2n+2 etc. Now, z2 terminate this p a t h in a point p ’ whi c h we shall put in M ,. The p a t h will have length l/2n and since this is done countably many times (once for each n) for p and there are only countably many such points p (when z' is countable), the process can be continued uncountably many times carrying on the inductive property. Obviously, no path intersects all b r anching levels for if it did there would be some natural number n such that its length between consecu­ tive levels would be 1/2n uncountably many times and hence after the first infinity it would be too long to have been continued because all paths have finite length, it being possible to b a c k up in a well-ordered sequence only finitely many times. Now let the space S be the union of all these uncountably many intervals w i t h "region" bei n g one of three types: 1. If a point p is of order 2 let a region be an open (connect­ ed) interval of points of order 2 containing it. 2.

If a point p

is a

point of order m

in a non-limit branching

level, let a region be the union of all of the half-open intervals that start from (and include) p and go i/2n of the distance toward the next br a n c h point in whatever direction one has chosen (either up or down). 3. If p is a point of order to in a branching level M z where z is a limit ordinal, p got into M z b y b e i n g the terminal endpoint of a pat h P starting from below. Let (x, p] be a half-open subarc of P. Add to (o, p] those half-open intervals wh i c h start at p and go up 1/2n of the distance toward the next bran c h point (as in (2) above), and to this add all regions of type (2) defined for this n whose center (or branch) point belongs to (o, p j . The reader should have no difficulty seeing what the topology of this space is whe n using the set of all regions as a topological basis. And it should be fairly easy to see that S is a Moore space wh i c h is con­ nected, locally connected, locally separable, but not perfectly separable. Such a space cannot be metric. My original thoughts (perhaps falacious) on normality went something like this:

REMARKS ON THE NORMAL MOORE SPACE METRIZATION PROBLEM

119

The space S corresp on d s to t h a t p a r t o f the space D below the Cantor s e t . Each p ath i n D - C w hich has p^ a s one o f i t s en dp o in ts and i s n ot a p rop er su b set o f an o th er such p a th i s a h a lf-o p e n a rc w ith P 1 a s one o f i t s e n d p o in ts; the o th e r endp oint i s in C. L ik ew ise each p ath in S w hich has pQ a s one o f i t s e n d p o in ts and i s n ot a p ro p er sub­ s e t o f a n o th er such p a th i s a h a l f open a r c w ith pQ a s one o f i t s end­ p o in t s . So i f one adds to e ach such h a lf-o p e n a r c a n o th er p o in t to become an endp oint o f t h a t h a lf-o p e n a r c , one o b ta in s a /lew space S. Denote S - S by K. Then the to p o lo g y o f S a t p o in ts o f K i s com pleted j u s t a s i t was f o r p o in ts o f C in the space D. Furtherm ore, the s e t K has a n a t u r a l li n e a r ( t o t a l) o rd er a s d id the p o in ts o f C ( le x ic o g r a p h ic o rd er o f the numbers n , w hich g iv e the le n g th o f the s t r a ig h t i n t e r v a l s a s l /2n form in g th e path) . A gain K K i si sn onto tthe thef ifri sr ts t c acte a te g ogroyr yinin the the open open i n i nt teer rvvaal l to p o lo g y to p o lo g y ooff the the llin in ee aa rr oord rdeerin ringg.. FFurth urtherm ermore, ore, K K iiss aa d disis c rc er teet e s es te t inin S.S. SoSoi f i f SS were were normal normal and and HH were were aa su subbset set oof f KK such suchthth a ta t H H and and KK -- H H a re a re dense toppoolo loggyy,, the the ccoovveerin dense in i n eeach ach ooth theerr in in the the lilinneeaarr oord rder er to ringg ooff HH andand KK -- H H by by ddiissjjooiinntt open open sseettss U(H) U(H) and and U(K-H) U(K-H) would would ggiv ivee rriissee to to two two rreeaall fu funncctio tionnss ff and and gg each each con contin tinuous uous aatt each each ppooin intt ooff K K where where the the ooth ther er ii ss ddis isccoonntin tin uuoouuss.. A Again gain tthh iiss would would ffoo rrccee K K to to be be oo ff the the ff ii rr ss tt ccaate teggoorryy.. Hence Hence SS iiss nnoott norm normal. al. However, in the a n a lo g y D - C was m e tr ic (and hence normal) . So perhaps S - K, t h a t i s , S i s norm al. I th in k I was n ev er a b le to s e t t l e t h i s q u e s tio n . I do r e c a l l p ro v in g t h a t K w ith i t s lin e a r o rd er to p o lo g y was n o t a S o u s lin space ( i t i s n o t sep a ra b le but i t does c o n ta in an uncount­ a b le c o l l e c t i o n o f d i s j o i n t open s e ts ) b u t t h i s d id n 't seem to h e lp .

REFERENCES [1]

R. L. Moore, F oun dation s o f P o in t - s e t T heory, A. M. S. Colloquium P u b lic a t io n , v o l . 13 (19 3 2 ).

[2]

F . B. J o n e s, M0n c e r t a in w e ll-o r d e r e d monotone c o l le c t i o n s o f s e t s , " J o u rn a l o f the E lis h a M it c h e ll S c i e n t i f i c S o c ie ty , v o l . 69 (1953) pp. 30- 31+.

TWO CONJECTURES IN POINT SET THEORY by J. N. Younglove Two conjectures in topology are discussed in this paper. One, r e ­ ferred to as the normal Moore space conjecture due to P. B. Jones [4] states that every normal Moore space (i.e., satisfies A xiom 0 and the first three parts of A x i o m 1 of [9 ]) is metrizable. The other due to C. H. Dowker [7] conjectures that every countably paracompact Hausdorff space is normal. A n example of R. H. Sorgenfrey [6] is shown to be an example of a paracompact 2 space S such that S is not countably paracompact. Traylor [ 1 ] has shown that every locally compact, normal, nonmetrizable Moore space contains an uncountable discrete collection of points w i t h respect to w hich the space is not coliectionwise normal. That is, there is no collection of disjoint open sets such that each open set cf the collec­ tion intersects the set in exactly one point. Bi n g [2] has shown that each separable, non-metrizable Moore space contains such an uncountable set. The following is a description of a class of separable Moore spaces whi c h is of considerable interest. Let H denote a subset of the x-axis In the plane. Let z denote the Moore space whose points are the points of the upper half-plane together w i t h the points of H and having the development ( £ ^ 7 sucJl that for each integer n, Gn is the collection of sets wh i c h are either (a) interiors of circles of radius less than 1 /n lying wholly above the x-axis or (b) the interiors of such circles tangent to the x-axis from above at a point H together w i t h the point of tangency. The space L will be said to be d e ­ rived from

H.

The set H is discrete in £ and the metrizability of z is e q u i ­ valent to asserting that H is a countable set. Thus, i f , there exists an uncountable subset H of the x-axis such that the derived Moore space is normal, then Jones' question [4, p. 6 7 6 ] would be answered. Jones [4] has shown that a set of cardinality of the continuum in a normal separable Moore space must contain a limit point of itself and, consequently, any set H whi c h gives rise to a counterexample as described must have cardinality less than C. 121

122

YOUNGLOVE

H eath [8] has shown that the existence of a separable, normal, nonmetrizable Moore space implies the existence of an uncountable subset H of the x-axis such that every subset of H is the intersection of H and a G 6 set in the reals. Bin g [33 has shown that if H is such a set, then the derived Moore space is a separable, normal, non-metrizable Moore space. By using a theorem of Mazurkiewicz [5 , p. 2 3 6 ] w h ich states that a G g set in the reals can be expressed as the union of two sets, one of w h i c h is empty or a homeomorphic image of the irrationals and the other is at most countable, we see that every uncountable G & set in the reals has cardinality C. Thus, to achieve a counterexample, the set H can contain only count­ able G c sets. o In the search for a counterexample to Dowker's conjecture, the above considerations assure the non-normality of a Moore space z derived from a n uncountable G & set on the x-axis. The question then arose as to whether any such Moore space could be countably paracompact and the following result was established. THEOREM. Suppose Moore space derived from coun t a b l e .

H is a G g set in the x-axis and Z is the H. If z is countably paracompact, then H

is

Proof: Let H denote an uncountable G. set In the x-axis. Let -----o E.j, Eg, Ej,... be an infinite sequence of open subsets of the x-axis whose intersection is H. Usi n g M azurkiewicz's theorem, there is a subset K of H w h i c h is homeomorphic w i t h the irrationals. This allows us to assert the existence of a countable subset A of H such that if x is in A and u is an open interval of the x-axis containing x, then u n H is uncountable. Let x 1, x 2 , x^,... be a sequential ordering of the points of A. Let G be a countable open covering of S, the set of points of Z such that if g is an element of G, then g is S - A or g is an e l e ­ ment of G 1 w h i c h contains some point of A. We m a y assume for convenience that no two sets of G w h i c h contain points of A intersect. Let W be a refinement of G. Let D 1 denote the union of the elements of W which intersect A and Dg denote the union of the remainder of the sets in W. There is a region g 1 of G 1containing x^ such that (i) g 1 _C and (ii) the closure of the vertical projection u 1 of g 1 into the x-axis is a subset of E ] . There is an open interval v 1 of the x-axis containing x 1 such that (u1 - v ^ n H is uncountable. Thus, (u1 - v ^ n H must contain a point of A. Let x^ be the first point of A in the order x 1, Xg, x^,... in this set. Let g 2 be an element of G 1 containing x^ such that the closure of its projection, u 2 , is a subset of (u1 - Vj) n Eg. As before, there is an open interval v 2 containing x| such that (u2 - Vg) n H is uncountable. Let x^ denote the first point of A in the given order w hich is contained in the set (u2 - v 2) . This process yields an infinite monotone sequence u 1, Ug, u^,... of closed intervals

TWO CONJECTURES IN POINT SET THEORY

123

w i t h a point p in their intersection. Since un C E , the point p must be a point of H. Due to our method of selecting the sequence xj, x^, x ^ , .. p cannot be a point of A. Thus, p is a point of H - A. D g.

Let g be a region of G 1 whi c h contains p and is a subset of The region g must intersect infinitely m a n y regions of the sequence

gi t compact.

811(1

contradicts the fact that

z

is countably p a r a ­

In Sorgenfrey's example, S is the real number system w i t h the half-open interval topology. The interval in the plane joining the points (o, i) and (1, o) becomes a discrete set in the S2 topology. By using a countable dense (in the plane topology sense) subset of this interval as we 2 used the set A in the proof of our theorem, we conclude that S is not countably paracompact. QUESTIONS 1. If a Moore space z is a coliectionwise normal with respect to each discrete collection of degenerate point sets, is it coliectionwise normal? 2.

The same as

l

but for a normal Moore space.

3. Is there a subset H of the x-axis such that the Moore space rived from H is normal and non-metrizable and further such that

z H

de­ is a

subset of a Cantor set in the x-axis? REFERENCES 1.

D. R. Traylor,"Normal separable Moore spaces and normal Moore spaces," Duke Math. J., 30 ( 1 9 6 3 ) pp. 485-493.

2.

R. H. Bing, "A translation of the normal Moore space conjecture,"

3.

R. H. Bing,"Metrization of topological spaces," Canad. J. Math., 8

Proc.

Amer. Math. Soc., 1 6 (1 9 6 5 ) pp. 6 1 2 - 6 1 9 .

(1951) pp. 6 5 3 - 6 6 3 . 4.

F. B. Jones,"Concerning normal spaces," Bull. Amer. Math. Soc., 4 3 (1937) pp. 671-677.

5 . W. Sierpinski,

General T o p o l o g y , 2nd

ed., U. of Toronto Press,

6.

R. H. Sorgenfrey, "On the topological Bull. Amer. Math. Soc., 53 (1947) pp.

7.

C. H. Dowker, "On countably paracompact spaces,"

1956.

product of paracompact spaces," 6 3 1 -6 3 2 . Canad. J. Math., 3

(1951) pp. 2 1 9 - 2 4 4 . 8.

R. W. Heath, "Screenability, pointwise paracompactness and metrization of Moore spaces," Canad. J. Math., 1 6 ( 1 9 6 4 ) pp. 763-770.

9.

R. L. Moore, "Foundations of point set theory,'1 Amer. Math. Soc., Colloq. Pub. vol. 13, rev. ed. 1 9 6 2 .

ALMOST CONTINUOUS FUNCTIONS A ND FUNCTIONS OF BAIRE CLASS 1 . by E. S. Thomas, Jr. The w o r k presented here grew out of a problem about almost continu­ ous functions (question 2 b e l o w ) . W o r king together F, Burton Jones and I solved that problem and in the process turned up some ideas w h i c h proved u s e ­ ful in investigating functions of Baire class i. Later on I'll try to give a glimpse of w hy the study of one class of functions led to the study of the other. A function f on a topological space X to I (the unit interval) is said to be of Baire class i provided f - 1 (F) is a G & in X whenever F is closed in I. The following fundamental result has been known since around 1 9 0 7 (cf. [i ]; p. 280 et seq.): If X is separable metric then f is of Baire class 1if and only if there is a sequence of continuous functions on X to I converging pointwise to f on X. We are interested in finding topological conditions on the graph of a function f wh i c h are necessary and/or sufficient for f to be of Baire class

1. As will be seen later

(Theorem 2 ) the graph of such a function is

a G e set in X x I (if X is separable metric this follows easily from the preceding characterization), however, it is easy to manufacture functions not of a Baire class 1 whose graphs are G ^ s . For example, let f : I -*■I be defined as follows: If x Is diadic rational in (0 , 1 ) write x = j/2k in lowest terms and put f(x) = 1 / 2 + 1 / 2 ^. Let f be identically 0 els e ­ where in I. The desired properties are easily verified. This example, and most of the examples one is liable to come up with in a hurry, works because the graph is disconnected. This common m i s ­ behavior leads to connected

Question 1 . If f G & in I x I, is

is a function from I f of Baire class 1 ?

to

I

whose graph Is a

A function f from a space X to a space Y is almost continuous provided that If U Is an open subset of X x Y containing the graph of f then U contains the graph of a continuous function.

125

THOMAS

126 Question 2.

If

f

is as in question 1, is

f

almost continuous?

To see that the two questions are related, suppose f is almost continuous and its graph G is a connected G g . Corresponding to any se­ quence , U 2 ,... of open sets whose intersection is G there is a sequence of continuous functions f 1, f 2 , ... such that, for ea c h n, the graph of fn lies in

Un .

A l t hough n"+1 U n = G does not a priori imply that the corresponding f converge to f pointwise, it is plausible that using connectedness of G we can manufacture a sequence of U ^ s w h i c h are nice enough (in some way) to yield this convergence. We remark that a special case of a conjecture of Stallings [2] is that the answer to question 2 is yes, even without assuming the graph is a G & . J. Cornette, in an as yet unpublished paper, has answered Stallings' conjecture in the negative. We shall answer b o t h questions in the negative b y means of a n e x ­ ample. The construction of the example is not hard but more easily u n d e r ­ stood w i t h a little motivation. Let f be a function from I to I whose graph G is a connected G b< E a s y arguments establish the following facts about G (the closure of G) : ^ (1 ) G is a nowhere dense subcontinuum of I x I. (2) 'There is a dense G 5 set L in I such that, for ea c h x in L, G n £x (I is the vertical line segnent (x) x I) is a single point and G (3)

is locally connected at this point.

G is almost irreducible from &Q to in the sense that there is a subcontinuum K of G wh i c h is irreducible from to and G - K is the union of vertical line seg­ ments each of w h i c h lies over a point of I - L.

In our example, will be irreducible.

L

will be the complement of a Cantor set and

G

To begin wit h we construct a Cantor set C in I x I whi c h inter­ sects the graph of every continuous function from I into I and such that C n £ consists of two points or one point according as x is or is not a diadic rational in (o, 1). C is obtained as the intersection of sets C^ C 2 ,... . Figures 1 and 2 show how C 2 is obtained from C 1.

AI>10ST CONTINUOUS FUNCTIONS AN D FUNCTIONS OF BAIRE CLASS 1

127

This process is repeated in the obvious w a y to get C^, C^,...; the desired properties of C are immediate. We next construct a function f whose graph is a connected G & lying in (I x I) - C. Let K denote the Cantor ternary set i n I, let I lf I 2 , ... be the collection of complementary intervals of K in I (no inter­ val listed twice), and for each n write In = (r , s ). For a fixed n, f is defined over [r , s ] b y specifying its graph as follows. Let zn and w n denote the points C n lr and ^ n ^sn respectively. Choose a point zn ’ on Hr w hich lies strictly below and within 1 /n of zn and choose a point wn ! on i Q which'lies strictly above and within i/n of w n . For the graph of f over [rn , sn 3 take any simple arc (simple = intersects each vertical line in r -*> [rn , sn ] e x ­ actly once) in [rR , sn l x I running from zn * to wn f, miss i n g C, and meeting the horizontal lines through 0 and 1 . (It is possible to miss C since there is a diadic rational between rn and sn ) . Figure 3 illus­ trates the process.

f(x)

=0

For x in I - I£=1 [rn , sR ] put f(x) = i if x < 1/a if x > 1 / 2 . This completes the definition of f.

and

Certainly the graph of f is connected and misses C. The fact that it is a G e in I x I follows easily once it is seen that the z ’ and & n w * were chosen to form a countable discrete set, hence a G 0 , In I x I. n o Since C meets the graph of every continuous function, f is not almost continuous. To see that f is not of Baire class 1, let e be a point of [0, 1) such that for x < 1 / 2 , the set C n S. lies strictly be—1 low the horizontal line through e. If the set f ([o, e]) were a G & in I, then its intersection w i t h K fi [0 , £] would also be a G e . But this intersection is countable and dense in K O [0 , ?], hence cannot be a G&. W e next state without proof some positive results.

THOMAS

128

The first theorem is a specialized result telling what extra con­ dition is needed to get a yes answer to question 1. THEOREM l. If the graph G of f : I «-►I is connected, then f is of Baire class 1 if and only if G = n“_ 1 where each is open and simply connected in I x I. (Theorem 1 fails if connectedness of G is deleted.) To generalize this result we use the following concept. If X is a topological spp.ce and G is a subset of X x I we say G has property B in case G = n ^ _ 1 where each is open in X x I and, for ea c h i and each x in X, n is connected. THEOREM 2. If X is a Hausdorff space such that X x I is normal and if f is a function on X to I whose graph has property B, then f is the pointwise limit of a sequence of continuous function on X to I. COROLLARY. If X is separable metric then class 1 if and only if its graph has property B.

f:

Q u e s t i o n . In order that Theorem i go through for functions m apping an n-cube into I, what geometrical condition on the should replace sim­ ple connectedness?

REFERENCES [1]

C. Kuratowski, Topologie I, Warsaw (i952).

[2]

J. Stallings,"Fixed point theorems for connectivity maps," Fund. Math. 47(1959), p. 249-265.

X -*

CHAINABLE CONTINUA by J. B. Fugate A.

DEFINITIONS A N D E L E M E N T A R Y PROPERTIES

A chain in a topological space is a finite collection (E1 , Eg,..., E n ) of open sets such that E^n E^ ^ 0 iff |i—J | < i. The members of the collection are called l i n k s . A chain in a metric space is an e-chain p ro­ vided each link has diameter less than e. A compact metric continuum M is chalnable (also called snakelike or arc-like) provided that for each positive number e, there is a n e-chain covering M. Note that there is no require­ ment that the links be connected. Examples of chainable continua are: an arc, the pseudo-arc, Cl((x, sin 1/ x ) : o < x < 1), Cl denoting closure. Proposition i . E a c h chainable compact metric continuum M is unicoherent (i.e., if M is the union of two subccntinua A and B of M, then A n B is a cont i n u u m ) . A circle is not chainable, since it is not unicoherent. Proposition 2. A chainable compact metric continuum is not a triod. (A con­ tinuum M is a triod provided M is the union of three proper subcontinua of M such that the common part of any pair is the common part of all three and is a proper subcontinuum of each.) A letter "Y" (the union of three arcs, having exactly one end point in common) is a triod, hence is not chainable . (Propositions i and 2 are not hard to establish. The basic lemma needed is the following: If M is a continuum and (E1 # ..., E n ) is a chain covering M such that M intersects E 1 and E , then M intersects each link of (E1,.. ., E m ) .) Proposition 3.

