BING'S DOGBONE SPACE AND CURTIS' CONJECTURE by JOHN.EDWARD HUTCHINGS M.A., U n i v e r s i t y o f B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of MATHEMATICS We a c c e p t t h i s required t h e s i s as conforming to the standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In p r e s e n t i n g an this thesis i n partial advanced degree a t t h e U n i v e r s i t y the Library s h a l l make i t f r e e l y f u l f i l m e n t of the requirements f o r o f B r i t i s h Columbia, I agree a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r extensive for by that copying of t h i s thesis s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r h i s representatives. I t i s understood that of t h i s thesis f o rfinancial written permission. Department o f Mathematics The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date Nay gain 25 1973 Columbia copying or p u b l i c a t i o n s h a l l n o t be a l l o w e d w i t h o u t my - iiABSTRACT Bing's dogbone space 3 sition space of E V i s an upper semi continuous decompo- which fails to be 3 E although the associated decomposition consists only of points and tame arcs. It has proved d i f - ficult to find topological properties of V which distinguish i t from 3 E . In this paper, we prove a conjecture of Morton Curtis i n 1961 that certain points of V f a i l to possess small simply connected neighbourhoods . - iii I wish to acknowledge my gratitude to my supervisor Dr. Whittaker for his unselfish and often indispensible aid during my graduate studies at UBC, and to Dr. Luft for his support and enthusiasm. I am grateful also for some conversations and a blizzard of letters from R. H. Bing. - iv- TABLE OF CONTENTS Introduction- ... .1 Chapter I Chapter I I 3 Bing's Dogbone Space and C u r t i s ' C o n j e c t u r e Chapter I I I G e n e r a l i z a t i o n o f a Theorem o f B i n g : Lemmas........ Chapter IV Main P r o o f Bibliography Appendix G e n e r a l i z a t i o n o f a Theorem o f B i n g : 24 56 78 98 .......99 - v - TABLE OF FIGURES Figure Page 1 6 2 6 3 7 4 11 5 13 6 14 7 15 8 . 9 15 16 10 f a c i n g page 17 11 " 17 12 " 18 13 21 14 22 15 23 16 a f a c i n g page 16 b " 17 18 25 26 26 _ " 19 26 26 20 " 27 21 » 27 22 29 T a b l e o f F i g u r e s cont'd - vi Figure 23 Page • •• f a c i n g page 24 34 11 25 34 " 41 " 41 27 " 41 28 " .42 29 " 45 30 " 45 31 Sr. " 45 31 b " 45 32 " 46 33 „ 46 34 " 47 35 " 47 36 " 49 37 " 52 38 " 53 39 " 53 40 " 54 41 " 55 42 " 57 43 " 57 44 " 57 26 • T a b l e s of F i g u r e s cont'd - viiFigure Page 45 f a c i n g page 58 46 " 59 47 " 59 48 " 60 49 " 62 50 11 65 51 " 65 51 a " 71 52 " 73 53 " 78 54 a,b,c,d " 84 55 " 88 56 " 91 57 " 92 58 " 92 59 " 92 60 11 92 61 " 93 62 „ 94 63 " 94 64 3. ^ • • » J •••••• S 6 6 Appendix 99 INTRODUCTION Bing's dogbone space (which i s denoted by is a decomposition space of E 3 V in this paper) which fails to be homeomorphic to E 3 even though the associated decomposition space is upper semicontinuous and point-like, and each element of the decomposition i s either a point • or a tame arc. The appearance of V in [12] caused some surprise since i t was thought at the time that a l l use point-like decomposition spaces of 3 E would turn out to be 3 E . Although V dates from 1955 and has become rather well-known, i t has been found hard to determine those top3 ological properties of the space which distinguish i t from E . Bing's original paper [12] showed that V i s a simply V i s a non-manifold; but connected homology manifold and locally simply connected. This paper contains a proof of a conjecture of Morton Curtis that V fails to pos- sess small simply connected open neighbourhoods about certain points. This property i s stronger than local simple connectivity (see our comments in II §1). A proof of Curtis' Conjecture was anounced in 1964 114]; however the detailed proof has not appeared. Only one other topological property 3 distinguishing V and E i s known: some points of V cannot be enclosed in 2-spheres [11], [13]. The general state of affairs seems to be that some points of V have no closed or open 3-cell neighbourhood systems, but do have systems of neighbourhoods bounded by double t o r i . Our arguments use elementary methods exclusively (except for an easily- circumvented reference to the Hopf property- of knot groups) and may well appear old-fashioned. We are less than proud of much of the exposition, which was intended to combine the detail appropriate to a - 2- thesis with the directness of a journal paper and somehow didn't. The reader w i l l probably share our pain at the length of the argument (the whole paper is essentially one theorem). The reader who i s unfamiliar with pathological decomposition spaces i s advised to read .13], which i s brief and exceptionally entertaining, and then skim Ch. II?.,. We w i l l mention some notational peculiarities: we follow common practise i n describing geometric constructions, even complicated ones, by the use of diagrams. "Theorem' i n this paper means 'working theorem'; thus 'theorems' appear i n the introductory-chaptereonly. - .3 CHAPTER ONE 0. Introduction. This f i r s t chapter gives preliminary material for the arguments in Ch III and especially Ch IV. The reader who wishes to skim the paper w i l l find that Ch I I , which contains the discussion of Curtis' Conjecture, is largely independent of this f i r s t chapter. In this paper, our approach to elementary topology is along the lines of the easier chapters of [101, in particular, we always assume a separable metric space. sections Ch IV, and 112]. 1 and 2 are elementary, In this chapter, §3 contains working theorems for §4 is essentially a comment on Bing's Theorems Section 5 6 and 7 of is part of the argument of Ch IV which is self-contained and has been smuggled into the preliminary material, althpugh i t could have been l e f t until i t appeared naturally in the main argument. 1. Notation. The arguments in this paper use elementary-methods exclusively, so that notation shouldrpr.esent.mo>prbblems. -Me use ,.0'.for.the null set and the symbol „'.-0in£or ithe"end-;.of-cthe^proof-nofda^numbered ?result>: expression 'Bd A' with-boundary- A, The may mean either the manifold boundary of the mainifoldor. the point-set boundary/of the set comment applies to the expression 'Int AT • A . A similar This reflects common practise; we w i l l comment whenever the meaning is unclear. As mentioned In the preface, our attitude to the construction of tame sets w i l l be cavalier; we w i l l construct many important tame sets simply by describing the set and perhaps giving a picture of I t . We advise against the intuitive approach of imagining our constructions as stralght-sided polyhedra whose structural detail is so fine that the polyhedra approximate the figures closely. Several of our arguments w i l l require extensive repair i f our geometric constructions are interpreted in this way-. If neccessary, methods in [4] could be used to - 4 - show that each of our constructions is in fact a curvilinear polyhedron. 2. Elementary Results. In this section we give some 'obvious' results which we have found hard to justify by simple references. This may be a matter of ignorance, especially in the case of and (2.1) (2.3) . We define an annulus to be a topological sphere with two holes. (2.1) . S^, Let be an arc which intersects two disjoint closed sets . Then there is a sub-arc and meets (2.2) . a a* of a only at the end points of U Any two annuli A^, of one boundary component of which connects and a* . are homeomorphic. Any homeomorphism A^ onto a boundary component of be extended to a homeomorphism of (2.3) . The proofs are omitted. A^ onto A^ may A^ . The union of two locally connected (lc) eontinua which intersect is a l c continuum. (2.4) . Let dary is l c . 0 be a bounded connected open set in the plane whose boun- Then any two points an arc which lies in (2.5) . Let A 0 x and y in 0 may be connected by except possibly for i t s end points. be a 2-manifold with boundary, and A . Then any two points of K K a continuum in may be connected by an arc in Int A cept possibly for end points) which lies within a distance e C2.6). E Let C. , C0 be disjoint simple closed curves in of (exK . 2 " . Then - 5one of the following exclusive alternatives is true: a) cz Int C 2 or equivalently Int c l n t C 2< b) C 2 cr Int c^ or equivalently Int c l n t C^. c) Each of 'C^, C 2 Int (2.7). lies in the others exterior, or equivalently D Int C 2 = 0 Let A be an annulus, and C a. simple closed curve in Int A which bounds no disk in A . Then C B2 are annuli. such that B1 U C and B2 U C separates A into components B^, - 6 3. S l i d i n g Curves on Spheres. C3.1). We w i l l often need to 'move' or 'deform' curves i n E 3 . This w i l l be done by s l i d i n g the curves on convenient spheres, disks and 3 annul! i n E f i g . 1. . A double ended lasso has loops may want to push p E move p , q z • over to the p o s i t i o n of so that i t looks l i k e 3 H: The sort of thing that may be encountered i s shown i n p' . and 'middle' z" z . We i n the figure or expand This can be done with a homeomorphism 3 -»- E 'z which c a r r i e s , say, to z". z onto z' and can thus be said to „ Suppose p U q U z homeomorphism sliding z H to 7 l i e s on a disk - 3 ACE . We ask what properties the should have i n order to r e f l e c t the i n t u i t i v e idea of z' on A while keeping p <j q One way this would be to construct a new disk contains and q except where they h i t carry z onto Then we H z u z' and misses could require that be the i d e n t i t y on p H A - D and on D C A fixed. to do (see f i g . 2) so that z' , Bd D , .(thus H[D] H D z U z' • = D , and will fix that p U q). It seems a good idea to specify a number of standard moves, prove that they can always be made and s t i c k to these i n the sequel. When, as commonly happens, an arc or loop moves only a short distance and has e x p l i c i t i n i t i a l and f i n a l l o c a t i o n s , then our idea of 'standard moves' i s probably too formal. However our standard moves are intended • case that the i n i t i a l p o s i t i o n of the-set i s unknown-. for the In this case the existence of the required move i s less obvious, e s p e c i a l l y when, as i n §5, Th 6 , a base point must be held f i x e d during the move. sphere with n holes i n E 3 , then a c o l l a r of S If S is a i s the image of 3 an embedding a c o l l a r of h S of may S * [-1,1] not e x i s t (S c o l l a r has been constructed of S into E so that could be w i l d ) . h(x,0) = x.. A set upon which a \ i s c a l l e d a c o l l a r e d set. i s not a neighbourhood of S . Evidently Note that a c o l l a r (3.2)_. A- B- and B' -moves. We g i v e t h r e e s t a n d a r d moves i n Theorems 1 and 2 . 3 Theorem^!. and a' , a Let D be a d i s k i n E , J two a r c s which have common end p o i n t s } a c o l l a r of D , ! and l i e i n Int D except f o r these end p o i n t s , which l i e i n Bd D . Then t h e r e i s a • 3 homeomorphism A C a j a ^ D j J ) of E onto i t s e l f which c a r r i e s a onto ~~3 a',.D onto i t s e l f , We moves a call and which f i x e s E A(a,a^,D,J) to -.a' . - J. 'the A-move' and say t h a t (Of course the f a c t t h a t a moves to one o f a number o f t h i n g s t h a t have to be kept i n mind. move as a f u n c t i o n o f and D J A(a,a',D,J) a^ i s only We w r i t e the t o emphasize t h a t the t r i c k o f u s i n g the move depends on the r i g h t d e f i n i t i o n o f D and J). 3 Theorem 2. E . Let simple c l o s e d curves which l i e i n the i n t e r i o r o f A and bound no d i s k s in A . Let B(c,c',A,Q) c onto Q be an anhulus i n c, c % be Then t h e r e i s a homeomorphism 3 , a l s o c a l l e d a B-move, of E onto i t s e l f which c a r r i e s 3~~ onto i t s e l f , c a homeomorphism h A be a c o l l a r o f c ' ,. A in addition, Let and c^ A . and which f i x e s have a common base, p o i n t .B'(c,c',A,Q) y and E - Q . If, y , then t h e r e i s and the f o l l o w i n g a d d i t i o n a l p r o p e r t y : i s the embedding a s s o c i a t e d w i t h the B'-move f i x e s Bd Q Q , so t h a t and i n f a c t a l l of Q = h[A x [^1,1]] , then hjy * [-1,1]] . The B'move i s a move 'keeping the base p o i n t f i x e d ' . p r o b a b l y f i x the base p o i n t by p r o v i d i n g if that One c u c" '. c o u l d h i t could Bd A C Bd Q - 9 - so t h a t y £ Bd Q '.(the B-move does n o t . p e r m i t t h i s ) , however the B'-move as g i v e n above f i t s prove. fix a misses want t o move a B'-move A F i g . 3 shows and t o ec" c b A while leaving i n which b hits. Q 'thin' and an a r c fixed. i s defined a U b l i e s i n " h[y E v i d e n t l y the u t i l i t y a such A . We We do t h i s w i t h so t h a t a l l p o i n t s i s perpendicular Q , the B^-move w i l l f i x b to h [ x x [-1,1]] i s s h o r t ) x 6 A with w i l l be f i x e d because Q . a U b (i.e. for x £ A , that each a r c H[x x [-1,1]] a sufficiently c, c', A, i s a s t r a i g h t arc perpendicular B^(c,c',y,A,Q) Q' l i e near and a p p l i c a t i o n s b e t t e r and i s e a s i e r to We w i l l g i v e an example which shows why we want the B'-move t o h [ y x-{-1,1]] . that of the i n t e n d e d to because x [-1,1]] A . For a c E - Q , wherever i t o f the B'-move i s l i m i t e d . subsequent use o f the B^-move w i l l be v e r y much along and so However the l i n e s o f t h i s example. 4. The Phragmen-Brouwer P r o p e r t i e s . The Phragmen-Brouwer P r o p e r t i e s sphere, b u t h o l d a l s o on a d i s k . S be a l o c a l l y connected m e t r i c of S are u s u a l l y given We quote from W i l d e r , space. f o r the n - [ I , I I 4.1]. Then the f o l l o w i n g Let properties are equivalent. (4.11) . If A, B such t h a t n e i t h e r a r e d i s j o i n t , c l o s e d subsets o f A nor A U B does n o t s e p a r a t e y S ' i s meant in The Z o r e t t i Theorem. (4.12) . If 'x B separates x and and y S = A U B , where y in x and y S . . (By S , in 'X and S , x,y 6 S then separates a r e i n d i f f e r e n t components o f A, B x and S - X'). a r e c l o s e d and connected, then - 10 A fl B is connected. (4.13). If b € B, - A, B are d i s j o i n t c l o s e d subsets of then t h e r e e x i s t s a c l o s e d connected which separates a and connected m e t r i c space. a l s o 9.2), D first (4.2). of From V I I , 9.3 w i l l have p r o p e r t i e s (4.11), B e t t i number i s z e r o ; thus We C ae A, S - (A U of ' I I ] s t a t e s t h a t these p r o p e r t i e s are alent i n a l o c a l l y its subset and B) b . Theorem I I 4.12 a disk S (4.11) ... of [1] (4.12), 6 and 7 (note (4.13), i f (4.13) h o l d on D . get the f o l l o w i n g important working theorems from These theorems resemble Theorems equiv- (4.11). of [ 1 2 ] . 3 Theorem 3. Let D be a 2 - c e l l i n E and F^, closed 3 d i s j o i n t s u b s e t s of end p o i n t s misses F^ that arc as i n Th 3 misses 'F^ p . E and q , pyq L e t ~pxq, - Theorem 4. Let D, F^, F^, Vi that arc px U a r c . misses either F^, or 1* yq pzq arcs F^ , arc the pyq p, q such misses D pxq, pyq be d e f i n e d F^, py are homotopic i n by a homotopy which f i x e s and Th 6 0 arc p, xq q, F^ . Th 4. D arc w i t h end p o i n t s P r o o f s of Th 3. and Th 3 which share . Then t h e r e e x i s t s an a r c f o r word to prove O. pxq D pzq. w i t h end p o i n t s pzq c D - except be a r c s i n and such t h a t a r c U s i n g t h i s f a c t , the p r o o f s of [12] pyq Then t h e r e e x i s t s an a r c such t h a t a r c misses and . Since and D i s s i m p l y connected, Th 7 p of [12] may Th 4 r e s p e c t i v e l y , reading D and pxq q . be used word for M tn 11 - (4.4). The P l a n e S e p a r a t i o n Theorem and t h e Z o r e t t i Theorem. We quote t h e s e r e s u l t s , s l i g h t l y s i m p l i f i e d , from [10,VI §3]; (4.41) . The P l a n e S e p a r a t i o n Theorem. Let A , B be compact s e t s i n 2 E w h i c h i n t e r s e c t i n a t most one p o i n t . and l e t e > 0 . Let a £ r " A - B , b e B - A , Then t h e r e i s a s i m p l e c l o s e d c u r v e J which separates 2 a and b A U B in E , lies within an e-neighbourhood The Z o r e t t i Theorem. If K i s a component o f a compact s e t M i n the p l a n e , then there i s a simple closed curve contains K ,. w h i c h m i s s e s K . 5. Annulus Dodging Theorems. A J whose i n t e r i o r M , and w h i c h l i e s i n an e-neighbourhood of Suppose i s an annulus and F i s a c l o s e d s e t i n A . When can we s a y t h a t a s i m p l e c l o s e d c u r v e w h i c h l o o k s l i k e e x i s t s so as t o m i s s F ? say t h a t A F bridges A , and m i s s e s A f) B . except p o s s i b l y a t the p o i n t (4.42) . of c The answer i s about what would be e x p e c t e d . i f f t h e two boundary components o f meets b o t h boundary components o f and m, of of and m . i s obvious. then no component o f [10] ( t a k i n g F '•'-«- ' A, i n t o compacta B, K F„, F t o be F I f no component o f meets b o t h I 0 F, such that £n m n F, F i n Bd A . are i n the F F components o f meets b o t h and ra n F, A £ and by 1(9.3) F ) , there i s a separation F» meets o n l y " t . F meets o n l y t m I * m E v i d e n t l y t h i s denies the e x i s t e n c e o f a connected subset of J m We A . We w i l l p r o v e t h e e q u i v a l e n c e . L e t t h e boundary Z A Bd A U F , o r e q u i v a l e n t l y , i f f some component o f same component o f be i n fig. 4 -•12 ^ F u £ U m which meets both (5.1). c If F and, m . f a i l s to bridge i n Int A . such that Proof: £ A, then there is a simple closed curve c bounds no disk in A We can assume that A is the set 2 2 Let D be the set x + y 1 . Let . £, 2 2 2 2 x + y = 1 , x + y = 2 respectively. Consider the component K connected set) F . 2 2 2 1 <_ x + y <_ 2 i n E m be the boundary components of £ o m v F which contains (the £ u m of £ . The set and c misses i s clearly- compact, and by the Zoretti theorem (4.4.2) there is a simple closed curve c which lies 2 in E - F - £ - m, contains K i n i t s interior and lies i n an e-neighbourhood of K . We w i l l show that by (5.1). To see that otherwise F. bridges K c Int A : K A . Thus c has the properties required contains "£ and misses K C (A D) - m = IntCA V D) . Since i s compact, K has an e-^neighbourhood i n Int (A u D) , assume that c lies i n this neighbourhood. Thus c K3£ encloses and hence = Int A . We know that D m, . since and we can c <= Int (A <J D) . But (by (2.6)); therefore c c Int(AU D) - D c bounds no disk i n A because, from the 2 "" Schoenflies theorem, c bounds just one disk i n E which is not a subset of A since i t contains This disk is .Int c D . Since c misses F (by construction), lies i n Int A, and bounds no disk i n A, the proof of (5.1) i s complete. Q . Remark: the converse of 5.1 i s true and easily proved. We w i l l look at some generalizations, the choice being influenced by later applications. - 13 Theorem 5. A . I f each of closed curve Proof: F2 F^, c in F^ be d i s j o i n t c l o s e d s e t s i n the fails to b r i d g e I n t h - F^ - F^ A , then t h e r e i s a such t h a t c A , then the p r o o f of Th. that fails t o bridge 5 i s completed by a p p l y i n g F^ U F^ no component o f F^ V F^ bounds no d i s k i n f a i l s to bridge £ U m U F^ £ U m U F^ [10], taking B , t h e r e i s a s e p a r a t i o n of K since i n t e r s e c t s both some component of A , A : would c o n t a i n . n , Once t h i s i s done, taking F^ F^ A . F t o be F^ U does n o t b r i d g e F^ A , £ and m , ( f o r otherwise £ and m). i n t h a t theorem t o be £ u m u F A . (5.1) annulus simple T h i s r e s u l t i s t r i v i a l once x^e show t h a t i f n e i t h e r of bridges To see L e t ; F^> - £ , m , By I (9.3) of £ o m u F^ i n t o d i s j o i n t compact s e t s U. , , 14 u" 2 Z c TJ^ , so that Z U m U m c. v 2 l . t o • Similarly d i s j o i n t compact sets U U V may U m c U2 , , V > 2 with Z c U V U2 misses U V . 2 Evidently O £ F^ U F and 'F^ U F^ 1 m a v and m fails 2 be replaced by.a f i n i t e * u n i o n d i s j o i n t closed sets with a few t r i v i a l changes i n the proof. 