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Bing's dogbone space and curtis' conjecture Hutchings, John Edward 1973

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BING'S DOGBONE SPACE AND CURTIS' CONJECTURE by JOHN.EDWARD HUTCHINGS M.A., Un i v e r s i t y of 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 the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y 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 e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p urposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i 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 w r i t t e n p e r m i s s i o n . Department o f Mathematics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date Nay 25 1973 - i i -ABSTRACT Bing's dogbone space V is an upper semi continuous decompo-3 3 sition space of E which fails to be E although the associated decomposition consists only of points and tame arcs. It has proved dif-ficult to find topological properties of V which distinguish i t from 3 E . In this paper, we prove a conjecture of Morton Curtis in 1961 that certain points of V f a i l to possess small simply connected neigh-bourhoods . - i i i -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. - i v -TABLE OF CONTENTS Introduction- ... .1 Chapter I 3 Chapter II Bing's Dogbone Space and C u r t i s ' Conjecture 24 Chapter I I I Ge n e r a l i z a t i o n of a Theorem of Bing: Lemmas........ 56 Chapter IV Generalization of a Theorem of Bing: Main Proof 78 Bibliography 98 Appendix .......99 - v -TABLE OF FIGURES Figure Page 1 6 2 6 3 7 4 11 5 13 6 14 7 15 8 . 15 9 16 10 facing page 17 11 " 17 12 " 18 13 21 14 22 15 23 16 a facing page 25 16 b " 26 17 26 18 _ " 26 19 26 20 " 27 21 » 27 22 29 Table of Figures cont'd - v i -Figure Page 23 • •• facing page 34 24 11 34 25 " 41 26 • " 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 Tables of Figures cont'd - v i i -Figure Page 45 facing 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 is denoted by V in this paper) 3 3 is a decomposition space of E which fails to be homeomorphic to E even though the associated decomposition space is upper semicontinuous and point-like, and each element of the decomposition is 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 3 3 of E would turn out to be E . Although V dates from 1955 and has become rather well-known, i t has been found hard to determine those top-3 ological properties of the space which distinguish i t from E . Bing's original paper [12] showed that V is a non-manifold; but V is a simply 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 is 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 is 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 tori. 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 will probably share our pain at the length of the argument (the whole paper is essentially one theorem). The reader who is unfamiliar with pathological decomposition spaces is advised to read .13], which is brief and exceptionally entertaining, and then skim Ch. II?.,. We will mention some notational peculiarities: we follow common practise in describing geometric constructions, even complicated ones, by the use of diagrams. "Theorem' in this paper means 'working theorem'; thus 'theorems' appear in the introductory-chaptereonly. - .3 -CHAPTER ONE 0. Introduction. This first chapter gives preliminary material for the arguments in Ch III and especially Ch IV. The reader who wishes to skim the paper will find that Ch II, which contains the discussion of Curtis' Conjecture, is largely independent of this first 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. In this chapter, sections 1 and 2 are elementary, §3 contains working theorems for Ch IV, and §4 is essentially a comment on Bing's Theorems 6 and 7 of 112]. Section 5 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 left 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>: The expression 'Bd A' may mean either the manifold boundary of the mainifold-with-boundary- A, or. the point-set boundary/of the set A . A similar comment applies to the expression 'Int AT • This reflects common practise; we will comment whenever the meaning is unclear. As mentioned In the preface, our attitude to the construction of tame sets will be cavalier; we will con-struct many important tame sets simply by describing the set and perhaps giving a picture of It. We advise