E a c h subcontinuum of a chainable continuum is chainable.

It follows from Proposition i that each subcontinuum of M is unicoherent (i.e., M is hereditarily u nicoherent) ; from Proposition 2, it follows that M contains no triod (i.e., M is a -trodlc) .

B.

EMBEDDING CHAINABIE CONTINUA

Proposition k. E a c h chainable compact metric continuum M p in the plane: i.e., there is a homeomorphism h: M -►E .

129

can be embedded

FUGATE

130

In [43, Bing shows that M can be embedded so that h[M3 does not separate the plane. Since the property of separating the plane is a topolog­ ical invariant, it follows that under any embedding g, g[M] does not se­ parate the plane. Indeed, M can be embedded so that h[M] can be chained wi t h connected links. However, this last property demands upon the embedding, and not upon the continuum M. Bennett has announded [1]: Proposition 5. If each of M and N is a chainable compact metric contin­ uum, then M x N can be embedded in E 3 .

C.

MAPS ON CHAINABLE CONTINUA

Proposition 6. A chainable compact metric continuum M has the fixed-point property [6]; i.e., if f: M -* M is a continuous function, then there is a point x € M such that f(x) = x. (One can prove this b y an easy argument using chains, once one e s ­ tablishes that if f has no fixed point, then there is a positive number d such that each point of M is moved more than d by f.) Since chainable compact plane continua do not separate the plane, Proposition 6 provides a partial answer to the following: Question i . Suppose K is a compact plane continuum w hich does not separate the plane. Does K have the fixed point property ? The name "a rc-likeM comes from the following characterization of chainability: Proposition 7. A compact metric continuum M is chainable iff for each p o ­ sitive number e, there is a continuous function f: M -*• [0, 1] such that each point in [o, 1] has inverse image of diameter less than e. M

If one adds the requirement that f to be locally connected, and hence an arc.

be monotone, then this forces

Proposition 8. E a c h chainable compact metric continuum is the continuous i m ­ age of the pseudo-arc [7, 8]. E v e n without this result, it follows from Proposition 7 that an arc is the continuous image of the pseudo-arc, thus, so is any locally con­ nected compact metric continuum. Clearly, then, the pseudo-arc can be mapped onto many continua w h i c h are not chainable. A characterization of those con­ tinua wh i c h are continuous images of the pseudo-arc can be found in [71.

CHAINABLE CONTINUA D.

131

A N APPLICATION OF INVERSE LIMITS

A useful characterization of chainable compact metric ccntinua is the following: Proposition 9 . A campac£ metric continuum M is chainable iff there is an inverse limit system I «-11^ I «• • •, where, for each positive integer i, f^ is a continuous function of the unit interval onto itself, and M is homeomorphic to the inverse limit set. For example, if each b onding m ap f^ is the same and is the "rooftop func­ tion" f, then the inverse limit is an indecomposable chainable continuum. In [9 ], Proposition 9 continuum U the following

H. M. Schori utilizes to construct a chainable whi c h is universal in sense:

Proposition 1 0 . If M is a chainable compact metric continuum, then be imbedded in U. U is called a universal snakelike c ontinuum.

M

can

N . B . Because of Proposition 8, the pseudo-arc has also bee n called "univer­ sal." Clearly, the continuum U above is not the pseudo-arc, since U con­ tains arcs, while the pseudo-arc is hereditarily indecomposable. Schori has indicated how U can be modified to obtain a chainable compact metric con­ tinuum U' w i t h the following properties: (i) (ii)

If M is a chainable compact metric continuum, then be embedded in U 1. If M is a chainable compact metric continuum, then the continuous image of

E.

M

can

M

is

U ’.

CONDITIONS IMPLYING CHAINABILITY

The preceding sections should demonstrate that if one knows that a compact metric continuum is chainable, then a great deal can be said about it. In this section, we consider the converse: What are sufficient conditions that a compact metric continuum be chainable? The first answer to this qu e s ­ tion is due to Bing [4]. Proposition 1 1 . tinuum, then M herent.

If M is an hereditarily decomposable compact metric con­ is chainable iff M is a-triodic and hereditarily unico-

A nother answer to the above question was announded in [5 ]:

FUGATE

132

Proposition 12. If M is a compact metric continuum w h i c h is the union of two chainable continua, then M la.chainable J_ff M is a-triodic and hered­ itarily unicoherent. In considering how one might attempt to extend Proposition 1 1 to continua w h i c h are not hereditarily decomposable, it is informative to con­ sider a continuum w h i c h is "nearly chainable." This continuum is the dyadic solenoid, S. S has the following properties: (i) (ii)

(iii)

S is indecomposable, hence it is (trivially) unicoherent and not a triod. S is circularly chainable, thus each proper subcontinuum of S is chainable. Hence S is a-triodic and hereditarily unicoherent. S is not chainable. Indeed S cannot be embedded in the plane.

In the following result, due to the writer, one eliminates such continua b y hypotheses. Proposition 1 3 . If M is a compact metric continuum such that e a c h indecom­ posable subcontinuum of M is chainable, then M is chainable iff M is atriodic and hereditarily unicoherent. This extends Proposition 1 1 , and yielc as a corollary, an extension of Proposition 1 2 . Co r o l l a r y . If M is a compact metric continuum w h i c h is the uni o n of count­ ably m a n y chainable continua, then M is chainable iff M is a-triodic and hereditarily unicoherent.

F.

THE HOMOGENEITY PROBLEM

Still unsettled Is the following: Question 2. Wha t are the homogeneous bounded plane continua? Bin g has shown [ 2 , 3 ]: Proposition U .If M is a chainable compact plane continuum and M is either homogeneous or hereditarily indecomposable, then M is the pseudo-arc Question 3. Is the pseudo-arc the only homogeneous compact plane continuum w h ich does not separate the plane? Question 1+. If M is a homogeneous compact plane able or circularly chainable ?

continuum,

is M

chain-

Question 5. If M is a homogeneous compact plane subcontinuum of M chainable?

continuum,

is each proper

CHAINABLE CONTINUA

133

REFERENCES [1]

R. Bennett, "Embedding Products of Chainable Continua," Notices of the A. M. S., vol. 11, no. 3 (196^). Abstract 612-9.

[2]

R. H. Bing, "A Homogeneous Indecomposable Plane Continuum,”Duke Math. J.,

15 ( 19^ 8) EP- 729- 7^ 2 .

[3]

________ ________,, "Concerning "Concerning Hereditarily HereditarilyIndecomposable Indecomposable Continua," Continua," Pacific J. J. Math., 11 ((1951 1951)) pp. pp. UU33--51 51..

U]

________ , "Snake-like Continua," Duke

[5]

J. B. Fugate, "On Decomposable Chainable Compact Continua," ^Notices of the A. M. S., 11, no. 5 ( 19 6 U ) . Abstract 6 U - M .

Math. J.,

18 ( 1951)pp. ^ 653- 663 .

[6] [6 ] 00.. H. H. Hamilton, Hamilton, "A "A fixed fixed point point Theorem Theorem for Pseudo-arcs for Pseudo-arcs and Certain and Certain other Metric Metric Continua," Continua," Proc. Proc. A. M. S., A.2 ( 1M. 9 5 1S., ) pp. 2(1951) 173-17U. pp. 173- 17U. [ 7 ] A. A. Lelek, Lelek, "On "On WWeeaakkllyy Chainable Chainable Continua," Continua," Fund. Fund. Math., Math., 51 (1962/63) 51 (1962/63) pp. 271-282. [8 ]

Mioduszowski, "A Functional Conception of Snake-like Continua," Fund. M a t h . ,LI 2 (1962) pp. 179-189.

[9]

R. M. Schori, "A Universal Snake-like Continuum," Notices of the A. M. S., n , no. 3 (196U). Abstract 612-5.

THE EXISTENCE OP A COMPLETE METRIC FOR A SPECIAL M APPING SPACE A ND SOME CONSEQUENCES by Louis F. McAuley

*

I ntroduction. Suppose that T is a complete metric space and that K is a compact metric space. The space T1^ of all (continuous) mappings f of K into T w i t h the compact-open topology is a complete metric space. There are important applications of special subspaces of T^. For example, see the w o r k of Dyer and Hamstrom [2], There they define the concept of a completely regular mapp i n g p from T onto a metric space B. And, they consider the space Gb of all home omorphi sms h. of K onto p ’1 (b) for each b in B. It is shown that the space G which is the union of the various spaces Gfe, b e B,is a complete metric space. This fact is essential in their proof that certain completely regular mappings are locally trivial fiber mappings. The author [ 3 ] uses a similar argument to show that a more general class of completely regular mappings have the w e a k bundle property I w i t h respect to a space K. In this case, the author replaces G^ w i t h the space of a ll mappings of K onto p ~ 1 (b) . And, G* is a complete metric space. W h e n one considers the reverse of this situation, that is, let G^ denote the space of all mappings of p _1 (b) into K, then it is no longer apparent that G possesses either a metric or a complete metric. We are considering special subspaces of neither T ^ nor be­ cause the m apping domain varies over the inverse sets p “1 (b), b € B. For example, one m a y w i s h to consider the case that p ~ 1 (b) is homeomorphic to a Cantor set and K to be the interval [0 , 1 ]. H e r e , the space G^ has the usef u l property of b e i n g locally n-connected (in the homotopy sense) for e a c h integer n > 0 . There are other interesting situations where it only m akes sense to consider maps from p “1 (b) to K -not the other w a y around. Yet, there seems to be n o information available concerning spaces of mappings from subsets of T to K. We show that It is possible to assign a complete metric to certain spaces of mappings from subsets of T to K w h i c h is related to the topology of b o t h T and K in a natural way.

*

R ese a r c h for this paper was supported in part b y NSF-GP-4571 . 135

MCAULEY

136

A Space of M a p p i n g s . Suppose that C is a continuous collection of closed subsets of T. That is, each element of C is closed and further­ more, if (g^) is a sequence of elements of C such that (g^3 g, a closed subset of T, then g € C. Notice that the elements of C m a y inter­ sect contrary to the usual notion of continuous collections. For ea c h g in C, let g into K, a complete metric space. G , g in C, and G denote the u nion O the set of all mappings f in G g for

denote the space of all mappings of Let G denote the collection of all of elements of G. That is, G is eac h g in C.

A metric for G*. If g € G*, let f denote the graph of f in T x K, i.e., the set of all points (x, f(x)) in T x K such that x € g * and f maps g into K. For ea c h pair of elements f , g of G where f €Ga and g € G^, let D(f, g) = H(f, g) where H denotes the Hausdorff metric on the space of all closed subsets of T x K. That is, let d be a bounded complete metric for T x K. Then H(f, g) = lubCe|Ng (f) 3 g and Ng (g) D f)where N £ (A) denotes the e-neighborhood of the subset A of T. Clearly, D is a metric for G but D m a y not be complete. To this end, we prove the following theorem. THEOREM 1 .

The

space G

is a complete metric space.

Proof. Let G denote the collection of all graphs g for the various elements g in G. Now, let Cl G be the closure of G in the space S of all closed subsets of T x K w i t h the Hausdorff metric H as defined above. As a subspace of S, Cl G is a complete metric space. We w i s h to show that G is a Gg-set (an inner limiting set) . For ea c h element A in Cl G, there is c in C such that c x K contains A. This follows from the definition of Cl G and the fact that C is a continuous collection. Let A i denote the collection, if it exists, of all elements A in Cl G such that for some x in c (where c x K 3 A), the set (x, K) n A has diameter > 1 /i. It follows that for each i, A i is closed. We w i s h to show that G = Cl G - U ^ _ 1 A i . Clearly, if g € G*, then g 4 A i for any i. To see that G 3 Cl G - U “ _ 1 A i , sup­ pose that (gi ) A where g^ € G , A is in Cl G - U “ _ 1 A^, g € G h (that is, g^ maps h^ into K, € C) . and (h^) -♦ h e C. If (x, k) € A and (y, k) € A, then x = y. Furthermore, h x K D A. For eac h x in h, there is a sequence (x^) -* x where x^^ e h^ such that (x^, g(xi )) lies in N 1 /j_(A) and N 1 yi (gi ) D A. Therefore, there is a sequence ( ( y ^ k ^ } * of points in h x K such that d[(x±, g(xi )), (yi , k^,) ] < 1 /i. Since d is a complete metric for T x K, we m a y suppose that Ck^} -*■ k without loss of generality. Thus, (yi ) x and (g(x1 )} -*• k. Let g(x) = k. Now, g is a mapping (continuous) of h into K whose graph g is A. We have G = Cl G - U ”=1 A i and G is a G &-set. Consequently, G possesses a complete metric w h ich leaves its topology invariant.

EXISTENCE OP A COMPLETE METRIC FOR A SPECIAL MAPPING SPACE A Condition Under W h i c h tions, it is useful to know that following:

G

is Lover Semi-continuous.

G

is lower semi-continuous.

137

In applica­ Consider the

D e f i n i t i o n . We shall say that the collection G Is regular iff for each g in G and e > 0 , there is a 5 > 0 such that if H(h, g) < 6 (where H is a Hausdorff metric for the space of closed subsets of T ) , then there is a continuous ma p p i n g f ^ w hich takes h onto g and moves no point as m u c h as e. If f^g is a homeomorphism in each case, then G is said to be completely r e g u l a r ,

That is,

THEOREM 2 . If G is regular, then G is lower semi-continuous. if (g^) -*• g where g., g € G, then G is in the closure of

Cl v Proof. Let f € G„. o that there is a m a p ping f ,

&ig

moves no point as m u c h as graph of

fg g

i/i.

There is no loss of generality in assuming for each i, w h i c h takes g. onto G and

1

Now,

is the set of points

m a y be viewed as the set of points quence of graphs

(t^)

f

and

t^ = ^ g ^ g x, ffg^g(x))

(f_ CT(x), ff gj_S

f € closure of

M P3

into

K.

The

whereas the graph of CT(x)).

f

Clearly, the se-

g-^g U ^ _ 1 Gg^ .

THEOREM 3. Suppose that for each g in G, G g is LCn (in the h omotopy sense) and that G is completely regular. Then G is equl-LCn . A proof of Theorem 3 is easy to construct if one follows that given for Lemma 1 . 2 [3] and observes the facts below. Suppose that 5 > 0 , t c G, and h is a homeomorphism of s onto t wh i c h moves no point as m u c h as 5. Let N g (f) be a 5 -neighborhood of f in G s . For eac h g in G s let cp(x, g(x)) = (h(x), g(x)). That Is cp induces a 1 - 1 m a p p i n g from G g n N & (f) into G t n N 2 & (f). Furthermore, 0 . Is S a c om­ plete metric space? Does the m apping m described above fail to raise di­ mension? Affirmative answers to these questions imply that there is no free action of the dyadic group on M n .

EXISTENCE OF A COMPLETE METRIC FOR A SPECIAL MAPPING SPACE

139

REFERENCES [1 ]

Bredon, G. E., Raymond, Frank, and R. F. Williams, "p-adic group of transformations," TAMS, vol. 99(1961) pp. 488-498.

[2 ]

Dyer, Eldon, and M. E. Hamstron, "Completely regular mappings," Fund. Math., vol. 45 (l95T)pp. 103-118.

[3]

McAuley, L. F., "Completely regular mappings, fiber spaces, the w e a k bundle properties, and the generalized slicing structure properties," This volume.

[4]

Yang, C. T., "p-adic transformation groups," Mich. Math. Jour., vol. 7 ( 1 9 6 0 ) pp. 2 0 1 - 2 1 8 .

RUTGERS UNIVERSIT Y

FINITE DIMENSIONAL SUBSETS OF INFINITE DIMENSIONAL SPACES by David W. Henderson

*

This paper gives a summary of some of the results related to the Qu e s t i o n : W h i c h spaces of dimension more than n have subsets of dimension n ? (Throughout this paper, "space" will m e a n "separable metric space.") We first state a few appropriate definitions. Let n always d e ­ note a non-negative integer. The empty set has dimension -1 (written d im 0 = -i) . A space X has dimension < n (dim X < n ) , if X has a n e igh­ borhood basis composed of open sets each w i t h boundary of dimension < n - 1 . D i m X = n (or X is n-dim) if "dim X < n" is true and "dim X < n-1 " is false. D i m X = ° ° (or X is °°-dim) if "dim X = n" is false for each n. A space X will be called countable-dimensional (c-°°-dim) if dim X = « and X is the u n ion of a countable number of o-dim spaces. A n co-dim space will be called strongly infinite dimensional (s-«>-dim) if It is not c-oo-dim. 1•

E v e r y space contains compact o-dim subs e t s .

E a c h point is such a subset.

2. E a c h n-di m space contains closed p-dim subsets for all p < n. follows directly from the above inductive definition of dimension.

This

3. (Mazurkiewicz) There exists n-dim spac e s , for each n, all of whose com­ pact subsets have dimension z e r o . These are the n-dim, totally disconnected spaces constructed b y Mazurkiewicz [16], because any compact totally d i s ­ connected space is o-dim. (See [ 1 1 ], page 20.) *o 4. (Hurewicz and Tumarkin) If 2 = ^ , then there is an s-°°-dim space w i t h no c-oo-d im or n-di m (n > o) s u bsets. Hurewicz showed [10 ] 'that there is a space in the Hilbert Cube ea c h finite-dimensional subset of w h i c h is countable. (Every countable set is O-dim [11, page 10].) Tumarkin (the proof is in [3 8 ] and an E n g l i s h summary is in [37]) extended this to show that in every s-«-dim space there is a n s-«-dim space ea c h of whose subsets are either countable or s-»-dim. 5.

(Tumarkin)

A compactum contains compact subsets of eac h finite dimension

* The author was partially supported b y NSF Contract GP-3857 during p repara­ tion of this paper. Lhi

HENDERSON

Ik2

If and only if quite easy and

it contains a compact c-»-dim s u bset. is contained in [38].

The proof of this

is

6 . (Hurewicz and Wallman) If a space is complete and c-00-d i m t then it has closed n-dim subsets, for eac h n. Hurewicz and W a l l m a n [11, page 50-5 1 ] show that every complete, c-°o-dim space has transfinite dimension (defined b y extending to all ordinal numbers the inductive definition given in this p a p e r ) . It is clear that any space w i t h transfinite dimension has closed n-dim subsets, for each n. 7 . (Smirnov) If X is c-00-d i m , and there is a map w i t h countable-mu l t i p l i ­ city of X into a complete s p a c e , then X has closed n-dim su b s e t s , for each n. Smirnov [33] shows that X has transfinite dimension. (See 6 .) 8 * If the homology dimension of a compact space X is n (< 00) but X is 00-d i m , then X contains no n+i-dim compact su b s e t . (The homology dimension of X is the largest n such that, for some compact set C C X, Hn (X,C;Z) ^ 0 (Cech cohomology).) In [11], pages 151 - 1 5 2 , Hurewicz and W a l l m a n show that the homology dimension of a compact n-dim space is n. (This theorem was originally due to Alexandrov [3].) Thus, if a compact space has a com­ pact n-d i m subset, then its homology dimension is greater than or equal to n.

9. There is an 00-dim compact space X with no n-dim compact subsets (n > 1 ) . A description of this space will appear elsewhere ([ 8 ]). It is known that some component of a compact n-dim space must be n-dim (see [11, page 9 4 ]). As was noted in 2 above, every compact n-dim space contains a compact 1-dim subset. Therefore, in order to show that the space X has no n -dim compact subsets (n > l) it is only necessary to show that X contains no 1-dim con­ tinua. (A continuum is a compact connected space.) In [ 8 ] it is shown that for every non-degenerate continuum K in X, there is a compact set P and a m ap f, such that the hypotheses of the following lemma are satisfied. IEMMA. If K is a compact subset of the compact space f : F -► I 2 (I 2 = [0 , 1 ] x [0, 1 ]) is a m ap such that (*)

P

and

if G is a compact subset of F that separates f ~ 1 (I x 1) from f - 1 (I x 0), then G *X contains a continuum joining G * f - 1 (o x I) to G - f 1 (l x I),

then

d i m (X) > 2 .