5 i s f a l s e for a non-compact union of sets and a c o l l e c t i o n F , F 2 > i n p o l a r coordinates and for ray • to AO. We remark that A , 2 Therefore Z <J m V F^ U Y^ are not i n the same component of bridge V be separated into the d i s j o i n t closed sets with ; Z c \] 2 there i s a separation of U V It i s e a s i l y checked that Z v m o' and n m c - 0 = 1/i . Although each ... 1=1, F^ such that F^, A F , i s the set F. 2, 3, does not bridge Theorem ... „ F i g . 5 2 of shows 1 ^_ r <_ 2 i s a subset of the A (nor does the union 00 {J F.) i=l , the curve c i n Th. 5 cannot be constructed. 1 We next look at the case where the curve i n Th. 5 but with the further property that point x . as F i g . 6 In this case shows. c c c i s constructed as contains a given.base cannot i n general miss either of F^,;.F 2 , - 15 - F 2 > We w i l l give so that c simple closed 'property following a c h a r a c t e r i z a t i o n o f those placements c a n b e made t o m i s s curve c one o f with base point n o t - P') w i t h respect x to closed F^, F . 2 of We .F , F , F^, say that has Property sets x ^ F a (read i f f one o f t h e 2 i s true: ^PCa): % c P(b): misses There and exists F^, F a point a decomposition arcs c^, c c^ n and F^ on o f c 2 2 misses > . 2 y & of with c - x c into c^ v = c = {x,y} , s u c h c^, F 2 misses that c 2 (see f i g . 7). This statement x and i s i s an u g l y 'c has Property by an a r c which misses we w i l l a n d awkward d e f i n i t i o n . ^ P i f f any p o i n t one o f use the e a r l i e r An e q u i v a l e n t i n c - x may b e j o i n e d t o F ^ , F ' ; h o w e v e r we w i l l statement 2 exclusively. and p r e t t i e r not prove this, T h e o d d name o f t h i s - 16 - property i s intended to r e c a l l Bing's Property P i s defined on double ended lassos (see f i g . 8). Property ^ P i n {12], Later we w i l l define on double ended lassos and i t w i l l turn out that the loops of such lassos, with the obvious base points, have Property present sense. The next theorem says that i f Property ^ P , then there i s a loop one of F^, Let A, the condition that neither in c A c' P i n the with base point which behaves l i k e F^, c x has and misses F^ F^ .be defined as i n Th. 5, including nor F^ bridges be a simple closed curve which l i e s i n and contains F , .F » c % F^ . Theorem 6. Let This property x . If c A . has Property ^ P A , has base point x £ Int A . Int A .and bounds no disk then .there ..exists .a .simple closed curve bounds no disk i h Let with respect to -c" x, which l i e s i n x , and misses one of F^, Int A, F^ . This r e s u l t cannot be improved so as to allow us to specify which of F^, F^ i s to be missed by Th. 6 cannot be made to miss and c F^ of . although exists with Property ^ P . which only F hits c . F i g . 9 shows a case where c" in F^ V F^ f a i l s to bridge A , (There are simpler counter examples i n One of these may be derived by removing F Facing page 17 - 17 from f i g . 9. However f i g 9 shows that matters do not improve i f we insist that both 'F in Z and F2 hit c .) 2 2 Proof of Th 6. We can assume that A is the set 1 <^ x + y <_ 2 2 , ' E .' The inner and outer boundary components of A w i l l be called and m respectively. Since neither of F^, F2, bridges follows from Th 5 that there is a simple closed curve bounds no disk in A and misses is completed by letting e be the further assumption that Y 'handle' of F^, Y e cf simple closed curve r c' x 6 Int e , we construct The loop of x . The whole of lies near Construction of x € e , then the proof Y by f i r s t is either e or and is constructed similarly, while the joins the loop to F 2 • The curve which x £ Int e ; i t turns out that this restriction as shown in f i g 10. a curve which behaves like If e C Int A ; thus we assume that x £ e . We make is easy to remove. Assuming that defining a lasso F^ U F^ • A ,it Y and meets Y . The lasso and an arc Y Y x misses one of as shown in f i g 11. consists of the union of a s , and is constructed so as to have the following properties: Y C Int A, Y misses one of '.:"'F , the circle r bounds no disk in A , the end points of and F2, s are x s - z misses and a point r . The construction of Y is divided into two cases. Case one: e meets erty ^ P ( b ) , since i f c c - x . We assume that satisfies Property z £ r , c satisfies Prop- P(a), we immediately let Facing page 18 - 18 - c Thus we take == c c to be the union of arcs at their end points, and for c^ 1 = 1, 2, (It w i l l do no harm i f e an arc F^ s which joins x meets both and e ri c^, (or take the obvious sub-arc of that Y ^ construction; and r Y c Int A we can assume that Case and s two: e misses c^, c^i and let let be be cted so that r misses bounds no disk in A . to r so that centre on c^ that F^ s misses because s bounds no disk in A by z . c c U F^ u F^ . b) F^ c which F^ which hit hit A 'Zoretti curve' lies in-. Int A, has Pro- to be the union plus those components of • r encloses is constru- K^c/ K^, and Some care needs to be taken to attach the t a i l (see f i g 12) so that enough to miss one F^, As usual we take c O ^-^U c) c^ c . As before, we assume that a) meets, say, C Int A . Finally, from (2,1), eve s 'misses one of the r, r = e plus those components of is a component of u If. e Y = r u s . To check only at a single point perty ^ P(b) . Outline of proof: of arcs r = e, misses one of misses both; r F^ . which meet only c^), then use (2.5) to construct c^). Let because meets misses c^ and lies so near has the required properties; Y misses one of c^ c^ c^, F^ . Construct a disk d is big enough to hit d C Int A U F^ . This is managed by a careful choice of the iated with the Zoretti curve, d) There is an arc s near u K with but small e Details of proof. Let that of = Let = c^ plus those, components of plus those components of U c U F^ \J F 2 is a component of F^ F^ which hit c V F^ U F^ . Let which contains the connected set which hit c^ . . We w i l l show K be the component U K^, assoc- U' d which has the required properties. a) s and suppose t h a t some p o i n t F^, p F^, say and c f> F^ compacta U^, U p exists in F^ . . Since By p ^ 1(9.3) F i s a compactum d i s j o i n t from c U F^ U 7^ = T h i s denies c o F U F u K ponent of Int e c u F^ U by we encloses 2 p l i e s i n one component of t n e p F^ respectively. whole of of meets b o t h F^ into Evidently c ; thus u U c t h e r e i s t h e r e f o r e a s e p a r a t i o n of l i e s with and c A, I n t c, encloses U K d i s k bounded by 2 r e c and p severally. i n a connected subset c u F misses the u s u a l argument. u F» of V K^, and a l l of 2 Since U can c o n s t r u c t a Z o r e t t i curve f o l l o w i n g argument shows t h a t no d i s k i n r p i n t o compacta c o n t a i n i n g 2 x 6 Int e Since c U F^ u F , The and and 2 u c U F * no Then . 2 lies in 2 U , the assumption t h a t b) U a n < 2 . t h e r e i s a s e p a r a t i o n of c 0 F^ but K^) U K.^* of [10], containing misses not o n l y 2 K - (K^ u r i s a comwhich misses l i e s withing a distance r bounds no d i s k i n A : which i s a d i s k , must meet p o i n t s of D c, by (2.6) r meets p o i n t s not i n encloses A . Int c . To see t h a t e of O K 2 since c bounds E . - A Since Hence the unique r C Int A : we 2 saw that c e n c l o s e s p o i n t s of E - A . These p o i n t s cannot be i n Ext m by (2.6), s i n c e m encloses I n t A o c . Hence I n t c meets I n t Z, by a connectedness argument, s i n c e c misses I n t Z, I n t c ^> I n t Z . Since r encloses compactum U K 2 I n t c, c I n t m, r misses r Int Z . Since r can be assumed to l i e i n lies and c l o s e to the s Int m . Therefore r C I n t m - Int|= I n t A . c) on r . We Clearly c o n s t r u c t a (closed) d i s k of r a d i u s d will hit (J K^. We show t h a t 2e w i t h c e n t r e anywhere d c Int A by showing . -20that dc Int m and chosen so that 4E separating c d misses Int £ . The distance ( i . e . the diameter of and Int £ , d) e could have been is less than the distance and the distance separating (j K2 and m; and we assume that this was done (the last distance is positive because K M K2 is compact and lies in by the choice of e . since points of d n r as we saw. d If d Int m). hits Int £ , By the choice of e, d d then l i e in the exterior of hits d c d C Int m K^U K^, must also hit which encloses cannot meet both c c, Int £ and Int £; thus misses "Int £ . We also assume that £ than the distance separating K^, K2> say K^, d misses diameter of F^ since otherwise The continuum V d miss Bd A, (2.5) U d be an arc in that s misses provides that 6 has the required properties: F^, F 2 from F^ . The disk (it does no harm i f d .hits both Using (2.5), let s so near was chosen so, that for, i = 1, 2., 4e is less F^ d must hit one of K^). Since d would be closer to hits than the d . d) of Since because s bounds no disk in A s has end points x Int F^ . s A x r and misses x and r (note that although misses F^ and which joins r U s c Int A misses one r meets and Bd A). and lies may To see that by construction; Y r F^ . not Y = r O s misses one misses both. The circle as we saw in b); and finally we can assume that and •• z 6 r with r n s - z =0 This completes"the construetion.of by (2.1) . Y. assuming.that x 6 Int e . - 21 - Construction during this Int £ to x Int Z u a - x . Let Q not disconnect each o f r, Int r s so t h a t misses o n l y a t the p o i n t (Int one lc Y F^, F^ . a continuum only at of a : since a r x . As suggested encloses c, Y - x x e Int r . i s connected because s - r Evidently and s i n c e which l i e s in Bd Q = Y, Using Q (2.4), does connect except f o r x . are continua in We now use (4.41) t o s e p a r a t e Evidently a c A theorem can then be used t o s e p a r a t e Int Z U a Z U a) - .x by a c i r c l e of Y continuum, so i s Y . and x . construct meets Int r - s . "Q by an a r c I n t Z, , a ([10, V I ( 3 . 4 ) ] ) . is a a point i n Int Z first Construction be t h e open s e t x 6 Int e We m a i n t a i n the assumption t h a t We w i l l f i g 13, t h e p l a n e s e p a r a t i o n and Y c' . construction. which j o i n s by of Int m Y - x c" which l i e s so c l o s e t o c" must pass through x Y x to Since which meet and t h a t i t misses (since otherwise 2 (Int Z U a) - x We know t h a t and Y - x c" C I n t A because c' l i e s so near c' bounds no d i s k i n A . T h i s means t h a t Int c ' C r u Z Y C Int m c^ a r e s u b s e t s o f a connected s e t i n that c" misses c " C I n t m. To see t h i s ; cannot e n c l o s e r, Int Z E - c')« by c o n s t r u c t i o n and I t remains t o show t h a t we know t h a t r encloses s i n c e t h i s would imply (from ( 2 . 6 ) ) , whereas we know t h a t c' Z . that separates r and Z - 22 Thus Ext c 1"'O r that c* and Int c'D £ . The fact that bounds no disk in A The construction of x £* Int e r- Int c ' ^ £ byxthe usual argument. C is now complete except that the restriction must be removed. Since the proof is easy i f x 6 look at the case that x f Ext e . Since we know that A £ and $[k n Ext e] = Int <f>[e] n A fact that A n Ext e, A A Int e if except that F^ bridges closed curve in point Let Int A A £ F , F2 while and misses respectively in <j>[A], using the be defined as in Th 5 and Th 6 F^ does not. has Property ^ P there exists a simple closed curve using the Q, Let which bounds no disk in A x . Then i f c bounds no disk in A, m, apply earlier arguments to A, o f A onto (j>[m] = £ . Then are connected to <j>(x) £ Int <j>Ie],- e t c Theorem 7. i.e. <)>[£] = m, we only f[A n Int e] F Ext <j>.[e] n A, and A - e . Then i f x 6 Ext e, fact that m, e, is homeomorphic to a nice annulus, i t is easy to construct a homeomorphism itself which exchanges implies c" be a simple and contains, the base with respect which meets F^ . c to x, F^ x, ' lies in F^, Int A, ».* Th 7 is proved in the same way as Th 6. At f i r s t glance one might think that one of Th 6, Th 7 is stronger than the other; but in fact this is not true. If pieces to, say, F^ F^, F^ f a i l to bridge so that the enlarged F^ A, one might wish to add would bridge A; this would obtain the conclusion of Th 7 which is stronger than that of Th 6 Csince i t predicts which this (F^U F 2 F is hit by c'). However i t may not be possible to do might be a number of circles concentric with Proof of Th 7. Use (5.1) with F taken to be F m in A). to construct - 23 a simple closed curve e which lies in Int A, misses no disk in A . If e meets x e, assume that x £ Int e A, e and m k F^ separates £ of F^ which hits both bridges Let as befdre. is the required Since £ and m b misses one of F^, F^ Y y' 6 e to x fact that misses Y there i s an arc b C c and connects a point of k F,, . Since and misses F2 k . bU k (as does b U k r misses to x . is a continuum e), we can construct for c U d and e F^ \j F^> whereas F 2 ; however following the procedure of the proof of Th 6 w i l l yield a lasso c" e meets some component c meets the same component r . In the proof of Th"6,. the curve to construct c'. If (there must be at least one since as in the proof of Th 6, reading e misses just and bounds e bounds no disk in by (2.7); in particular k C F , evidently b u k misses the lasso here e A). For similar reasons, which connects for then y' 6 k *~> e . Because c has Property ^ P, such that Since x, F2» Y which misses F^ • The lasso Y is used precisely as in the proof of Th 6, keeping in mind the The assumption that F^, so that the resulting c' also misses F^ . x € Int c i s removed just as in the proof of Th 6 • . CHAPTER TWO: 1. BING'S DOGBONE'SPACE AND CURTIS' -CONJECTURE. An upper semicontinuous decomposition G of E i n t o compact 3 sets 3 (or s i m p l y a decomposition - of E ) i s a c o l l e c t i o n of d i s j o i n t compact 3 A sets of E s u c h - t h a t • t h e u n i o n o f the elements o f the decomposition 3 is E , neighbourhoods space A 6 G and each element G which a r e unions o f elements associated with i s as element p o s s e s s e s a-system A 6 G, G of G . o f open The d e c o m p o s i t i o n i s a t o p o l o g i c a l space i n which each p o i n t and the open s e t s a r e j u s t those s u b s e t s o f G 3 the u n i o n o f whose elements i s open when c o n s i d e r e d as s u b s e t o f E Thus each p o i n t A of G has a system o f neighbourhoods each o f which 3 G i s open 'both i n and i n E ' . One can use t h i s i n t u i t i v e i d e a to get a c e r t a i n geometric grasp o f the t o p o l o g y o f that G s i m p l y by remembering 'some p o i n t s a r e s e t s ' and keeping an eye on the neighbourhoods; f o r example one o f t e n does geometry on a t o r u s o r K l e i n b o t t l e by l o o k i n g at the e q u i v a l e n t d e c o m p o s i t i o n space o f a r e c t a n g l e identified'. then A A I f an element A €: G i s c a l l e d a b i g element i s a s m a l l element b i g and s m a l l p o i n t s . are a l l p o i n t l i k e t of G . c o n t a i n s more than of In 'with c e r t a i n G, G . If A one p o i n t o f the c o r r e s p o n d i n g p o i n t s a r e c a l l e d The decompositions G i n which we w i l l be i n t e r e s t e d which i s t o s a y t h a t the complement o f each introduction. A is The c l a s s i c a l to decompositions and d e c o m p o s i t i o n spaces may be found i n Ch V I I of [ 1 0 ] . " Our approach w i l l be more a l o n g the l i n e s o f [3,§6]. use two main c l a s s i c a l r e s u l t s : space A 6 G is We d e f i n i t e l y assume some a c q u a i n t a n c e w i t h these i d e a s and do not r e g a r d the p r e s e n t t e x t as an adequate approach E , i s a s i n g l e t o n , then t o p o l o g i c a l l y e q u i v a l e n t t o t h a t o f a p o i n t ; i n p a r t i c u l a r , each connected. sides 3 (i.e. i) an upper semicontinuous We w i l l decomposition the d e c o m p o s i t i o n space a s s o c i a t e d w i t h an upper semicontinuous 3 decomposition) o f E i s a s e p a r a b l e m e t r i c space, i i ) t h e r e i s an Facing page 25 obvious way of expressing G as a quotient space. In this case the quotient topology turns out to be the decomposition space topology, and the canonical mapping the corresponding point <p[A] if A 0 of the quotient space carries each <j>[A] in G . We w i l l often write 3 i s a subset of E . A€ G A* for In the sequel, 'decomposition space' w i l l mean 'pointlike upper semi continuous decomposition space of is onto E . An important question i s : i f G is a decomposition space, 3 3 G homeomorphic to E ? That G i s homeomorphic to E is Wardwell's conjecture (in [8]) and i s known to be false. R. H. Bing showed this in 1957 with a celebrated example ([12]) which reinforced everyone's worst 3 prejudices against the analytic topology- of E . In Bing's example, the dogbone space of our t i t l e , most of the elements of the decomposition are small. Each big element is a tame arc (so that the example refutes a very strong form of Wardwell's conjecture), and the big points in the decomposition space form a totally disconnected set. Detailed construction of the dogbone decomposition. We w i l l describe an infinite sequence of compact sets whose elements intersect to form the set of big elements of the dogbone decomposition G . Our construction differs slightly from Bing's, but we assume an acquaintance with the original construction i n [12] and w i l l not prove, for example, that the various embeddings to be described can be assumed to be polyhedral. Dogbone space takes i t s name from the distinctive shape of the double handlecube A depicted in f i g 16a. We imagine A imbedded in 3 E . A path ter Int A, which makes one circuit of the circle marked I. Facing page 26 CO - 26 in f i g 16a is called the upper eye of one circuit of the curve marked of A m m <= jnt A A . A path which, makes in the figure is called the lower eye 3 (we imagine the dogbone placed vertically in E so that i t makes sense to talk about 'upper' and .'lower' here). One could imagine A to be a closed r-neighbourhood of a planar double ended lasso consisting of the eyes arc a £ and (with r m laid out as nice circles plus a straight connecting of course, taken sufficiently small, say less than one- third of the common diameter of the nice circles £ v m u a the centre of £ and m) . We c a l l A . The centre of a dogbone w i l l not be impor- tant in this chapter (but w i l l be needed in Chapters I I I , IV). The idea of A as an r-neighbourhood of i t s centre pin down the embedding of k k is introduced mainly to in A; we usually draw k and A as in 1 2 fig 16b. Fig 17 shows four short solid .'cylinders B , B , B , B 2 , which are subsets of A and cut into the eyes of A as the figure sug1 gests. The removal of one of 2 B , B and one of B^, B2 from A leaves a set whose closure is a cube. A dogbone can be imagined in the topologically equivalent form of a thick double ended lasso as shown in fig 18. In a sense, we are pictorially confusing the dogbone with i t s centre. Let A , . A , A^, fig 19 by embeddings h..: A^ be four dogbones embedded as shown in A -> A, j = 1, 2, 3, 4 so that the A_. = tu [A] are mutually disjoint and l i e in Int A . In f i g 19, two double twisted bands g"*" and g^ are placed so that g^(g^) lies in the 1 2 interior of the upper (lower) component of the obvious way, the centre of A^ upper loop m... parallels on £_. and lower loop A - B^ ^ B2 - B is called The g^ . The connecting arcs £^ k.., B . j = 1, 2 , 3 , 4 , In with are placed so as to l i e as a_. are laid out i n a peculiar Facing page 27 (r,&) dish c/cftn eS S. As 0< <p< 1TT/ which f * point on /ncreascs (f SlVec/>s is a. figure B.bout the. phTur of the in a toroid revelation ctrc/e. C- . A - 27 - way which i s c h a r a c t e r i s t i c o f the dogbone c o n s t r u c t i o n . coordinates Using B^" (which we- r e c a l l i n f i g 20) , we c o u l d d e f i n e be a p p r o p r i a t e a band w i t h an even double t w i s t . and 9 = < f > and thus t r a n s l a t i o n s o f the s e t r <_ 1, toroidal 3^ to construct However the bands i n the drawing a r e t r a n s l a t i o n s o f the s e t r <_ 6 = 0: 1, r <_ 1, IT/3 <_ <f> <_ 2TT 6 = 6<fi: 0 <_ cf> <_ TT/3 T h i s g i v e s a ' f l a t t e r ' band and a b e t t e r p i c t u r e . art a p p r e c i a t i o n i s the p l a c i n g o f p a r t s l i e on the p l a n e o f the a^ imposed on a. n m., 3 3 3^(3-^) 3"^ near a. a. 3^ • and t h a t the p a r t o f a. 3 except a t lying within a. n JL. and 3 £j' s o n l ^2* ^3' ^4 s w i n m.'s on j 3, > due to the unusual embedding o f the A., 1 j The and B. B 1 l o c a t e the A. A The l A 2 A 3 A 4 B (3 ) 1 1 i s called Int;(A - B 2 - lies in Int(A - B 2 - lies in Int(A - B 1 l i e s i n Int K (jy . to l e of 3^(3^) • (A - B A - B -- B 1 - 1 2 F i n a l l y we l e t - i s m. , m., m „ , m..' 1 3 2 4 A V V V V B l U A B 2 Note the o r d e r o f e i n the f o l l o w i n g way (see f i g . 