against the intuitive approach of imag-ining our constructions as stralght-sided polyhedra whose structural detail is so fine that the polyhedra approximate the figures closely. Several of our arguments will 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 (2.1) and (2.3) . We define an annulus to be a topological sphere with two holes. The proofs are omitted. (2.1) . Let a be an arc which intersects two disjoint closed sets S^, . Then there is a sub-arc a* of a which connects and and meets U only at the end points of a* . (2.2) . Any two annuli A^, are homeomorphic. Any homeomorphism of one boundary component of A^ onto a boundary component of A^ may be extended to a homeomorphism of A^ onto A^ . (2.3) . The union of two locally connected (lc) eontinua which intersect is a lc continuum. (2.4) . Let 0 be a bounded connected open set in the plane whose boun-dary is l c . Then any two points x and y in 0 may be connected by an arc which lies in 0 except possibly for its end points. (2.5) . Let A be a 2-manifold with boundary, and K a continuum in A . Then any two points of K may be connected by an arc in Int A (ex-cept possibly for end points) which lies within a distance e of K . 2 " C2.6). Let C. , C0 be disjoint simple closed curves in E . Then - 5 -one of the following exclusive alternatives is true: a) cz Int C2 or equivalently Int c l n t C2< b) C2 cr Int c^ or equivalently Int c l n t C^. c) Each of 'C^ , C2 lies in the others exterior, or equivalently Int D Int C2 = 0 (2.7). Let A be an annulus, and C a. simple closed curve in Int A which bounds no disk in A . Then C separates A into components B^, B2 such that B1 U C and B2 U C are annuli. - 6 -3 . Sliding Curves on Spheres. 3 C3.1). We w i l l often need to 'move' or 'deform' curves in E . This w i l l be done by sliding the curves on convenient spheres, disks and 3 annul! in E . The sort of thing that may be encountered is shown in f i g . 1. A double ended lasso has loops p , q and 'middle' z . We may want to push z • over to the position of z" in the figure or expand p so that i t looks like p' . This can be done with a homeomorphism 3 3 H: E -»- E which carries, say, z onto z' and can thus be said to move ' z to z " . „ 7 -3 Suppose p U q U z l i e s on a disk A C E . We ask what properties the homeomorphism H should have in order to reflect the intuitive idea of sliding z to z' on A while keeping p <j q fixed. One way to do this would be to construct a new disk D C A (see f i g . 2) so that D contains z u z' and misses p and q except where they h i t z U z' • Then we could require that H carry z onto z' , H[D] = D , and that H be the identity on A - D and on Bd D , .(thus H w i l l f i x p U q). It seems a good idea to specify a number of standard moves, prove that they can always be made and stick to these in the sequel. When, as commonly happens, an arc or loop moves only a short distance and has explicit i n i t i a l and f i n a l locations, then our idea of 'standard moves' is probably too formal. However our standard moves are intended for the • case that the i n i t i a l position of the-set is unknown-. In this case the existence of the required move is less obvious, especially when, as in §5, Th 6 , a base point must be held fixed during the move. If S i s a 3 sphere with n holes in E , then a collar of S is the image of 3 an embedding h of S * [-1,1] into E so that h(x,0) = x.. Evidently a collar of S may not exist (S could be wild). A set upon which a \ collar has been constructed is called a collared set. Note that a collar of S is not a neighbourhood of S . (3.2)_. A- B- and B' -moves. We give three standard moves i n Theorems 1 and 2 . 3 Theorem^!. Let D be a disk i n E , J a c o l l a r of !D , and a' }, a two arcs which have common end points and l i e i n Int D except f o r these end points, which l i e i n Bd D . Then there i s a • 3 homeomorphism ACaja^DjJ) 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 , and which f i x e s E - J . We c a l l A(a,a^,D,J) 'the A-move' and say that A(a,a',D,J) moves a to -.a' . (Of course the f a c t that a moves to a^ i s only one of a number of things that have to be kept i n mind. We write the move as a function of D and J to emphasize that the t r i c k of using the move depends on the r i g h t d e f i n i t i o n of D and J ) . 