P r o o f . Suppose that dim (K) < 2. Then there is a m a p g: K -►Bd I 2 , such that g|K'f”1 (Bd I 2) = f l K ’ f ' ^ B d I 2). (See [11, page 83].) We m a y e x ­ tend g to G: TJ -► B d I 2 , where U is a neighborhood of K + f ”1 (Bd I 2) in F and G | f _ 1 (Bd I 2) = f|f“1 (Bd I 2). Let A =

(I x 0) + (1 x to,i)) + (0 x

B -

(0x (j, 1]) + (to, 1) x 1),

C =

(1x (j, 1]) + ((0, 1] x 1)..

(See figure.)

FINITE DIMENSIONAL SUBSETS OF INFINITE DIMENSIONAL SPACES

143

G _1 (A) is a neighborhood of f- 1 (I x 0 ) and thus N = (Boundary of G " 1 (A) in F) is a compact set separating f “1 (I x 0 ) from f ”1 (I x 1 ). It can be seen that N - f “1 (o x I) C f “1 (0, i) C G _1 (B) and N - f - 1 (i x I) C f " 1 (l, i) C G “1 (C). Also G “1 (A)*G”\ b ) * G “1 (C) = 0 since A - B * C = 0 . Therefore, R = N - (G- 1 (B) + G ”1 (C)) is a closed (possibly empty) subset of N that separates N * f _ 1 (o x I) from N * f - 1 (i x I). But this contradicts (*), since K C G _ 1 (A)*G~1 (B)*G“1 (C). This completes the proof of the Lemma. 10. W r o n g Results. In this section we will mention two results whi c h have been claimed in print, but whose proofs are faulty. V a n Heemert claimed ([40]) in 1 9 4 6 to have shown that eac h compact space contains 1 -and 2-dim compact subsets. The example mentioned in 9 shows that his result is incorrect. The mai n error is V a n Heemert's use ([40, pages 5 6 8 - 9 ]) of a theorem ([7], Seite 2 2 9 , Satz 4) of Fre u d e n t h a l ' s . Freudenthal's theorem deals w i t h inverse limit (Rn -adic) sequences w h i c h have the property that all finite compositions of the bond i n g maps are normal and i r ­ reducible (see [ 7 ] or [40] for definitions). V a n Heemert assumes only that the b onding maps (between consecutive polyhedra in the inverse limit sequence) are normal and irreducible. It can be shown that (a) V a n Heemert's modif i c a ­ tion of Freudenthal's theorem Is false and (b) V a n Heemert's arguments will not w o r k if the correct version of the Freudenthal theorem is used. The author recently published an abstract ([ 9 ]) in whi c h he claimed to have proven that every compact non-zero-dimensional space contains a con­ nected l-dim subset. However, his proof was incorrect. 11 .

We finish the paper w i t h a conjecture.

C o n j e c t u r e : E v e r y compact space contains connected subsets of every dimension less than the dimension of the s p a c e . INSTITUTE FOR ADVANCED STUDY

HENDERSON

144

R EFERENCES (These references include most of the published works pertaining to the d i ­ mension theory of infinite dimension spaces. The square brackets at the end of some of the references refers to the review of the article in Mathematical Reviews (MR) . Russian journals are given b y the abbreviations used in MR and also b y the standard Russian abbreviations.) [1]

Alexandrov, P. S., "The present status of the theory of dimension," Amer. Math. Soc. Translations, Series 2 , vol. l (1955) pp. 1 - 2 6 .

[2 ]

___ , M0 n some results concerning topological spores and their con­ tinuous mappings," General Topology and its Relations to Modern Algebra and Analysis (Proc. Sympos., Prague, 1 9 6 1 ), pp. 41-54. Academic Press, New York; [MR 2 6 (3 0 0 3 )].

[3 ]

__________ j "Dimensionstheorie," Math. Ann. 1 06 (1 9 3 2 ) pp. 1 6 1 - 2 3 8 .

[4]

Anderson, R. D., Connell, E. H . , "Concerning closures of images of the reals," Bull. Acad. Polon. Sic. Ser. Sic, Math, Astrom, Phys., 9 ( 1 9 6 1 ) pp. 807-10,

[MR 2 4 (A 2 3 6 9 ) ].

[5]

A r h a n g l e 'skii, A., "The ranks of systems of sets and the dimensions of spaces," Soviet Math. Dokl. 3 (1 9 6 2 ) pp. 456-459 [MR 24(A2368)]. (The details in Fund. Math. 5 2 ( 1 9 6 3 ) pp. 257-275.)

[6]

Erdos, Paul, "The dimension of rational points in Hilbert Space," Ann. Math. 41 (1940) pp. 734

[7]

Freudenthal, H . , "Entwicklungen v o n Raumen and ihren Gruppen," Compositio Mathematica 4 (1937) pp. 1 0 - 2 3 4.

[ 3]

Henderson, D. W . , "An infinite dimensional compactum w i t h no positive dimensional compact subsets, (submitted).

[9 ]

_________ , "Every compactum contains a connected 1 -dimensional subset," Abstract 6 2 2 - 6 9 , Notices of AMS, 12 ( 1 9 6 5 ) p. 347. (The proof claimed in this abstract is faulty.)

[ 1 0 ] Hurewicz, W., "Une remarque sur 1*hypotheses du continie," Fund. Math. 19 (1932) pp. 8 - 9 . [ 1 1 ] Hurewicz and Wallman, D imension T h e o r y , Princeton University Press, Princeton, N. J., 1 9 4 1 . [12]

Levsenko, B. T., "Strongly infinite-dimension spaces " (Russian), V e s t n i k Moskov Univ. Ser. Mat. Astr. Fiz N i m . , (Becthmk M.Y.C.M.A.®.H.)

[1 3 ]

______________ , "On 00-dimensional spaces," Soviet Math. Dokl., 2 ( 1 9 6 1 ) p. 91 5 [MR 2 4 (AI 708) ] .

(1959), no. 5, 219 [MR 22(4053)1.

REFERENCES

1^5

[14]

Mardesic, Sibe, "Covering dimension and inverse limits of compact spaces," Illinois J. of Math., 4 (i 96 0 ) pp. 2 78 - 2 9 1 .

[15]

Morita, K . , "On closed mappings and dimension," Proc. Japan Acad., 32 (1 956) pp. 161-165.

[ 16]

Mazurkiewicz, "Sur les problemes

k

et \ de Urysohn," Fund. Math., 1 0

( 1 9 2 7 ) pp. 3 1 1 - 3 1 9 . [17]

Magata, J., "On the countable dimensional spaces," Proc. Japan Acad. 34 (1958) pp. 146-149.

[1 8 ]

________ , "On the countable sum of O-dimensional metric spaces," Fund. Math., 48 (1960)pp. 1-14 [MR 22(5028)].

[19]

________ , "Two remarks on dimension theory for metric spaces," Proc. J apan Acad., 36 (i 96 0 ), 53 [MR 2 2 (9 9 6 3 )].

[20]

Nagama, Keto, "Finite-to-one closed mappings and dimension I," Proc. J apan Acad., 34 ( 1 9 5 8 ) pp. 503-506 [MR 2 1 ( 86 2 )].

[21 ]

________ , "Finite-to-one closed mappings and dimension II,"

Proc.

Japan Acad., 35(1959) pp. *+37-439 [MR 22(4025) (1 961)]. [2 2 ]

________ , "Mappings of finite order and dimension theory," Japan J. Math., 3 0 ( 1 9 6 0 ) pp. 25-54 [MR 25(5494)]. (This paper contains a gen­ eral development of dimension theory for metric spaces.)

[2 3 ]

Pasynkov, B., "Absence of polyhedral spectra for bicompacta," Soviet Math. Dokl., 3 ( 1 9 6 2 ) pp. 1 1 3 - 1 1 6 .

[24]

________ , "On the spectra and dimensionality of topological spaces," Ma. Sv. (N.S.) (Ma. C. H. C.) 57 (99)(1962), pp. 44 9- 4 7 6 [MR 26(1856)]. (This paper contains details and proofs of the preceeding paper and m o r e .)

[25]

Sersnev, M., "Strong dimension of mappings and the associative charac­ terization of dimension for arbitrary metric spaces," Soviet Math. Dokl., 1(1961) pp. 1267-1269 [MR 23(A2865)3. (This details in Ma. Sb. (N .S .) (M a . C. H. O ) 60 ( 1 0 2 ) ( 1 9 6 3 ) pp. 207-21 8 [MR 28(585) 3 .

[ 26]

Sklyarenko, E. G., "Two theorems on infinite-dimensional spaces," Soviet Math. Dokl., 3 ( 1 9 6 2 ) pp. 547-550 [MR24(A2367)3.

[27]

________ , "On dimensional properties of infinite dimensional spaces," Izv. Akad. N a u k SSSR ser Mat (NAFICCCP) 23 (1959) p. 197 MR 21 [5 t 79 J (translation: AMS Trans. ( 2 ) 21 (1 9 6 2 ) p. 35-50).

[2 8 ]

________ , "Some remarks on spaces having an infinite number of dimen­ sions" (Russian), Dokl. Akad. N a u k SSSR (DAHCCCP) 1 2 6 (1959) p 1 20 3 MR 22[ 1 885].

[2 9 ]

________ , "Homogeneous spaces of an infinite number of dimensions," Soviet Math. Dokl., 2 ( 1 9 6 1 ) p. 1569.

HENDERSON

146

[30]

Smirnov, Ju. M . , "Some remarks on transfinite dimension," Dokl. Akad. Nauk SSSR (DAHCCCP) 1 4 1 ( 1 9 6 1 ) p. 814 [MR 26(6934)3. (Translated in Soviet Math. Dokl., 2 ( 1 9 6 1 ) p. 1572.)

[31]

________ , "On dimensional properties of infinite dimensional spaces," Gen. Top. etc., (Proc. Symposium Prague, 1 9 6 1 ) pp. 334-336.

[32]

________ , "Universal spaces for certain classes of 00-dim spaces,” Izd. Akad. Nauk SSSR, Sem. Mat. (NAHCCP) 23(1959) p. 185 [MR 51783. (Translated in AMS translations ( 2 ) 21 ( 1 9 6 2 ) pp. 21-33.)

[33]

________ , "On transfinite dimension" (Russian), Mat. Sb. (W.S.) C. H. C.) 58 (100)(1962) pp. 4 1 5 - 5 2 2 .

[3M

Smirnov and Sklyarenko, "Some questions in dimension theory" (Russian) Proc. 4th All-Union Math. Congress (Leningrad 1 9 6 1 ), vol. I, pp. 2 1 9 2 2 6 . (T

[35]

Toulmin, G. H . , "Shuffling ordinals and transfinite dimension," Proc. London Math. Soc., 4 ( 1 9 5 4 ) pp. 177-195.

[363

Tumarkin, L. A . , "On ber of 0 -dimensional Mek, (Becthnk M. Y . ,

[373

________ , "Concerning infinite-dimensional spaces," General Topology and its Relations to Modern Algebra and Analysis (Proc. Sympos.,

the decomposition of spaces into sets" (Russian), V e s t n i k Moskow, C. I) 1960, no. 1, pp. 2 5 - 3 2 [MR

(Mat.

a countable n u m ­ Univ. Sec.I Mat. 2 5 (5 5 4 ) 3 .

Prague, 1 9 6 1 ) pp. 352-353. [383

________ , "On strongly and w e akly infinite-dimensional spaces" (Russian), Vestnik. Moslow Univ. Ser I, Mat. Meh. (Becthnk M. Y. C. I . ) ( 1 9 6 3 ) no. 5, pp. 24-27 [MR 27(5230)3.

['393

________ , M0n infinite-dimensional Cantor manifolds" (Russian), Dokl. Akad. N a u k SSSR (DAHCCCP) 115(1957) pp. 244-246.

[403

V a n Heemert, A., "The existence of 1-and 2-dimensional subspaces of a compact metric space, A m s terdam Acad, v a n Wetenschappen-Indagations Mathem a t i c a e , 8 (1946) pp. 564-569* (This paper is incorrect.)

[41 3

Alexandroff, P. S., "Results in topology in the last 25 years," Russian Math. Surveys 1 5 ( 1 9 6 0 ).

[423

________ , "On some basic directions in general topology," Math. Surveys 1 9 (1964) # 6, pp. 1-39.

[433

Sklyorenko, E. G., "Representations of infinite-dimension compacta as an inverse limit of polyhedra," oviet Math. Dokl. 1 (1 9 6 0 ) pp. 1 1 4 7 - 4 9 .

[4 4 ]

Nogata, J., Modern D i m ension T h e o r y , Interscience, New York, 1 9 6 5 .

Russian

TYPES OF ULTRAFILTERS by Mary Ellen Rudin

*

Introduction. then and

If S is a set and a is a collection of non-empty subsets of is said to be an ultrafilter on S provided that: (a) (b)

the intersection of any two members of ftl is maximal with respect to (a) .

ft belongs to

S,

fn, t,

There is a close connection between ultrafilters on N (the set of all positive integers) and the Cech compactification PN of N as the fol­ lowing construction of £N (see [1]) shows: (i) Let Let the the ultrafilters ultrafilters on on NN be be the the points points of of PN. PN. (ii) Identity Identity each each nn € €NN with with the the ultrafilter ultrafilter on on NN which which con­ con­ sists of all subsets of N which contain n. (iii) For For every every set set EE CC N, N, declare declare the the set setof of all all fftl € €PNPN such such that EE €fift to to be be open open in in PN; PN; and and every every open open set in£N £Nis is aa union union ofof such sets. My concern will be with the space N* = 3N - N and a classifica­ tion of the points of N*. The points of N* N* are are called called free free ultrafilters ultrafilters on N ; two free ultrafilters on N are said to be of of the the same same type type if if there there is a homeomorphism of N* onto N* which carries carries one one of of these these ultrafilters ultrafilters onto the other. While the homeomorphisms home omorphi sms of £N are easily described (they are induced in a natural way by the permutations of N; see [l]), the situa­ tion is more complicated in N*. The hypothesis of the continuum will be assumed throughout this paper and and cc will will denotedenote the power thepower of of the the continuum. continuum. A point pp of of N*N* isiscalled calleda aP-point P-pointofof N* N* if ifthe the intersection intersection of every countable collection of neighborhoods of p (in N*) contains a neighborhood of p. It is shown in [1] C1] that N* has 2C P-points if the hy­ pothesis of the continuum is true, and that all P-points are of the same type. points.

This paper will classify the limit points of countable sets of PThe classification is in terms of ultrafilters on the P-points

*

Work on this paper was supported by the National Science Foundation under grant GP-3 857. 857.

1^8

RUDIN

themselves. Also it will imply that there are 2 C distinct types of ultra­ filters among the limit points of each infinite countable set of P-points. (The results of [13 showed only that there are at least two types of free ultrafilters.) However, the problem of classifying all of the points of N* is still far from complete. For instance, I know that there are non-P-points which are not limit points of any countable set of P-points. And I do not know whether there are any points of N* except P-points which are not lim­ it points of any any countable countable set. set. Also Also II do do not not know know if if there there are are any any points points of N* which are not limit limit points points of of any any countable countable discrete discrete set set but but are are limit points of some countable set. Classification of 2 C Types of in N*. N*. of Points Points in 11 . Definition. For For every every E C N,E put C N, W(E) putW(E) = {Q = € {Q€ N*: N*: EE €€a } . a } . The sets W(E) are the open-closed sets of N* and form a basis for the open sets in N*. Observe that W(E1) C W(Eg) if and only if E 1 - E 2 is finite. 2 . LEMMA. LEMMA. If If XX UUYY is is aa countable countable discrete discrete subset subset of of N* N* and and is in the closures closures of of both both XX and and Y,Y, then then n n is in is the in the closure closure of of X n Y. Proof. Suppose on the contrary that there is an E € a such that W(E) n n (X n Y) =Y)0 .=Since 0. Since no point no point of ofXX UUYY is isa alimit limitpoint ofpoint X Uof Y X U and X X U Y isiscountable, countable,there thereare aredisjoint disjointsets sets EE00 ee 00 for for each each0 €e X€ UX Y. U Y. Then define E^ = (n € N: n c (E0 n E) for some 0 € (X n W(E))} W(E))) and define Ey = {n € N: n € (E(E 0n 0 E) nE) for some 00€€ (Y n W(E))}. Since Sincea ais is a limit point of X and and n n€ W(E), €W(E), nn is is aa limit limitpoint pointof of XXnnW(E) W(E) ; ;and since W(E^) is an open closed set containing X Xn n W(E), W(E), aa€€W(E^) W(E^).. Similarly n € W(Ey) . But since Ex E^ nnEy Ey ==0, 0 , W(EX) nW(Ey) W(Ey) == 00which which is is a a contradiction. 3 . Definition.If IfX Xis is a a countable countable discrete discrete subset subset of of N* N* andand p € X, then then define define n^(p)n^(p) = CY = (Y C X:C X:pp€€ Y} Y}.. By By I^mma I^mma 22,, Ji^(p) Ji^(p) is an an ultra­ ultra­ filter on X whenever p isisaa limit limit point point of of X. X. k. LEMMA. If X is aa countable, countable, infinite, infinite, discrete discrete subset subset of of N*, N*, then XX= =ftXPX, , X X isishomeomorphic homeomorphic to to PN, PN, and and XX -- XX is is homeomorphic homeomorphic to to N*. N*.

Proof. Let tttt be any one-to-one correspondence from XXonto onto N N and, for p € X define f(p) = (E C N: 7r(E) € n^(p)}. Clearly ff is a a and f(X) f(X) == N. N. homeomorphism of X onto PNPN and sets 5. Definition. IfIfnn11 and and n 2 are are ultrafilters ultrafilters on on countable countable sets X and and Y,Y, respectively, respectively, then then nn11 and and ft2 are are said said to to be be similar similar if ifthere there is aaone-to-one one-to-one correspondence correspondence ff of of XX onto onto YY such such that, that, for for each each EE CC X, X, E € ft if and only if f(E) e ftg. The main theorem can now be stated.

Y

TYPES OF ULTRAFILTERS

1^9

6. THEOREM. THEOREM. Suppose Suppose that that X and X Y andare Y countable are countable sets of sets P-of Ppointsof of XX and and Y, Y, respectively. respectively. points and that p and q are limit points andonly only if n^-(p) and ^(q) Then p and q are of the same type ififand n^-(p) anc^ are similar. Observe that every countable set of P-points is Is discrete. Then let us immediately point out one consequence of this theorem. Observe that that XX has has 2° 2° limit limit points points (since (since N* N* ha has .s 2C 2Cpoints points this this follows follows from Lemma U) . Also, since there are only c permutations of X, each ultrafilter on X is similar ,to at most c other ultrafilters on X. There­ fore there are 2C distinct types of limit points of X. 7* Proof Proof of of the the necessity necessity for for Theorem Theorem 6. 6. Suppose Suppose that that hh is Is aa homeomorphism of N* onto N* suchthat that h(p) h(p) == q. Select subsets X1 and Y1 ofof X X and and Y, Y, respectively, respectively, such such that that X - X 1 and YY --Y1 Y1 are are infinite infinite and and pp €€ X1 X1and and qq €€Y1 Y1.. Let Let Y2 Yg== hfX^ n Y1 and X2 == h_1 hence € Xg. h(X^) h_1(Y2) (Y2).. By By Lemma Lemma 2, 2, qq €€ Y2and and hencep p € Xg. Clearlyh hinduces a a one-to-one correspondence one-to-one correspondence X 2: induces between (Z Cbetween X 2: p e(ZZ)C and (V C Y2: q € V) . Let g be any one-to-one correspondence from X - Xg on­ to Y - Yg. Now define f as the one-to-one correspondence from X onto Y such that f(x) == h(x) h(x) for for x € X2 and f(x) = g(x) for x € X - X2. If E n X2; hence E C X, then pp €€ EE if if and and only only if If p is a limit point of p € E if and only only if if qq is isa limit point of E n Y2. Therefore, ^x (p) is similar to since since if if E C X, p e E if if and only if y e f(E) 8. I think the following theorem is interesting In in itself and it will be used In in the theproof proof of of the the sufficiency sufficiency for for Theorem Theorem 6. 6. The The proof proof of of Theorem 8 is similar similar to to the the proof proof of of Theorem Theorem ^.7 ^.7 of of [1] [1] and and especially especially makes makes use of Lemma 11 which is proved in [1] although stated in a different form. Throughout the rest of the paper all open-closed subsets of N*.

ffi is used to denote the set of

A subcollection of ffiffiwill will be be called called aa ring ring if if it it is is closed closed under under finite unions, finite intersections and complements. 9 . THEOREM. Assume that for each n e € N, is a home homeomorphism__ omorphi sm_ of N* onto N* and that CUn) and aTe aTe countable countable families families of of disjoint open-closed open-closed subsets subsets ofof N*. N*. Then Then there there is is aa homeomorphism homeomorphism hh of of N* N* onto N*N*such such that, that, for for each each nn €€ NN and and x € xUn, € Un, h(x) = h(x) ^ ( x=) ^ .(x).