21): lies in c l o s u r e o f the component of 3 a distance consists of a s i n g l e s t r a i g h t arc perpendicular t h a t the o r d e r o f 'flat' The a d d i t i o n a l c o n d i t i o n s a r e 3^ U 3-, 1 misses 3 3-j, so t h a t t h e i r and T h i s n e c e s s i t a t e s a r i g h t angled bend i n and a g a i n near that 3 k . 3^ Another c o n c e s s i o n to U A 3 U A^ - 28 - Now since each dogbone '.A embed four dogbones are embedded i n 16 A jk , j , k, i s homeomorphic to A.., A.„, A.„, A.. jl j2V j3' j4 in each. A. A, we can just as the A. 3 1 A . We could write A., = h.h, [A] . The union of the jk j k 1, 2, 3, 4, i s called 01 . The construction chosen from I proceeds as in [12] with the definition of h.h, h 0 embeds A j k -t in A., jk just as A„ t 64 A^^ = h^h^h^IA] is embedded in where A . The union is called CL^ ; The construction proceeds in this defining at each m-th stage 4™ dogbones whose union is Ct^ . the intersection A n n $ n ($^ of the 64 A ... = AQ n 2 . The components of are compact and are defined to be the big elements of G way, Let AQ while the re- 3 maining points of E are the small elements. The dogbone space the associated decomposition space of Remark 1. In k^0 ... u k^, V is G . each upper (lower) eye f a i l s to shrink to a point in the complement of any other u upperflower)^eye. iThis is easily checked using, say, Ch XV of [ 6 ] . Remark 2. We are sure that the construction of that given by Bing in [12"]. upper part of struction. k^U ... fJ \ V here is the same as In the Appendix we show a deformation of the to look like the upper part of Bing's con- We think that the reader w i l l see the p l a u s i b i l i t y , but we give no strict proof that our embedding of A^U ... V A^ i s the same as the corresponding embedding in [ 1 2 ] , and our attitude i n this paper w i l l be that Dogbone space has been redefined. Remark 3. We know l i t t l e about the hu except that they embed A in certain ways. . We cannot, for example be sure of the location of the 64 hjh^h^Ik]^. However the various subsets of a r e images of sub- Facing page 29 - 29 sets of Ajy, are images of subsets of A and continue to be related to each other in a l l the ways which are preserved by a homeomorphism of A; and we w i l l usually apply results obtained for A without further justification. Note that not preserved by homeomorphism: a^ wherever i t lies near these sets. stage of the dogbone construction. of to any A jK • • •r k_. has a property which i s 1 i s perpendicular to g or g^ This property is lost after the f i r s t This does not prevent the construction V, but further comment w i l l be required when we use the property in Chapters III and IV. Remark 4. Partly out of adherence to the traditional representation in [3] and partly because the use of and g^ w i l l not become apparent until Ch III (apart.from the fact that they cause the eyes to link together) we w i l l often use the picture in f i g 22 to describe the embedding of k^ U ... V k^ in A . We w i l l use pictures like f i g 22 in which the crossovers of the links are ignored, whenever the exact manner of linking is unimportant. In this chapter, the only thing which needs to be kept in mind concerning the linking of the Z^ and m_. i s that no w i l l shrink to a point in the complement of any other Z.^fja..,) 3 £^ (m ) . Another 3 pictorial abbreviation shown in f i g 22 i s the omission of much of the boundary of A, even though the figure purports to describe the embedding of the four centres in A . As in f i g 22, we w i l l often show only the holes of A which w i l l be represented by- the symbol # . Intuitively i t often helps to see a decomposition space as 3 E with certain sets identified. One typically finds the small elements distributed so that i t i s easy- to define a neighbourhood system for the - 30 r- big elements. Thus a lot can be. learned about the topology- of the decomrposition space by looking at elements of the associated decomposition. However i f we try to approach. V of AQ, i n this way, we find that the components which constitute the big elements of G , are hard to see. To find a big element, note that each big element of sequence of dogbones A, A., A Evidently each big element may be specified by an infinite sequence from 1 , 2 , 3 , 4 ; and the A, JK. system of this big element. Because A j , k, ... of integers chosen A., -A.,, ... constitute a neighbourhood 3 is a big element of G i s the limit of a G , then i f A A i s compact, we know that i f A lies in an open set V, of the neighbourhood system lies in V.. (see I, 7.2 some member of [10]). It J.K. • • • IT is known that each big element of canonical mapping G $ G i s a tame arc (see [ 1 2 , § 2 ] ) . The i s a local homeomorphism near small elements of (because A Q i s compact) but not of course i n general. The fact that <f> i s monotone means that [10]). cf> ^ preserves connectedness (VIII ( 2 . 2 ) of Simple connectivity properties are more complicated. As w i l l appear later, any open set V*c V which lies in A* big point of V cannot be simply connected. We must expect a proof of this property to be delicate since i t i s known that connected. and contains a ([5]). V i s locally simply (Roughly, what happens i s that any mapping of into small neighbourhood V* of a big point of in the second smallest dogbone which, contains V w i l l shrink to a point V* . Thus one can satisfy the definition of 'locally simply connected' by taking a smaller neighbourhood V* although V* i t s e l f w i l l never be simply connected.) For the rest of this section we w i l l prove a result which relates simple connectivity in V 3 to the same property in E . A mapping - 31 f of S"^ topic in into a space X shrinks to a point in X iff f is homo- to a constant mapping or represents the identity in TT^(X) X for an appropriate base point. A third equivalent statement i s : S"*" to be the boundary of a disk iff f can be extended to a mapping (1.1). Let V* <j> i s the canonical mapping of Corollary: then f i f V* frS"*" -> X of be an open,set in V . consists of small points, then ' A : A If into f:S 1 consider shrinks to a point X . V* so that mg -1 <|> f w i l l shrink to a point in V, 3 E onto V . i s simply connected then so i s V . f where •-'*"•". i - • We cfan-^uSe:':tjhis'<reS^ilehtS'Sexa®'ine.;S,§.tls '^V* -which we suspect not t(? be simply-.,connected, by looking at the associated V C E . The result CL.ll and i t s corollary are not new and are particular cases of Lemma 1 of 12]. . The proof, the .(1.1) introduces methods which, will, recur frequently in the sequel, and we w i l l complicate the (pretty easy) proof slightly by introducing more generality in the method than i s needed for the present argument. Outline of proof, disk A 7: A -* V* into V* . such that Since a) f f Assume that = f . Recall that V . The set holes such that Q c A, (open) holes of Q A* i s the union of the 0 f ^[Aj5j] i s compact. Let the outer boundary of contain maps the boundary of a shrinks to a point, there is a mapping I BdA big points of f f "*/A*J . b) Q is The mapping Q be a disk with BdA, f and the maps Q into r small points of. V; thus < ) > ^"f = f on Q . Let the (open) holes of Q be u. , ... u . For each u extend d> ^fi„, to a mapping v ^ 1 n r 'r 1 Bd u r into V by shrinking <f> ^ f i ^ ' ^ to a point in a certain cube in V . r - 32 c) Glue the Yr> r = 1, ... n, < to t>~lf|q t o f o r m a mapping of into A V . Details of proof, A o a) - -We know, that ' f :A + V*' 'so that" f = f. is compact, so is A* o because f[BdA] and , ' f ~ [ A * ] . o Note that 1 f [A*l o wl Since misses consists of small points, from the hypothesis. the disk with holes ' Q, . BdA To obtain we use the following result which w i l l be needed several times in the sequel. (1.2) Let there exist A n be a disk in disks E 2 W., ... W I n - i ) W n W = 0, r s i i i ) Each point of and S S e a compact set in W r A . Then A and - —- r j= s . 1 distance S such that -- ii) s c w . u ... M iv) If and n . Bd W^_ of lies withing a positive S . S c Int W, u Int W„ a . . . u Int W , 1 2 n' A - Int W. - ... - Int W is a disk with holes. If 1 n hits misses BdA, then BdA, then S misses Bd W - BdA for each r = 1, ... n . S ^ 0 Proof of (1.2). We can assume that an equilateral triangle. Triangulate A and that A has the form of into a finite number of 2 - simplexes (i.e. closed triangular disks) whose diameter is less than e/2, and whose edges are parallel to the three sides of the big triangle A . Note that the three vertices of only so that the three vertices of of 2 - simplexes. A A each belong to one 2 - simplex cannot be cut points of any union The only properties of the 2 - simplexes which w i l l be used are that each 2 - simplex has an edge of length less than e/2, and i f two 2 - simplexes meet, they meet either along the whole of one A edge or only at a vertex. Let S be the finite union of those 2 - simplexes - 33 A which meet lc S . Evidently A S is lc and each component of continuum. For later reference, note that S S is a cannot meet Bd S A at a point interior to A; for assume that S meets Bd S at a vertex A v £ Int A . Then by construction of A S S, /S and the entire star of S cannot meet Bd S by a similar argument. We alter —_ at the interior of an edge in S, ^ S to a set :S which has no cut po r> nts in this;way: a cut point of in lies in A v ^ Bd S . A. Int A v S cannot l i e in the interior.of a 2r-simple nor in the interior of an edge belonging to one "2-rSimplex, nor in th , interior of an edge belonging to two 2-simplexes. Thus the cut points of are a (finite) subset of the Vf.rtexes. cover each and centre (thus b tg lies on BdA t, , . . . t , , and and is not a vertex of the big triangle is a 'disk relative to s three remaining points of A A'). We do hot define b s for the since these points are never cut points A of A S . Note that the \b are disjoint. Define S to be s A S u b, U . . . 1/ b, . 1 k A A components of It w i l l turn out that the A ponents of Bd W are some of the r - A A Bd S , We know the following facts about A A S : the com-^ , S are Every point of A A S lc continua and are consequently bounded apart. (in particular every point of A A ' Bd S ) lies within a A distance S t g with a set b "s • = 1,.. .k, which is.a "disk of radius - e/6 t i f t C Int A: and is a semi-disk of the same centre s s and radius i f A Let the cut points be A e of S; the boundary of S' consists of the union of a finite number of straight arcs (which are either edges of 2 - simplexes or edges minus the interior of one or two bg ) and a finite number of seg- ments of circles (i.e. proper subsets of various Bd b ). S.uch a subset s is precisely Bd b - BdA intersected with a connected subset of St t ; s s a suitable upper bound for the number of segments is the number of b s Facing page 34 The. be a-rigfc oc wny rt(Air--}Xm- - 34 - times the number of s u b s e t s of 2 - s i m p l e x e s ) . Bd S meet as i n f i g 23; a s t r a i g h t Segments never meet because the no first n each W straight arcs i n a r c meets a segment as i n f i g 24. are d i s j o i n t . g Evidently cut p o i n t on i t s boundary and hence no cut p o i n t a t a l l . components of IV b Two S be ••• W^j . components w i l l W, and VI (2.5) These w i l l be r e o r d e r e d so t h a t the lc Since continuum w i t h no cut p o i n t , by of 110],' the unbounded complementary domain of i s bounded by a s i m p l e c l o s e d curve which w i l l be c a l l e d a ently c a c W/ . a c. , ... c I n c a Let c . a satisfy Reorder the W^ a and corresponding c c . n+1 M a c , a* ... c , i f they e x i s t , i s c o n t a i n e d i n the i n t e r i o r m J > W a = Irit c , a i ) , i i ) ,i i i ) , a = 1, ... m . i v ) of the statement P r o o f of i ) . r f If i n the i n t e r i o r of the o t h e r . s We w i l l show t h a t Int c Evid- while of , ... W I n of (1.2). then n e i t h e r of Then . so t h a t a are c o n t a i n e d i n the i n t e r i o r of no o t h e r c i r c l e each of has L e t the l i e i n the d i s k s r e q u i r e d i n ( 1 . 2 ) . a =.il, ... jn.> *-ts a (9.3) S c , r = W O Int c r c s i s contained s O W r D F0 Y s 1(1.6). P r o o f of i i ) . a = 1, W a for . ... m, Hence n'+ since S c W n 1 Ext c u 1 <^ a <_ m, 1(1.6), I n t c = W C ' a a s = w;w ... ( y f . Each W c I n t c , 1 m a a i s the unbounded complementary domain of S c .. . u c W W . m And l i e s i n some r i n fact Sc W. u I n t c , r = 1, ... U I ... n, W n because and by- . At P r o o f of i i t ) . Each t h a t a l l of (the c l o s e d s e t ) Bd W S r is a c C r l i e s near S . Bd S. * and we saw e a r l i e r ' . Proof of i v ) . S to BdA if S Take misses e BdA less than the distance from (compact) . The rest follows from the definition of a disk with holes, from 1), and from the fact that a point of and hence a point of Bd S misses- S wherever i t lies in We now return to a) in the proof of CUI). is compact, from (1.2) there are disks and such that distance e W^, ... W^ Int A D. f" 1 [A*] Since which l i e in A _1 f [A*] C W. U ... U W and each Bd W l i e s within a L 0 1 n r of f - 1 [ A § ] . Since f^I-A*]' misses BdA, by (1.2)iv, the set A - Int W.. - ... - Int W = Q is a disk with holes. I n be the 'holes' of q, r = 1, ... n, i.e. u r == Int W^_. b) F"1[A*] Since lies in the holes consists only of small points. Thus when restricted to — f[Q] and find cubes in which to shrink _ f[A] O. A* Bd S -1— <}> f |Q <f) ma ur of Q, Let u r flQ] is a well defined mapping ps <f> ^ f | B ( j ' r >r Q = into Vc E 1> ••• n • 3 . We now Since is compact and the dogbones (considered as sets in V) evid- ently form a neighbourhood system of A*, f[A] n A* by a finite number of dogbones (J there is a covering of J * , ... J* each of which lies 1 q 3 in V* . Look at the corresponding J . , ... J in E . If J , 1 q s s = 1, ... q, belongs to the mth stage of the dogbone construction,define J ,, si J J s2 lying in ' ° J s s3 TJ . s4 to be the four dogbones of the . Note that since J C V, s each m + 1st stage J . lies in a cube M . sj S3 which is a subset of AC V etc.) J . and hence of V ( i f J were the dogbone s s then J , C M -, = A - B2 - B 0 , J _ lies in ti „ = A - B2 - B. sl si 2 s2 s2 . • 1 Now in a) above, we could have chosen e so small that each f[Bd W'-] = flBd u ] lies in some J* r r sj Cfor V is a separable metric - 36 space, and there i s a minimum distance i n the dogbone metric separating the compact set f { A ] ^ A* from the complement of the union of.the J* .). SJ U <> j f[Bd u^] is defined and lies i n the union of the J g ^ . We Clearly can assume the J . are dis-joint because we could have removed from the S covering J J ^ , ... of the covering. any J g which was contained i n any other member Since J S J. are separated,I T (j)" ^"f [Bd u j" i s connected and the closed sets J <> j ^f[Bd ur] lies entirely i n some one sj J . and L d shrinks to a rpoint i n M . c V . Thus there is an extension T> f i „ , sjJ 1 Bd u r Y 'r Y of T d) "*"f i „, I Bd u^ to a l l of u , i . e . y : u r r r M . C sj V and r|Bd u = *"lf|Bd u * 1 1 r r c) In view of the set-theoretic definition of function we can express the idea of 'mappings glued together' by unions of mappings. ij) ^ f i . V y,y .. . U y . This i s a well-defined |Q 1 n mapping of Q v dom y^v ... v dom Yn = Qu u-^ u . . . c u into V because Consider the union each mapping in the union has i t s image in V. and because where the domains intersect the intersection i s closed and the mappings agree on the intersection; i n fact every point of domain intersection occurs on a Bd u r of where'we know that . Finally we note that the new mapping agrees with < $ > y agrees with r -1— < f > f l g u^ ' r by construction dT^f | ^ U T± u • •• V T n <f> "*"f on BdA C Q and is thus a homotppy which shrinks to a constant mapping into V . This completes the proof of (1.1) C » We w i l l record the argument i n this paragraph as a separate result. (1.3). that Let A, W^, ... Wn be .defined as i n (1.2) including the fact A - Int W. - ... - Int W is not neccessarily a disk with holes. J 1 n Let g:A - Int W, - ... - Int W 1 and i f g u , 1 Bd W . Then each g i ^ , IT Bd W is defined; r shrinks to a constant mapping in a space §|BdA Proof: 3 r then there i s a mapping of icular, E n A into P , r = 1 , ... n, r g u P^ u .. • " mg w i l l shrink to a point i n In part- g 0 P^ o ... V P^ . mg The argument of c) in the proof of . (1.1) is valid is-3used;and even i f A - Int W- - ... - Int W i s not a disk with holes. It i s easy I n to see that BdAC A - Int W, - ... - Int W since W c A; then g is 1 n r defined on BdA .. Since Bd W C 4, , g | H is always defined Q, M Proof of the Corollary to connected, we can use V only i f rng To construct cover (1.1), . AQ IJKS1 -»• V . if) ' - ' If V* is simply shrinks to a point i n Evidently in order to apply i|> is homotopic in is sufficient to show that use r to show that (1.1)' misses i^rS"*" -> V - AQ .We Let (1.1). ' V (1.1), it to a mapping to mean ? i s homotopic i n V to' . TJJ': using an argument like that of b) in the proof of ^[S"^] D with dogbones AQ and l i e in V . Dogbones J ^ , ... which are disjoint J '., j =. 1 , 2 , 3 , 4 , are defined just as in SJ b) of so that ^ J . covers T!»IS"*"] O A and each J . lies in r r] r j 0 sj a cube M . C V (the construction of the J . here is not identical rj sj to that of the J . in b) of ( 1 . 1 ) , but the construction here i s easier sj (1.1) n since we need not consider sets in V). exists in and the tp[S'] n(E3;- - U Int J Int J . are separated, r sj ) , for otherwise, since rng il; lies in one ° the proof i s concluded by shrinking choose § > 0 the distance so that i f x 6, then and i|>(x) and We assume that some point y z S 1 - i s connected J . sj M «, sj' and i|> to a point i n Mg.. V . Now are closer together on i|>(y) a r e S"*" than closer together than the - 38 Y[S''] P A . to E - .v< J . (remember that 0 sj sj sj sj neighbourhood of ip[S"] O A ^ so that this distance is positive). distance from point of S 1 inition of ip' = ip . lies closer than 6 no point of S"*" maps under If some point of If every - 1 3 / / ip {E - \s J .],' then by the defsj sj to 6 is a ip into A ^ , and we can let fails to l i e withing { i|i "'"IE3 - ^ of "'sj''5 then there is an open interval: e' in ip ^[^, Int J .] such that the ' • •' • '1 sj sj length of e^ is greater than 6 . Let e^ be the largest open interval e' c e. c il "*TM, Int J .] . Then 1 1 sj sj such that length greater than 6 whose end points by the usual continuity argument. p^ (Since an easy allowance for the possibility that the J .. are separated and sj of just one Bd Rx. ^ij^ J • • - Define the mapping i s a 'Path i n ip-rS (connected) (this is well defined because ip "and ip^ are paths in V ± Bd j^j J g ^ is a c i r c l e , we must make ip(p^). = ipCq-^) = z.) - R , while Kp-,) J• . 1 so that and = ^ q^ map into Because lies in the interior i ipCq-',). l f e in • . 1 s1 - e o n with end points. ip.^) p 1 ' and and S^" - + E Bd R and -q^ map into i s connected, tple.l x tple..] x which we w i l l c a l l ^ . sj is a closed interval of e, 1 Bd R while and i p ^ ) under ip).. Both ip |— -.tp-^—. because they share end points and both map into the, same cube M . D L with M . <Z V. Evidently ip - ip . ' • • SJ 1 "- J -W, ' ^ ^1 3 Since rng ^ j — C Bd R^C E - A Q , the homotopy has moved images of points in .1 _ .......... i — e^ away from A Q . We now look for an open interval in S e^ where e^ is of length greater than 6 and such that e£ maps into : S U i n t 3 . under sj sj If , e^ does not exist, let ip^ =• ipf , open interval C into ip, (and in fact under 1 sj I n t e2 J ip, since If e^ of maximal length; such that s j • The end points p2» q 2 on ip =. ip, 1 §^ ^ e_). 