3 Theorem 2. Let A be an anhulus i n E . Let c, c % be simple closed curves which l i e i n the i n t e r i o r of A and bound no disks i n A . Let Q be a c o l l a r of A . Then there i s a homeomorphism 3 B(c,c',A,Q) , also 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~~ c onto c' ,. A onto i t s e l f , and which f i x e s Bd Q and E - Q . I f , i n a ddition, c and c^ have a common base, point y , then there i s a homeomorphism .B'(c,c',A,Q) and the following a d d i t i o n a l property: i f h i s the embedding associated with Q , so that Q = h [ A x [^1,1]] , then the B'-move fi x e s y and i n f a c t a l l of hjy * [-1,1]] . The B'move i s a move 'keeping the base point f i x e d ' . One could probably f i x the base point by providing that c u c" '. could h i t Bd A C Bd Q - 9 -so that y £ Bd Q '.(the B-move does not.permit t h i s ) , however the B'-move as given above f i t s the intended applications better and i s easier to prove. We w i l l give an example which shows why we want the B'-move to f i x h[y x-{-1,1]] . F i g . 3 shows c, c', A, and an arc a U b such that a misses A and b i s a s t r a i g h t arc perpendicular to A . We want to move c to ec" while leaving a U b f i x e d . We do t h i s with a B'-move B^(c,c',y,A,Q) i n which Q i s defined so that a l l points of Q' l i e near A ( i . e . f o r x £ A , h[x x [-1,1]] i s short) and so that each arc H[x x [-1,1]] with x 6 A i s perpendicular to A . For a s u f f i c i e n t l y 'thin' Q , the B^-move w i l l f i x a because a c E - Q , and b w i l l be f i x e d because b l i e s i n " h[y x [-1,1]] wherever i t hits. Q . Evidently the u t i l i t y of the B'-move i s l i m i t e d . However subsequent use of the B^-move w i l l be very much along the l i n e s of t h i s example. 4. The Phragmen-Brouwer Properties. The Z o r e t t i Theorem. The Phragmen-Brouwer Properties are usually given f o r the n-sphere, but hold also on a disk. We quote from Wilder, [ I , II 4.1]. Let S be a l o c a l l y connected metric space. Then the following properties of S are equivalent. (4.11) . I f A, B are d i s j o i n t , closed subsets of S , and x,y 6 S such that neither A nor B separates x and y i n S , then A U B does not separate x and y i n S . . (By 'X separates x and y i n S ' i s meant 'x and y are i n d i f f e r e n t components of S - X'). (4.12) . I f S = A U B , where A, B are closed and connected, then - 10 -A fl B i s connected. (4.13). If A, B are d i s j o i n t closed subsets of S and a e A, b € B, then there e x i s t s a closed connected subset C of S - (A U B) which separates a and b . Theorem II 4.12 of 'II] states that these properties are equiv-alent i n a l o c a l l y connected metric space. From VII, 9.3 of [1] (note also 9.2), a disk D w i l l have properties (4.11), (4.12), (4.13), i f i t s f i r s t B e t t i number i s zero; thus (4.11) ... (4.13) hold on D . (4.2). We get the following important working theorems from (4.11). These theorems resemble Theorems 6 and 7 of [12]. 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 subsets of E . Let ~pxq, pyq be arcs i n D which share the end points p and q , and such that arc pxq misses F^ , arc pyq misses F^ . Then there e x i s t s an arc pzq. with end points p, q such that arc pzq c D - - . Theorem 4. Let D, F^, F^, Vi 1* arcs pxq, pyq be defined as i n Th 3 except that arc px U arc yq misses F^, arc py 0 arc xq misses 'F^  . Then there e x i s t s an arc pzq D with end points p, q, such that arc misses e i t h e r F^, or F^ . Proofs of Th 3. and Th 4. Since D i s simply connected, pxq and pyq are homotopic i n D by a homotopy which f i x e s p and q . Using t h i s f a c t , the proofs of Th 6 and Th 7 of [12] may be used word for word to prove Th 3 and Th 4 r e s p e c t i v e l y , reading D for M tn [12] O. 11 -(4.4). The Plane Separation Theorem and the Z o r e t t i Theorem. We quote these 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 Plane Separation Theorem. Let A , B be compact set s i n 2 E which i n t e r s e c t i n at most one p o i n t . Let a £ r " A - B , b e B - A , and l e t e > 0 . Then there i s a simple c l o s e d curve J which separates 2 a and b i n E , l i e s w i t h i n an e-neighbourhood of A , and misses A U B except p o s s i b l y at the p o i n t A f) B . (4.42) . The Z o r e t t i Theorem. I f K i s a component of a compact s e t M i n the plane, then there i s a simple closed curve J whose i n t e r i o r contains K ,. which misses M , and which l i e s i n an e-neighbourhood of K . 