Proof. Define R1 as as the the ring ringgenerated generatedby by (Un). {Un). That That is is AA belongs to R 1 if and only if A is 0 or N* or the finite union of Un 's or the finite intersection of (N* - Un)'s. Let f, be the function from R1 into ffi which preserves unions, intersections, and complements such that f 1(0) = 0 and f1 (Un) = . Using the hypothesis of the continuum, index W = (Wa) by the countable ordinals. Lemma 10 which follows will show that f1 can be extended

p e Z)

RUDIN

150

by transfinlte induction to a permutation f rton © such that ( 1 ) f prec c serves intersections, unions, and complements and (2 ) if V C U n , f c (V) = h j j V ) . Hence fQ Induces a homeomorphism h of N* onto N* such that h 0 1r X - Y

and g € such that

Hom(Xk , X k - Y; E n )

are given, choose a

d ( f ( x ) , g(x)) < 6 for all x € X - Z. B y using standard extension theorems and general p o s i ­ tion arguments, one can extend f |XK - Z to an imbedding f 1 of X k into E n so that f ’|Z is piecewise linear and d(f*, g) < e. B y definition, F'€ Fj hence, F is dense. To see that F is solvable, choose &F (e) = e for a ny arbitrary positive number e, and suppose that fQ , f 1 €F with d(fQ ,f 1) < e . Notice that f and fagree w i t h f except on a polyhedron Z in k X - Y , and that b o t h fQ and f 1 are peicewise linear on Z. B y a suit­ able modification of a theorem of Bin g and Kister [13, one obtains a n epush h of (En , fo (Z ) ) such that 1) 2) Thus

F

hfQ |Z = f,|Z h|P U f(Xk - Z)

and =1.

is solvable.

5.

A special c a s e .

THEOREM 5. Suppose Y is a subpolyhedron of X k imbedding of X k into E n (2k + 2 < n) such that f |Y and f |Xk - Y is locally tame. Then f is e-tame.

and f is an is locally tame

Proof. Since Gluck's results [7, 8] and Lemma 3 allow one to assume that f|Y is piecewise linear and f |Xk - Y is locally piecewise linear, the theorem follows immediately from Lemma 4 and Theorem 2, for the set of piecewise linear extensions of f |Y in Hom(Xk , X k -Y; E n ) is known to be dense and solvable. Using Theorem 5, one can prove, b y induction on k, the following theorem, w h i c h has been established for k = i and k = 2 b y Cantrell [33 and Edwards [63, respectively. T HEOREM 6. A k-complex K in only if, each simplex of K is tame.

En

(2k + 2 < n)

is

e-tame if, and

156

BRYANT

Prom this and a theorem of Klee [ 1 2 ], one obtains a n alternative proof of a theorem of Bing and Kister [ 1 ].

En+k

THEOREM 7. If f is an imbedding of X k into the n-plane (n > k + 2 ), then f: X k — E n + k is e-tame.

En

in

6 . Proof of Theorem 1 . Case 1 . M 11 = E n . Let Y = f ” 1 (P) . One may assume that P is piecewise linearly imbedded in En , b y Gluck's results [ 7 , 8 ], and that f|Xk - Y is locally piecewise linear from Lemma 3.

En .

Let For each 1) 2)

3)

denote the set of all piecewise linear imbeddings 9 € to be locally flat in Mn + ^c whe n k = 2 , then any imbedding of X ^ into if1 is an e-tame imbedding of X ^ into Mn + ^. lr It follows immediately from Theorem 1 that an imbedding of X Into M 11 (2 k + 2 < n) cannot fail to be locally tame at precisely one point. k Applying this to a polyhedral neighborhood of a point of X , one obtains 2 k + 2 ) and if M 11 is assumed

THEOREM 1 0 . If S is the set of points at wh i c h an imbedding f of X ^ into M 0, (2 k + 2 < n). fails to be locally tame, then S contains no isolated points and must, therefore, be uncountable or empty. If M and N are topological manifolds, of dimensions m and n and if f is an imbedding of M into N, then f is said to be locally flat at the point x e M if there is a neighborhood U of f(x) In N such that the pair (U, U n f(M)) is homeomorphic to the pair (En , E m ) . A n imbedding f of a k-cell D into E n is said to be flat if there is a homeomorphism of E n onto itself carrying f(D) onto the standard k-cell in the k-plane E k in E n . Similarly, f is locally flat at x € E if x lies in the interior (relative to D) of a k-cell D' in D on w hich f is flat. If 2 k + 2 < n, then an imbedding of D in E n is flat if, and only if, it is tame. Thus Theorem 1 implies

2

< n)

IEMMA 1 1 . such that

If

f

is an imbedding of the k-cell D into f|D - f " 1 (E^)( 2 i + 2 < n)

E n (2 k + is locally flat, then

is flat. Cantrell and Edwards [4] have shown that there exists no "almost locally fla t ” imbedding (that is, an imbedding which is locally flat e x ­ cept at a countable number of points) of a topological m-manifold Into a topological n-manifold whenever 2m + 2 < n. Using Lemma 1 1 , one can show that the following more general situation exists. THEOREM 1 2 . Suppose that M, N, and Q, are (not necessarily triangulable) topological manifolds of dimensions m, n and q, respectively, wit h m, q < § - 1 > such that Q is a locally flat submanifold of N. Let f be an imbedding of M into N such that f|M - f _1 (Q) is locally flat. Then f is locally flat.

BRYANT

158

REFERENCES [1]

R. H. B i n g and J. M. Kister, ’ taming complexes in hyperplanes," Duke Math. J., vol. 31 (1964) pp. 4 9 1 - 5 1 1 .

[2 ]

M. Brown, "Locally flat embeddings of topological manifolds," Annals of Math., vol. 75 ( 1 9 6 2 ) pp. 331-341.

[3 ]

J. C. Cantrell, "n-frames in Euclidean k-space," Proc. Amer. Math. Soc., vol. 15 (1964) pp. 574-578.

[4]

J. C. Cantrell and C. H. Edwards, Jr., "Almost locally flat imbeddings of manifolds," Michigan Math. J., vol. 1 2 ( 1 9 6 5 ) pp. 2 1 7 - 2 2 3 .

[5 ]

J. Dugundji, "An extension of Tietze's theorem, " Pacific J. of Math., vol. 7 (1951) pp. 353-357.

[6]

C. H. Edwards, Jr., "Taming 2-complexes in high-dimensional manifolds," Duke Math. J., (to appear).

[7 ]

H. Gluck, "Embeddings in the trivial range

Bull. Amer. Math. Soc.,

vol. 69 (1963) pp. 824-831. [8]

_______ , "Embeddings in the trivial range (complete," Annals of Math., v o l . 81 (1 9 6 5 ) pp. 1 9 5 - 2 1 0 .

[9 ]

C. A. Greathouse, "Locally flat, locally tame and tame embeddings," Bull, Amer. Math. Soc., vol. 69 ( 1 9 6 3 ) pp. 82 0 - 8 2 3 .

[10] V. K. A. M. Gugenheim, "Piecewise linear isotopy and embeddings of elements and spheres I, II," Proc. London Math. Soc. (3), vol. 3 ( 1 9 5 3 ) pp. 2 9 - 5 3 , 1 2 9 - 1 5 2 . In]

T. Homma, "On the imbeddings of polyhedra in manifolds," Yokohama Math. J., vol. 1 0 ( 1 9 6 2 ) pp. 5 - 1 0 .

[1 2 ]

V. L. Klee, Jr., "Some topological properties of convex sets," Trans. Amer. Math. Soc., vol. 78 (1955) pp. 30-45.

[1 3 ]

E. C. Zeeman, 'iJnknotting combinatorial balls," 78 (1963) pp. 501-526.

Annals of Math., vol.

SOME NICE EMBEDDINGS IN THE TRIVIAL RANGE by Jerome Dancis

*

1. Introduction. This paper is concerned w i t h criteria for determining when k-complexes w h i c h are embedded in combinatorial n-manlfolds, 2 k + 2 < n, (the trivial range of dimensions), are tame.

Let n, and + B where M, K, k, n, 2
0 is given, then there is an e-homeomorphism h of M onto itself such that h(B) is a subpolyhedron of M and h is the identity outside a n e-neighborhood of B. Gluck (Th. 9 . 1 of [6 ] and Th. 4.1 of [ 7 ]) has improved Homma's result b y showing that the above result holds whe n M and M 1 are possibly noncompact combinatorial n-manifolds. Also, if K, is a subcomplex of K and f |Kj is a piecewise linear then one can find an e-homeomorphism h of M onto itself such that h | f ( K 1) = 1 and h(B) is a polyhedron. Gl u c k [6 ] and [7] and Greathouse [8 ] use the above to prove Theorem 1 : If f(K) = A then A is tame, i.e., locally tame k-complexes are tame in combinatorial n-manifolds, 2 k +2 < n. Bin g and Kister [ 1 ] have shown that when is contained in an (n-k)-hyperplane of E n , then

M = E 11, f (K) = B f(K) =B is tame.

and

B

J ohn Cobb (unpublished) has shown the following: Let P be a com­ plex, Q a subcomplex and N a combinatorial n-manifold with boundary. Let g: P N be an embedding such that g |Q is p.w.A., g(P-Q) is local­ ly tame, g(P-Q) C Int N and dim (P-Q) = k, 2k + 2 < n. If e > 0 is given, then there exists an e-push T of g(P-Q), such that T°g is p . w .1. and the isotopy of the p u s h does not move any point of g ( Q ) . W o r k on this paper was partially supported b y the National Science F o u n d a ­ tion. 159

DANCIS

l6o

In their theorems, G l uck and Bing-Kister have shown that f(K) may be moved onto a subpolyhedron of the space by an e-push of f (K) . This paper is a proof of: Theorem 2: A necessary and sufficient condition that a k-complex K, wh ich is a closed subset of a combinatorial n-manifold (without-boundary) n > 2k+2, be tame in M is that K lie in the union of a countable number of locally-tame (n-k) simplices in M. (Note n -k > k+2) We shall establish two special cases of Theorem 2. The first special case (Theorem 3) and Theorem 5 will be used to establish the second special case (Theorem 10). Corollary 2 of Theorem 10 will then be used to establish Theorem 2. Corollary 2 of Theorem 10 will then be used to establish Theorem 2. Corollary 2 of Theorem 10 is dependent upon some of the "immediate consequences" (in section 5) of the first special case and Theorem 5. 2.

Definitions

A complex L, topologically embedded in a combinatorial manifold is a polyhedron if the injection map i: L M is piecewise-linear.

M

A complex L, topologically embedded in a combinatorial manifold M is tame if there is a homeomorphism h of M onto itself such that h(L) is a polyhedron. Let L be a complex, embedded in a combinatorial manifold M. A n open subset U of L is locally polyhedral if for ea c h point x in U, there is a compact neighborhood C of x in U such that C is a s ub­ polyhedron of b o t h L and M. A n open subset V of L is locally tamp if for each point x in V, there is a compact neighborhood C of x in V and an open neighborhood N of C in M such that C is a subpolyhe­ dron of L and C is tame in N. A n ambient e-isotopy of M topological space such that if for eac h h t (x) then h t is a homeomorphism of M < e, for each x e M, t € I. A n e-push d(x, A) > e

P

of

A

is said to p u s h

3.

Motivation

=

onto itself;

A

onto

=

H:

H(x, t) h = 1,

is a n ambient e-isotopy of h t (x)

P

is a map t € [o, 1],

x/

for all

h 1 (A),

P(x)

M

and

d(h^(x), x)

such that if

tel. =

h 1 (x) .

The proof of Theorem 3 was motivated b y the following problem set wh i c h was assigned b y R. H. Bing.

M x

161

SOME NICE EMBEDDINGS IN THE TRIVIAL RANGE

(NOTE: A stable home omorphi sm is a homeomorphism wh i c h is the composition of a finite number of hoemamorphisms each of w h ich is the identity on some open set.) Problem 1. Suppose A is an arc in E11 (n > k) w hich is the sum of a countable number of segments and points. Then there is a line L in E n such that no line parallel to L intersects A in two points. Problem 2 .

Generalize

1 to suppose that

A

is a set in

E 11 w h i c h is the

sum of a countable number of simplexes of dimension less than

k.

Problem 3. If A, L arc as in 1 , there is an (n-1)-plane P and a stable homeomorphism h: E n -►E n such that h(A) C P and h is invariant on each line parallel to

L.

Problem U. If A is an arc in the plane E n ~ ] of E n , then there is a stable homeomorphism h: E n -►E n such that no two points of h(A) have the same last coordinate. Problem 5.

E a c h arc in

En ’1

is stably flat in

En .

P roblem 6.

Generalize 5 to topological k-complexes in m-planes of

En .

(Klee proved such an extension.) U sing the methods suggested b y the above problems one can easily p r o ­ vide proofs for the following problems too. Problem 7. Suppose that A 1 is a compact set contained in the hyperplane E n 1 of E n . I^t g: A 1 -*>E n (g(A1) = A 2) , be a homeomorphism. Then there is a stable homeomorphism h: E n -♦E n such that the last coordinates of a point x i n h ( A ^ has the same value as the last coordinate of g ' h ”1 (x) in A 2 . Problem 8. Suppose A is an arc in E n , n > 4, w hich is contained in a countable number of k-dimensional hyperplanes, 2k + 2 < n. Then there is a stable homeomorphism of E n onto itself such that h(A) is a canonical interval I . This method of showing that an arc A in E n , n > 4, wh i c h is the sum of a countable number of straight line segments and points, is a tame arc, is a modification of the original method of proof due to Cantrell and E dwards [3]. h. The first special c a s e . Suppose is a subset of a simplex then we shall say that f: t -► M is linear w.r.t. if flT r has a linear extension (f)r :7 ar - M. Let f: K M. We shall say that f is countably linear if K may be written as the countable union of subsets t , 7 = 1,2,..., where each is contained i n a simplex a^ of K such that f|t^ is linear w.r.t. for each 7 .

162

DANCIS

THEOREM 3. Suppose f: K -*• E where K is a finite k-complex, 2k+2 < n and f is a countably linear embedding. Then f is a tame embedding of K into E11. Al s o if e > o is given, then there is an e-push p of f(K) such that p-f: K -♦E n is p.w.l. Furthermore, under the isotopy of the push, ea c h point of E11 m a y be made to move along a polygonal path, having length less than e. Proof: A rewording of Kister paper [1] yields:

Lemmas 3.3, 3.4 and 3.5 of the Bing-

LEMMA 1 (Bing and Kister) : Let n be the natural projection of E n = E k x E n _ k onto E k . Let K be a finite k-complex, f^:K -*• E n an embedding, 2k + 2 < n. Let i be a p.w.l. embedding of K into E n such that tt•i : K -*■E k is 1-1 on each simplex of K. If iT-i(x) = 7r*fk (x), for each x c K, and if e > o is given, then there is a n e/3k-push p of fk (K) sending ^ ( K ) onto a polyhedron such that, under the push, each point of E n moves along a polygonal path of length less than 6 /3k. We obtain a p.w.l. injection i:K E n such that ir-i: K -*■E k is 1-1 on each simplex of K by taking a piecewise-linear, e/3k-approximation of f such that the vertices of the approximation are in general p o s i ­ tion and such that the images under tt, of the vertices of each simplex of v the approximation, are in general position in E . Hence, thanks to the Bing-Kister lemma, the proof of Theorem 3 is reduced to finding e/2-push P of f(K) such that ir*i(x) = ir-P«f(x), for e ach x € K. That is we must give f(K) a push P wh i c h will make the first k-coordinates of P-f(x) agree w i t h those of i(x) for each x € K. We will adjust the coordinates, one coordinate at a time (in order) and this of course we will do b y induction. In order to simplify the machinery we shall assume that the triangulations of K and E n had small m e s h and that i: K -*•E n plicial. We will identify K w i t h its image under i. Let t?\: E n = E 1 x E n_i projection map. Note: ir^ = tt.

E1,

(i = l, 2,...,

original is sim­

k) be the natural

Theorem 3 will be established once we have verified the following. Inductive c l a i m : and a set of

There is a set of homeomorphisms : K — En )

e/2k-pushes

CP,,--

■pk ;

fQ = f

and

2.

T^f.U)

= ir±(x), for each

3.

f± is countably linear w i t h sion of each

xc

fjJT a,

f±+1

a

Pi+1 °fi ;

defined in a=

is a push of f±(K) in E n )

pi+ >

such that: 1.

x € (fjL)a a& , and

1,2 ... .

and denoting the linear e x t e n ­ 7ri(f’ :j_)a (x ) =

for

SOME NICE EMBEDDINGS IN THE TRIVIAL RANGE

163

Assume that f. and all of the (f’ i )a a = 1 ,2 , are known and satisfy conditions 2 and 3 above. The existence of an appropriate Pi+1 is established b y the two lemmas that follow. Proof of inductive claim:

Keeping in mind the following picture m a y help the reader juggle dimensional indices.

rk-i

In the figure, Z axis, respectively; planes respectively.

En _ k , E k_i and E 1 are analogous to the X, Y and E11'1 and E k are analogous to the X Y and YZ

LEMMA 2. Suppose that f±(K) is contained in the sum of a countable such that dim(7/\( aj)) > number of simplices a^, eac h of dimension < k i - (k-dim a . Then there is O a, line X,i L *+., 1 such that: l. L

. 3.

2

then L n (U , a ’ ) and If L is any line, parallel to Ji + i , " vwr=i L n f. (K) are each sets that contain at most one point; i n-i. C EJ Ji + 1 goes through the origin, and the angle between L. and 1+1 Li+ the (i+i)3t coordinate axis is arbitrarily small, and furthermore (and hence all lines parallel to it) We can coordinate L,1+1 in a continuous manner so that if x € L and L is parallel to then the coordinate value of x shall be the value of st coordinate of x . the ( i + 0

16k

DANCIS

R e m a r k . If we let = (*j_)a (ffa )> dim = r < k, then the induction h y p o ­ thesis says that tt^(JT^)a (x) = ^ ( x ) , for all x c a& , i.e., ir^(a . Now

tr

is 1-1

on

Therefore is satisfied.

and hence

dim(Tr^(o& )) > r - (k-1) = i - (k-r) .

din^TT^a^)) > 1 - (k-r)

and the hypothesis of Lemma 2

Proof of Lemma 2: Let p e and suppose that p Is the origin. This supposition is for convenience and is temporary. Now since dim(-n\(cM)) > I - (k-r) there is a linearly independent set of vectors { v ^ . . . , v j__(k-r)^ 1. 2. Notation:

such that:

each vector is contained in. oa*, and < v ±,..., > n E n " = the origin. < A >

means the hyperplane spanned b y the elements of

Let be an s-simplex, listed imply that dim(
n E n_i)

A.

Then the two properties just < r + s + i - [ i -

(k-r)]

Now r + s + and since

k > s,

1 - [i - (k-r)] = k + s +

n > 2k + 2,

1-

i

we see that

dim(
n E 11"1 ) < n-i

Now the lines in a hyperplane H, if H intersects E*1-1, wh i c h are parallel to lines in E n-i are also parallel to lines in H n E n ~^. Therefore the subset of the vector set {v:

v goes from a point

a

to a point

b, for all

a, b €

w hich lie in the vector space E n_i span a vector subspace through the origin) whose dimension is strictly less than n-i. (aM, Cv:

Now since f^(K) is contained in the where dim < k, it follows that the v goes from a point

a

U

(a hyperplane

countable union of simplices subset of the vector set

to a point b, for all a, b €

aJ)

w h i c h lie in E n ”^, lie in . H * w h i c h does not contain any open set n —1 '> ~ '> in E . (Courtesy of the Baire Category Theorem.) Hence we can find a line and the lemma is established.