1 exists, then there is an e^c of e2 e2-C S,^ - e ^ and map under ip E 3 - ^/ Int J ,, either because of the maximal!ty of sj sj 1 e„ l i f the ^ 39 end point Is i n S 1 - or, i f the., end point is i n Bd (S 1 - e1) = Bd e ^ if) [Bd e ^ c Bd R^ . By a continuity argument, ^ [ e ^ lies i n because the interior of some one J in called 1 3 Bd R2 . Define T\>2'.S ^2\e that ± s a Path i n B d R so that -* E (note that this means that and ^ 1 (p 2 ) i ^ 2 and ^ C q ^ ) l agrees with w i t n e n d Points ^(P^) a n d ^1^2^ ' if* - ip 2 and - e^ - e 2 e 1 on S - e 2 lf^ By a previous argument, ip - ip^ - if) . Note that the fact that on i agrees with ' if) on S 1 - e^ - e 2 ) , and so ^ 2 R2 2 means that . ip = if) on the end points of both 2 e^ e2 . In general, suppose that mappings into V, intervals .... R^, ... R r _-^ ° f e^, ... er_-^ and components have been defined so that each ' e C-S''' - e_ s 1 1 '' ° f S1 ifi^ * ... * . "• — e s-1 . .' ' ': '" " , J s j s = 1, ... r - 1, 3. if) , -if) ~ = ij) 1 onS - e , ip"-[e"'] C, Bd R:? d E - A . Now s s s-1 ... . s s s s . o look for an open interval e' c - e - e^"- ... - e • ' such that the r r 1.-2 r-l and for each length of e^ i s greater than alentlyJ 6 and ij^ -^te^] CL ^ ib , [e 1 CL Int R , where r-l r r R i s a single 0 r Int J g . , or equivJ . (and hence sj lies i n a cube M . C V). If there i s no such interval, l e t i> , be sj . ' r-l ty' . If e^ exists, then let e r S^" - e, - . .; - e , such that 1 r-l i^ _^ r \b , [e ] C Int R r-1 r r . We know that 3 p^, q^ of e^ into E - Int R^ by i f the%end point i s i n S''" - e ^, carries the end points the maximality of e T be the largest open interval i n or, ,if the end point i s i n Bd(S^ - e, - ... 1 because \p . ' carries e, U ,., l i e . into r-l 1 r-l if) ,Ie ] lies in some Bd J .: for s = l , . r^l s sj by construction. ri> ., 0agrees with J s+l - e .) C e, (j ... U e .. , r-l 1 r-l {J. Bd J . . (To see that sj sj . . r - l , il) [e ] C Bd R s s s if) on .S^" - e , - D e s s+1 s since lies i n S 1 - e x - ... - e g ; eg+1 S l " a+2 e S i n C e S Z) o n with e s 'i> = > » 6 s+2C e t c > that the end. points "of i|T •-rjf-r^]. Minis'JU, r-l r r r - ^ g + 2 .agrees with .. . - 7 . o n $r_2 if)^ ^ e^ are mapped By e • Since s outside of Int R^, and r-l r * . ....... . r-l ^r-l^r^' E v i d e n t l y - -. , , - i er r a n d e 1 • r" 6 and r6 must be less than the circumference of S"*", ( i t is easily checked that the e^ are disjoint) if)' be ij)^, . . . i f ^ , .... ends at if)^ . Let the sequence if), . We know that :S"^" -* V because each ijj does this; if) - ip - . . . - if), = i p ' i n V . Before we can show that and 1 X misses mg if)' AC A Q , we must show that if) = if)' = if^ on S"^ - e^ - ... - e^ . To see this: I(J£ = if) on ^2 = ^1 o n if)^. = i f ^ - i e^ , 1 S ~e2 a n d carries every x£ y J on . S^" - e^ and if)^ = if; on S^ - e^ - ... - e^ . yc — eg into rng if)' = mg if; misses A Q , for I(J^ 3 Bd R g C E - A Q by a previous result; and i f S"*" - e- - ... - e , then such that 1 *2= ^ '~on"! ? ~ e l ~ e 2 ' It i s now easy to show that x lies within a distance 6 of a point if;, (y) € E^ - U Int J . . We can assume that k sj sj - e^ - e 2 - ... - because otherwise y £ e^u ... U . with end points "~ * r - l ' r is of length greater than r . -.. . . . . . * r | e '~.. * r _ i | - Since each r S - e - . and \b i — be a path i n Bd R r '•• - - -"re •. - r r , ' a n d agrees e , if) . [e ] = if, [e ] C Bd R ) . • Since we know - - i / ' / 1 ^r_l(Pr) °n s + 1 - 7a+1; _ i agrees.with r ^ bv continuity, > - ( p _). a f i d ~ i|> - lq ) l i e in Bd R \1>-r = \b•, , on r r - l ~- Let S1 - e ^ u n t i l - . . . = if) , on if) and Facing page 41 Ay. may lc shrunk^3L}AtjAf WW m/ss a.t least one Di. U yitist Jut loth. A lA m*y. he 1 t ezch of 4j,A}. trusses one But 7)ew/Ij Vied- loth. - 41. - y" of Bd(e^O some point to c x than y does. Since ^ in and S 1 - e^ - ... - e^ lies closer carries y* G Bd(e^V ... U e^) ^,Bd J . C E 3 - 1/ Int J ., we could have sj sj sj sj' e- U ...v e, into 1 k originally chosen ... <J e^) y" instead of y . But i f both 1 S - e^ - ... - e^, then since x and y' l i e 1 ip = ip^ on S - e^ - ... - e^ ip, (y-*) ^ F? Int J ., ip(x) = rip (x) lies so near to r k^ sj sj k tyiy") 2. = ^(y - *) that ip(x) misses A^ by our definition of 6 P e In his paper [12], Bing was concerned with an interesting property of G which we w i l l make use of here and i n Ch. IV. The formidable aspect of G lies i n what might be called i t s 'topological idiom', as shown i n f i g 25: four double ended lassos strung i n a special way inside a 2-holed torus. struct Bing's intent i n using this idiom to con- G was to u t i l i z e this property: l e t D^, D^, be the planar disks shown i n f i g 25. Then, no matter how the four lassos are deformed (provided that they remain linked and stay i n the interior of the 2-holed torus), some one lasso w i l l h i t both and D2 • Figs 26, 27 show unsuccesful attempts by the lassos to avoid this necessity, and there i s a proof of a very similar idea i n §7 of [12]. Bing hoped to show that this property was induced through the construction of G i n the following sense: assume that f i g 25 shows D^, i n relation to the f i r s t stage of the construction of G, then, no matter how A^,. A 2 , A^, A^ were deformed, one of these, say A^, would h i t both of D^, D2 . Additionally, however, i t might turn out that for any deformation of inside A, one of the 16. A ^' would hit both and D2, and so on for the 64 - ^ j j ^ etc. Bing found that there was no easy Facing page 42 - 42 proof of this (see §7 of [12]); however he was able to define a pro- perty which he called that A Q on the dogbones of the decomposition and show had this property. If a dogbone had property t r i v i a l l y that i t intersected both of could be shown that i f a dogbone B B^; D^, had property Q, this implied at the same time i t Q, then one of the four dogbones of the next stage of the dogbone construction lying in B would have property Q . Evidently there would be a descending inter- section chain of dogbones each with property chain would be a big element of G Q which touched both We can express this idea in a slightly different (2.1) ^.^ (Bing). Let l i e in A G and . way: be topological disks whose boundaries and link the upper and lower eyes respectively of shown in f i g 28. of and the limit of the Then either meets both and metts in A, g^, A as or some big element T)^ • We w i l l refer frequently to f i g 28, which shows the relationship of C^, to A . 1 be an embedding of Strictly speaking, we take c^, i = 1, 2, to 3 S in E ; however we frequently w i l l confuse the embedding with the circle which is i t s range (at^-the same : t i i i i e T r e s e r v i n g the right to write mg c^ when we wish to make the distinction clear). Bing showed that (2.1) was inconsistant with the existence of 3 a homeomorphism between V and E (Th 12 of [12]). In this paper we w i l l be interested in this conjecture: 3 (2.2) . so that E 3 - A Let A be a 2 - simplex. For f.i-r,,. = a \ i|BdA 1 and f„i„,A = C „ 21 BdA 2 i = 1, 2, let f : A -> E are vpaths whose ranges l i e in & and which w i l l not shrink to a point in the complements of the - 43. upper and lower eyes respectively of A . Then either f^lA] and f^[A] and intersect i n A, or some big element of G meets both f^[A] f 2 [A] . In (2.2), we replace the disks disks of (2.1) with singular f^[A] . The conjecture is plausible and lacks earthshaking sur- prise. It i s interesting because i t leads directly to the following topological property of V: (2.3). If (2.2) i s true, then V fails to possess arbitrarily small simply connected open neighbourhoods about any big point. The conclusion of (2.3) i s called Curtis' conjecture (see 3, §6), and (2.3) reduces i t to the somewhat more plausible, conjecture (2.2). The remainder of this chapter consists of a proof of (2.3). The pleasures of (2.2) w i l l be deferred to Chapters III and IV. Proof of (2.3). that iii . V, .We bones A i s a big element of G and A* = <|>[g] lies i n a simply connected open neighbourhood V* such that 3 E Supppose that A* c V* c A* • Clearly could write 3 A (1 A j k ^ *** A=A A, A'., A AC V c A, and V s o m e s e c l i s open i n u e n c e ° f dog- By the Corollary to (1.1) (of lemma 1 of [2]), Jk V . is simply.connected i f V* i s . Thus our assumption implies that V is simply connected. We w i l l demonstrate that this i s false by showing that t AC V C A with V and the lower eye m simply connected, implies that the upper eye of A shrink to a point i n A . We define an upper (lower) principal path of A_. to be a mapping of S"*" into which is homotopic i n A. to the upper eye t. of Int A^ (the lower eye m.) A. . Upper and lower principal paths of other dogbones, including - 44 - A, a r e d e f i n e d a n a l o g o u s l y ; t h i s a mapping o f which i s homotopic i n A A . 3k to Jk h., [&] into Int A J K- i s an upper p r i n c i p a l p a t h o f Jk As u s u a l , we w i l l o f t e n confuse the mapping w i t h i t s range. know t h a t A, A_. , ••• by a p r e v i o u s remark, i s a neighbourhood system o f t h a t some member A., j in S^" V . iC A and, o f the system • • • We lies ITS However t h i s f a c t p l u s the f o l l o w i n g lemma l e a d s t o a c o n t r a d i c t i o n . Lemma f o r (2.3). upper p r i n c i p a l p a t h e^ and l i e i n V, A intersect then I f one o f A^, j = 1, 2, 3, 4 and a lower p r i n c i p a l path c o n t a i n s an which intersect c o n t a i n s upper and lower p r i n c i p a l paths which I n g e n e r a l , i f A. contains i n t e r s e c t 3 ... r s • i n g upper and lower p r i n c i p a l paths which l i e i n V , then so does A. j and l i e i n V . . .. r To a p p l y the lemma we l o o k a t the neighbourhood which we know to be a neighbourhood o f V D A. 3 ••• of r s contains i n t e r s e c t i n g A. s i n c e any i n t e r s e c t i n g j • • • rs The lemma i m p l i e s t h a t the dogbone A in V . A. 3 •• • r s Obviously upper and lower p r i n c i p a l paths p r i n c i p a l paths w i l l A.. 13 ••• r contains qualify, intersecting p r i n c i p a l paths i n V ., Repeated a p p l i c a t i o n o f the lemma l e a d s t o the conclusion that Since A V c A, V A c o n t a i n s an upper p r i n c i p a l path which l i e s , i n V . i s s i m p l y connected, and the upper p r i n c i p a l paths o f a r e a l l homotopic t o in A . t in A , therefore T h i s i s c l e a r l y f a l s e from f i g 16a. Z must s h r i n k to a p o i n t Thus the p r o o f o f (2.3) w i l l be complete when we have proved the lemma. x that P r o o f o f the lemma f o r (2.3): e^ and Simplified version. Suppose l i e i n A^ . • The f o l l o w i n g o u t l i n e r e f l e c t s our o r i g i n a l i n t u i t i o n o f the p r o o f . A l t h o u g h the ' p r o o f we g i v e now i s Facing page 45(1) /// Facing page 45 'di) A nox-pl«n*r dL Tndy Tffect A3 [t is neitf ConsTriut pL Cd fivsilh px so £t*>? t/ees -n*t y . t/>e U/fzr v /4 l„uir A»Je <?/ 4 ?e qui re J. Fig 31b. curve- pj way censtruotcU 31*.. und f* l'°f> once aiout iS is - 45 - simple minded and needs much patching, we give the crude version because we think that i t clarifies basic ideas which tend to.be submerged in the final version of the proof. &2 Suppose that by good fortune the paths take the form of the double ended lasso of circles C^, and connecting arc J a in f i g 29. J consists as shown. The circle w i l l not shrink to a point in the complement of the upper eye of Similarly eye of A^ . We also pretend that d^ bounded by lies in V C^, V 3 d^ contains every element of ' G V (remember that meets both of A^ d^ and a, d^> d^, g« p^ shown in and p^ U P so that 2 g By (2.1) lies in that i t intersects We can now f i g 30 from parts A similar procedure using w i l l yield the lower principal path intersect in A d^l is the pre-image of an open set in V). V construct the upper principal path lying in and that disjoint also l i e in V . g since in A^ J some big element V A^. w i l l not shrink to a point in. the complement of the lower planar disks of e^, p 2 . The paths A^ instead p^, p2 is the set required by the conclusion of the lemma. The above 'proof is far to easy and w i l l f a i l i f we allow d2 to be non-planar, for then show-. We ensure that by trapping P l ° ^dl ° ^ d2^ = may not be a principal path as figs makes one .circuit about the upper hole of 2 in the cube A - B p^ C\ a g p^ p^ q i n tlie C U b e A 2 - B ~b1 ~ B 2 (which is easy) and (see fi § 32) • T n i s last step is hard since one would fear that the connectivity would be spoilt by parts of d^ u d^ projecting from the cube. The trick of controlling the homotopy class of d^, p^ by constructing certain arcs in 31a, b A Facing page 46 A - 46 l i e in A - B 1 - TS^ . But i f we use the cubes only works i f the obvious candidate for J, v i z . C. = e. 1 with arc a degenerate, 1 then f i g 33 shows that this may not happen, and in fact usually be e^U cannot • However we show that, provided that intersecting principal paths exist in A^ n V, with singularities) in AO assigned to J V there is a double ended lasso (perhaps which has just the properties which we J . We w i l l now give, the final version of the proof of the lemma for (2.3). This proof uses the ideas of the earlier crude version, but incorporates the various improvements suggested in this paragraph. Outline of final version of proof. assuming that. e l ^ 2 -*--*- ^ e s e & n A 2~ < ^2' A 3 lower principal paths of l' °r A^ t n e n A We f i r s t give the proof indicate alterations in the case that 4 " a ^ L e t e l* which l i e in V 2 & ^ e u PP e r a n d and intersect at least at p . We follow the sketch of the 'proof already given, but as previously explained, we cannot use J = <J a U we w i l l regard e^ U e 2 for J in f i g 3j2.. We construct so as to satisfy five properties i ) , ... v) . Sometimes as a mapping (not necessarily an embedding) of and sometimes as the range of this mapping. The set following properties: i) ii) for J must satisfy the i = 1, 2, C±[BdA] c V r> Int (A - B1 - B2 mg misses A^ . 3 iii) iv) S^ (C2) fails to shrink to a point in E There is a point y^ € rag ry mg e^ n y 2 i n rng C2 O mg e 2 / l A - B - B2 - ^ 3 - £^(E - m^) and a point (recall that is the topological cube which i s the closure of the upper 1 component of A - B 2 - B , see f i g 34) . Facing page 47 - 47 v) the arc a C V ^ and the points -y , y 2 of iv) are the end points of a . The idea of i i ) and i i i ) i s that we want the Gi C^, C 2 to act like the circles i n f i g 28 with respect to A^ . Property iv) provides the end- points of a 'above and below B^ ' . This plus v) and the fact that the are trapped i n A - B^ - B2 alows us eventually to construct an upper principal path of A which winds one around the upper hole.of. A . This happens because we w i l l join y^ and y 2 in f i g 35. b) For i = 1, 2, l e t f. :A -»- V x Using (1.2) and (1.3), obtain a new mapping on A - Int in A; i n particular 1 by a path like so that f. . ' = C. . l|BdA I which agrees with ."f - ... - Int W , where Int W . r = 1, ... n, are holes 7 n r ' f = f^ i n BdA . The f i [W r ] may leave ± but this does not harm the proof, c) • By (2.2), q from and lies i n V in f1 [ A 3 , an to a n passes from The path f ^ A ] to f 2 [A] i n A 3 a n C2 by. means of f 2 t A ] • d ) travels to a A G 2 on q T n e q q in f i g 35 travels to A 3 either at the inter- section of f^[A] and f 2 [A] or using the element to g i n A^ ,. There which resembles and i n Int (A - B 1 - B 2 ). V (!) f f A ] and f"2[A] either intersect i n A^ or hit the same big element is a path q ^ A - B^ - B2 g, and then proceeds path which begins at a n C^, and returns to aa on a, i s an upper princ- ipal path of A which lies i n V , e) The lower principal path of A in V may be constructed as in a), b), c ) , d) above, using A and 2 1 A - B1 - B instead of A and A - B - B2 . f) If k = 2, 3, 4 the 2 3 lemma remains true. Details of Proof. Suppose that e.^, e 2 are upper and lower - 48 - principal paths respectively of p . For i = 1, 2, mappings e^:A -> V since e. and, intersect at shtinks to a point in V, there are e such that e ^ • We'do not claim that e^ e,. 1 e.^IA - B"*" - B„] . Thus• there are disjoint disks 1 z l i e s near 1 x^BdA W which miss BdA r . e" ^ - B if r A . We use (1.2), taking Those W 1 - B J, which h i t r BdA are W ,,, ... W n+1* e, to be the r e s t r i c t i o n of e r = n+1, ... m, misses Thus for = i|gd W BdA for to 1 Wn, 1 misses to be ... W m are called L - B'] 1 1 W. , ... W ; 1' n 2 those J We now apply (1.3) with 1 Ps and of (with obvious adjustments of one m faa x in e " [A - B A - Int W. - ... - Int W ^Ifid W r > n x6 Bd and 2 or the other class does not e x i s t ) . Bd W r f , S, e l 1 [A - B 1 - B„]c:W- u W„ U ... u W , each point 1 z 1 z m such that Bd W^ or even = ^| mg A lies in A^ which l i e in V i g taken . For m n t o Ext(A - B^ -•B^) • because hence misses — e,1"-1 [A - B 1 - B_] 2 . e. |„, TT shrinks to a point in Ext(A B''" - B_) 1 1 Bd W z r 3 1 which is the exterior of a cube in E ; and we can let Ext (A -• B - B2) be e r = n+1, ... m, P n+1 ,-=...= Pm C l|Bd W' in the hypothesis of (1.3). Jf v i ^or We suspect that r > n ' is an s i n c e -^3 misses e^lgd y ^ o r r — r e i l t j j TT J. 1 b d w §IT>J = 1 r letting P ± taken to be mg TT Bd W = P w i l l shrink to a point in r z = ... = P P ... = ...= n+1 n be E 3 There is no chance that Ext(A - B^ n B2) = . • Assume that every 3 E - -£„i . Use (1.3) 'again, ....... - I. . Then with J Ext (A - B 1 - Bj z P , gi„,, = e, w i l l shrink to a rpoint in m' °IBdA 1 g U P. U ... U P u E u ... u P . Each 1 n n+1 .m by-de^tnit-ipri of -because\'P misses A - B"^ ~ B P 2 r misses • And 3 either rng g misses Facing page 49 - 49 l as w e l l ; f o r 3 g = e ig _ _ 1 A - Int W^ - ... - Int W^ and from (1.2) i i , i v , m which can map into 1 the only points of A - B"*" - are those i n A rng g C A BdA . Such points are i n 8|T>JA I BdA Hence = e i 1 U Ext (A - B shrinks to a point i n contradicts the fact the — f a l s e that every e^ - <=• E rng g U P, y 1 - £ . • Therefore 3 ...UP C E — £_, which m j i s an upper, p r i n c i p a l path. Thus i t i s 3 E 3 6J. J B ( ^ , •ei , 1 Bd W r <_ n, and map into 3 1 . dom e^ shrinks to a"point-in - £^ . Let r C be one of the 1 1 As regards C^: which f a i l s to shrink to a point i n 3 E - £„ . r _ the above argument plus the fact that rng 3 C rng e ^ c V shows that i i i ) i s true; i i ) i s true because from. (1.2) i v , every point 3 1 x i n Bd W^_ i s either i n Int A, i n which case C^(x) € E - (A - B - B^) 3 3 C E - A, or x 6 BdA, when (^(x) = e ^ x ) 6 A± C E - A . In 3 3 general,i) i s not true because some candidates f o r C^(x) A - B^ - B lies in as we have just seen. 2 Int (A - B"*" - B^) assume that ej [A - B 1 that 1 dom C. 1 - B] = g ^ e^ have replaced of K (which i s one of the on A - B lies in However we,can assume that by the following argument: that, rng 2 1 - B 2 . . 2 that Bd W r Int (A - B rng C± c Int (A - B misses S 1 2 - B) 2 of A - B 1 - B 2 (remember . such that an e-neighbourhood Such a cube i s shown i n f i g 36. rng - B) 1 £ K => A^ by a cube Int (A - B"*" - B ) i . e . rng By (1.2), we In this paragraph a) so f a r , we could this had been done, we would have K, C^[Bd W ] Bd W ) l i e s so near r l i e s within dom C^) l i e outside of i n the If e-neighbourhood of . We assume that this was done and Proof of i v ) : ( i n (1.2)) except where Bd W r In (1.2) i v , each hits BdA . In the present context, with A - B 1 - B 2 ), misses - 50 -r. for S e^fK] the domain of ( i . e . continuing to. use is a . To show that there is a K Bd Wr and C^Bd y^ 6 mg \ mg • C,[Bd W ]/) e, [BdA] n K. " C. [Bd W ft BdA] f) K_ , 1 r 1 1 1 r 1 C [BdA n Bd W ] Z).^ = 0 . J Then . C^Bd 1 because C^[Bd Wr of . mg <- E 3 - BdA] - assume that misses and 1 5 D K ; and ^ as we just saw. K K2 2 . - BdA] /) fC^ = C.jBd Wr This means that a l l shrinks to a point in which contradicts the choice of K , m„ C K„ c K <0 K„ . and 1 Wr f) BdA] . 