5. Annulus Dodging Theorems. Suppose A i s an annulus and F i s a closed s e t i n A . When can we say th a t a simple close d curve which looks l i k e c i n f i g . 4 e x i s t s so as to miss F ? The answer i s about what would be expected. We say t h a t F b r i d g e s A i f f the two boundary components of A are i n the same component of Bd A U F , or e q u i v a l e n t l y , i f f some component of F meets both boundary components of A . We w i l l prove the equivalence. Let the boundary components of A be Z and m . '•'-«- ' i s obvious. I f no component of F meets both £ and m, then no component of F meets both £n F and ra n F, and by 1(9.3) of [10] ( t a k i n g A, B, K to be I 0 F, m n F, F ) , there i s a s e p a r a t i o n of F i n t o compacta F„, F such that F» meets only " t . F meets only t m I J * m m i n Bd A . E v i d e n t l y t h i s denies the exi s t e n c e of a connected subset of -•12 ^ F u £ U m which meets both £ and, m . (5.1). If F fails to bridge A, then there is a simple closed curve c in Int A . such that c bounds no disk in A and c misses F . 2 2 2 Proof: We can assume that A is the set 1 <_ x + y <_ 2 in E 2 2 Let D be the set x + y 1 . Let . £, m be the boundary components 2 2 2 2 x + y =1, x + y = 2 respectively. Consider the component K of £ o m v F which contains (the connected set) £ . The set £ u m of is 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 in its interior and lies in an e-neighbourhood of K . We will show that c has the properties required by (5.1). To see that c Int A : K contains "£ and misses m, . since otherwise F. bridges A . Thus K C (A D) - m = IntCA V D) . Since K is compact, K has an e-^neighbourhood in Int (A u D) , and we can assume that c lies in this neighbourhood. Thus c <= Int (A <J D) . But c encloses K 3 £ and hence D (by (2.6)); therefore c c Int(AU D) - D = Int A . We know that c bounds no disk in A because, from the 2 "" Schoenflies theorem, c bounds just one disk in E This disk is .Int c which is not a subset of A since i t contains D . Since c misses F (by construction), lies in Int A, and bounds no disk in A, the proof of (5.1) is complete. Q . Remark: the converse of 5.1 is true and easily proved. We will look at some generalizations, the choice being influenced by later applications. - 13 -Theorem 5. Let ; F^> be d i s j o i n t c l o s e d s e t s i n the annulus A . I f each of F^, F^ f a i l s to b r i d g e A , then there i s a simple c l o s e d curve c i n I n t h - F^ - F^ such that c bounds no d i s k i n A . Proof: This r e s u l t i s t r i v i a l once x^ e show th a t i f n e i t h e r of F^ , F2 bridges A , then F^ V F^ f a i l s t o b r i d g e A . Once t h i s i s done, the proof of Th. 5 i s completed by a p p l y i n g (5.1) t a k i n g F to be F^ U F^ To see that F^ U F^ f a i l s to b r i d g e A : s i n c e F^ does not b r i d g e A , no component of £ U m U F^ i n t e r s e c t s both £ and m , ( f o r otherwise some component of £ U m U F^ would contain. £ and m). By I (9.3) of [10], t a k i n g A , B , K i n that theorem to be £ , m , £ o m u F^ , there i s a s e p a r a t i o n of £ u m u F n i n t o d i s j o i n t compact s e t s U. , 14 -u"2 so that Z c TJ^  , m c • Similarly there is a separation of Z U m U l n t o disjoint compact sets V , V 2 > with Z c , m c. v 2 . It is easily checked that U V misses U2 U V 2 • Evidently Z v m o' U may be separated into the disjoint closed sets O and U V 2 with ; Z c \] U , m c U2 U V 2 . Therefore £ and m are not in the same component of Z <J m V F^ U Y^ and F^ U F 2 f a i l s to bridge A O . We remark that 'F^ U F^1 m a v be replaced by.a finite*union of disjoint closed sets with a few t r i v i a l changes in the proof. Theorem 5 is false for a non-compact union of sets F^, F 2, ... „ Fig. 5 shows A and a collection F , F 2 > ... such that A is the set 1 ^_ r <_ 2 in polar coordinates and for 1 = 1 , 2, 3, F. is a subset of the ray 0 = 1/i . Although each F^ does not bridge A (nor does the union 00 {J F.) , the curve c i n Th. 5 cannot be constructed. i=l 1 We next look at the case where the curve c is constructed as in Th. 5 but with the further property that c contains a given.base point x . In this case c cannot in general miss either of F^,;.F 2 , as Fig. 6 shows. - 15 -We w i l l g i v e a c h a r a c t e r i z a t i o n o f t h o s e placements o f x , F^, F 2 > so t h a t c can be made to miss one o f F^, F 2 . We say t h a t a s i m p l e c l o s e d c u r v e c w i t h b a s e p o i n t x has P r o p e r t y ^ F ( r e a d ' p r o p e r t y n o t - P') w i t h r e s p e c t t o c l o s e d s e t s .F , F 2 i f f one o f the f o l l o w i n g i s t r u e : ^PCa): c mi s s e s on o f F^, F 2 . % P ( b ) : T h e re e x i s t s a p o i n t y & c - x and a d e c o m p o s i t i o n o f c i n t o a r c s c^, c 2 > w i t h c^ v = c and c^ n c 2 = {x,y} , such t h a t F^ m i s s e s c^, F 2 m i s s e s c 2 (see f i g . 