Lj_+1

wh i c h

satisfies conditions i to

Now we can find an e/2k-push Pj_+1 ^ l i c h will adjust the (i+i)3t coordinate of f^(K) without disturbing the first i coordinates, i.e.:

4,

165

SOME NICE EMBEDDINGS IN THE TRIVIAL RANGE

LEMMA 3. Given the line L i+1 of the preceding lemma, and the i n ­ duction hypothesis, there is an e/2k-push P ^ +1 where: 1.

P i + 1 (L)

2-

W

3.

f i+1 =

K —►E n

^i+l^i+iM^

=

i

=

L, for all lines

+ ,f ^ x )

=

7ri + 1 (x),

L

parallel to

for all

Li+1;

x e X, and

is countably linear and

^i+i^'

for each

x € aa »

a = 1,2,...

.

P r o o f : Let E n ~ 1 be a hyperplane wh i c h meets in exactly one point. Let p: E n - * E n ' 1 be the natural projection m ap w hich sends each line L, parallel to L ^ + 1 , onto the point L n E n _ 1 . Let h be a solution to problem 7 for A 1 = pf^(K), g ’1 = p|A2 such that ph is the identity on E n “1.

A g = fj_(K),

Now we define a new m ap r:

A, - E 1

such that r(x) = (i+i)3t coordinate of Next extend

r

f ^ 1 g(x)

the L i+1 coordinate of

to r 1: E n_1 -♦E1

g(x ) , x € A 1 =

such that either

|r'(x)| + d(h(x), A 2 )
2k + 2 (the "trivial" range of dimensions). Mainly, we shall show that in many cases the conditions about "global" triangulations of complexes and m a n i ­ folds m a y be replaced b y conditions about "local" triangulations of sets and manifolds. The mai n results of this paper are: THEOREM 1 (Locally-flat approximations) . L e t g be a map of a ir compact, topological k-manifold-with-boundary M into a topological nmanifold-with-boundary M n , n > 2k + 2. Then given a n e > 0, there exists a homeomorphism h of M ^ into Mn such that d(g, h) < e and h d ^ ) is locally flat. COROLLARY 1.1 (Locally-flat embeddings). cal k-manifold-with-boundary m a y be embedded in set.

Every compact, topologi2k+2 E as a locally-flat

THEOREM 2 (Locally-tame approximations I ) . Let g be a m ap of a k-complex K into a topological n-manifold-without-boundary Mn , n > 2k + 2. Then given a n e > 0, there is a (strongly)-locally-tame e m bed­ ding h of K into M n such that d(g, h) < e. THEOREM 3 (Locally-tame approximations II) . Let L be locally a k-complex. Let g be a m a p of L into a topological n-manifold-with­ outboundary M, n > 2k + 2. Then given an e > 0, there exists a homeomorphism h of L into M such that d(g, h) < e and h(L) is (weakly) locally-tame. THEOREM 4. Let L be locally k-complex. Suppose that fQ and f 1 are two embeddings of L in E n , n > 2k + 2 such that f0 (L) and f 1 (L) are (weakly) locally-tame. Then there is a p u s h P such that

171

DANCIS

172

COROLLARY 4.1. Suppose that f and g are two embeddings of a 1/r-) compact topological k-manifold-with-boundary M into E , n > 2k + 2, such that f(Mk ) and g(Mk ) are locally flat. Then there is a homeomor­ phism h of E n onto itself such that hf = g and h is the identity off a compact set. We shall present a summary of some of the known results on tame k-complexes in combinatorial n-manifolds n > 2k + 2, in section 2. In section 4 we shall show that a "weak” definition of locallytame and the definition of locally flat are equivalent for k-manifoldswith-boundary, which are embedded in n-manifolds, n > 2k + 2, (Theorem 18), and that the "weak" and "strong” definitions of tame and locallytame are equivalent for k-complexes In n-manifolds, n > 2k + 2, (Theo­ rems 16 and 1 7 ). tion 5.

We shall set the stage for proving Theorems 1, 2, 3, and 4 in sec­ Here, we shall also establish

IEMMA 5. Let f be a m ap of an r-complex K into a combinator­ ial n-manifold M. Let X be the finite union of tame s-complexes in M, r + s < n. If e > o is given, then there is a piecewise linear map g:

K

- M

such that g(K) n x = «f , and

d(g, f) < e . In section 6 we shall establish Theorems i, 2, and 3, and in sec­ tion 7 we shall establish Theorem 4. 2.

Background V. K. A. ’ M. Gugenheim (Theorem 5 [8]) proved:

THEOREM 6. Let K be a finite k-polyhedron whi c h is piecewiselinearly embedded in E n , n > 2k + 2, and let f be a piecewiselinear homeomorphism of K into E n . Then there is a piecewise linear homeomor­ phism h of E n onto itself such that h |K = f. A s Homma observed, (also see p. 506 [1]), Gugenheim's proof of Theorem 6 induces the following, more general theorem: THEOREM 7. Under the hypothesis of Theorem 6 if K* is a subpoly­ hedron of K such that f|K* = ^ , then for any e > o, there is a piecewise linear homeomorphism h of E n onto itself such that h |K = f; d(h, 1) < d(f, 1) + s,

173

APPROXIMATIONS A N D ISOTOPIES IN THE TRIVIAL RANGE and h|En - U £ ( ^ where

f*t (x) = (l-t)x + tf(x),

Notation:

U i f t (K-K*)) =

1 ,

t € [0 , 1 ].

U £ (A) means a n e-neighborhood of the set

A.

Homma uses Theorem 7 to prove the following theorem (see the Corol­ lary to Lemma 2 [9 ]) and his main theorem [9 ]: THEOREM 8 . Let f be a piecewise linear homeomorphism of a finite k-polyhedron K w hich is in the interior of a compact combinatorial nmanifold w i t h boundary M, into Int M such that f is homotopic to the identity and n > 2 k + 2 . Then f can be extended to a piecewise linear homeomorphism h of M onto itself. A special case of Homma's main theorem [9 ] is: THEOREM 9 . Let K be a finite k-complex, topologically embedded in a combinatorial n-manifold M, n > 2 k + 2 . If K is tame and if e > 0 is given, then there is an e-homeomorphism h of M onto itself such that h|K is piecewise linear wi t h respect to K and h|M - U £ (K) = 1. Homms's proofs of Theorems 8 and 9 may be combined to prove: THEOREM 1 0 . Under the hypothesis of Theorem 9, if K* is a subcomplex of K, f is a homeomorphism of K into M such that f(K) is tame, f|K = 1 and there is a homotopy (ft |ft :

K-M,

ft |K*

= 1 , t € [0 , 1 ]},

then there is a homeomorphism h

of

fQ = 1

M onto itself

and

f1 =

f,

3 u c h that

h| K = f and h|M - U_( U f . (K - K * )) = 1 . EX t € I ^ 7 B i n g and Kister (see section 5 [ 1 ]) have sharpened Gugenheim's r e ­ sult (Theorem 7) b y showing that one may obtain an h wh i c h is ambient isotopic to the identity namely: THEOREM 1 1 . Under the hypothesis of Theorem 6 if K* is a sub1 * complex of K such that f |K = 1 , then for any e > d(f, 1 ), there is an e-push P of (K-K ) such that P|K = f, ■ Jf y\ P does not move any point of K , and e a c h point of E is moved along a polygonal pat h of length n o more than e b y the isotopy of the push. Furthermore, if 5 > 0 is given, then one m ay find a n e-push P such

DANCIS that P|En - U .( U f.tK-K*)) = 1 8V t e I '

where

f t (x) = (l-t)x + tf(x),

x € K,

t € I = [0 , i].

Remark. In order to adapt Bin g and Kister's proof for this more general * case, one should reorder the vertices of K, so that if K has r vertices then {v-,..., v„} C K*. Now let f(x, t) = f(x, o) for all x c K , t c [o, 1 ] and omit the first r-parts of h^ on pages 5 0 1 , 50 3 [l], and then section 5 ( 1 ] will prove Theorem 11. Suppose we used Bing and Kister's w o r k (Theorem 1 1 ) instead of Gugenheim's result (Theorem 7 ) in the proofs of Homma's theorems [9] as G luck does in [7 ]. Then we may establish the following two theorems whi c h are similar to some of Gluck's results [ 7 l. THEOREM 12. Suppose K is a k-complex topologically embedded in E n , n > 2 k + 2 , and K* is a subcomplex of K. Let f: K E 11 be an e-homeomorphism such that f|K = 1 . Suppose that K and f(K) are * tame. Then for any e 1 > e, there is an e-j-push P on K - K such that

and

P

P |K = f, * does not move any point of K .

THEOREM 13. Suppose a combinatorial n-manifold Let (ft |ft : K - M ,

K M,

is a k-complex topologically embedded in * n > 2 k + 2 , and K is a subcomplex of K.

f t |K* = 1 ,

t € [0 , 1 ],

fQ = 1 }

be a homotopy. Suppose f = f 1 is a homeomorphism, K and tame and e > 0 is given. Then there is a n ambient isotopy that (i) (ii) (iii) (iv)

1, h,|K = f, h t |M - U e (

U j f t (K-K*)) =

f(K) are (h^) such

1,

h t |K* = 1 .

Gluck [5], [6 ] and Greathouse [7] use many of the results w h i c h are listed here in order to establish: THEOREM 14.

A k-complex, in a combinatorial n-manifold,

n > 2k +

2 , wit h a (strongly)-locally-tame embedding is tame. 3.

Definitions

of

K

A subset of a complex K is a polyhedron in K or a subpolyhedron if it is the point set of some subcomplex of some subdivision of K.

APPROXIMATIONS A N D ISOTOPIES IN THE TRIVIAL RANGE

175

A n embedding f of a complex K into a combinatorial manifoldwith-boundary M is (s trongly) tame if there is a homeomorphism h of M onto itself such that h«f is p i e c ewiselinear. A subset K of a combinatorial manifold-with-boundary M is (w e a k l y ) tame if there is a homeomorphism h of M onto itself, such that h(K) is a polyhedron in M. W e shall say that a complex embedded in a combinatorial manifoldwith-boundary is tame If it has a (strongly) tame embedding. Let f be an embedding of a complex K Into an n-manifold-withboundary M and let A be an open subset of K. We say that f|A is (strongly) locally-tame if for each point x € A, f(x) has a compact neighborhood U x in M and if there is a homeomorphism h^ sending Ux onto In such that is pie cewise l i n e a r . A subset U of a topological n-manifold-with-boundary M is (w e a k l y ) locally-tame t if eac h point x e U has a compact neighborhood Nx and if there is a homeomorphism sending Nx onto In such that h ^ U n Nx ) is a subpolyhedron of In . We note that the "strong" definitions used here are equivalent to the definitions used in the preceding paper [4], A q-manifold-with-boundary M 1 , topologically embedded in a top­ ological manifold M, is locally flat in M if each point x € M 1 has a neighborhood U in M such that the pair (U, U n M 1) is homeomorphic to (En , E q ) or (En , E^) where E ^ is a q-dimensional ho If-plane in En . A compact set L is locally a k-complex if each point x €L has a compact neighborhood w h i c h is homeomorphic to a k-complex. We note that every compact, topological k-manifold, every finite k-complex and every compact, (weakly)-locally-tame set of dimension k is locally a k-complex. A n ambient isotopy of a that if we set

space M

h t (x) then

hj.

=

is a homeomorphism of

M

is a m ap

H: M x I -► M, such

H(x, t), onto itself, for each

t € I, and

h 0 . i. We say that

ha

is "(ambient) isotopic" to

A n ambient e-isotopy of the additional condition: d ( h t (x), x)

M

h^, for a, b € I.

is an ambient isotopy wh i c h satisfies

< e, for each

x € M,

A push is an ambient isotopy of a space M compact proper subset A' of M

t €I . such that for

sane

DANCIS h t (x) = x, and h Q = i . A n e-pu s h that

for a ll P

of A

x €M - A 1

1 . We say that

t € I.

is an ambient e-isotopy of

h t (x) = x, and h Q =

and

wh e n

d(x, A) > e,

P "pushes" or "moves" P(x) =

M such

t €I

A onto

h^ (A),

and

( x).

Also, "P does not move any point of a set for all x e B and t e l .

B"

means that

b^(x) = x,

We define two rules of "composition" of pushes as follows. Sup­ pose P, P ’ and P" are three pushes w h i c h represent the isotopies (ht , t e l ) , (h|) and th'^}, respectively, on a space M, and h and h*

are nomeomorphisms of M onto itself, then * * (i) P 1 = h Ph means h£ = h h^h, t e l , and (ii)

Notation:

P" =

(i)

P'-P

means h" = t

r k 2t> ^ € 2 ], \ ,, . * * r1 n ^ h 2t-i h i ' t e [ 2 , 1 ].

U p (S)^= (x, d(x, S) < r)

(ii)

Let

f, g:

X — Y, then

d(f, g) = lub x e X

(iii)

pwl

means piecewise linear.

d ( f ( x ) , g(x)).

Manifolds are manifolds-without-boundary, and all complexes are finite. k.

Equivalence of definitions

among

In this section we shall establish several equivalence relations the various types of "nice" embeddings in the "trivial range." We b egin by stating

a very useful lemma.

L EMMA 15. Let f be an embedding of a k-complex K into a topo­ logical n-manifold M, n > 2k + 2 . Suppose U is an open subset of K such that f(U) is contained in some (weakly) locally-tame set Y, dim Y < k. Then f |U is (strongly) locally-tame. Furthermore, if K = U and M has a combinatorial triangulation, then f is (strongly) tame Lemma 1 5 is a special case of Corollarys 2 and 5 of Theorem 1 0 [U] also it is essentially a corollary of Bryant's ma i n theorem [5 ]. We now show the equivalence of the "strong" and "weak" definitions in the "trivial range" o£ dimensions.

APPROXIMATIONS AN D ISOTOPIES IN THE TRIVIAL RANGE

177

THEOREM 1 6 . Let K be a k-complex, M a combinatorial n-ma n i ­ fold, n > 2 k + 2 and f an embedding of K into M. Then f is (strongly) tame iff f(K) is (weakly) tame. THEOREM 1 7 . Let f be a n embedding of a k-complex K into a topological n-manifold M, n > 2 k + 2 . Let U be an open subset of K. Then f|U is (strongly) locally-tame (with respect to any triangulation of K ) , iff f(U) is a (weakly)-locally-tame set. Proof of Theorems 16 and i 7 . (i) The "strong” definitions always imply the "weak" definitions, (ii) That the "weak" definitions imply the "strong" definitions follows directly from Lemma 1 5 . Theorem 17 generalizes a result of Gluck (Theorem 6.1 of [5] or Theorem 11.1 of [6]) wh i c h says that if an embedding of a k-complex K into a combinatorial n-manifold, n > 2k + 2, is (strongly) locally-tame w i t h respect to one triangulation of K then it is (strongly) locallytame wit h respect to any triangulation of K. Theorems 17 and 14 have the following corollary: COROLLARY 1 7 .1 . (Weakly) locally-tame k-complexes in combinatorial n-manifolds, n > 2k + 2, are tame. The remainder of this section is about some relationships between local flatness and local tameness. Let f be an embedding of a combinatorial q-manifold-with-boundary into a topological n-manifold M 11. Greathouse [7 ] has used Morton Brown's bicollaring theorem [2] to show that f is (strongly) locallytame if f (M^-) is locally flat and dM^ = 0 (for all q < n) . One m ay combine Greathouse's method wi t h an (unpublished) technique of R. H. Bing (for proving that locally-flat q-cells are flat in E n , q < n) in order to show that Greathouse's result is true without the condition dM^- = 0 , i.e., for embeddings of combinatorial manifolds-with-boundary, locally flat implies (strongly) locally-tame. vP-

THEOREM 18. A topological k-manifold-with-boundary M k , embedded in a topological n-manifold M n , n > 2 k + 2, is locally flat iff it is (weakly) locally-tame. Proof:

(i) (ii)

The definitions imply that locally flat always implies (weakly) locally-tame. (Weakly) locally-tame = > locally flat:

The definition of (weakly) locally-tame says that each point x € M k has a compact neighborhood Nx in Mn and that there is a homeomor­ phism h- of Nx onto In such that h 1 (Nx n M k ) is a subpolyhedron Q of I (with dim Q = k) . The definition of manifold-with-boundary

DANCIS

178

jr tells us that the point x has a compact neighborhood B that B k C Int h " 1 (Q) and there is a homeomorphism h 2 of k-cell Ik .

in M such B k onto a

Therefore we m a y now use Lemma 1 5 to show that h 1 h ~ 1 : is (strongly) tame. Therefore there is a homeomorphism h^ of onto itself such that hjl^hg1 :

Ik

-

Ik -» In Int In

Int In

is piecewise linear. We ma y now use Theorem 6 in order to find another homeomorphism h^ of Int In onto itself such that h^h^hj h ^ 1 (Ik ) h^h^hj (Bk ) is a canonical k-cell Ik in Int In . Thus, Theorem 1 8 is established.

5.

The set-up

In this section, we will establish a "general position" lemma (Corollary 1 9 .1 ) and a "canonical representation" for compact sets which are locally complexes. These results and Lemma 15 are the basis of the proofs of Theorems 1 , 2 , 3 and k. LEMMA 1 9 . Let: (i) K be a (k+1)-complex; (ii) K* and K" be subcomplexes of K; (iii) K' U K" = K; (iv) f: K -►E n , n > 2k + 2 be a map; (v) f | K ! be a homeomorphism; (vi)

X be a subset of E n w hich is contained in the finite union of tame k-complexes or a (weakly) locally-tame set of dimension < k; (vii) f ( K ’ ) n X = 0 ; (viii) d ( f ( K ’n K"), X) > 0, and (ix) e > 0 be given. Then there is a m ap

g:

K -+ E n

such that

(x) g|K* = f |K*; (xi) d(g, f) < e; (xii) g(K) n 0 and (xiii) d(g(K"), X) > 0.

x=

Note:

(vii), (x), (xiii) and (iii) imply (xii).

COROLLARY 1 9 .1 . If K* is a k-complex and f | K T is a piecewiselinear homeomorphism in Lemma 1 9 , then we m ay find a g wh i c h is a piecewise-linear, general position map and satisfies (x)-(xiii) above.

APPROXIMATIONS A N D ISOTOPIES IN THE TRIVIAL RANGE

179

Proof of Corollary 1 9 .1 : We obtain the desired m ap by taking a general-position, piecewise-linear approximation of the map obtained from Lemma 1 9 wh i c h agrees w i t h f on K * . Proof of Lemma 1 9 * First, we note that every (weakly) locallytame set is contained in a finite union of tame complexes and hence X is contained in the union of m tame k-complexes, for some integer m. This proof wil l b e a n induction on " m " . Case 1.

m s i, i.e., X is contained in a tame k-complex. Step 1. Send the tame k-complex (which contains X) onto a polyhedron by a space homeomorphism h. Step 2. There is a 5 > 0 such that if p, q € h [ U 1 (f(K))] and d(p, q) < 5, then d ( h “1 (p), h _ 1 (q)) < e. Step 3. Now we can find a m ap g 1 : K -* E n such that g 1 is piecewise linear outside of a small neighborhood of K 1, d(g'(K"), h(X ) ) > 0 , d ( h * f , g«) < 5 and g« |K = h*f|K'. Step 4. The desired m ap g is h ^ g * .

Case 2 . Suppose Lemma 1 9 is true whenever X is contained in the union of (m-1) tame k-complexes, and now we will use this supposition to prove that the lemma is true whe n X is contained in the union of m tame k-complexes. Step 1. W e m a y decompose X into two sets X 1 and X 2 , where X 1 is contained in a tame k-complex and X 2 is con­ tained in the union of (m-1) tame k-complexes. Step 2 .W e use Case 1 on X 1 in order to obtain a map K

E

such that g,|K'

-

f|K\

and d ( g , , f) < e/2. Step 3. Let e, Step 4. We note on X 2

= d(g, (K"), X,) . that our supposition enables us to use Lemma 1 9 in order to obtain a map g2 : K -► d ( g 2 (K"), X 2 ) >

0

and d(g2 , g ^ Step 5.

< min(e/2, e,) .

A little checking shows that (x)-(xiii). This completes the proof of Lemma 1 9 .

g 2 satisfies conditions

R e m a r k . The proof of Lemma 5 is essentially the same as the proofs of Lemma 1 9 and Corollary 1 9 . 1 for the case K 1 = j

Int

K

711

.

namely:

Lj = Closure (Lj_1 - Nj) U Kj . A brief checking of the above equation and inequalities (and F i g ­ ure 1) shows that Conditions (4),(5) and (8) are satisfied. We have shown that all the conditions of a canonical representa­ tion are satisfied and therefore Lemma 20 is established.