0 ^ = 0 because - BdA] e^ - In f i g 36, note the two cubes £_ C K c K n K. which are placed so- that Wr K for 3 E 3 - E - . We repeat the entire procedure of this paragraph a) taking e2, e 2 , m1, m^, K 2> A - B1 - B2 - K^, .K e±, e±; l±, for ¥L± . ±i This is just the preceding argument 'upside down' and constructs the path C2 3 y 2 as required. The only unexpected thing is the use of the cube A - B"*" - B2 fact that y^ for the original cube should be found in B^ 1 . but merely 'in A^ and below by i v ) . mg Let To construct e^ C A^ O V; .a = a C C V, i • f^:A -> V a, join y2 to not necessarily ihr\ K2 We now have y^ p y2 to p this reflects the y^ and by a path by a path a2 in a^ mg y2 as required lying in e^c A^H V . (j a ^ . b) mg and and K^; We can assume that C. i = BdA, shrinks to a point in V such that us a big element dom g' ^jedA = in V/1 C i * i = 1, 2 . Since and there is a mapping * t ^aPPens t o ^ e true that (2.2) gives which hits both f^A] and f 2 [A] - 51 (unless they i n t e r s e c t ) , but we a r e not sure t h a t t h e r e i s a connected set f £ A] in that w i l l j o i n ± %' and, y so t h a t i t i s n o t y e t p o s s i b l e to b u i l d f i g 35. By are d i s k s (1.2), t a k i n g WI\ ^ 1 Since in S t o be A such t h a t (A - B^ qC'V H Ext(A - B- - B ) 2 1 i i A - I n t W - ... - I n t W i s a disk with holes. 1 n Bd W l i e s so near fT [Ext(A - B - B j ] that r l 2 1 Ext(A - B 1 - B ) . 2 l i k e , we- can c o n s t r u c t that N then f ^ [ B d W*] N = P 1 i s simply = P 2 such t h a t f,, = f . i l connected. shrinks on BdA = f.[A - Int l . N to a p o i n t i n and element f [A] 2 and looks 2 2 f.[Bd W ] C l r N; 1 g = f., i „i , .there, i 1 A - I n t W. - ... - I n t W 1 n A y T fc ... U'.P I T T =f.[A1 1 I t i s important t h a t n . fc I n t W?" - ... - I n t W^UN n" In p a r t i c u l a r r f . [ A - I n t W* - ... - I n t W ] l 1 n 1 1 Since --rng C.^ = f [BdA] = f^tBdA] in A^ f [ A ] either intersect i n 2 . A3 misses A^ A must l i e i n in A^' Since rng f . - N c i f ^ [ A j c: A - I n t W* - ... - I n t W^, A^f by o r h i t the same b i g f o r a small 'f^[A] A and a l l o w i n g f.[A - I n t i and f a i l s eyes o f We can combine these ideas by s a y i n g a b i g element o r a s m a l l element. I - B and by (1.3), t a k i n g N; A - I n t W, r- ... - T n t W meet the same element A O rng 1. 1 - ... - I n t W ] C V . I n f ^ [ A ] and X 2 Ext (A - B"*" - B ) so of to s h r i n k t o a p o i n t i n the absence o f the a p p r o p r i a t e (2.2), - B )l, 1 1 A - B e, and t o be e i t h e r N misses A^ , - ... - I n t W ] c V,, n 1 1 . n j We can assume t h a t each f.:A -> r n g g WP. f . = f . • on i i c) E-neighbourhodd 1' f .[BdA] C I n t (A - B S i n c e we know e x a c t l y what = . .. = P , n' i s a mapping 1 an U-..-. \J W We assume t h a t each f.[Bd W ] l i e s close l . r n 1 to there ... 2 (because 2 as i n T - B„)]CH f~ [Ext(A - B - B ') ] 1 2 2 l f~ [Ext(A - B - T> > B ) , 1 n BdA. misses A -, B and s t a y i n ± which we saw was a d i s k w i t h 7 Facing page 52 V' 7 - 52 c ^^y^) c holes and which contains and l i e - in the image under into V,. there.is a path ^ more V'^C A - B"*" - B 2 , d o m c x f\ ^B d A • Since X n f^lA] of a disk with holes which, maps which, joins y^ and A in V . Futher- because v'^ may be constructed in f". [A - Int W. - ... - Int W ] = f.[A - Int W. - ... - Int W ] which i 1 n i 1 n misses Ext(A - B"*" - B_) by the construction of the W . Hence . . . . . 2 r 1 \>/> C v H (A - B - B2) • Let q be a path joing y^ and ; y^ i n J Hi U A U v q c y ^ (A - B 1 - B ^ . . clearly d) We w i l l show that the path to y2 of A in V to £ in A . Let £ £" c B 1 in q and returns to y by showing that and z^, z 2 respectively. z^ y^ y^ is an upper principal path c A <"> V and that 5 ^ i s homotopic be decomposed into two paths Z' • and £ £" Bd B 1 pierces such that in as shown in fig 37 . We can do this because i s horizontal near Construct arcs which travels from E 3 - B 1 . We assume that Z' just two points Bd in -a . £ and because and z 2 y 2 £ can be a nice c i r c l e . in the cubes 1 2 A - B—B2 - B - and z^ y^ w i l l l i e i n A - B^ - B 2> (The idea here i s that both 2 the cube which locates locates A^ Z3 a). through z^ y^, a, A^ D q , The path which begins at and z2 y2 £" . The path which begins at and z^ y 2 and in A - B z^ y^ u lit- z 2 y 2 ^ z^ the cube which and travels to z 2 2 is homotopic in the cube z2 and travels to is homotopic in the.cube A - homotopies, the path - B 2> which.begins at and returns to - B2 A - B - B z^ via 2^2' z 2 to ^' to Z" .' Combining z^, travels to z 2 in z^ in z 2 y 2 y a u z^ y^ is Facing page 53 - 53 Nomotopic in A Note that ^ to I . The path is evidently homotopic to passes through the point p £ a . Eventually be the ' o f f i c i a l ' intersection point of the principal paths ?2 of p ^ . will ^ and A . e ) There is no difficulty in altering, the argument to construct a lower principal path o n e ^ 5' 2 keeps in mind the fact that 'the pictures are different' and that everything in the construction of mus t be repeated. We cannot, for example, use the they were defined with respect to A - B 1 - B2 A - B 1 - B2 (the cube which located A3 ^ from a) because and we must replace and 'shaped' the right side 2 of ?1) with e^ and e2 A - B-B 1 which locates as before, but to use A2 A2 . The idea is to start with rather than A^ in f i g 38 which, in a sense, is a replacement for f i g 35. turns out to contain p 6 e^ n e 2 just as 5^ as suggested The new £2 does; this establishes that C ^ ^ ^ ^ 0 . We begin by finding a new lasso J ' = C£ V V a' 2 so that C: U c; C V A Int (A - B - B.. ), C'C A , and C. contains 1 2 1 l 3' I 2 7 2 point y^ y 2 £ mg points such that! y£ 6 mg C2'^i mg y^,' e 2 n K2 . C£ n mg The arc e^ f> (A - B a' - B^ - K2) lies in V .0 A^ and and has end y 2 . One finds u C2 V a' by adapting the procedure in 2 a); there is very l i t t l e more involved than reading m, m., B^, B , K2J A - B2 - B± - K2 for , B 1 , B 2> A - B 1 - B2 - K±, K2> and priming every new construction. It w i l l be found that the arc tains p just as the original a K£, , replace f i g 36 by f i g 39. a' does. For the construction of conK', It i s quite easy to adapt b) and Facing page 54 - 54 c) by keeping In mind that the important cube i s A - B replaces A - B^ - i n the construction of £^ . - B^ which (The point i s that in b), c ) , one must use a cube whose boundary encloses the 'important' dogbone A 2> joins see f i g 38).: Finally we construct a path y£, in A - B - B be combined into the path m 1 . This plus etc. The path - B^ a' C A - B - B can 2 -£> which can be shown to be homotopic to by adapting d) above, decomposing vx" C. E q' which ^ m into paths lies i n V m"C B^ and by an argument which should appear naturally from the adaptation of a), b), c) to construct ?2; and and a' 1 S ^ and hence in both and lies i n both a ?2 ; , therefore 5:fiK ^ 0. 2 lies in A^, i f 'j••-= 1 . The i s no difficulty i n constructing a proof for the lemma when j = 4 the %^ p The proof is now complete i f e^ V f) i.e. clearly i n A . Since the point in view of the symmetry of the construction of A.. . We w i l l give only an outline of the proof for'- j =v2 symmetry for j = 3) for these reasons: in along the lines of a), ... e) a) ... e) 1) (and by the details can be f i l l e d above, and, 2) the argument in i s sufficient to prove the 'meat' of (2.2), viz. that there are uncountably many big points of V "which f a i l to possess arbitrarily small simply connected open neighbourhoods, these being images of elements of the form AHA. l from 4. • To construct the principal path 5 ^ when j = 2, 1 or the paths A - B 1 e^, e2> A.. ij A... xjk ... where i , j , k, ... are chosen which we now assume to l i e in A2 f\ V, and the cube - B^ (see f i g 40) which acts toward acts toward A^ use i n a) (i.e. A2 just as A - B^ - B^ separates A2 A - B 1 - B 2 just under the Facing page 55 A - B 1 - B2 upper eye while of a), construct a lasso e l ^e 2 ' A 4 ' e l u G2 2' e A 3' a n d A A does the same for A ^ . o au J = F i s shrink to a point i n V ^ s n o w s arc the n e w J • When Cj. and \ of A^ ( A may q joining the end points of i n V ^ (A - B"*" - B^) may be constructed by adapting the argument of b), c ) , and of i n a) was related to V, they hit the same element be a big element or a point). The path a which is related to V, ~ B 1 - B ^ just as J ~ B l ~ B2 ' Using the argument ^ = a V q may be shown to be an upper principal path A by an argument like that of d). a contains a point the lower principal path pe 5 Just as i n the case n e 2 . Thus when j = 1, the p € ^ . . To construct j = 2, we start as before with 2 e 1 u e 2 c A 2> but we use the cube A - B - B 2'OA^ and construct J ' = C ^ a ' U C ^ ' so that J ' i s related to V, e 1 v; e 2 , A . 2 A - B 2 - B2, ' A - B - B with just as J A2> , see f l g 0 41 . Fig^ 41 also shows q'~ which ts used a' to form to contain i n e) i s related to V, e^'U e2', £'2 . The arc a" and hence the path p ; hence £ n £2 ± 0 as before, 0. K2 turns out - 56 CHAPTER THREE: GENERAL,IZ ATI.ON OF A THEOREM OF BING: LEMMAS. 1. In this chapter, we give two lemmas for the proof of II (2.2), the generalization of Bing's theorem II (2.1). In proving II (2.1), Bing defined a property A h a dP r Q such that °Perty o n eo f jk ... r \. », A , had Property ... r j j K ... r^f J K a chain A 3 A^. Q. A had Property A j k ... r i> Q, and i f a dogbone A k ... r2> Q . This meant that there was Aj^^ 3 ... of dogbones with Property possession of Property Q . Since the Q implied intersection with both disks II (2.1), the limit of the chain was a big element and j (see the discussion in I I § 2 ) . in A which hit both We follow Bing's proof closely (in spite of the fact that we alter Property has to be applied to a whole B.. Q to a property which to be of any use) and in fact depend on the reader's familiarity with [12] for the motivation in this chapter and the next. In the remainder of this paper, i = 1, 2, and j = 1 , 2, 3, 4. In the proof of II (2.1) i n [12], i t i s evident that the crucial part of the argument i s the proof of [12, Th 10], where i t i s shown that i f the four centres of A^, A^, A^, A^ f a i l to have Property P, then some set homotopic to the centre of A have Property P, (The precise definition of Property also fails to P i s unimportant until Ch IV). In Ch IV we w i l l prove just this result with the disks D. i n [12§7] replaced by the singular disks l f [A] i in II (2.2). Our proof w i l l differ from the proof of [12, Th 10] in that whereas i n [12 Th 10] the disks remain unchanged during the proof, in our proof of the analogous result the f^[A] are replaced by new singular disks f^lA] which retain the desirable properties of the f^[A] . Although Facing page 57 & ) Facing page 57 (li.) - 57 this i s a considerable change, i t turns out that our Ch IV resembles the argument of [12§7] very closely. In the present chapter, we prove an important lemma which shows that i f each f^[A], f2[A], only does each J arcs ^ then the new (in f i g 19)'misses one of f^[A] may be constructed so that not k. miss one . f7['A], but both j x and f1[A] miss each of the x l i e in g shown i n f i g 42. The and g^ and tie the upper and lower loops of the k_. together as shown i n the figure. If we can obtain such singular disks for then parts of the k^ set each b^ the reward i s considerable, can be erased as shown in f i g 43, leaving the .b^ U b 2 u b^ \j b^ u K^U ? 2 K C k_., f^[A], shown i n this figure. misses one ~f^[A] while Since each £^ v £ 2 misses both. One can now apply Part II of the proof of Th 10 of [12] to h^u ...Vb^ u Z^U ? instead of to O'pq.r.s i n [12, f i g 2], This can be done with very l i t t l e change i n the argument of [12] and results i n the construction of a centre of A which fails to have Property P . We say that mappings 3 3 g^A ->- E are Z^disjoint i f f Z C E and mg g^ D rng g 2 n Z = 0, i.e. i f f .the ranges are disjoint at least in Z . Lemma One. Let ZO A Consider A, A_., g \ 3 E - Z . Let g. :A x and let C.. :BdA 1 mappings such that S^EdA = C i " L e t consisting of the cylindrical annulus caps. Each S C Int A pierces each and let N k. as defined in Ch II (see f i g 19). ^ ^ et 3 E be Z-disioint ' i es P n e r e shown i n f i g 44 Q with disks d^ exactly once and d^, d 2 for end g^" misses be an n - neighbourhood of S such that £2 . Let Nd"Int A . The arc x.^ shown i n f i g 42 lies i n Int A - N . Then there exist Z — 3 disjoint mappings g^:A -> E such that Facing page 58 - 58 - i) g± = c A oh BdA , ii) g :A (rng g i - Int S) u N , If £ . U Q misses iii) 3 Corollary. Let rng g., ^~ then L J misses rng g. . be the cube defined in Ch II (see'fig 21Y. Then i i ) and i i i ) in Lemma One may be replaced by i i ) " , g^'A -* rng S V & > ± x i i i ) ' If k. V fi misses j rng g., i then k. misses j rng g. . •a i The proof of Lemma One i s delayed to §2, which may be read after Ch IV i f desired. We give a second lemma which i s intended to r e p a i r a gap-which would otherwise appear in the proof in Ch IV. This lemma i s quite specialized, but appears here because i t s proof i s just a variation of the proof of Lemma One. As before, the proof is delayed to §2 and may be omitted on a f i r s t reading. Lemma Two. Let Consider A, Z, E be the sphere shown in f i g 45. The sphere as defined in f i g 42. E together with an W of £ lies i n Int A; g"*" Cl Int E - n, n - neighbourhood of k_., a.., g \ and each 3 a^, a^y a^ pierces E as shown. Let mappings g_^:A -> E be Z-disjoint with g . = C. , where c. is defined as i n Lemma One. i|BdA i i rng g^ miss the set ^ u k^ \j y k 3 . Let U ^ J be arcs Both in 1 g 1 which join a^rtg respectively and miss 1 1 and and rt 1 1 g , a^ O g rng g^ . Let ^ 2 ' 1 3 v rt 3 , a 2 Cl g V 1 join a^rtg miss rng g 2 • The arcs ^2» 1 3 ' 1 2 ' 1 3 a u U V loe a r c s 1 and a^rtg V r e n o t 1 and a^rtg i n ^ which respectively and necessarily Facing page 59 - 59 disjoint. Then there e x i s t Z - d i s j o i n t mappings such t h a t S^'" C-^ i k BdA, n Corollary. One o f g^:A rng g i Although orderly arcs r n g g^, U r n g g^, u^2 a^ rt I n t E, i n I n t E, ab i n a cube K with r n g g^ misses with a 2 O I n t E, 2. V b^u b^U b^ y £ > 2 E by the the a r c ( a ^ y u ^ w a^) rt I n t E 2 as a few moments experiment w i l l show (an a r c ab c Bd-D = a U b ab c ;BdQDerBd<CKU ab) . i s knotted To be k n o t t e d ; than one c i r c u i t on t h e t w i s t e d annulus. 13' misses l i e s i n an annulus and i s j o i n e d t o be k n o t t e d U r n g g^ . may D c K and one o f V V u and o g7:A -*• (rng g.. - IntE) U W i f t h e r e i s no d i s k t h e a r c must make more A s i m i l a r comment a p p l i e s t o 12' 13 * V Proof o f Lemma One. 3 (2.1). As a p r e l i m i n a r y , we d e s c r i b e an u n t w i s t i n g f u n c t i o n y:E 3 -> E which i s onto and one-to-one and which unwinds the t w i s t i n g\ i . e . (/(g^) i s the p l a n a r annulus shown i n f i g 46. For w e l l known reasons y cannot be a mapping» b u t we ensure t h a t on t h e curved c y l i n d r i c a l s u r f a c e z end caps o f z a r e c a l l e d M^, M , 2 y y w i l l be d i s c o n t i n u o u s only shown i n f i g 47. I n f i g 47, t h e and the cube I n t ( z U U M) is 2 called K . Eventually w i l l be composed w i t h a mapping whose range misses z . Thus t h e E e s u l t o f t h e composition w i l l be a mapping. The 3 function To d e f i n e y y i s d e f i n e d t o be t h e i d e n t i t y on i n I n t K: means o f a c u t on Z Imagine and on M, 2 K E - K and on b o t h t o be c u t f r e e o f t h e space by remaining a t t a c h e d o n l y on . . Faing page 60 - 60 K may--.be • thought of as a stack of circular disks of infinitesmal thickness. These disks span the cylinder a straight arc. z and each meets in Fig 48 shows M^, which is called the i n i t i a l disk; M2, which is called the final disk; and a 'typical disk' i n the stack between and M2 . Now apply a twist (which may be thought of as. an isotopy of K) to M2 so that M2 rotates once (i.e. through an angle of ' 2ir) i n place. When this happens, the disk M^, which i s attached to the space, necessarily remains fixed and does not rotate. Each disk intermediate between and M2 which i s close to zero for disks close to disks whose location approaches that of rotates through an angle and approaches for . The rotations of the various disks i n the stack can be contrived so that onto the plane which contains 2TT B ^ H K i s carried B"*" - K, and so that the final result i s homeomorphic to K . In f i g 48, the 'typical disk', which i s located half-way between and M2 w i l l rotate through an angle of TT . This carries i t s intersection with M2 B " * " on to the desired plane. Since has returned to i t s original position, we can restore the cut at M2 Evidently y i s one-to-one and continuous in Int K, Ext K, and on U M2 . The fact that we cannot sew up the cut on z appears i n the definition of y as- a discontinuity on z . Clearly y carries B ^ into the plane containingft'*'- K . (2.2). We w i l l prove a simpler version of Lemma One to show the general approach. (2.21). bourhood 3 Let S be a sphere i n E having a simply connected neigh3 N . Let g:A E so that glBdA] C Ext S . Then there - 61 exists a mapping g:A -> (rag g - Int S) u N which agrees with g on BdA . When simplified i n this way, (2.21) is insignificant, for there are easier proofs of stronger results, as the reader doubtless sees. However our proof is intended to show how 11(1.2) is used in the proof of Lemma One. Proof. Apply 11(1.2) to obtain disks W^, ... Wn i n A such -I -1 that g [Int S] c W1u ... U . Since g [Int S] misses BdA, A - Int W., - ... - Int W i s a disk with holes and g-1,lint S] c 1 n Int W^ ... Int carries each Bd W r . If e in 11(1.2) is sufficiently small, then g into N, for g[Bd W ] lies close to, but not in r Int S and hence close to S . In 11(1.3), take (simply connected) N to be P = P 2 = ... = P n (j J Y 2 o ... J Yn;A either in to obtain the mapping I = g|A _ I 1 rng g V N . Since each point A - Int W^ - ... - Int Wn in some W^_, in which case rag g u W2o g(x) £ E „ _ i n t s, or Int S - N . Thus g = g on BdA because the ...tfW^, which misses BdA . We w i l l now give a formal proof of Lemma One. Case one: g^[A] g(x) £ N, rng g misses lies in (rng g - Int S)U N . Finally two mappings differ only in (2.3). in which case _ _ _I 1 x in A lies n t meets Ext S, neither 8-^^] meets S . Let g^ = g^ • Since a connectivity argument shows that g^A] misses Int S . The rest of the requirements of Lemma One are clear. In the next two cases we insist that one of the rng g^ touch while the other does not. n tw n Facing page 62 - 62 Case two: meets S exactly one Q . Assume that also meets j*2 = rng g^ meets rng g^ S . The rng g i Q <= S . meets • Evidently -i)'., i i ) , i i i ) of Lemma One are true of Apply the argument of 2.2, taking a mapping g in 2.2 to be g^:A -* (rng g^ - Int S) U N With regard to g^, g^, which Let g^ . and construct which agrees with on BdA . i) and i i ) are satisfied, and i i i ) is vacuously satisfied since g.[A] hits fi,. The g are Z-disjoint because we I i could have taken N small enough to miss rng g 2 -. Thus 0 = Z n rng g1 n rng g 2 = Z r> Crng g 1 U N) r\ rng g 2 D Z rt rng g 1 n rng g 2 . Case three: meets Q; the other both rng g^ (rng g2) meet does not. S . One rng g^, say rng g^, The aim of the proof w i l l be k g to construct an intermediate pair of mappings k rng g 2 such that K misses S although rng g^ may not. The argument then reduces to an easy variation of either case one or case two. Outline of proof, It i s important that, since a) Choose a component rng g 2 misses Using this fact, we construct a circle Q, of rng g 2 D rng g 2 n S c Int d^ u Int d 2 . c^ c Int d^ o Int d 2 - rng g^ - rng g 2 which encloses (on. one20f '.the .;. d^) points of exactly one of rng g 2 n S . Although we choose enclose points of so that w u 6^ v <5 2 c^ is the equator of co V S^u c^ may S, turn out to 6~ 2 Assume that Int c^ C Int d^ . Construct in the shape of a pill-box as shown in f i g 49 c) using z^ c rng g 2 n S, rng g^ n rng g / i . S • b) a sphere S . instead of to u <5^ v &^ • An argument like that of case two but S yields a pair of Z-disjoint mappings 1 g.:A rng g. u N 1 1 rng g.^ y rng g^; such that 1 g. R C... on and i f rng g i misses BdA, Qu Int c. misses , . then so does 1 rng g^ . The argument of case two i s used virtually as is i f c^ encloses points of rng g^n S . If c^ encloses points of rng g 2 f\ S the argument must be modified somewhat, since the method of case two would not ordinarily ensure that rng g^ would miss a l l the t. d) If some component that It turns out that 1 cr rng g 2 n S rng g 2 misses. rng g^ misses 0. . exists i n , say, Int d^, then we 2 1 repeat steps a), b) c) to define mappings g^:A -> rng g^u N with proper- ties analogous to those of the g^ . We continue to construct mappings 4 r S 3 i ' g i ' '*" ' f i r s t finding a component z ^ ^ c rng Clnt d^U Int d2> r+1 r r+1 and then constructing the pair g^ :A -> rng g_. u N such that g^, = on BdA and i f rng g^ misses 2 rng r g^, ... rng g ^ Q, o Z^, rng gr+1.We i T T then so do rng t. ^ ^ show that r+1- (\ S rng r n „S O rng g^. and.that the sequence of mappings ends at a pair k rng g 2 H S - 0 . although g^ for which k rng g^ may hit S . e) The situation now reduces to case one or case two. An additional argument shows that i i i ) of Lemma One i s satisfied. Details of proof, a) hits or S, z^ d^'). then there i s a component Irit d 2 , say Int d^ . By in Int d^ which misses that i t misses If If we assume that rng g^ rng g 2 misses Q but of rng g 2 n S lying in Int d^ the Zoretti Theorem, there i s a circle rng g 2 , encloses z^, and lies so near to (by 'encloses' we mean 'encloses relative to encloses points of just one of rng g^ n S, rng g 2 r\ S, -64 - ^e then let rng g^ r> S both points of in c i • We must expect that and rng g^ n S). Int x-^ C Int d^ (g^I'A]'C rng - w e x <- I n t X-j_ which misses 2 g^A] n S in of Int X-j_ .• Int x g2IA] ^ Since (rng g^ u rng g2> n S . rng g^ n S rng g 2 n S, and If s may be in contains points of 2 but by the Plane Separation Theorem we know that closes points of both of cannot predict that 2 g 2 n S; In this case use the Plane Separation Theorem and separates the component z' Int x » (z^ could be i t s e l f a circle enclosing to construct a circle A]) <n S from a component Int rng g 2 n S x-^ w i l l enclose .points of x X2 en- encloses points of 2 Int x ^ (rng g 1 u rng g2> then separate 2 s t i l l further by means of another application of the Plane Separation Theorem. We repeat this procedure, defining circles being constructed in Int Xr_-^ whenever Int Xr_^ x^> X^> ••• > Xr contains points of both r-l rng g_^ f\ S . The following argument shows that the sequence X,l., Xo> ••• ' must be finite:, z. points of each annulus Int xr ~ * n t X r+l>i contains (rng g^ *J rng g2) o S . Without loss of generality in the x> construction of the r rng g^n S we could have replaced the sets with 'thickened' sets obtained by covering the )^rng g^ r> S disks of area a (from compactness, the thickened by small rng g^ n S assumed to remain disjoint from the thickened rmg can be g 2 r» S) . But since each of the disjoint open annuli must therefore have area at least the number of- x Since and Xt+1 r could be defined i f x rng g 2 n S, rng g 2 n S . Let rng g^ n S, must be finite.and the sequence ends at some x x encloses points of both must enclose points of only one of t x fc fc be rng g 2 n S a, t • rng g^ n S rng g^ n S, c^ . We repeat that we do not know which of is intersected by Int x = t ^nt . Facing page 65 - 65 b) Now assuming to be horizontal, we build a small sphere in the shape of a p i l l box consisting of verti c a l cylinder OK and end caps <5 ^, < $2 , (see f i g 49). which are parallel to d^ The cylinder co intersects equal distances above and below d subset of d^) lies i n Int(co o rng g^ u rng g^ we build d^ only at c^ and extends . Thus Int c^ (considered as a u <$2) • Since to so near c^ misses c^ that rng g^ u rng g 2 . co misses Fig 50 shows <fru 8^ <J <$2 and part of d^ . We assume that c^ has %^) n d^ . been moved slightly, i f necessary so as to miss (JJ^ u #2'c/ We also assume that co misses every making co smaller or even curving . The sphere although this may necessitate co slightly to follow the curve of toi/6-^ u <$2 is constructed as in f i g 50 so that i f some l. meets d- then % . (misses co and) pierces each Int 6. J i j just once. Let be the simply connected neighbourhood of 6\ shown in f i g 51. v . so that l deformation retract of v . - J l . x 1 2 6. - I- - H- - I, is a x 1 2 3 4 and so that v . misses 3 4 x S => Int c^ and any & Int c^, misses We construct which misses co u S^u 6 • We assume that 2 constructed so that c^ c: Int d2) so that cot/ co u v §2 u u v Int(co u U § 2 ) c/ c) 2 i.e. any u 6 u 2 which uv 2 lies i n N, and (since u v o d2 . misses 2 Assume for the moment that contains points of rng g^ n S . Because has been rng g^ hits Int c^ Q, we can ignore — ^ i i i ) in Lemma One as far as g^ is concerned, i.e. the g^ which we are about to construct need not miss any &. . We assume that lies- so near Int c^ that co u u <$2 u u v 2 misses co u 6^ u 6' u 2 rng g 2 . Since u v2 - co also misses 66 - rng g^ , rng g^ meets to U 6^ o fi^ only i n 6^ or Apply an argument like that of 2.2 to corru 6 w 6^ , i . e . use 11(1.2), 62 taking S i n 11(1.2) to be & u 8^ a n d N t ob e v 17 v -|_ 2 ' t o obtain disks W.^, ... Wn i n A such that g [Int (co ^ '5 ^ & 2 )] lies i n Int W-.f ... U Int W . Since gg,[Bd W ] lies near 1 n •1 r 1 rng g^rt(co iy <5^ u 6,,) therefore each g-^lBd W^] as we saw in 2.2, and since lies near in U v 2 . Evidently . g,[Bd W ] lies 1 r in a U v 2 , we can construct the mappings 11(1.3) and, following the argument of 2.2, define a mapping 1 g^:A -* [rng g 1 - Int(co c ^ u 6 ) ] u 2 gj = 1 g^ that on BdA, and i f rng g^ misses (this last property is only true because which can miss some QU t., Int c^ on d^ misses because Int c, misses 1 rng g 2 1 g^ = on BdA 1 g^A -*• rng g^ U N, then so does Q, O rng g^ hits rng g^ 0, and because i s identical to g^). Additionally because g2 = g2 and misses rng g^ v. U v„ . The %~. are Z-disioint because J 1 2 ° i rng g 2 = rng g 2 , and rng g^ exceeds misses (By 'rng g^ exceeds rng g 2 . such that u 1 g 2 = g 2 . Then i t is true of both Let g„, rng g^, ; 6^ U <$2, i.e. i n g,[Bd W ] lies i n one v. . Since each 1 r l simply connected subset of co misses rng g^ only i n which U rng g^ only i n N', we mean rng g 1 u N .3 rng g* .) that Suppose that instead of rng g^rt S, Int c^ encloses points of: rng g 2rtS . Let g^= in b) , this time so that misses a n d construct co u <5^ u <$2 U rng g 2 . Constructing g2 co u u v2 misses various V £j that rng g 2 £^ . Some <-> u as rng g^ and co is harder than constructing we did i n the last paragraph for we must ensure that set v rng g 2 g^ as misses any misses. We must take careful account of the are not missed by rng g 2 and can be ignored. 67 Some £^ are missed by co o 6^ v &2 a isses mis but do not meet Int c^; we note that has been constructed so that any y Int c^ rng g 2 also misses co u <5^ u 6 U £ which y v 2 • With this pre- 2 caution, i t i s safe to ignore those Z^ which miss miss £^ rng g 2 and also Int c^ . In the remainder of this paragraph we w i l l assume that and £ are those £ . which miss rng g_ and h i t Int c .We think that the procedure in the general case that some subset £ . , J l £j , ... £„ of 2 s w i l l be evident. as before. £^, £ 2 » £^, £^ misses rng g 2 and hits Int c^ -• We proceed to define g 2 using 11(1.2) and 11(1.3) The only difficulty occurs when we wish to shrink to a point so as to define 82|Bd W r t o S^ig^w ' r in the course of defining y r • It was easy to shrink a point in one component, say , of v v2 t o g^ . But in this case we must shrink 8213^ y a point i n - £^ ~ £ 2 » r otherwise that rng Yr an 1 rng g 2 w i l l h i t £^ v Z^ • The reason d hence w i l l shrink to a point on v- - Z. - £» i s that g0[W ] r £^ U Z^ (because rng g 2 does) and can be assumed to miss goi^j T7 1 misses z i n f i g 47 without loss of generality. There is a retract 3 not a deformation retract) of E Additionally i t turns out that R - z - £^ - £ 2 onto restricted to R (though v»". 6^ - £^ - £ . 2 - £^ - £ 2 is a de- formation retract of - £^ - £ 2 onto <5^ - £^ - £ 2 . This means that R82|Bd W i S homotopic to S2|Bd W i n V i ~ ^1 ~ ^2' a n d s i n c e • ' r ' r Rg 2 [ Bd W shrinks to a point i n Rg^W^] c - £^ - £ 2 > therefore ' r ^2[Bd Wr shrinks to a point in description of the retract z R as required. We delay the and the proof that until the end of this proof. ^2lBd W ' r - £^ - £ 2 g2[Bd Wr] Except for the use of shrink to a point, the construction of g2 R can miss to make i s like that of g^, - 68 and we have g2:A •> [rng g 2 - Int (to v $^ ^ <$2)] v (v^ o - t^ - £2) . 1 The g^ are Z-disjoint by an argument like that in the previous para1 1 graph, and g^:A -*- rng g^ o N, g_. = c\ on BdA as before. We know that 1 rng g 2 misses u v £j Si . If £ which is remote from 2 meets 1 rng g 2 exceeds SI because Int c^, in which case misses t^ misses one of t- , £„ above; or else t. misses 1 2 j misses rng g 2 because £^ i s remote from Note that the 1 Evidently we can write then either £^ i s Int c, , in which case t. 1' j v v . 2 Int c^ n rng g^ = 0 rng g 2 . Since rng g* n S <=• rng g± n S 2 rng g 2 , rng g 2 because g^ and g^ is that Int c^ hits one of rng g^, rng g^ \j v V v , only in g^ satisfy the hypothesis of Lemma One. The important difference between whereas rng g 2 (because rng g"|" a v u x> misses (rng g^ v rng g2) n S C (rng g^ 2 S) . rng g 2 )O S where the inclusion i s proper. d) Since the of Lemma One, we look for a component Int d^ U Int d 2 2 1 g^:A ->• rng g^ U N with 2 g. = c 2 c Int d^ \J Int d 2 on BdA, and 1 2 rng g_^ and rng g,. also miss Si U t^ . 2 2 1 1 (rng g1 J rng g2) n S c (rng g 1 u rng g2) ,o S C (rng g u rng g2) f) S rng g^ misses Furthermore of rng g 2 n S in and repeat a), b), c) to obtain a circle and Z-disjoint mappings if g^ satisfy the hypothesis SI U t^, then both inclusions being proper. We can continue in this way, defining mappings z 4 g^, g^, ... and components z^ c rng g 2 D (Int d^O Int d2> , c rng g 3 o (Int d u Int d 2 ) , ... so that g^ = ± on BdA, and i f rng g^ misses g*;A -> rng g^"1 SI U £^, N, then so does rng g_^ . - 69 v TC ir—1 i r ~ l (rng g^ u rng g2) H S c (rng g^ C rng g 2 ) O S where Furthermore the inclusion is proper. An argument from compactness like that used i n a) to show that the number of x r was finite can be used to show that there must be a final pair of Z-disjoint mappings could be defined i f z-^j_ existed in must miss g^+"*" g^ . Since k rng g 2 n S, rng g^ therefore k S . r Since : rng r-l e rng g_^ U N, evidently k k-1 k-2 k g. :A -> rng g. V N c. rng g. u N C ... C rng g. U N: and g. = c. x 1 on x x BdA, while i f ft V t. misses some rng g., k rng g^ . Since one or Case two. 3 k rng g 2 misses i then ft f £ ^ x misses j S, the argument reduces to either Case In the course of the argument of Case one or two, g2, whose range already misses S, w i l l be set equal to g^ . This means — k that g 2 has the properties of g 2 ; thus i ) , i i ) , i i i ) of Lemma One are true for g 2 . The argument of either Case one or of Case two w i l l — k — now construct a new g^:A -* rng g^ u N C rng g^ \j N with BdA, so that for g^, g 2 are Z-disjoint. g^ = on This proves i ) , i i ) of Lemma One g^, while i i i ) is vacuously true by the Case three assumption. Case four: Exactly one rng g^ meets S but misses ft . The reader w i l l find that the method of case three works here almost word or word i f i t is assumed that the rng g^. which hits ' S i s rng g 2 . When we arrive at the point in Case three where k let g^ is defined, we can — g^ be g^ immediately (or go to Case one). Actually the retract R works on S in Case two just as well as on co U & o <5' in Case three. 2 Thus a quick proof is possible by adapting Case two. Case five: Both rng g^ h i t S; both rng g^ miss ft . Proceed as In Case three to the point where the are defined, allowing for the fact that i i i ) in Lemma One applies to both g^. rather than only to g2 as in Case three (thus one may have to use the retract R to x construct both g^, whereas in Case three, R was used only to construct r g 2 ). k When the g^ are defined, the argument reduces to Case four or Case one. Case six: Both rng g^ h i t S; both rng g^ h i t fi. This case is not used in the applications of Lemma One, which always require 9, U every set to miss one rng g^ . It is not hard to prove Case six using the ideas of the other cases. (2.4). The retract R . This retract was used in 2.3 Case three c). We w i l l show how 3 to define when R: E - z - - •+ & - I - I ; the definition of R 6^ is replaced by <52 is similar. of Lemma One requires a retraction onto Strictly speaking, the proof d, - £ . 1 Z. J l Z. J s are some subset of Z Z 2 the assumption in 2.3 Case three c) that > - ... - Z , where l Js -£o> Z,; however we continue J rng g 2 misses Z^ V Z^ . Assume that the unique boundary component of (3^ which is a planar circle lies on the the origin. Y - Z plane and that the centre of this circle i s 1 The idea is that i f we untwist g by means of y, a l l the circles y[ZV\ w i l l be nice circles on the the origin. We assume further that half-plane. We describe 3 lied in sequence to E • R Y - Z plane with centre 6^ lies on the left-hand X - Y in terms of several mappings which are app- z - Z^ - Z^ . Each mapping w i l l leave Facing page 71 - 71 - 6 1 " l l ~ ^2 f i x e d First: Ct/ » a n d ttie untwist l a s t W i l 1b e g1 °nt° 5 1 ~ ^1 " ^2 * £/j 3 _ by applying the mapping E z _ £ „ £^ becomes a mapping by restricting i t so that the domain misses the 'bad' set z ) . Each y[Z^] i s a plane circle with centre the origin. 3 Second: using the symmetry of E - y(Z^ U Z^ reflect the right-hand half-space minus half-space minus across the X - Z plane, y(Z^O Z^) onto the left-hand 3 * E - z - yiZ^ U y(l U Z^ . This reflection carries 3 into those points i n E l i e on Z^U Z^ • Third: with non-positive coordinates which do not retract the left-hand half-space minus £ ^ 0 Z^) y(Z^ (J Z^) (which i s the same as the left-hand half-space minus onto the left-hand £^ U Z^ . This i s easy be- X - Y half-plane minus cause the remaining parts of Z^> Z^ are nice semicircles with centre the origin; one could imagine the X - Z plane hinged along the X axis. Using this hinge, topple the upper half of the X - Z plane onto the left-hand X - Y plane; simultaneously bring the lower half of the X - Z plane up to meet the left-hand X -Y plane. These movements define a (deformation) retract which crushes the left-hand half space minus £^ V Z^ onto the left-hand Finally retract the left-hand X -Y X -Y half-plane minus half-plane minus <S^ - £ ^ - Z^ • The four successive mappings define acts on Z^U Z^ . £^ y Z^ onto R . Note that R as a deformation retract (this was used i n §2.3 Case three c) ) . Finally we w i l l show that g2lWr] be assumed to miss the curved cylinder i n 2.3 Case three c) can z . Since g^lA] misses fi, 3 by the case three assumption, we can construct a mapping g^iA E whose range misses the curved cylinder Z i n f i g 51a and which agrees - 72 with g2 on every point of A which maps under g2 outside of a small nelghr- Q •*>• X and bourhobd of Z:. In f i g 51a, z. i s constructed so as to contain £ ^ U t^. mxs£ JS v2 Thus rng g 2 can be assumed to miss .-£•]_" ^ ^2* ^ es e t s i 1 (see 2.3 Case Three b)) miss ft and could have been constructed so as to g2 = g2 miss a l l of z.; Hence we assume that °n Bd W^ since §2^^ ^r""'C V ] A, by the d e f l a t i o n of W^ . We do not intend that since g2 should replace g 2 g^, g 2 may not be Z-disjoint; but i f g2[W ] hits apply the retract 8 2|Bd W ' r s h r i n k s not to S2IW R t oa P°int i n v ^ U tt 0 l "£ 1 " l 2 g 2|w l f f a n dU S ^2lBd W1 et z, we can ^ i e *-act d 0 e S r t n a t * Essentially the same argument applies to the construction of the other mappings i n the sequence 3. g2> g ^ , g2^, ... D. Proof of Lemma Two. We w i l l modify the argument of §2 so as to serve as a proof of Lemma Two. We assume familiarity with §2 in what follows. Modifying the argument of §2 to f i t f i g 45 presents a small and a large d i f f i culty. The small problem is that we cannot build pillboxes according to the nice picture i n f i g 49, where considered to be horizontal while c^ is planar and the 6^ can be co is vertical. It w i l l be appreciated that the problem is more apparent than real; we have room to construct Z c Int A with some obvious smoothness conditions so that i f c i s any circle on Z and Int c is defined, then a sphere be constructed together with neighbourhoods v^, co u §^ u <$2 can so that u v 2 behaves like the corresponding set In §2, i.e. 3 co o Z. fr c, <S^, 5 are disks i n E - Z which meet co only at i t s co o v v 62 f 2 Facing page 73 0 - 73 two boundary components, while the hoods of the S. which miss i then a. pierces each 3 misses Int c, then require that when are simply connected neighbour- E . Furthermore, i f a. hits j Int <S. just once and misses i a_. misses to; while i f a. J Int(co u 6^ \j S^) u Int c and hence Int c, u • We 6_. hits just one a^ , - a^ is homeomorphic to the structure shown i n f i g 52. then This require1 ment i s easy to manage; for E can be made to meet the a^ near g , where (according to the definition on Ch II) the a^ are straight and parallel and perpendicular to g"^ (in fact i t i s easy to make a_. n W a straight arc perpendicular to E). The hard problem is that i n Lemma Two we cannot use the retract R, which was crucial to the proof of i i i ) of Lemma One. that we permit the arcs (a^ V L/ n The reason i s Int E, etc. to be knotted, and in general deviate from the specialized geometry i n f i g 19. R was used to show that certain mappings g. \ T> ,T T 1 point i n ~ £3 - - _ ^ Recall that w i l l shrink to a r • Instead of R, we use the following easy but very weak result. (3.1). Let be the usual neighbourhood of 5 . Let intersect 3 only a^ as shown i n f i g 52. that f= F on BdA that L3 . The curve point i n a l * v j a i • Proof. u A Let f :BdA -> - a^ . Let F:A -> E and rng F misses a simple closed curve L may be knotted. Then L such f shrinks to a similar result i s true i f a^ or a^ replaces Let u be the small circle shown:in f i g 52. can be considered to represent the sole generator y of so Then - a^, - 74 and also (by consulting, say, the definition in [6 Ch VI]) a generator 3 E -L of the Wirtinger presentation of presentation only to be sure that u does not represent a t r i v i a l 3 TT^(E - L) i s the inclusion homomorph.ism, generator,). If i:ir^(v^ - a^) f G ym then, with a change of basepoint, l(y) i s an element of TT^(E^ (we specify the particular - L) . Then for some integer m, and f £ i ( y m ) = ( i ( y ) ) m , which 3 is the identity of TT^(E - L) because f shrinks to a point i n 3 E - L . Since i(y) i s a non-trivial element (in fact a Wirtinger 3 generator) of i r ^ ( E - L ) , finite order. either m = 0 or i(y) i s an element of It is known ([7, (31.9)] ) that the fundamental group of the complement of a knot has no element of finite order; therefore m - 0, . and f represents the identity ~ aj) ^ ° y^ i n To prove Lemma Two, we w i l l apply arguments like those of to a disk D rather than the sphere S . We w i l l f i r s t define some simple closed curves to play the part of L i n (3.1): Let be the unique simple closed curves which are subsets of ? 2y ^2 a U a lU U'13U a U P e c t i v e x y except that 13 i2' . r e s 3 'r e s P e c t l v e x y ' L e t ^12' i 3 ^ u e v -_2>y '^ 3 miss ' a a i3 lu U12U a 2 ' identical to i s replaced by From the hypothesis of Lemma Two, i t is clear that rng g^ while §2 and u ^ by u.^ and LL miss rng g^ . Now i f D i s a disk which is a subset of Zh and Bd D misses rng g^ U rng g^ u a^ua^o rng g^ U rng g^ meets 'D, we can define a circle a^, then i f c^ just as in §2 so that c^ C Int D and c^ encloses points of just one of rngig^/) £ , rng g^ n E . Then a pillbox w u 5^ u ^ can be constructed as usual, and finally a pair of mappings rngthe g^ ugl u v^, properties where thelike the are g the hoods of the g^:A ^ •*•and have "!" usual i n §2,neighbour^ Case 1 three. 1 If Int c^ and hence co U 1 <52 me e t s just one of a^, a 2 , - 75 say a^ , then we use (3.3) instead of the retract to shrink the various mappings a^ because rng g_£ misses And by the usual arguments, u let L be as L Int c^, R o r be Int c^, 0 1 rng g^ a while g^ - g., . rng g^ . In applying (3.1), we T.2 ( a s s u m i n g that hits a^ because as and a^, which are remote from Int c^ hits U rng g^ or rng or rng g} misses continue to miss both O v. u v Int c^ meets only one of a^, a.