6.

The app r o x i m a t i o n s .

The b u l k of this section is devoted to the Proof of Theorem 3. The compact set L has a canonical representation (see Lemma 2 0 ) such that Kj (j = 1 ,2.,..., r) is a k-complex and g(Kj) ^as a euclidean neighborhood E*? in M. Let s 1 = m i n {e, d[g(Kj),

M - E j ],

(j = 1, 2, ..., r ) }.

K^)

DANCIS

182

The desired approximation will be constructed b y induction, n a m e ­ ly; we will find a series of homeomor p h i s m s : (h,,..., V

hj :

Lj

-M)

such that (i)

is contained in a (weakly) locally-tame set of dimension < k, and d ( hy g|Lj) < e 1 / 5 r ">* .

(ii) Mote:

d(hj_,(Kj), M-E*) >

e, - e 1 /5r _ J"'.

Let ^ = L 1 -*• E ^1 C M be a piecewise-linear, homeomorphic e 1 /3r _ 1 -approximation of g|Kj . (Here h 1 is piecewise linear w i t h r e ­ spect to K 1 and E^, and d ( ^ , g | K ^ < e |/3r _ 1 w i t h respect to the metric on M.) Assume that h ^ : L j _ 1 -*• M is known (and satisfies the two con­ ditions listed above). We note that h ^ _ 1 (Kj) is contained in a (weakly) locally-tame subset of and therefore we may use Lemma i 5 to show that h j _ 1 (Kj) is tame in E*J. We now use Homma's result (Theorem 9 ) in order to obtain a homeomorphism of E ^ onto itself and hence a homeomorphism h of M onto itself such that h*h j _ 1 |Kj:

Kj

E^

C M

is piecewise linear

and d(h, 1 )
0

Therefore we may use Corollary 1 9 .1 , where

Kj,

. Kj, hj,

h * h j _ 1 (Lj- Kj) and of this proof correspond to K, K ’ , f, X and e, respectively, of Corollary 1 9 .1 , in order to obtain a piecewise linear homeomorphism h j':

kj

-

ej

APPROXIMATIONS A N D ISOTOPIES IN THE TRIVIAL RANGE

183

such that

d lhj'(K p , h -h j_ , ( L j - K j ) ) > 0;

hJ|Kj

=

h-hj.jKj

,

and

d ( h j , h^) < e 1 /5i' ' j_1 Now we m a y define

h j : Lj hj(x)

-*•

. M

as:

h - h j ^ C x ) , - x e Lj - Kj I h" hj(x),

x € Kj

.

The reader may easily verify that hj is a well defined homeomor­ phism w h i c h satisfies Conditions (i) and (ii). This completes the induction. We note that

d(g, hjJ
0 . We observe that g m a y represent a homotopy connecting k^lKj and Pj hfj,-] IKj w h i c h moves no point of hf (Kj) . Therefore, we m a y apply Theorem 1 3 in order to obtain a pu s h Pj such that

( s.

of set

DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN SPACE Remarks:

(i)

(2 )

(Rn )o set a 1 1 ordered triples of distinct points i n Rn .(Rn )^— 7— is the set of all ordered triples of v 5 /2 points in Rn w h i c h form the vertices of a n equilateral t ria n g l e . If 5 < 5 », then (Rn )^ D (Rn )^, .

(3 )

(Rn )g is closed in (rI1)q because a is a continuous function on (Rn )o, but it is not closed in (Rn )5 .

(k )

For any value of 5, 0 < 5 < 3 / 2 , the closure of (Rn ) ^ in (Rn ) 5 will contain the diagonal a = { (x, x, x) : x € Rn ) . The closure of (Rn )g will be precisely (Rn )gU 5 > >/2 / 2 . But whe n 0 < 5 < >/2 / 2 , the closure A when of (Rn )g will be (Rn )g U a *, where A* = { (u, v, w) € (Rn ) ^ : u = v or v = w or w = u).

(5)

F ix 5 > 0 . If (u, v, w) € (Rn )g and each edge of the triangle uvw makes an angle < e w i t h a plane P , then the plane P of the triangle makes an angle a w i t h P* subject to the bound sin a < £ i n J L 5 Thus as

6.

205

e -+ 0,

so does

a

. 0 .

The distribution of triangles of ffood shape on a two-dimensional C° manifold

LEMMA 6 .1 . Let M be a two-dimensional C° manifold in Rn , x a point of M and W a neighborhood of x Q in M. Then there is a neighborhood U of x Q in M such that if u and v are distinct points of U, there is a thrid point w € W such that the largest angle of the triangle uvw is < 9 0 0 , i.e., such that a(uvw) > s/2 / 2 . Let M, x Q and W be given as above. essary, we may assume that W is connected. of x Q in M subject to the conditions: (i) (ii)

Making W smaller if n e c ­ Let U be any neighborhood

U C W diam U < diam W.

Now let u and v be any two distinct points selected from U. We will construct the set A C R n of all points w, distinct from u and v, w hi c h together w i t h u and v form a triangle uvw whose largest angle is < 90°. To prove the theorem, we will simply show that the original neighborhood W must intersect the set A. n al to

Draw through the point u the n - 1 dimensional hyperplane orthogo­ the line uv, as shown below. The third vertex w must lie on or

206

GLUCK

"below" this hyperplane (see figure) if the triangle uvw is to have an angle diam U > d(u, v ) , while the diameter of B U {u, v) is just d(u, v ) . W cannot be entirely contained in C U D U {u, v ) , because u cannot be connected to v in C U D U (u, v ) , whereas it can in W. So, under the assumption that W n A = 0, we have W contained entirely in B U C U D U (u, v ) , partially in B, partially in C U D and (of course) partially in (u, v) . But then removal of either u or v from W disconnects W, whereas this is patently impossible in a two-dimension­ al manifold. So W must meet A, and therefore there is a point w c W such that the largest angle of the triangle uvw is < 9 0 °, proving the lemma. R e m a r k : The above lemma is best possible, i.e., if we substitute for 9 0 0 any smaller angle, the lemma becomes false. Let (M)^ = M x M x M denote the set of ordered triples of points in the two-manifold M. Let (M)* = (M)^ n (Rn ) \ denote the set of oro 0 dered triples (u, v, w) of distinct points of M such that o(uvw) > 5. Let a denote the diagonal of (M)^. COROLLARY TO LEMMA 6.1. Let M be a two-dimensional C° manifold in Rn and 5 a real number satisfying 0 < 5 < 2 / 2 . Then the clo­ sure of (M)g contains the diagonal A of (M ) ^ .

DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN SPACE 7.

Characterization of two-dimensional C 1 manifolds in

Let secant map

M

be a two-dimensional C 1 manifold in Z:

207

Rn

Rn , and consider the

( M ) 3 - O n>2

w hich associates w i t h the triple (u, v, w) of distinct points in M the plane P Q (u, v, w) through the origin in R n and parallel to the plane P(u, v, w) through the three points u, v and w. The map Z is cer­ tainly continuous but, as we saw in section 5, there is no hope of e x ­ tending it to a continuous ma p over (M)^ U A. But we can now get around this predicament.' Pick any real number 5 subject to the condition o 2

So we have established

for each real number 5, 0 < 5 < ^ 3 / 2 , the secant map £: (M) o' -► OLn, nd has a continuous extension over (M)g U a by defining Z(x, x, x) = P Q (x), the plane through the origin parallel to the tangent plane to M

at

x.

This is the generalization of item (3) of section 3. We are ready now to give the characterization of two-dimensional C 1 manifolds in R n . THEOREM 7.1 . Let M be a two-dimensional C° manifold in Rn and 5 a real number, 0 < 5 < V 2 / 2 . Then M is a C 1 manifold if and only if the secant m ap Z: (M)^ G 0 admits a continuous extension over x O ii, c. m l o u a.

208

GLUCK

We have just observed, in item (4) above, the necessity of the condition under the more liberal bounds: o < 5 < >/~3/2. To prove sufficiency, let M be a two-dimensional C° manifold in R n whose secant map admits a continuous extension over (M)* U a, 0 < 5 1 O < V2/2. We will show that M is a C manifold by showing that M sat­ isfies conditions ( 1 ), ( 2 ) and (3) of Theorem 4.1, and then invoking that theorem. For each x € M, let P(x) be the plane in Rn passing through x and parallel to the plane E(x, x, x) . To show that P(x) is the tan­ gent plane to M at x, we must show that for y € M - (x) sufficiently close to x, the vector x-y makes a small angle wit h P ( x ) . To do this, use the continuity of Z on (M)g U a to choose a neighborhood W of x such that if u, v and w are three distinct points in W and (u, v, w) € (M)g, then Z(u, v, w) makes a small angle wi t h Z(x, x, x ) . Now use Lemma 6.1 to select a smaller neighborhood U of x such that if u and v are distinct points in U, there will exist a thrid point w € W such that cr(uvw) > si 2.I2 > 5 , and hence (u, v, w) € (M)g. Finally let u = x and v = y. Then there is a third point z € W such that (x, y, z) € (M)g. Hence E(x, y, z) Makes a small angle with E(x, x, x) and hence wi t h P(x). But then the vector x-y, being parallel to an edge of the triangle xyz, makes at least as small an angle w i t h P(x) . So P(x) is indeed a tangent plane to M at x. This verifies condi­ tion (1) of Theorem 4.1. Condition (2 ), the fact that P~(x) = E(x, x, x) varies continu* ously w i t h x, follows directly because Z is continuous on (M)g U A, and hence surely on a. Condition (3) follows in the same fashion as did condition ( 1 ). Given x c M, pic k a neighborhood W of x in M so small that if u, v and w are three points in W wit h (u, v, w) € (M)g, then £(u, v, w) makes an angle s w i t h E(x, x, x) such that

5

Following the pattern of the argument for condition (1), we use Lemma 6.1 to get a smaller neighborhood U of x such that if u and v are distinct points in U, there will exist a thrid point w € W such that (u, v, w) € (M)g. Then b y Remark (5) on page , the plane £(u, v, w ) , and hence surely the vector u-y, makes an angle < 9 0 °w i t h the plane £(x, x, x) . This implies that the orthogonal projection vx of Rn on­ to the tangent plane P(x) is one-one on U, verifying condition (3). Theorem 4.1 is now applicable, and completes the proof of the theorem.

DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN SPACE REFERENCES [i]

L. V. Toralballa, Surface Area (to appear).

HARVARD UNIVERSITY

209

WHITEHEAD TORSION AND h-COBORDISM by R. H. Szczarba* Introduction In this lecture, we will be concerned wi t h the properties of hcobordisms, particularly the relationship between h-cobordisms and W h i t e ­ head torsion. Most of what we say here holds equally well in either the differentiable or piecewise linear category so we will use "manifold" to mean either differentiable or piecewise linear manifold. Two manifolds will be termed isomorphic if they are either diffeomorphic or piecewise linearly homeomorphic and we write M « M*. Finally, all manifolds will be compact, oriented unless explicitly stated otherwise, dM will denote the boundary of M, and -M will denote M with its opposite orientation. The material here is divided into three sections. The first r e ­ calls the h-cobordism theorem of Smale and its generalizations, the se­ cond deals wit h the Whitehead group and the notion of torsion, and the last contains some properties of h-cobordisms.

1.

The h-cobordism and s-corbordism theorems.

A triad of manifolds (W; M, M ’ ) is an h-cobordism if dW is the disjoint union M U (— M 1) and eac h of the inclusions M C W, M 1 C W is a homotopy equivalence. (It will sometimes be convenient to refer to W as an h-cobordism.) We say M and M ’are h-cobordant if there is an h-cobordism (W; M, M'). The fundamental result of Smale [8] is the fol­ lowing (see also Milnor [7]). "h-cobordism" T h e o r e m .

Let

(W; M, M') be a differentiable h-cobordism

w i t h 7T1W = 0 and dim W > 6. Then W is diffeomorphic to [0, 1]. In particular, M is diffeomorphic to M 1.

M x I,

I =

To demonstrate the strength of this result, let Z be a differen­ tiable homotopy n-sphere and W = £ - (D1U D 2 ) -where D 1 and D 2 are disjoint n-discs in Z. It is easily checked that W is an h-cobordism t The author was partially supported b y NSF- G P - 1i-037. 211

SZCZARBA

212

so, if n > 6, W is diffeomorphic to Sn_1 x I, Sn “] = dDj . Z (Sn_1 x I) U D 1 U D 2 wh i c h is clearly homeomorphic to Sn have proved the Poincare conjecture in dimensions > 6.

Thus, and we

If the hypothesis ^ W = 0 is omitted from the h-cobordism theo­ rem, it no longer is true. In order to deal with the non simply connect­ ed case, it is necessary to define the Whitehead group Wh(-ir) of a d i s ­ crete group ir » and to associate w i t h any h-cobordism (W; M, M ’ ) a tor­ sion element t(W, M) in Wh(ir 1W) . We will call an h-cobordism (W;M,M') an s-cobordism if t(W, M) = 0 . The following is due to Mazur [5 ], Barden [1], and Stallings [9 ]. "s-cobordism” T h e o r e m . Let (W; M, M 1) 6. Then W is isomorphic to M x I. R e m a r k . Since Wh(7r) = 0 if ir = 0, alize the h-cobordism theorem.

be an s-cobordism wi t h

dim W >

the s-cobordism theorem does gener­

We close this section with a strengthened form of the s-cobordism theorem due to Stallings [9 ]. A n application will be given in Section 3. THEOREM. Let M be a manifold of dimension > 5 and a any e l e ­ ment of Wh(7T.jM) . Then there exists an h-cobordism (Wff; M, M a ) unique up to isomorphism such that T (Wa , M) = j^a where j^: i^M ffjW^ is induced b y j : M C W fl.

2.

The Whitehead group and the notion of torsion

In this section, we define the Whitehead group of a rin g and of a discrete group and outline the procedure for defining the torsion of a homotopy equivalence and of an h-cobordism. Two general references for the material in this section are Whitehead [ 1 0 ] and the excellent set of notes of Milnor [6]. Let R be an associative ring w i t h unit having the property that any two bases for a finitely generated from R-module have the same n u m ­ ber of elements. Let GL(n, R) be the group of Invertible n x n Rmatrices and GL(R) = Un ^ 1 GL(n, R) . We do not distinguish between an element of GL(n, R) and”the element of GL(R) determined by it. Let M be the subgroup of GL(R) generated by those matrices w h i c h are the identity except for a single non-zero off diagonal entry together wit h the single 1 x 1 matrix (- 1 ). The reduced Whitehead group of R, K ] (R), is defined to be the quotient GL(R)/M. According to Whitehead [ 1 0 ], K 1(R) is an Abelian group. Let

M'

Let ir be a discrete group and R = ZC 7r] be the subgroup of GL(R) generated by

its integral group ring. M and the 1 x 1 m a t r i ­

WH I TEHEAD TORSION A N D h -COBORDISM

213

ces (g), g € tt. The Whitehead group of 7r, Wh(7r), is defined to be the quotient GL(R)/M*. Clearly Wh(7r) is a homomorphic image of K^ZtTT]). The following facts are known about Wh(Tr). (i) Wh(7r) = 0 if 7r « Z 2 , Z^, Z^, or Z (see Higman [3]) and Wh(Z^) ~ Z. Here Zm = Z/mZ. (ii)

If tr if finite Abelian, Wh(7r) is a finitely generated free Abelian group. (See Bass [2].)

We now define the torsion of an h-cobordism in four steps. Step 1. Let C be an R-complex w i t h the property that each C^ is a finitely generated free R-module wi t h a prescribed R-basis and C^ = o for q > N, some N. Such a complex will be called a based R-complex. If C is a based, acyclic R-complex, the torsion of C, t (C) e K 1 (R) is uniquely determined by the following two properties. a)

If Cq = 0 for q ^ n+1, n, then t (C) is the element of K 1 (R) determined by the matrix defining the isomorphism d: Cn+1 — Cn (with a sign (-1)n ).

b)

If 0 -► C 1 C C" 0 is a short exact sequence of based acyclic R-complexes w i t h the obvious compatibiltiy condition on bases, then t (C) = t ( C ’ ) + t (Cm) .

Step 2. If cp: C -► C* is a chain equivalence between based R-complexes, then the algebraic ma p p i n g cone of cp, C^, is the based, acyclic Rcomplex defined b y (Cq,)q = C ^ ^ © C 1 , C‘ ) = (dC, cpC - d ' C ’ ). We define the torsion of

cp b y

x(cp) =

t

(C^) .

Step 3. Suppose X and Y are finite simplicial complexes and f: X -*■Y a simplicial homotopy equivalence. If X and Y are the universal covering spaces and f : X -*• Y the homotopy equivalence induced by f, then f^: C(X) -*• C(Y) is a chain equivalence of Z [ir1Y ] - m o d u l e s . (Here C(Y) becomes a Z [it 1Y ]-module by identifying tt^Y wi t h the group^ of covering transformations cn Y and C(X) b y using the isomorphism f m .: tt.X -► tt Y.) Now if we choose for each q-simplex a of Y a qtr • ^ ' rs j /v simplex a of Y lying over it, we obtain a Z[7r1Y]-base for C(Y) . D o i n g the same for C(X), f^ becomes a chain equivalence of based com­ plexes so we can define T (f#) € KjCZtu^Y]). We define the torsion of f, t (f ), to be the element of W h ^ Y ) determined b y x(f^) . It is easily checked that x(f) does not depend on the choice of the simplices o'. and

Y

If T(f) = o, we say that f is a simple equivalence and that have the same simple homotopy t y p e .

X

21k

SZCZARBA

Rema r k 1 . The geometric m e aning of simple homotopy type is as follows. Let a be a simplex of X w i t h exactly one free face. The operation of passing from X to X - interior a is called an elementary contrac­ tion and the inverse operation an elementary ex p a n s i o n . Whitehead proves in [ 1 0 ] that X and Y have the same simple homotopy type if and only if Y can be obtained from X b y a sequence of elementary expansions and contractions. Remark 2 . It is known that Y. However, the topological question.

x(f) is unchanged if we subdivide X and invariance of T(f) is an important open

Step k. Finally, if (W; M, M 1) is an h-cobordism, we choose triangula­ tions such that the inclusion map j : M W is simplicial and define (If W is differentiable, we choose the t (W, M) = t (j ) in W h ( it, W) . essentially unique differentiable triangulation.)

3.

Some further properties of h-cobordisms W e beg i n this section wit h a result of Stallings [9 !. THEOREM.

Then

Suppose

(W; M, M ’ )

is an h-cobordism with

d im W > 6.

W - M 1 * M x [0 , 1).

A n easy consequence is the following. COROLLARY. W - ( M U M 1) = » M x

For any h-cobordism (W; M, M !) (0, 1) and M x (0, 1 ) « M f x

with

dim W > 6, (0, 1 ).

To prove the theorem above, let t = t(W, M) and use Stallings' theorem to find an h-cobordism (W1, M 1, M 1) w i t h t(Wj, M 1) =- t . Now we construct a new h-cobordism (W U^, W 1, M, M 1) b y attaching W ] to W a long M ’. It is easily shown that t

(W Um , W 1# M) =

t

=

T

so W U^, W 1 « M x I W * M» x I.

and

M 1 =» M.

(W, M) + -

T

=

t

(W,, M»)

0

In the same way we show that

Now consider the infinite union V = W UM, V,

UM w UM, V,

u •••

.

On the one hand, V

- W UM, (W,

UM W) UM, (W,

UM W) u

- W UM , (M'x I) Um ,(M' x I) U ••• - W UM , (M'x [0, 1)) ~ W - M 1 .

...

W1

215

WHITEHEAD TORSION A M D h-CORBORDISM On the other hand, V = (W UMIW , ) UM (W UjyjjW,) U ... «

M

«

M x [0, 1)

x

IU

m

M

x

I U * “

and the theorem is proved. lowing.

Another result in the same spirit as the theorem above is the f o l ­ (See Kwun and Szczarba O ] . )

THEOREM. Let S 1 the circle. Then

(W; M, M')

be an h-cobordism with

W x S 1 « M x S 1 xI In particular,

dim W > 5

and

.