^ a^; if to U to a point in r §2, Case three. Thus i f rng g^ hits was done in misses g . i _ , TT R a^) depending Int c^ . If Int c^ hits g a i n » i f Int c^ hits a^ only, let a^ only, let L be u^^ . Unfortunately, as the reader doubtless sees, i f c^ encloses more than one of a^ n £, a^ O Z, a^ r\ £ , then the present argument f a i l s (because the argument with (3.1) is weaker than the original argument in so that §2 which used the retract R), and gj cannot be constructed rng gj misses a l l of... a^, a2» a^ . The trick of proving Lemma Two is to apply the argument of §2 so that none of Int c^, Int C2» ... ever hits more, than one of a^, a^, a^ . By extending 2 the above ideas to further pairs 3 g^, g^, ... and using methods from §2, we can prove (3.2). In the context of Lemma Two, let D be.a disk such that and let Bd D miss exist circles rng g^ u rng g2 u a^ u c^, C2> ••• c 1 2 m , J_T_ i_ g^.. . g±, ••• g± such that m i n l n t u D C. £ , Then there D and Z-disjoint mappings r * r^-1 , , _N 1 2 m g^A -*• rng g^ U In - £;) , . 8 ± . g <" ••• ~ %± ± H =c ± r-1 on BdA, c^_ encloses (relative to D) points of just one of rng g_. rng g 2 O £, and rng g^ misses be constructed so that D . I f , additionally, each n E} c r can c^ encloses just one of a^ n E, a 2 n E, - 76 - A E, then a^u rng g™ can be constructed so as to miss Proof. o a^ . (3.2): is proved in the same way as Lemma One. We can, ignore Cases two and three in the proof of Lemma One because the fact that Bd D misses rng g^ \J rng g 2 evidently takes the place of the condition in Lemma One that ft misses we cannot have because D rng g™ miss rng g^ \J rng g 2 . Clearly Int E in this version of the argument is a proper subset of E . The only part of the proof which does not have an exact counterpart in §2 is the statement that r r—1 rng g^ c rng g^ 0 (n - E). The reason that only in (3.3). misses n - E i s , as usual, that Corollary. rng g^, then u r rng g r—1 rng g_^ exceeds is remote from E. I f , additionally to the hypothesis of (3.2), D rng g™ misses a^ u U a^ regardless of the number of a^, a 2 » a 3 which are hit by the Int c r . Similarly, i f ^ m rng g 2 misses D, then rng g 2 misses a^ \J a^ y a^ . Proof. According to the argument of §2, i f rng g^ misses D, then we let g^ = g™ immediately. We will-now give the proof of Lemma Two. The following question does not look like a simplification at f i r s t glance: Z-disjoint mappings on Do there exist f^:A -*- (rng g_^ u vi) - a^ - a^ - a 3 with BdA, and a decomposition of E into disks D-^» D2 D^U D2 F E and D1 n D2 =. Bd D^ - Bd D2, and so that one rng f ^ , say rng f ^ , and hits a2 f± ~ so - that Int D2 and a^; while misses Int D^. hits - 77 Case one: the mappings at the decomposition fj. and such that tion that Int c exist as described above. Look E == D^u T>^ •• Apply (3.2) to to Z-disjoint mappings BdA. f f^:A rng rng f^ misses a^, miss u (n - E) with a^ and 1 2 3 Int c^_ . Furthermore 3 rng f i since i|BdA i rng f ^ does, and f ^ evidently satisfies the hypothesis of (3.2) and (3.3). rng g^ misses a^ and a^ g'|_JA '= C . . We know x rng f ^ misses rng g 2 misses to replace the f ^ with g':A -> rng f . O (n - E) with Z-disjoint mappings although on . In (3.2), the condi- by the usual argument. Now apply (3.3) to D2, that = hit at most one of a., a_, a_ is satisfied because r a2> to convert the By (3.3), probably does not. Since rng g^ misses a^ u a 2 i> a^, rng g_T now misses a l l of E, Int E except perhaps i n ft . Case two: no mappings f exist as described. Let d c E be a disk pierced by a^ which is small enough to miss both rng g^ and a 2 u a^ . Let D = E - d . Using (3.2), construct a sequence of 1 2 3 circles and mappings c^» g^, c2» g^, c^, g^, ... as described in (3.2), ending i n the construction of Z-disjoint mappings 8 i = 8 i misses of : A r n g8 D iU ( n "E ) W i t h g i= C i ° n B d A and a 2 u a^ . We know that .every and '~<such E - Int c r the Case two assumption. Evidently d u a^, so that r n gg i for otherwise satisfy the definition of f ^ , D^, X>2 r-1 given above, which means that (since _g but also . c^ encloses at most one a^ n E, a 2 o E, a^rt E, as required by (3.2): r-1 g^ , Int c r , t h a t exists) c^ contradicts rng g£ misses not only D v a^ U a^, rng g^ misses a^ y a,2 u a^ and-all of E, etc CJ. Facing page 78 - 78 CHAPTER FOUR. GENERALIZATION OF A THEOREM OF BING: MAIN PROOF. 1. We w i l l use Lemma One, Lemma Two and I§5 to prove 11(2.2). The organization of the proof i s much like that of [12 §7] and we depend on the reader's familiarity with I12] for orientation (although a detailed reading is required only of the section called 'Part II of Proof in [12 § 7 ] ) . As in [12 § 7 ] , we f i r s t give a (somewhat altered) definition of Property Q, then induce Property of the dogbone construction. of this chapter. Q through'.the steps This argument occupies most of the length As in [12-17], i t follows immediately (and for more or less the same reasons) that some big element of the decomposition hits both singular disks f t A] in 11(2.2). S t i l l following [12], we w i l l not present a formal induction, but w i l l show that i f A (Bing proves that one has Property Q, A^. has Property is only useful when applied to a^, then so does Q; &2, A^\j A^ U A^U A^ our version of Property ^» > ° f Ex 2 in §2). proof of this i s divided into Part I and Part II as i n [12 §7]. I, we look at the set each rng g". f^ M pq^r^s in 112, fig.2]. in 11(2.2) can be replaced by mappings :>'g^ misses one b. and both t a e cradle of In 112], L/ pq..^s We such that . We call the set A, as in f i g 53, which preserves the embedding of A . In Part ± t,^ U b^ U b 2 U b^ U b^U t,^ in The ^'u b u b 2 u b 3 U b^ U r,^ (see f i g 43) which serves a purpose like that of the set show that the Q and later represent i t t,^U b^ V b^ U b^ U b^v l behaves like the cradle of A in that each pq.r.s misses one of the disks D. in 11(2.1) . In Part II J j i of our proof we follow [12] very closely and require a detailed reading of the corresponding part of [12, -Th 10] . There are a few alterations; these are required by the fact that some homotopies are replaced by isotopies. 2. Properties P and Q . We w i l l define a Property with respect to closed sets of circles Z and m P on double ended lassos Y^, Y2.. The lasso connected by an arc Z uao m Zu a U m consists a . In Ch II we often specified constructions only up to homotopy (e.g. the intersecting principal paths of Ch I I ) . The consequence was that we ignored singularities in .these constructions. In this chapter, this practice is emphatically not allowed; in particular, i n the lasso and m Z V a u m, the circles Z are disjoint simple closed curves and a meets Z\J m only at its end points. One of the things that make the present chapter harder than Ch II is that geometric constructions have to be moved isotopically, whereas in Ch II homotopy was good enough. Properties P and Q are defined i n terms of their negatives, which we write Property 'VP and Property ^Q. ZU a Vm A double ended lasso has Property ^P.with respect to closed sets Y^, Y2 o n e of the following two conditions obtains. <vp(a): Z u a u m misses Y^. or Y 2 Cor both), fVP(b).; Z ^ a V m meets both Y '' and Y 2 . The set a {J m misses 1 Z contains a point Y^ U y f Z a such that of the two distinct arcs in Z with end points misses y and Z n a, one. Y. while the other misses Y„ . - 80 We intend that Property'VP(b) should be symmetric, i.e. a u Z may miss Y^ v Y^ less of whether Z U a <j m each of m and Z has and the point has y may be in m - a . Regard- Property 'VP (a) or Property 'VP(b), Property "VP as defined in I §5 with base point (the base points here are taken to be for circles Z r> a, m n a) . This statement, which i s important, i s easily checked. Evidently Z l> a U m may have both Property 'VP (a) and Property 'VP(b) . Property 'VP is the negative of Bing's Property P It i s easy to see that our Property 'VP implies the negative of Bing's property, i.e. our Property 'VP implies that i f x^ £ Z and Z u a u m and x^ 6 m has Property 'VP (by our definition) , then there is an arc in Z U a U m • in 112]. with end points x^, x^ which misses one of Y^, We w i l l neither use nor prove the complete equivalence of the two definitions here, although a proof w i l l be found to be straightforward. Property Q« i s defined on dogbones. If a dogbone X "> *~2_' *"2 has Property ^Q^ , this means roughly that the centre of X has ' 1' 2 Property 'VP with respect to the ranges of certain mappings f ^ , • 3 To be. precise, let Z z> X and for i = 1, 2, e. : BdA -> E - Z . Then A X has Property ~Q such that • g. :A • 1' i f f there exist Z-disjoint mappings ,g1, 3 'E , g. R C , on Property'VP with respect to Property -x-Qz.,c.^, We define X i 2 BdA, rng g^, rng with respect to g.,, x to have Property Q ' and the centre of X has • We also say 'X has l g ' with the obvious- meaning. £ tff X fails to have Property pings (i.e. with respect to every qualified pair of map-"»'-•^' 2 g.) . Note that a statement like 'X has Property Q„ X g„ Z , , C-2 - 81 ^2 with respect to m e a n s very l i t t l e . Example 1). Suppose shown in f i g 28. Then A Z = X = A has and c^, C2 . For i f c. Property Q L are the two circles , c.^, c 2 (d'--). y -- shrinks to a point, i t must hit the upper (lower) eye £ (m) Thus i f f rng f ^ hits £ and i s an extension ot c\ rng f 2 hits to a l l of A, then of A ., m . This ' k i l l s ' Property n/p for k = 1 v au m with respect to rng f ^ , rng f ^ , since Property'vp would require either that one m miss both f.[Al miss both l f^[A] . Example 2). has £ Let Z = A^; and m c^, c 2 or that one of £ or as in Ex. 1). Then A^ Property ; for the c. can shrink to a point so as to , ^ A-^ > ^2 miss Z and A . We emphasize that 'X has Property Q ' 1 ~2 does not imply that C^, link the eyes of X . 1 ( Evidently i f X has Property Q < and "» ~-^» ^"2 any Z-disjoint mappings of A the centre of X f^fA], into fails to have E"^ with f^ = f , f on BdA, are then Property 'VP (a) with respect to f-Xh], and consequently both f^[A] and f 2 [A] meet (the centre of), X . This suggests that the obvious way to attack the proof of 11(2.2) i s to l e t Z = A and l e t C^, C 2 and. we w i l l eventually do this. , ... has Property Q. jK. in fact every dogbone ... such that each, of A, A^ , , with, the c - defined" as-in H:C2.2); " » C--^ > c - 2 X f, A in I1C2.2), But i t turns out that in this case there i s no sequence A ^ A., o A ^ D A be the has J- Property • We overcome c this difficulty with the next definition. 2 - 82 A set T- {X , ... X } of dogbones has Property <\,Q each X , r - 1, 2,'. ... m has Property r a n dt n the same pair of mappings f ^ , If L , , *~2 e iff with respect to same triple Z, c.^, . {X-, ... X } fails to have Property H L , then we w i l l say J. m z, Cj., ^2 that {X , ... X } has Property 1 m some 0. has Property Q„ o z,,i_^, AyC-, i i_2 G„ . I f the set of components of z . 3 and i f g : A -> E i is an extension of C\, i = 1, 2, and the g^ are Z-disjoint; then some component X of a fails to have Property s z,c^, saw earlier, this, means that both a has Property Q s , C-2 L*, Property Q z,, Z, 3. g^[A] meet X . We w i l l say that i f f the set of components of a has s /,--•' Eventually we w i l l show that each of a., a_, <X_, .. •-- , has Property Q_ with respect to the g. . As we l 1 'G1» c 2 2 J . We now give our version of [12, Th 10]. 3 (3.1). Let Z 3 A and C^, In particular, the c {A , A , A_, A.} -L Z j by any circles whatever i n E - Z . do not necessarily link the eyes of A . Then i f has Property 'vQ , so has A . Z , C ^ ' , Ll^ H We remark that i n [12], the proof of Th 10 does not use the fact that Bd Dx, Bd T> 2 (in f i g 1 of 112]) link the eyes of A, even though a short proof of [12, Th 10] can be constructed along the lines of the second paragraph of [12 §7]. The. reason i s that in later applications of the argument of the proof of [12, Th 10] (which i s a disguised induction step) to, say, A 1 and , A^2> ^13* ^14' tIie B d D i d o n p ti n fact link the eyes, of A^ . For a similar reason we state (3.1) for very- general circles c. rather than the c... i n f i g 28. We assume that - 83 Z, c^, have been chosen once and for a l l before the proof of (3.1) begins, and w i l l now write Property 'v-Q for Property We w i l l not refer to Bing's Property Q again in this paper. We w i l l continue the convention in Ch III that i = 1, 2, and j = 1, 2, 3, 4 . Proof of (3.1): Part I In this part of the proof we assume that has Property with respect to mappings g^ can be replaced by Z-disjoint mappings on BdA b^ misses one rng g^ By the definition of Property ^Q, each in I §5. rng g . Look at g1U g"*" or g^ g7:A -* E while rng g^ with g^ = £^ u b^ o b^o b^ j t,^ u t,^ has Property M? ^ misses both. with respect divides into three cases depending rng g n ( g 1 U g^) rng g^ Ct g"*" bridges g^" is bridged once by k and show that the 3 g^, and recall the definition of bridging The construction of the on the way that the sets rng g^n g^ and with the property that in the cradle (see f i g 43), each to the g^, ( A ^ , A^, A^, A^} or g\ rng g^ bridge g^ and g 1 . If but not both, then we say that respectively. If both sets rng g^ n g \ rng g^ Ci g \ bridge g \ then g"*" is said to be bridged twice The bridging of g^ is defined anologously. The three cases (not exclusive) Case one. Each k_. has Property 'vP(a); neither of g\ g^ is bridged twice. 1 Case two. Some kj have Property 'VPCb); neither of is bridged twice. Case three. One of g\ g^ is bridged twice. g , g^ Facing page 84 - 84 These cases are clearly exhaustive (taking 'one' In case three to mean 'at least one'; however the reader has probably noticed that i f one 1 of 3 , 3^ i s bridged twice, the other cannot be bridged even once). Case one. Since each k. misses one gests an immediate application of Lemma One. rng g.., this case sug- It is easily seen that the hypothesis of Lemma One is satisfied except for the fact that the rng g^ may hit . If this happens, we alter the the following argument: rng g 2 ( i f another pair of the same rng g^, r assume that ng g^> k. miss the there is a circle rng g^ rng g^ £ by means of and Since £^ misses £ rng g^ . We imagine which i s free of the arcs 1 g . By sliding onto i t s e l f such that and carries misses Z rng Mg^, misses M is fixed on E 3 - K.. 1 to a position directly over Z^ misses rng Mg2 k'1? k^k rng Mgi M[£] and 3 contains a simple closed curve misses both misses and sliding on Z on the side of Z we are assured that 3 that i t s boundary components are a £^ stays 3 1v can move without 3 a. . This shows that there is a homeomorphism M of E touching the a., misses 3^ and eventually coming to rest directly Z^ . Although we use the term 'slide', we intend that close to but does not touch k^ or i f a l l four miss 3 1 £ C E — 3 which lies near the surface of the twisted band that misses the method is similar or easier). approximates i t so that over k^ g^ £ and on S 1 3 u a, u a„ v a_ u a, ; 1 2 3 4 M[£^] = £^ . Clearly M[£] . Construct a small annulus and k^ - £^ . By Th 5 (in Ch I ) , which bounds no disk in without hitting a so Z^ . This can be done so . Figs 54a, . ,.,'d show how location of an equator of Z 1 3 u £ Int a ct and which may be moved to the o k 2 U k 3 v k^ - 85 This shows that there is a homeomorphism H 1 fixes 3 , every k^ , E 3 of E 3 onto i t s e l f which A - K^, t and carries onto the location A W[l] shown in f i g 54d. Evidently in fact we can assume that a l l of misses both 0, misses both M'Mg^lA]; and M^Mg^fA], since other- wise an obvious homeomorphism can be used to push, the M^Mg^ away from 0, . Note that the M'Mg^ BdA, has Property "vP'(a) with respect to the WMg^f'A] while each because both kj M' and M continue to be Z-disjoint and are fixed on each Lemma One to construct Z-disjoint mappings rng g^ C~ rng M'Mg^ U such that rng g_^ misses every both rng g^ miss k^ £^ . Since M rng g_^ c rng g^ u end' of {J kj rng g^ M' g^ = rng g^ G rng g± V K^,, k^ that rng g^ misses. Since which J rng M^Mg^) are fixed outside of and rng g^ g^ such that misses ?2 K^, M, M' Case two. In this case we allow some of the X k. misses one to have rng g± . We reduce this case to with the usual properties such, that k^ to Property 'vPGa) = g^, 7 G ^ > or, more accurately, we w i l l define mappings rng G2^, with rng g7 misses both 1 Case one by converting the . k.. with Property ^p(h) rng G^., g\ above, in order to make rng g^ rng g. misses. Evidently Property-VP (b) with respect to the g^ = as well as any ?. and every k. that rng g. misses. Since each k, l J i j rng g^» the cradle of A has the required property. respect to BdA, Int S - N, It may be necessary to alter the homeomorphisms which act like G^ on JJ misses both to construct Z-disjoint mappings BdA, k. with . Now apply a result like Lemma One to 'the 3^ on miss those g^ misses. and on k.. . We can now apply and (since that M'Mg.j. e t2' ^t3' has Property 'VP(a) with and in fact each k_. misses one rng G ^ , - 86 - Th_e argument t h e n reduces t o Case one. We w i l l show how G . , i s c o n s t r u c t e d and i n d i c a t e t h e c o n s t r u c - xl G.. . t i o n of the other I f k.. has P r o p e r t y M ? ( a ) . w i t h r e s p e c t t o 1 xj the r n g g^,- then l e t G r n respect to rng ^ D » r n g G. v r n g G Io g± *= G.Q - G ^ . g the p i c t u r e upsidedown'. one o f r n g G^ f) g , 1 q misses has P r o p e r t y 'VP(b) (as d e f i n e d i n I §5 I for a c i r c l e s with basepoint has P r o p e r t y 'vP-Cb) w i t h a ^ u m^ > we assume t h a t and t h a t 0 /.o If k1 £^ a ^ ) , s i n c e o t h e r w i s e we s i m p l y Now b y Th 6 o r Th 7 ( i n I § 5 ) , s i n c e a t most rng G n g , 1 2 q w h i c h bounds no d i s k i n g \ g bridges 1 . there i s a c i r c l e q . We now have a c e n t r e ( o r a t l e a s t a double ended l a s s o ) w i t h P r o p e r t y 'vP(a) s i n c e rng ^ v G Q £/ u U m^ b u t £ ' i s l i k e l y t o be a v e r d i s o r d e r l y c i r c l e and U among o t h e r d e l i n q u e n c i e s , p r o b a b l y h i t s that £ / C Int r\ a^ and c o n t a i n s t h e base p o i n t s a y , rng G ^ m i s s e s one o f t h e r n g misses 'turn i." U a^U m^ u k^ (which means c a n ' t be used i n Lemma One ( t h e c o n s t r u c t i o n o f R i n Lemma One a b s o l u t e l y r e q u i r e s d i s j o i n t £ ^ ) . We g e t d i s j o i n t loops and a p i c t u r e l i k e f i g 44 by t h e f o l l o w i n g p r o c e d u r e w h i c h r e c a l l s t h e manipulation of t i n Case one. l i e s n e a r and approximates a connects (\ a ^ one p o i n t . Now s l i d e side of t o a base p o i n t on Au a X g"^ . A s h o r t s t r a i g h t a r c so t h a t misses a rng g \ X a t only rng G = r n g g^ . k e e p i n g the base p o i n t f i x e d , so t h a t t h e f i n a l A i s d i r e c t l y over g^\ to s l i d e meets and we now r e g a r d as a double ended l a s s o w h i c h m i s s e s A over position of be a s i m p l e c l o s e d curve w h i c h £ ' b u t misses We can assume t h a t A u a u a^ c m^ Let X £ ^ . As b e f o r e , we choose t h e ' r i g h t ' A on so t h a t A w i l l miss a^ u a^u a^U a^ . - 87 We now have a double ended lasso which looks .like k^ except that the 1 g , and i t remains only to telescope upper loop rides near but not on A moves to the location of i.^ . We 3 conclude that there is a homeomorphism M" of E onto Itself which a o X so that . a collapses and 3 fixes E - K^, Evidently misses k^, k^, k^, M"G^g A u a v a^o m^ and carries has the required properties of G-Q » a n d since rng G ^ n k 2 = rng G ±Q to. have Property 'VP with respect to the ment applies to k^ . k^, Since how rng G^ g"*" . oncethe by the rng G bridge ^, since just as i t was bridged by the k2 k3 misses one of k^ G„_ ii g^ is rng G-Q». rng G , , ^ , with respect and k2 misses one (note that we may have to work at the lower end of the g^ is bridged twice by the is used in the second application of Th 6 or Th 7). Evidently misses one of the so that so that rng G_^ G.„ so that i2 figure; the fact that neither band, g^ or rng G ^ g \ has Property ^P(b) n (k^ \J k^ u k^) = rng G ^ n (k^ U k^ u k^) , 2 we must ask rng G ^ . Thus neither of rng G rng G^ g \ g^ is bridged by the rng G,_, and we construct il of the continues and a similar argu- to the I 2 k^ It which i s clear that rng g"^ is bridged at-most misses separates the . Otherwise 2 n k2» = k2 is not fixed on rng G ^ . If then let G ^ = G ^ , k^ . G ^.. Clearly rng G _ ^ , g"*"; and of course boundary components of bridged twice by the M onto k0 3 rng G misses one I 3 rng G rng G^ . Proceeding in the same way-, we define misses one rng G.