M x S 1 =» M'x S 1 .

Remark 1. One might conjecture that for any h-cobordism (W; M, M 1), W is topologically homeomorphic to M x I. Note that a non-trivial hcobordism (W; M, M') w i t h W homeomorphic to M x I would provide a counterexample to the hauptvermutung for manifolds and to the topologi­ cal invariance of torsion. Re mark 2. The question raised in Remark 1 is a special case of the fo l ­ lowing. Suppose two manifolds have isomorphic interiors and isomorphic boundaries. Are they homeomorphic? Notice that two manifolds may have isomorphic interiors and isomorphic boundaries and not be.isomorphic. For example, if (W; M, M') is a non-trivial h-cobordism wit h dim W odd and ir^W finite Abelian, W UM ,W can be shown to be a non-trivial h-cobordism (using Milnor's duality theorem, the fact that "conjugation” in Wh(7r) is trivial for it finite Abelian, and the fact that Wh(ir) is torsion free for ir finite Abelian (see Milnor [6] and Bass [2])). Thus W UM ,W and M x I not be Isomorphic.

have isomorphic interiors and boundaries but can­

216

SZCZARBA

REFERENCES [ 1 ] D. D. Barden, Barden, "The "Thes tr s tr u cutu c tu r er e o fo fmm a na ifo n ifo ldld s ,"s ,"T h T ehseis s is , , Cambridge CambridgeU U n iv n iv er­ er­ s i t y , 1963. [2]

H. H. B a ss,B a"The ss, "The s ta b le s tasbtr leu cstu trruec tu o fr eq uoite f q ugite e n e ra g eln elin ra el alin r gero a ru pg s," ro u p s," B u ll . Amer. Math. S o c ., 70 (1961+) pp. 1+29-1+33. 1+29-103.

[3]

G. G. Higman, Higman, "The "The u n it su no it f sgroup o f group r i n g sr,i"n gPsro , " c. P ro London c. London Math.Math. S o c ., S o c ., 1+6 (19^0) pp. 231-21+8.

[i+] [ 1+]

K. W. Kwun and R. H. S zcza rb a , "Product and sum theorems f o r W hite­ head t o r s io n ," Ann. o f M ath ., 82 (1965) pp. 183-190. 183- 190 .

[5]

B. Mazur, Mazur,"R"R egular egular neighborhoods neighborhoodsand andthethetheorems theoremso f oSm f Sm ale," ale," Ann. o f M ath., 77 ( 1963) 2 3 2 - 21+9 . 1963) pp. 232-21+9.

[[6] 6 ] J . M iln or, M"Whitehead iln or, "Whitehead t o r s i o n t,"o r smimeographed io n ," mimeographed n o te s , nP orin te cs eto , Pnrin c eto n U n iv e r s it y , 196*+. [73

J . M iln or,

h -cobordism ,

P rin c e to n U n iv e r s it y P r e s s , P1965. ress,

[[8] 8]

S. Smale, "On s tr u c tu r e o f m a n ifo ld s ," Amer. J .

1965.

M ath., 81+ (1962)

pp. 387 - 399 . [9]

J . S t a ll l l i n g s , "On i n f i n i t e p ro c e ss e s le a d in g to d i f f e r e n t i a b i l i t y in the complement o f a p o in t ," D i f f e r e n t i a l and C o m b in ato rial To­ p o lo g y , (to a p p e a r), P rin c e to n U n iv e r s it y P r e s s .

[ 10 ] J . H. C. W hitehead, "Sim ple homotopy t y p e s ," Amer. J . M ath ., 72 ( 1950) pp. 1- 5 7 .

A D DITIONAL QUESTIONS ON N-MANIFOLDS (Richard Goodrich) It is known that if K is a knot in E^, then there exists an arc A whose intersection w i t h K is just its e n d ­ points and the resulting pair of simple closed curves are unknotted [i]. Can the same thing be done in higher dimensions?, i.e., let 2 4 S be a piecewise linear, locally flat embedding of a 2-sphere in E . 2 14Does there exist a disk, D , piecewise linear, locally flat in E 2 2 2 w ith D n S = Bd D such that the resulting two spheres bound 3cells? Is the locally flat condition necessary? Reference 1. S. Kinoshita and M. Terasaka, "On unions of knots," Osaka Math. J., 9, pp. 131-153. (Burt Casler) Is the link of every 1-simplex in a triangulated 5m anifold simply connected? (Burt Casler) Is there a line spanning E^ so wild that no 3-sphere links the line? (Burt Casler) Does there exist a line in Y? so wild that every 5ball containing an interval of the line has a boundary w hich inter­ sects the line in a Cantor set w h i c h is wild wit h respect to the boundary of the ball? (Burt Casler) If the open star of a 1-simplex in a triangulated 5-manifold is homeomorphic to E 5 , will the image of the open 1simplex (under this homeomorphism) be tame in E^? (A. C. Connor) Let C be a 4 -cell and S its boundary. Suppose that h is a crumpled cube in S such that S-h is a real cell. If you sew two copies of C together along h, will the result be a 14--cell? (A. C. Connor) tial?

Is every monotone m apping of

(Herman Gluck) S 1 x Sn _ 1 , n on S 1 x Sn “1

Consider the manifold S 1 x D11 and its boundary > 3. Conjecture: every locally flat (n-1)-sphere bounds a locally flat n-cell in S 1 x D 11.

217

Sn

onto itself e ssen­

218

SZCZARBA.

10.

(Mike Yohe) Bing has described wild 2-spheres whi c h are the fixed point sets of involutions of S^. Are there wild 3-spheres in S^ which are the fixed point sets of involutions of s S

11.

(R. H. Bing) Let A be a n arc in E 5 and E 5 /A be the decomposition of E 5 whose only non-degenerate element is A. Could E 1 x A be the fixed point set of an involution of E^ = E 1 x (E5 /A)?It can if A is either tame or symmetric wi t h respect to some point.

COMPLETELY REGULAR MAPPINGS, FIBER SPACES, THE W E A K BUNDLE PROPERTIES, AND THE GENERALIZED SLICING STRUCTURE PROPERTIES by Louis F. McAuley* Introduction The various concepts of a fiber space are generalizations of a C ar­ tesian product B x F where each of B and F is a topological space. Indeed, products are trivial fiber spaces. The projection mapping p from T = B x F to B i s a fiber m a pping w i t h fiber F and base space B. In some cases, G is a topological group. However, G is isomor­ phic to a group of homeomorphisms of F onto F where F is the fiber and G operates transitively on F. The reader m a y consult Steenrod [ 8 ], Hu [4], and F a d e 11 [2 , 3 ] among others for a variety of definitions of fiber spaces and their relationships. It appears that these concepts are aimed towards the development of the 3 0 -called exact sequence of a fiber space ( or fibering ). This sequence is one of the fundamental results in fiber space theory and it is clearly a very powerful tool. Some definitions of a fiber m apping imply that p: T B is an open mapping, i.e., p(U) is open relative to p(T) for each open set U in A, while others require reasonable conditions on B and T in order that p be open, e.g., consult the papers [2 , 3 ] b y Fadell. Although fibers need not be homeomorphic under the most general concept of a fiber space, this is clearly the case for all fiber bundles and locally trivial fiber spaces where the base space B is connected. We think that the following conjecture is plausible: CONJECTURE. Suppose that (T, B, p) is a fiber space such that p has the polyhedral covering homotopy property (PCHP) [cf. k ) . F urther­ more, ea c h of T and B is a finite dimensional Peano continuum (local­ ly connected compact metric contin u u m ) . If each, fiber over a point is *

The research for this paper was supported in part by NSF GP-4571 219

220

MCAULEY

homeomorphic to a fixed Peano continuum erty (local t r i v iality).

F, then

p

has the bundle p r o p ­

In this paper, we are primarily concerned w i t h the following problem. P r o b l e m . Under what conditions is an open mapping fiber mapping?

p:

T -*■B a

This seems to be a rather natural question to raise since fiver m a p ­ pings are essentially open mappings wi t h rather strong conditions, e.g., the various covering homotopy properties and slicing structure properties. There are a few answers to this question available to us. In the paper [1], Dyer and Hamstrom have defined the completely regular mapping. And, they have shown that under rather strong conditions (local n-connectedness for the space of homeomorphisms of the inverse of a point onto i t ­ self, a mong others) such a m a pping is topologically equivalent to a m a p ­ p in g w i t h the bundle property. We show that a more general class of co m ­ pletely regular mappings have at least one of two w e a k bundle properties as defined here. Furthermore, we show that certain completely regular mappings have one of two properties wh i c h we call generalized slicing structure properties. D e f i n i t i o n s . In this paper, the word m apping means continuous mapping onto unless stated to the contrary. A metric space T is said to be locally n - connected where n is a non-negative integer iff for each i n ­ teger k, 0 < k < n, eac h point x in T, and each e > o, there is a 5 > o such that each ma p p i n g f of a k-sphere Sk into the 5 -neighbor­ hood N 5 (x) of x, there is a n extension F of f taking the k+i-aisk D k+1 into N £ (x ) [cf. 1]. A m apping p of a metric space T onto a metric space B is homotopically n -regular iff for each integer k, ° < k < n, each x in T and each e > o, there exists a 6 > 0 such that each mapping f: Sk into N & (x) n p ”1 (b) for b in B can be extended to a m apping F: D k+1 into N g (x) n p “1 (b) [cf. 1, 7]. Completely Regular Mappings and the W e a k Bundle Property I. The bundle property [4] for fiber maps is rather nice but it is u n ­ fortunately too m u c h to expect for open mappings. We define a consider­ ably weaker property w h i c h preserves in a sense some of the desirable properties of a bundle or local Cartesian product. Definition. A mapp i n g p: T -»■B is said to have the w e a k bundle property I wi t h respect to a space K iff for each point b in B and mapping g^ of K onto p - 1 (b), there exists an open set U^ containing b and a m a pping cp^j : U^ x K -► p ~ 1 (Ufe) such that

MAPPINTS, FIBER SPACES, W E A K BUNDLE PROPERTIES ( 1)

Pu

(2)

cpTT | (u, K) b cp-g is an

(3)

k)

= u maps

for

(u, k)

in

221

U^ x K,

(u, K) onto p ”1 (u),

and

extension of gb .

Note: We could require in (2) that stead of onto p “1 (u).

(u, K)

be mapped

into

p -"1(u) i n ­

D e f inition (Dyer and Hamstron) [ 1 ]. A mapping p: T -*• B where each of T and B is a metric space is said to be completely regular iff for each e > o and each point b in B, there is a 5 > o such that if x € B and d(x, b) < 5, then there exists a homeomorphism h. of —1 —1 p ” (b) onto p ” (x) wh i c h moves no point as muc h as e. In the theorem wh i c h follows, we show that a class of completely ular mappings has the w e a k bundle property I.

reg­

THEOREM 1 . Suppose that p: T B, each T and B is a metric space, T is complete, covering dimension B < n+1, and p is complete­ ly regular. Furthermore, there is a metric space K such that for each b in B, there is a m apping of K onto p ~ 1 (b) and that the space G^ of all such m appings is locally n-connected (LCn ) . Then p has the w e a k bundle property I w.r.t. K. [If K is homeomorphic to p " 1 (b) and G^ is the space of all homeomorphisms of K onto p ”1(b), then p has the bundle property — Dyer and H a m s t r o n . ] The basic ideas for the proof of this theorem came from the paper [ l ]. There are enough subtle differences so that our presentation of the d e ­ tails seems justifiable. The proof depends (as those proofs in [ 1 ]do) on an important and powerful selection theorem of E. A. Michael [5, 6] w h i c h we state below. T HEOREM M. If each of A and B is a metric space, A is complete, covering dimension of B < n+1, Z is a closed subset of B, F is a ma p p i n g of A onto B such that the collection of inverses under F is lower semicontinuous (defined below) and equi-LCn (as defined below), and f is a m a p ping of Z into A such that for z in Z, f(z) € F - 1 (z), then there is a neighborhood U of Z in B such that f can be e x ­ tended to a m appi ng f* of U into A such that for b € U, f*(b) e F ”1 (b ) . If each inverse under F has the property that its homotopy groups of order < n vanish, then U m a y be taken to be the space B. D e f i n i t i o n . A collection G of closed poipt sets filling a metric space X (i.e., the union of the elements of G is X) is said to be equi-LCn iff for each s > o, g in G, and x € g, there is a 5 > 0 such that if h € G and f is a ma p p i n g of a k-sphere Sk , 0 < k < n into h n N & (x), then there is an extension F of f to the k+i-disk D k + 1 , into h n N £ ( x ) , [cf.i].

222

MCAULEY We shall assume that

thermore, thermore, the metric metric for for in Ln di

Gb ,

T

Gb Gb

has a bounded complete metric

D (f , g) g) = = m max(d[f ax(d[f(x), ( x ) , g(x)]} g(x)]}

to to denote denote aa metric metric for for B. B.

and ind let

d.

is is understood understood to to be be as as follows: for for xx

in in K.K.

Fur­

For follows: f, g

We We shall shall also also useuse

No No confusion confusion should should arise. arise.

Notation. Let G G denote denote the the collection collection of of aall ll Gv. -----b G* denote the space of those mappings of K into

the the union of the elements of

G.

for for T

b' b' in in BB which is

The metric for this space is defined as

ibove. above. LEMMA. 1 LEMMA i ..1 i. Proof:

The metric metric space space G* G*

is is complete. complete.

Suppose that Cauchy sequence in that {z^} {z^} isis a a Cauchy sequence in GG**..

is Ls complete and the space

Z

of of all all mappings mappings of of

{zi Now, ziziisisa amapping mapping of of [Zj_)) -►-► z z€ €Z.Z. Now, b}

in

B.

KK

K K

onto onto

into into

Since T T Since

TT is is complete, complete,

p”11(b^) (b^) .. Also, p” Also,

Cb^} Cb^} -*•*•

It follows follows from from the the hypotehsis hypotehsis that that iiff e e >>o,0, there there exists exists mm

so 30 that i f

f, ,, of of p-1(b n > > m, m, then then there there is is aa mapping mapping f, p-1(b )) onto onto p” p”11(b) (b) n e/3 and d (z , z ) < e /3 . Observe that

which fhich moves no point as much as gn = fb bzn and and

is a mapping of

cKg^, z) > 00

K

onto

p~1 p“ 1 (b)

zz € € Gb Gb and and

such that

G* G*is is complete. complete.

The collection collection GGisis equi-LCn equi-LCn . .

Suppose

that

g e Gb and and ee >> o. o.

such such that that each each mapping mapping rr

of of Sk Sk, ,

Since Since Gb Gb isis LCn LCn, , there there

oo oo

into into

such such that that iiff

c; € B, then there isisa ahomeomorphism homeomorphism hcb hcb of of p_ p_11(c) (c) noves no point as much moves muchas as y1 € Gc G q nn N (g), then then

5 S.,/2. ^ 2.

Choose Choose 5, 5,

into into

Gb nGb n

Gb fln Ng^2 Ng^2(g (g).). Gb d(b, d(b, c) c) < a, < a,

onto onto p-p1 (b) 1 (b)which which

0 0 0, and integer k with ° < k < n, there is a 5 > 0 such that for each mapping f of a k-sphere Sk into the 8 -neighborhood N g (x), there is a n extension F of f m a p ­ ping the k+1-cell D k+1 (where S^ is the boundary of D k +1) into N £ (x ) [1]. A mapp i n g p of a metric space T onto a metric space B is homotopically n-regular if and only if for each x in T, e > 0 and integer k with 0 < k < n, there is a 5 > 0 such that if f is a m a p ­ ping of Sk into Ng(x) n p ”1 (b) for b in B, then f has an e x ten­ sion F m apping D k+1 into N £ (x) n p “1 (b) [5 ]. A mapping p of T into B is said to have the covering homotopy property (CHP) w i t h respect to a space X if and only if, given a mapping f of X into B, a lift*

R e search for this paper was supported in part by NSF GP-4571 . 229

230

MCAULEY A N D TULLEY

ing f* mapping X into T wit h pf*= f, and a homotopy H taking X x I into B such that H(x, o) = f(x) for each x in X, there is a homotopy H* taking X x I into T such that H*(x, o) = f*(x) for each x in X and pH* = H [2j. In the situation just described, we shall say that H* is a lifting of H relative to f*. If p has the CHP for each polyhedron (finite), then p is said to have the polyhedral covering homotopy property (PCHP) or, equivalently, (T, p, B) is said to be a Serre fiber space. The following fact should be noted. Suppose that p: T -+ B, has the CHP wit h respect to X, and B is_ a metric spa c e . F u r t h e r m o r e . suppose that f, f*, and H are as a b o v e . Then there is a lifting H* of H relative to f* such that i f , for x in X, H(x, t) is indepen­ dent of t, then H*(x, t) is also independent of t, i.e., if x in X has the property that H(x, t) = H(x, o) for each t in I, then H*(x, t) = H(x, 0) for each t in I. This fact is crucial to our proof of Theorem 1. It can be proved by an argument similar to that given in [3] where it is shown that any Hurewicz fiber space w i t h a m e t ­ ric base space is regular, i.e., has a lifting function taking constant paths to constant paths. Lifting Small Homotopies to Small Homotopies Suppose that p: T -*■B has the CHP with respect to a space X and that each of T and B is metric. We shall say that small hom o t o ­ pies of X can be lifted to small homotopies if and only if for each s > 0 and each e in T, there is a 5> 0 such that if H is a m a p ­ ping of X x I into N 5 (p(e))and f* is a mapping of X into N g (e) w i t h pf*(x) =H(x, 0 ) for x in X, then there is a lifting H* of H relative to f* such that H* maps X x I into N £ (e). Theorem 1 states that small homotopies of an n-cell can be lifted to small h o moto­ pies for certain Serre fiber spaces. This is the fact needed in our proof of Theorem 2. However, the restriction that X b a an n-cell is not n e c ­ essary. After proving Theorem 1 we will indicate how it can be general­ ized to the situation where X is any polyhedron. THEOREM 1 (Tulley) . Suppose that p:T -*■B has the PCHP, each of T and B is metric, T is LC° and X is an n-cell. Then small homotopies of X can be lifted to small homotopies. Proof: Suppose that the theorem is not true. Then, since B are metric, there are sequences (H^) and (g^) of mappings that: (1) for ea c h i, H ^ : X x I -*■B, gi : X -*■T,and = H^(x, 0 ) for each x in X, (2 ) (g^(X)) -► e for some e in T, and

T and such pgj_(x)

p

FIBER SPACES AND n-REGULARITY (3)

(Hj.CX x I*) - b

=

231

p(e) .

Also, there is no sequence {H^*} relative to g^ for each i and

such that H^* is a lifting of (H^*(X x I)) -*• e.

Let Z denote C(X), the cone over X, i.e., the quotient space obtained from X x I by collapsing X x {0 } to a point v, the vertex of the cone. We identity Z - (v) wit h X x (0, 1 ]. Let X^ = { (x, s) € Z|s = 1 /i) for each positive integer i. Choose any x Q in X and let Y i = X i + 1 U X ±U ({X Q ) x [i/i + 1 , i/i]) for each i and let i=! E a c h Y^ is contractible and therefore is an absolute retract. lows that there is a retraction r: Z -* A.

It f ol­

We define a mapp i n g g of Z into T as follows: Let g(x, 1 /i) = g^(x) for each i and each x in X. This defines g| li“=1 X^. Since T is LC°, for sufficiently large i we m a y extend g to such a w ay that (gCY^)} -*> e. We assume without loss of generality that these extensions can be made for each i. Letting g(v) = e we complete the definition of g|A and then we define g b y g = (g|A)r. We define a mapping define K: (U X^) x I U Z ((x, s ) , 0 ) for x in X K((x, 1/i), t) = H ±(x, t) foreach i, we let be

H of Z x I into B as follows: First we x {0 } -►B b y K( (x, s ) , 0) = pg(x, s) for and s in (0, 1 ], K(v, 0) = b and for ( (x, 1 /i ) , t) in (U”=1 X ±) x I. Now, a retraction of [i/i+1, 1 /i] x I onto

( d / i + 1 ) x I) U ([ i /i+1 , i/i] x {0 }) U ({1 /i) x I) and for any (x, t) in (0 , 1 ] x I let ( s ’ , t’ ) be the point r ^ s , t) for any i for w hich 1 /i+i < s < 1 /i. We define H( (x, s), t) = K((x, s'), t r) for C(x, s ) , t) in (Z - (v)) x I and H(v, t) = b for each t in I. It is easy to see that H is a (continuous) mapping of Z x I into B and that pg(z) = H(z, 0 ) for each z in Z.

1/i -h1

1/i

1/i+1

1/i

Y^ in

232

M CAULEY A N D TULLEY

Since the PCHP holds for p: T -*■B, there is a mapping Z x I into T such that H*(z, o) = g(z) for each z in Z pH* = H. Furthermore, since B is metric, we can suppose that = e for ea c h t in I.

H* of and H*(v, t)

Letting H^*(x, t) = H*((x, 1 /i), t) we get a sequence, of homotopies m apping X x I into T such that H^(x, o) = g^(x) for each x in X, pH^* = H^,and, b y the continuity of H,( H ^ X x I)) -+ e. This is a contradiction and the theorem is proved. R e m a r k s . W i t h Theorem 1 and the techniques used in [2] (pp. 6364) to show the equivalence of various lifting properties for Serre fiber spaces, the following stronger theorem can be proved. One must be care­ ful to obtain sufficient Msmallness” at each stage of the proof but this is easy to do. THEOREM 1 '. Suppose that p: T -*• B has the PCHP, X is a po l y ­ hedron, each of T and B is metric, and T is LC°. Then small h o m o ­ topies of X can be lifted to small homotopies. Also, it should be noted that if (T, p, B) is a Hurewicz fiber space, i.e., p has the CHP for every space, and each of T and B is metric, then small homotopies of any space X can be lifted to small homotopies. The requirement that T be LC° is not needed. In fact, a proof is easily obtained b y following the procedure used in the proof of Theorem w i t h (U*_1 X^) U (v) p laying the role of Z. In this case, the appropriate mappings g and H are trivial to define. A Fiber Mapping w h i c h is n-Regular THEOREM 2 (McAuley) . of T and B is metric, T homotopically n-regular.

Suppose that is LCn and B

p:T -*> B has the PCHP, each is LCn+1 . Then p is

P r o o f : Suppose that e € T, e > 0, b = p(e), and Then there exist 5^ for i= 1,2,3, and 4 such that: (1)

if

g

is a ma p p i n g of D k+1

into

1

N

(e)

0 < k < and

H

n. is a

homotopy taking

D k+1 x I into N (b) such that pg(x) = H(x, o) l for each x in D k + 1 , then there is a lifting H* of H relative to g such that H* maps D k+1 x I into N (e) ; (we use Theorem 1 to get 51 >

(2) m a p ping of (3)

any m apping of Sk+1 into D k+2 into N (b) ; Np(b) 3 p ( N

3

(e)) j

and

(b) 2

can be extended to a

FIBER SPACES A M ) n-REGUIARITY

233

(k) any m apping of Sk into N & (e) can be extended to a mapping D k+1 into N ft (e) n N (e ) . k 1 63 Let 5 = 6. and consider f: S^-'-N^Ce) n ^ ( b 1) for some b 1 k+1 in B. Let F be an extension of f ma p p i n g D into N (e) n 3 N (e) . Clearly, pF is a m a p p i n g of D k+1 into N (b) such that 1 2 pF(Sk ) = b'. We consider Sk + 1 as D k+1 w i t h Sk identified to a point z. Then pF yields a mapp i n g g of Sk+1 into N & (b) such that g(z) = b'. Let G be a n extension of g m a pping 2 D k+1 into N (b). ®1 lc+1 B y shringkin S to the point z in D while keeping z fixed and following this shrinking by the m apping G, we obtain a homotopy taking Sk+1 x I into N 5 (b). This induces a homotopy H taking D k+1 into N (b) w i t h the properties that H(x, o) = pF(x) for each x in D k+1 of

1

and H((Sk x I) U (Dk+1 x {1))) = b'. Now, there is a lifting H* of H relative to F m apping D k+1 x I into N £ (e). It follows that H*(x, o) = f(x) for x in Sk and that p ”1 ( b 1) n N e (e)

H * ((Sk x I) U (Dk+1 x C D ) ) .

Let A = (Sk x I) U (Dk+1 x C D ) and let F = F|A. A i 1 Since a k+1-cell, it is clear that F yields a m a p ping of D into p~1 (b’ ) n N £ (e) w hich extends f. This completes the proof. R e m a r k . If LCn . Thus, Theorem fact that if p: T connected then each

A

is

p is homotopically n-regular, then each fiber is 2 implies a local result analogous to the well-known -►B has the PCHP, T is n-connected and B is n+1fiber is n-connected.

MCAULEY A N D TULLEY

234

REFERENCES [1]

Dyer, E. and Hamstrom M.-E., "Completely regular mappings," Fund. Math., vol. 45 (1957) pp. 10 3-118.

[2]

Hu, S.-T., Hcmotopy T h e o r y , Academic Press, New York, N. Y . , 1959.

[3]

Hurewicz, W., "On the concept of fiber space," Proc. Nat. Acad. Sci., U.S.A., vol. 41(1955) pp. 956-961.

[4]

Michael, S. A., "Continuous selections III," Annals of Math., vol. 6

[5 ]

5 (^957) pp. 357-390.

Raymond, Frank, "Local triviality for Hurewicz fiberings of m a n i ­ folds," To p o l o g y ,

[6]

vol. 3 ( 1 9 6 5 ) pp. 43-57.

Serre, J.-F., "Hcmoiogie singuliere des espaces fibres," Annals of Math., vol. 54(1951) pp. 4 2 5 - 5 0 5 .

RUTGERS UNIVERSITY

FIBER SPACES W I T H TOTALLY PATHWISE DISCONNECTED FIBERS* by Gerald S. Ungar** Two outstanding problems in topology involving light open mappings are as follows: I. Does there exist a light open mapping of a manifold onto a manifold such that the inverse of some point is uncountable? II. Is there a finite to one open m apping f wh i c h is not the identity mapping of the n-cube In onto itself such that f is the iden­ tity on the boundary of In (bdry In ) and f ^ f C b d r y In ) = bdry In ? It can be shown (Corollary 1 of Theorem 3) that there are no Serre fibrations with the properties of either I or II. Hence, a method of attacking these problems might be the following: assume that there exists such a mapping and prove that it must be a fibration. The object of this paper is to begin a study of conditions on light maps so that they must be fiber maps. For references see [1], [ 3 ], and [ 7 ]. cases B.

The following are some definitions wh i c h will be needed. In all p is a m ap from a topological space T onto a topological space 1)

p

is a - light if

p - 1 (b)

contains no arcs for every

b

in B.

2 ) p has weak local cross sections at every point if given any b in B arid any y in p -1 (b) there exists a neighborhood U of b and a map cprj of U into T such that cpTT (b) = y and pcp^. is the

identity on b

y

y

U^.

y

3) p has strong local cross sections at every point if given any in B there exists a neighborhood U of b such that if y is in

The author would like to express his gratitude to Professor McAuley for his interest, encouragement, advice and assistance. The author was partially supported b y an NSF Academic Year Extension.

2 35

UTJGAR

236 p ”1 (U) on

U

there exists a map and

y is in

4) If E is have the E covering space X € E into = pf (x) then there = f (x) and pK(x, t)

cp^:

U -► T

such that

b) c) d)

f)

If

p

p is said to ma p f of a that H(x, o) that K(x, o)

has the ECHP

and

E is the class of all topological spaces, then

p has the ACH P (absolute covering homotopy property, see Hu [k]). E is the class consisting of the unit interval, then p has the I C H P . E is the class of compact locally arcwise connected spaces, then p has the C L A C H P . E is the class of finite polyhedra, then p has the P C H P .

If p has the PCHP, the sense of Serre.

e)

is the identity

a class of topological spaces; then homotopy property (ECHP) if given any T and a homotopy H: X x I -►B such exists a homotopy K: X x I -*■T such = H(x, t) .

W e will use the following notation. a)

pq>^

B by p(x, y) = x. It would be interesting to know when the O-CHP implies that p has strong (or even weak) local cross sections at every point. In general it does not. Keldys [6] has reported an example of a light open m ap of a one-dimension­ al Peano continuum onto a square. However, suppose that either T is a m anifold or that p does not raise dimension, (or even suppose that p is a f i bration), then does the O-CHP imply that p has local sections of some kind. Theorem 3 gives a partial result if p is a fibration. A l o n g these same lines, the following theorem is obtained.

and

p

THEOREM 2 . If p has w e a k local cross sections at every point has the uO-CHP then p has the CLA.CHP. This theorem suggests the following question: Is the CLA.CHP equivalent to the PCHP?

If not, when is it the case?

_ Theorems 2_and 3 give partial answers to some of the questions w h i c h were raised above. However, the author has not been able to show that these are the best possible. THEOREM 3. If p is a-light and has the PCHP and B is first countable, locally arcwise connected and semi-locally simply connected, then p has strong local cross sections at every point.

UNGAR

238

COROLLARY 1 . If T, p and is locally arcwise connected, and the bundle property (B.P.).

B are as in the theorem above, T B is arcwise connected then p has

COROLLARY 2. If T, p and B are as in the theorem above and p ”1 (b) is compact for all b € B, B is locally compact, arcwise con­ nected, and p has the FCHP, then p has the B.P. The following question can be raised w i t h regard to Corollary 2. W h e n do the PCHP and FCHP imply the B.P.? In general they do not since there are one to one fiber maps w hich are not hone omorphi sms, and there is an example [5] of a map w hich has the ACHP but does not have the B.P. The following example shows that in general as the FCHP must be assumed to obtain the B.P. Example:

Let

T = ( (x, y)|y =

T

be the following subset of the

x + 1,n = 1,2,... and

U C(x, y)|y = o

some condition such

and

plane:

0 < x < 1}

o < x < 1}

U ( (x, y) |y = -

n=i ,2,... and < x < i} n-i 2 ~~ U ( (x, y) |y = 2n”1x - -— , n = 1,2,,.. and

0 < x
) = co(o), and set q Q = ^ | N Q . Furthermore, let (F, F Q ) = (q- 1 (b o'* ), q.Z1 (b0 for some base point b o ^< M. vuo )) " (N, N 0 , , M) has the following properties; (1)

(I, lQ)

(2)

(F, P 0 ) ~ (Rk , R k - 0),

Then, (I, |Q so') =

is a locally trivial fibered pair wit h fiber

(F, F Q ) ,

(3) Let G(S, M) denote the topological group of stable ([i ]) homeo morphisms w hich leave M invariant and G Q (S, M) the subgroup leaving b Q fixed. Then (|, | ) is a Steenrod bundle pair wit h struc­ ture group G q (S, M) and in fact the associated principal bundle is just f: where

G(S, M)

M

f(g) = g(bQ ) .

2.1. Q u e s t i o n . Is there a euclidean bundle (n, n0) * (E, E Q , M) w h i c h is fiber homotopy equivalent to (|, |Q ) . Here (n, t)q ) has fiber (RK , R k - o) and group H Q (Rk ) , the group of homeomorphisms on ]/• R w hich leave the origin fixed.

,

2.2. Remarks. First of all, observe that the existence of a tub­ ular neighborhood for M obviously implies an affirmative answer. On the other hand, it is unlikely that an affirmative answer to Question 2 . 1 will imply the existence of a tubular neighborhood. Also, it m ay be w orth recalling the Na s h tangent fiber space of a manifold. Given a manifold M, let T q denote those paths ec in M such that co(t) = m ( o ) ,if t = 0, 0 < t < 1 > and let T denote T Q plus all the constant paths in M. D e ­ fine t t : T -♦ M b y tt(oq) = oj(o), set ttq = 7rlT0 , and let ( F 1, F 0 ‘ ) = (tt"1 (bc ) , t ' 1 (b0 )). Then (t, t q ) = (T, T Q , t , M) has the following properties. (1)

(t,

t

)

is a locally trivial fibered pair w i t h fiber

(F‘ , F0 ‘ ), (2)

( P f, P c ») ~ (Rn , Rn -

o),

n = d im M,

(3) Let G(M) denote the group of stable homeomorphisms of M and G 0 (I-I) the subgroup leaving b Q fixed. Then, (t , tq) is a Steenrod bundle pair w i t h structure group G Q (M) and associated principle bundle f: G(M) M where f(g) = g(bQ ) . The question analogous to 2 . 1 wit h (t, t q ) replacing ( £ , s ) is answered affirmatively b y let­ ting (ti, t^q ) be the "core" of the tangent microbundle of M (see Kister [4 ]) . 2.3. Quest i o n . What favorable topological properties do the spaces (N, N Q ) , (F, F Q ) , (T, T Q ) , ( F 1, P *) possess? It is known that all these spaces are ULC (uniformly locally contractible [5]). This

QUESTIONS IN THEORY OF NORMAL FIBER SPACES FOR TOPOLOGICAL MANIFOLDS

243

result is due to D. C. West. Are these spaces A N R (metric)? If not, we would have an answer to the following question raised b y Dugundji. 2 .4

.

2 .5 . Remark 2 .2 . of a 1 -field k-fields, k

3.

Question.

Is a metric ULC space A NR (metric)?

Q u e s t i o n . Consider the tangent fiber space ( t , t q ) in Then, a section for 7rQ : T Q M is the natural analogue on a differentiable manifold. Wha t might be analogues of > 2?

The Normal Fiber Space of an Immersion

Let f: M -► S (dim M = n, dim S = n+k) denote a locally flat immersion, i.e., for some open cover (va ) of M, -*• S is a locally flat imbedding. Because f(M) m a y have self-intersections, the constructions the construction of a normal fiber space for

o, together with all trivial normal paths (b, a ^ ) , b € V. Let fl> denote the family of all such sets A(V) and the c-o topology for N. 3.4. D e f i n i t i o n . The w e a k topology for N is given b y taking the topology generated b y $ U n. N wit h this w e a k topology is de s i g ­ nated N*. N* wil designate N Q wit h the weak topology, q* = q: N* -► M, (F*, F Q*) is (F, F Q ) w i t h the w e a k topology. One determines the w e a k homotopy type of the pair (F*, F Q*) as follows. Choose a sequence of "nice" neighborhoods V 1 V2 ••• each containing b Q such that for any set G open in M, b Q € G, then Vj C G for some j . Let (p*(V±),

= (f * n A f ^ ) , F 0* n A ( V i

then, (f*(V ,

F 0* ( V , ) }

C •••

C ( f * ( V ± ) , F 0* ( V t ) ) c . . .

gives (F*, F Q*) as a n inductive limit of open subsets. A l l the inclu­ sion m ap can be shown to be homotopy equivalences and (F*(V1), F * ( V 1)) V v' ° 1 can be shown to have the same homotopy type as (R , R - o ) . It then follows that (F*, F q*) has the same wea k homotopy type as (Rk , R k - o) thus one obtains 3.5. PROPOSITION. (5*, tQ*) = (N*, N Q*, q*, M) is a locally trivial fibered pair w i t h fiber (F*, F Q*) w hich is of the w e a k homotopy type of (Rk , R k - 0). (I*, |Q*) is called the normal fiber space of the locally flat immersion f.

M)

3.6. Q u e s t i o n . Is there a euclidean bundle (t|, which is fiber homotopy equivalent to (£, |Q )?

nQ) =(E,

EQ,

,

Obviously, a necessary condition for an affirmative answer to this question is that (F*, F Q*) have the same homotopy type as (Rk , R k - o ) . 3.7.

Question.

Is

(F*, F Q*) ~ (Rk , R k - o)?

3.8. Wha t properties does F* possess as a topological space? is clearly Hausdorff. Is it UI£, ANR(Q) for some category Q ? 3.9. R e m a r k . If, for example, one desires to apply Stasheff's classification theorem ([6]) to q *: N * -*• M, one would need to know that F q* ~ SK . 3.10. Does (§*, g *) have the structure of a Steenrod bundle? If so what is the structure group?

It

QUESTIONS IN THEORY OF NORMAL FIBER SPACES FOR TOPOLOGICAL MANIFOLDS 2 U 5

3 .1 1 . E v e n though (t, t q) as defined in question 2.2 serves well as the tangent fiber space of a manifold, there is an alternative w h i c h works as a somewhat more natural "mate" to the normal fiber space of a n immersion. Given a manifold M, let T Q r denote those paths cd in M such that for some s > 0 (depending on cd) m(t) ^ 0 0 ( 0 ) for 0 < t < s. Let T 1 denote T ' plus the constant paths of 'M and give T ’ the c-o topology. As usual we have a m a p tr1: T ’-*■M, where 7r'(a>) = a>(o) . Let 7rQ * = tt* |Tq * and (F*, F 0 f) the fiber over b Q € M. Then (t 1, tQ f) = ( T 1, T Q !, tt1, M) is easily seen to be a locally triv­ ial fibered pair w i t h fiber ( F l, F Q f) • However, just as in the case o the n o r m a l fiber space of a n immersion F 1 has the wro n g homotopy type. 3 .1 2 .

LEMMA.

Fq 1

3.13.

COROLLARY.

is c o n tractible. tto

1:

TQ !

M always admits a section.

3.14. R e m a r k s . Corollary 3.13 in the language of popular m a t h e ­ matics says that it is possible to cover the head completely w i t h hair (well plastered down w i t h that "greasy kid stuff") provided there exists at least on ingrown hair. Thus, ( t 1 , t ' ) is not acceptable as a tangent fiber space to M. We can, however, weaken the topology in T 1 in order to make things work. For fixed s, 0 < s < 1 , let A(s) = {as € T q 1 :

cd(

0)

^ oa(t),

0

< t < s} .

Then, we have Tq . A ( 1 )

C A(i)

C •••

C A(i)

C •••

w i t h Un A(i/n) = T Q '. We introduce a new topology in T 1 so that each A(i/n) is open as follows. Let * denote the family of all sets A(s), 0 < s < 1 , and ft* the c-o topology for T ’. Let ft*denote the to­ pology generated b y ft1 u Y and let T* denote T ’ w i t h topology ft*. Furthermore, let T Q*, F*, F Q* denote, respectively, the sets T Q !, F 1, F 1 w i t h topologies induced b y ft*, and p*: T -► M is just p'. Now, it is not difficult to prove the following. 3.15. PROPOSITION. F* h anotopy type of Sn _ 1 , where

is contractible and n = dim M.

3 .1 6 . PROPOSITION. ( t* , t q* )= (T*, T Q*, trivial fibered pair w i t h fiber (F*, F Q*) w h i c h mo t o p y type as (Rn , R n - 0 ), n = d im M. 3 .1 7 .

space of

M.

Definition.

(t* ,

t q*)

F Q* has the wea k

p*, M) is a has the same

locally weak ho­

is call the extended tangent fiber

FADELL

246

3.18. R e m a r k , ( t , t q ) and ( t * , t q * ) are weakly fiber homotopy equivalent, i.e., the inclusion map a : ( t , t q ) - ( t * , t q * ) is a wea k homotopy equivalence when restricted to fibers. If we know that (F*, F q*) ~ (Rn , R n - 0), then we could assert that a is a fiber hom o ­ topy equivalence. 3.19.

Que s t i o n .

Is

(F*, F Q*) ~ (Rn , R n - 0 ) ?

3 .2 0 . Q u e s t i o n . What topological properties are possessed by the spaces F*, F q* ?

REFERENCES [1 ]

M. Brown and H. Gluck, "Stable structures on manifolds I," Ann. of Math., 79 ( 1 9 6 4 ) pp. 1 - 1 7 .

[2 ]

E. Fadell, "Generalized normal bundles for locally flat imbeddings," Trans. Amer. Math. Soc., 114 (1 9 6 5 ) pp. 488-513.

[3 ]

E. Fadell, "Locally flat immersions and W h i tney duality," Duke Math. J., 32 (1 9 6 5 ) pp. 37-52.

[4]

J. Kister, "Microbundles are fibre bundles," Amn. of Math., 80 ( 1 9 6 4 ) pp. 1 9 0 - 1 9 9 .

[5 ]

J. P. Serre, "Homologie singuliere des espaces fibres," Ann. of Math.,

[6]

J. Stasheff, "A classification theorem for fibre spaces," Topology 2 (1963) pp. 239-246.

54 (1951) pp. 425-505.

UNIVERSITY OF WIS CONSIN

'78 D 6 9

0 8 0 567"