„, and since i3' G..i3 n (^ u k^) = rng G ^ n (k^ u k^) each, of I 3 . Finally define rng G ^ . Evidently the to construct the 2 G^ G^ so that each k^ can be constructed k^, k2 misses one can be used in the argument of Case one g^ . When altering the g^ to G^^, G ^ to G^ 2> Facing page 88 - 88 etc., we preserve 'Z-disjointness' because we adjust only.points in Similarly each G.. = a.. on Z . BdA . 1 In this case we know only- that one of g , 8'^, 1 is bridged twice. It i s easy to see that i f g is bridged Case three. say- 1 g twice, then no k^ can have Property 'vPGa) . For this would mean 1 that some t. misses, say, ° 1 cannot bridge i f each k. rng g^ n g g , so that the number of bridges is at most one. has Property 'VP(b), then in every case, m. rng g^ o. rng g^ _3 £j,c: E : rng g^; then by 1(1.7), and But must miss £ . must have Property 'vPCb) . For evidently i f any - . rng g^ - rng g^, then there can be no bridges at a l l . We are thus led to the conclusion that when case three holds, there is just one possible configuration (assuming that g^" is bridged twice): g^ is bridged twice, g^ is bridged not even once, and each k^ has Property 'VP(b) with respect to rng g^, rng g^, with 3 irij v a^ C E miss - rng g^ - rng g^ • Except for the fact that rng g^v rng g^, one. ^2 misses ?2 a n d U-^v etc.). We f i r s t alter the k_. and a sphere m j • S * n ^ 8 55, the 'k do not have Property^ X^ means that misses every rng g.^ _j rng g^ nu that rng g^ misses X, 4. mtsses-. Thts stmply misses each, m^ . A n method of Case one shows that i f rng Using the g. ;A.-*-(rng g v o K „ ) i - method of Case one, construct Z-dtsjoint mappings rng g^ g^ placed tn the usual way wtth, >P(a), so that we use Lemma One.itself and not the corollary. such that n o t rng g^ v rng g^ •• This is done just as tn Case Fig 55 shows the respect to y the picture begins to resemble f i g 45,- (though we s t i l l must construct the arcs so that ma examination of the ... a j> so does rng g^; c. « 89 thus rng g. xn. u a rng g„ misses a l l four each point of which misses . We also know that rng g^ also misses rng g^; t. has Property r"P(b) with respect to rng g., means that Therefore the four this rng g ... with respect to the rrng "g. k^ have Property- 'VP On the other hand, the fact that the inclusion rng g^c^rng g^ u J ^ ) " C 2 maybe proper means that the number of bridges on B^" with respect to — 1 — 1 rng g^H 6 , rng g^ n 3 may not be two, but may be one or zero. If this happens, then, since the number of bridges on 8^ with, respect to rng g^ n By rng g 2 A 8^ is zero (because of the presence of, say-, 3 m^cE _ r n g g^ „ r n g g using a previous argument) we have reduced the situation to either Case one or (Case two, i.e.' we have each, k with Property M? with respect to the rng g^ and neither 3 1 is bridged twice. However in the,'worst case', 3 nor 3 ^ continues to be bridged twice. If Lemma Two. construct 1 3 i s bridged twice by the — i rng g^ n 3 , then we use The hypothesis of Lemma Two is satisfied except that we must u 1 2 , u13> v i2» V — 1 13 ' S i n c e rng g^ n B bridges — i there is a component Q of rng g^ o B 1 f? , which connects the boundary- components of 3"*" . Q is cdmpact and misses rng g^ . By- the definition of Property ^P(b), Q meets a continuum e^c in and misses one of rng g^, £ 2 such that rng g 2 . Since e^ contains e^ hits the whole continuum construct an arc misses a^ O Q C rng g 2 , e^u Q U e^ must miss misses u^2 which, joins rng g 1 . The constructions of and a continuum e 2 rng g 2 , n a^ and u 13 > v rng g^ . Since we use 1(2.5) to n a^ i n B^ ^2' V13 a r e s imilar« and - 90 g,T:A -^(rng g^ - Int z)u n Now by-Lemma Two there are Z-disjoint mappings (where on Z, n are the sets described in Lemma Two) with g_T = BdA, and such that one rng g^ , say rng g^ , •' misses while both rng g^ miss = b^ u b^ U b^ rng g^ c rng g^ y ^ u ? 2 • Evidently . In the argument of Case three we did not suceed in constructing the g^ so that ?T. ^ ^2 one rng g^; instead m i s s e s rn 8. g i ^ r t,^ u £ 2 misses both n gg 2 a n de a c ^ ^-j rng g^ and three m i s s e s b^ miss the same rng g^ . In Part II of the proof of (3.1), i t turns out that it rn is sufficient to define the gl so that, three "b. g miss the same (the same thing happens i n the proof of [12, Th 10]). With some additional complication, i t is possible to improve the argument of Lemma Two so as to yield the usual result, i.e. to construct each g^ so that bj misses one rng g^; however we omit this argument. We have now completed the three cases of the proof of Part I of (3.1). Note that in,each Case, we constructed g^' so that rn rng g^ c rng g.D A . g g^ C rng g^ u U K2 . Thus we can write This w i l l be important when we apply the argument of (3.1) to the components of OL^y etc. To summarize the situation: i f {A^, A 2 , A^, A^} has Property mappings with respect to g^, g 2 , then there exist Z-disjoint 3 g_T:A -> E such that g i F C i ° n BdA ~ ' rng g£ c mg g^U k , if U X^y b^u b 2 u b^ U b^ i s the cradle of A, then both rng g£ miss Z^O X rng g' or three 2 a n d either each b^ misses one b. miss the same rng gT • Facing page 91 - 91 Part II of the proof of (3.1). We remind the reader that we are proving a result much like Bing's Th 10 of 112], which is also divided into a Part I and Part I I . Our Part II is very similar to Part II in Bing's proof and we absolutely require familiarity in detail with Bing's Part II (this i s only a matter of half a page). We think i t likely that the reader sees from the proof in 112], how to complete Part II here, and instead of a formal proof, we w i l l give what amounts to a gloss on Bing's method, plus a few comments required by the fact that our Property Q is not quite identical to Bing's. We begin by replacing iy^pq.r.s J J 1 shown in f i g 56. pq^.r^s misses one rng g_T b ^ f ... V U ^ by the figure This can be done so that either each arc or three pq^r^s miss the same Our terminology is now like that of [12] except that rng g.T rng g^ . replaces in [12]. We follow the division into cases found, in [12], We w i l l not prove that the three cases given in [12] exhaust the p o s s i b i l i t i e s , but remark for plausibility that the case division ... 1) Three miss one plus rng g^, pq^r^s P.q-2r2S Plus 2) misses P^3r3S the possibility: P.^'4r4s hisses Plus PI-J^T8 rng g^, 3) m i s s e s pq^r^s rn plus Pq2r2s P^^s m i s s e s Plus g §2 " ' s e e m s pq 3 r 3S misses rn a t P^1^ 8 pq^r^s pq^r^s misses rng g^ ; f i r s t glance to ignore rng g£, pq 2 r 2 s plus rng g^ • However this last variation i s just Case Two with the diagram inverted. We w i l l now describe how Bing's Part II can 3 be. altered to show that there exist Z-disjoint mappings such that F^ = with respect to on rng F^, BdA and the centre of rng F^ . A F_.:A E has Property 'VP Facing page 92 - 92 Case One: same rng gC any three of • If pq?r.s 1 pq.r.s (j - 1> 2, 3, 4) miss the rng gl i s ah arc which f a i l s to miss , 1— mC mC • o . . o then the structure shown in f i g 57 lies near pq-rs V p q ^ s - U pq,r,s -and 1 this ses 1 3 3 4 4 rng gf . The structure in f i g 57 can-be moved $6% fcKe°j5osition of the o 3 k of A by a homeomorphism M,. which fixes E - A . , Evidently x centre M^g^, M^g2 are the required F^, F 2 . If pq^r^-s is aim arc which f a i l s to miss rng gl , then one uses the structure in f i g 58 which lies o near pq^r^s u pq 2 r 2 s u pq^r^s and misses rng g«, . If pq^r^s or opq^r^s f a i l to miss rng g^ , the method i s like one of those already o given. If a l l four ^q.r-ss miss rng g' , then 'forget' one of them. J Case Two. pq^r^s misses J J X pq^r^s plus o pq 2 r 2 s rng g 2 • We replace misses Li pq^r^s rng g^, pq^r^s plus with the more compli- cated construction i n f i g 59. In f i g 59, s has been replaced by s,, s „ , s „ , s. which l i e near s so that the s. and arcs s.s„, 1 2' 3 4 j 1 3 2 S 4. m i s s ' a r n g g^ U rng g 2 . Abusing the notation slightly, 3 we have arcs p q j r j S j w i t n P ^ i r i s i ^ p q 2 r 2 S 2 C E ~ r n g S l * 3 s^^, S pq^r^s^ U pq^r^s^ c E which lies near lies near carries pq^r^s^., - rng . We build two new arcs: p'q^r^s^ , pq^r^s^ and misses pq^r^s^ and also misses s^^q^p'q^r^s^ s the circle ^s3» S 2 S 4' rng g 2 > and p'q^r^s2 rng g£ . Apply a move which which to the location shown i n f i g 60 and fixes a n d S 1S4 * IjOOK a t a disk in A bounded by pq 1 r 1 s 1 s 3 M 6 CrpM 6 (qpM 6 (p^)M 6 CqpM 6 Cr^)s 2 r 2 q 2 p . We w i l l c a l l this disk T and assume that i t i s just the obvious disk suggested by the'figure. Thus T misses a l l but the end points of Later we w i l l need the fact that miss- a l l but the end points of T pq^r^,s^s2 . can be constructed so as also to pq^r^s^ (in Case 3). There i s an arc Facing page 93 - 93 ACT with end points because arc and arc s and s^ which misses both s^gCrpMgCqpMgCp^MgCqpMgCr^^ 3 i i9iP92 2 2 s r r S m ^ s s e s r n S ^gS^ • abbreviate our arc nomenclature). to the p o s i t i o n of arc s s^s^s 3 misses rng M g g which moves A and fixes each pq^r^sy and . Although we do not know the location of A 2 can be done by means of the A - move defined i n I §3. rng M^Mgg^ misses rng M^Mgg^ miss the c i r c l e P.9]_-j_j_" r s pq^r^s^, rng M^Mgg s s s 3 and t' ( i t i s easy to give with M s V p o s i t i o n of the centre s s pq^r^, r e pi a c e by d a s e t m"v a' misses both f U a'u m' Case three, of A . I f this i s accomplished £O a u m pq^r^s misses that of Case two. Mg pq^r^s rng g 2 can coincide can be moved to the has Property ^P(b) with respect to the rng MgM^M^g^, which we define to be the required plus and both U a' O m' this property since much of V then the centre of A Mg, Evidently has Property ^P(b) with respect to the rng M^Mgg^ s^r^q^pq^r^s^). Evidently by a move this 2 which l i e s very near the f i r s t set so that rng M^Mgg^, i n T, s^s Mg(p')s s^s^ . F i g 61 shows <J P.9^ 4 ^ u i 3 6 ^ P ^ ^ 2 4 l r misses 2 2 (We w i l l now begin to Define a move s^MgCp'is^ rng MgS^ plus pq^r^s misses F^ . rng.gj, pq r s 2 2 . The mechanism of this case resembles We repeat the construction i n f i g 59 and define p r e c i s e l y as i n Case two, so that we arrive once more at f i g 60. However, since the rng g^ are related d i f f e r e n t l y to the various parts of the figure, we have this time: 3 s^^a^s,, c E - rng l ^ g ^ ~ rng Mgg as usual, but p q ^ r ^ w pq^v^s^u M, (p')M Cr^s^ c E pq r s 2 2 2 g 3 U p q r s t / MgCp'iMgCrps^ E - rng M g 3 3 3 & 2 3 - rng M ^ , . In this case we 2 Facing page 94 e - 94 must use a fact that we stated but did not completely use i n Case 2, v i z . that Mg fixes a l l four placed so as to miss pq^r^s . We assume tht the disk T is pq^r^s^ . We use Th 4 from I §4 at-this point; at the analogous place i n [12], Th 7 of [12] i s used. By Th 4, since s3M6(qpMg(p') S 3SiriqlP s^, s m i s s and p q 2 r 2 s 2 r n g M 2 ' such that 6gl' miss t n e r e rng Mgg2 i sa na r c X misses either a move M', similar to M.,, to move / / and M6(p')M6(q^)s2 and ACT with end points rng Mgg^ or rng Mgg2 . Apply X to s„M/.(p')s_ . This can be 3o 2 done by an A - move as before; but some care should be taken so that 7 fixes every J pq.r.s. ]j (as well as, of course .'s„s,.s.s„) : - the 3 1 4 27' reader might f i r s t prefer to move pq^r^s^ cannot interfere to a new location where i t with the collar of T used i n the A - move. The proof i s now completed along the lines of the previous cases,: using the fact that i f s^MgCp^s,, misses f i g 62 lying near s^MCp^s^^ v s^q^q^r^ while i f s3Mg(p'')s2 misses near pletes rng M^M^g^, then the set shown! in s^s3Mg(p')s2s^s^V rng M7Mgg2> misses rng M^Mgg^; then the set i n f i g 63 lying s 3 r.jq 3 pq 2 r 2 s 2 misses rng M^Mgg,, . This com- part II of the proof of 111(2.1)£J. Corollary to 111(2.1). If {A.^ ... A^} has Property- M} with respect to mappings to mappings F^, F2 Proof. g , g 2 , then such that We know that A has Property rng F c.r n g g. with respect , rng g^ c rng g± U A . And a l l the moves given In Part IT of the proof of 111.(2.1) can be defined so as to fix E3 - A O . - 95 4. - Proof of 11(2.2). We have now shown that i f then A has Property Bing's i n [12]. f^ Suppose that has Property Q; i.e. has Property Q, 3 such that and JA c to be assigned 2 a^, We a^, {A^, ... A^> cx^, ... has Property Q. (3.1) does not imply that some f o r the luminous reason that each as the argument i n §1 Ex 2 shows. for each then each of then by (3.1), ct^ has Property Q . Property Q c^, Z, f. i „ = C. i|BdA i the proof of 11(2.2) follows d i r e c t l y from this f a c t . has Property Q, Property Q has Property according to the remark at the beginning of §3). w i l l show that i f A A f.:A-> E i are Z-disjoint (we continue to take arbitrarily If ... A^} .. Our argument now diverges somewhat from r r the {A^, A^ A^ has has Property ^Q, However we can show that has by adapting the argument of the proof of (3.1) to show that A^, if ••• j 4 ^ A n a s Pr °perty then so does A^ . This i s easy to do since the proof i s simply restated i n terms of images under the embedding proof of h_. of various subsets of A . Occasionally i n the 111(2.1) we constructed arcs which were perpendicular to certain surfaces. While h^ does not preserve this property, the reader w i l l appreciate that we used such constructions for topological purposes, e.g. to make one arc l i e along another, or to miss certain subsets, and these properties are preserved by h^ . We do not re-define Z, ]_> ' 2 C c of course, since we intend to show that the same Property Z i s possessed by each of contain to l e t A a^, so that we have Z F A eventually. a2> Z Di^ P~y ... • C C We o r i g i n a l l y defined as required. To show that ' i'' 2 Z to We intend of course CL^ has Property Q, assume that - 96 the set of components of mappings 8^' § & " ^PP-^ 2 has Property 2 a r e s u x with respect to q u a l i f i e d l i k e the c o r o l l a r y of (3.1) to t {A^^, ... A ^ } to obtain Z-disjoint mapping that on F^ - the F^ . BdA, and A^ F^:A -> rng g^ U A^ has Property We can see that since rng F.^ such. with respect to does not exceed rng f.. i n 3 E ~ A^, the dogbones A^^' ••• 2 4 ' A A 3 1 ' **' 34' A tinue to have Property • M}- with respect to the A 41' E^^, A 44 c o n ~ f o r as we saw e a r l i e r , possession of Property 'VP depends on the fact that rng g^ misses certain continua i n various dogbones, and this property i s inherited by rng F.. 3 at least f o r dogbones i n E r n g i l 2 F U A S U c h t h a t F i2 = C - A^. i o n B d Construct Z-disjoint mappings; F A a n d A 2 ^ a s P r °P e r t y : A i 2 with respecl "3 to the F ± 2 . Once again, dogbones i n ' E respect to the - A . ^which. have Propertiy/v,Q with 2 F^. v.-, continue-to have . P r o p e r t y , . w i t h respect to the '.F. . This means that not only 'A 0 2> A^ but Property 'vQ with respect to the F ^ 2 . 'A'31, ... A^,- on M}. BdA ... A ^ F^:A -> E and with respect to which, a l l of Assume that the set of components of which agree with A^, ... A^ have Property have Property . an argument.like that of (3.1) Corollary can be applied to each (perhaps lexicographic) order to show eventually that . Q If A and has Property Q, have Evidently we can continue i n this 3 way and f i n a l l y derive Z-disjoint mappings c\ A41, Then A., in . has Property then by induction, a^, have Property must also have Property Q . We think that i t Is now evident hpw^ to proceed i n the case that m = 4, 5, . . . . We w i l l show how the induction argument above implies II(_2.2). If the f^ In the hypothesis have ranges that Intersect tn 11(2.2) ts true; thus we consider only the case that A , then rng f ^ f\ rng f 2 n A = 0, 97 i . e . the case t h a t the we f are A - disjoint. showed t h a t f o r a f i x e d c h o i c e of Q„ , ^Z,a^, G t h e n so does each a 2 c\ - l i n 11(2.2) then has P r o p e r t y Q A c Z c: E 3 m I n the p r e c e d i n g Z, G^, c ^ , . If - rng if Z = A and - rng G 2 CL m means t h a t b o t h has P r o p e r t y Q„ z,c^, rng f . l G . has Property C, ,. c „ 1' 2 are the as r e q u i r e d , and by an argument l i k e t h a t of §1 Ex 1. argument, e v e r y A argument A By the i n d u c t i o n As we saw earlier, this 2 a h i t some component of for m however l a r g e . m F i n a l l y we w i l l show t h a t b o t h rng f ^ must h i t a b i g element A. A of the dogbone d e c o m p o s i t i o n G . Let G be the s e t o f a l l elements <X^, a^, o f the dogbone c o n s t r u c t i o n ( i . e . a l l components o f w h i c h meet b o t h r n g f ^ and rng f 2 . by the arguments^of t h i s r t c h a p t e f , each ••• ) Evidently G i s i n f i n i t e , for a must c o n t a i n an element of m /\ r, G . C l e a r l y one f o r the f o u r s u b s e t of A. 3 G, subset of of A^, c o n t a i n a l l of then one G . ... A^, of A.,, jl must c o n t a i n an i n f i n i t e s u b s e t o f G . If ... A.., There i s a sequence A. c o n t a i n s an 3 say A., , jk' j4' A 3 ••• w h i c h c o n t a i n s i n f i n i t e l y many dogbones w h i c h meet b o t h Obviously section each member of the sequence meets b o t h An A. n A 3 n A m e e t s 3^- both V, and . the inter- can a l s o use i 3 the dogbone m e t r i c to show t h a t i f the images of the in One infinite each of rng f rng f , rng f . . infinite c o n t a i n s an 3 A ^ r> ^ j j ^ G, rng then t h e r e i s a neighbourhood s y s t e m of the p o i n t s of are disjoint consisting V o f s m a l l 3 - c e l l s around the s m a l l p o i n t s and images o f dogbones about the b i g p o i n t s such t h a t no neighbourhood of d i a m e t e r s m a l l e r t h a n dogbone m e t r i c ) meets b o t h images o f the some a.^ has P r o p e r t y rng . e Ctn the This implies that ^Q, cf>;proof o f Th 12 o f [12] D. - 98 BIBLIOGRAPHY [1] R. L. Wilder, Topology of Manifolds. A.M.S. Colloquium Publications 32(1949). 3 [2] T. M. Price, Upper semi-continuous decompositions of E , Thesis, University of Wisconsin (1964). [3] R. H. Bing, Decompositions of E , Topology of 3-manifolds and Related Topics. Prentice-Hall (1962), 5 - 21. [4] R. H. Bing, Locally tame sets are tame, Ann. Math., 59 (1954), 145 - 158. [5] M. L. Curtis and R. L. Wilder, The existence of certain types of manifolds, Trans. Amer. Math. Soc. 91 (1959), 152 - 160. [6] R. H. Crowell and R. H. Fox, An Introduction to Knot Theory. Boston: Ginn and Co., (1962). [7] C. D. Papakyriakopoulos, Dehn-s lemma and the asphertcity of knots, Ann. Math. 66 (1957), 1 - 26. [8] J . F. Wardwell, Continuous transformations preserving all.topological properties, Amer. Jour. Math., 58 (1936), 709 - 726. [9] S. T. Hu, Homotopy Theory. Academic Press (1959). [10] C. T. Whyburn, Analytic Topology. A.M. S. Colloquium Publications 28 (19.42). [11] L. 0. Cannon, Another property that distinguises Bing's dogbone space 3 E-= , Notices Amer. Math. Soc. 12 (1965) p. 363. 3 R. H. Bing, A decomposition of E into points and tame arcs such 3 from [12] that the decomposition space i s t o p o l o g i c a l l y d i f f e r e n t from E , Ann. of Math. 65_ (1957), 484 - 500. 3 113] tt. M. Lambert, A topological property of Bing's decomposition of E into points and tame arcs, Duke Hath. J . , 34 (1967), 501 - 510. 114] S. Armentrout, A property of a decomposition space described by Bing, Notices Amer. Math. Soc. 11 C1964), p. 369. - 99 - - 100 - - 101 - - 102 -
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Bing's dogbone space and curtis' conjecture Hutchings, John Edward 1973
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Title | Bing's dogbone space and curtis' conjecture |
Creator |
Hutchings, John Edward |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | Bing's dogbone space D is an upper semi continuous decomposition space of E³ which fails to be E³ although the associated decomposition consists only of points and tame arcs. It has proved difficult to find topological properties of D which distinguish it from E³. In this paper, we prove a conjecture of Morton Curtis in 1961 that certain points of D fail to possess small simply connected neighbourhoods. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-04 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080386 |
URI | http://hdl.handle.net/2429/32041 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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