7). T h i s i s an u g l y and awkward d e f i n i t i o n . An e q u i v a l e n t and p r e t t i e r s t a t e m e n t i s 'c has P r o p e r t y ^ P i f f any p o i n t i n c - x may be j o i n e d t o x by an a r c which m i s s e s one o f F^, F 2 ' ; however we w i l l n ot p r o v e t h i s , and we w i l l u s e the e a r l i e r s t a t e m e n t e x c l u s i v e l y . The odd name o f t h i s - 16 -property is intended to re c a l l Bing's Property P in {12], This property is defined on double ended lassos (see f i g . 8). Later we w i l l define Property ^ P 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 % P in the present sense. The next theorem says that i f c with base point x has Property ^ P , then there is a loop c' which behaves like c and misses one of F^, F^ . Theorem 6. Let A, F^, F^ .be defined as in Th. 5, including the condition that neither F^ nor F^ bridges A . Let x £ Int A . Let c be a simple closed curve which l i e s i n Int A .and bounds no disk in A and contains x . If c has Property ^ P with respect to x, F , .F » then .there ..exists .a .simple closed curve -c" which l i e s in Int A, bounds no disk ih A , has base point x , and misses one of F^, F^ . This result cannot be improved so as to allow us to specify which of F^, F^ is to be missed by of . Fig. 9 shows a case where c" in Th. 6 cannot be made to miss F^ although F^ V F^ f a i l s to bridge A , and c exists with Property ^ P . (There are simpler counter examples in which only F hits c . One of these may be derived by removing F Facing page 17 - 17 -from fig . 9. However fig 9 shows that matters do not improve i f we insist that both 'F and F 2 hit c .) 2 2 Proof of Th 6. We can assume that A is the set 1 <^  x + y <_ 2 2 , ' in E .' The inner and outer boundary components of A will be called Z and m respectively. Since neither of F^, F 2, bridges A , i t follows from Th 5 that there is a simple closed curve e C Int A which bounds no disk in A and misses F^ U F^ • If x € e , then the proof is completed by letting e be ; thus we assume that x £ e . We make the further assumption that x £ Int e ; i t turns out that this restriction is easy to remove. Assuming that x 6 Int e , we construct c' by first defining a lasso Y as shown in fig 10. The loop of Y is either e or a curve which behaves like e and is constructed similarly, while the 'handle' of Y joins the loop to x . The whole of Y misses one of F^, F 2 • The curve cf lies near Y and meets x as shown in fig 11. Construction of Y . The lasso Y consists of the union of a simple closed curve r and an arc s , and is constructed so as to have the following properties: Y C Int A, Y misses one of '.:"'F , F 2, the circle r bounds no disk in A , the end points of s are x and a point z £ r , and s - z misses r . The construction of Y is divided into two cases. Case one: e meets c - x . We assume that c satisfies Prop-erty ^P(b), since i f c satisfies Property P(a), we immediately let Facing page 18 - 18 -c =  c Thus we take c to be the union of arcs c^, c^ which meet only at their end points, and for 1 = 1, 2, c^ misses F^ . If. e meets, say, c^ (It will do no harm i f e meets both c^), then use (2.5) to construct an arc s which joins x and e ri c^, and lies so near c^ that s misses F^ (or take the obvious sub-arc of c^). Let r = e, Y = r u s . To check that Y has the required properties; Y misses one of F^, F^ because s misses one of ^ and r misses both; r = e bounds no disk in A by construction; and Y c Int A because eve C Int A . Finally, from (2,1), we can assume that s meets r only at a single point z . Case two: e misses c . As before, we assume that c has Pro-perty ^ P(b) . Outline of proof: a) As usual we take c to be the union of arcs c^, c^i let be c^ plus those components of F^ which hit c^ and let be plus those components of F^ which hit • u is a component of c U F^ u F^ . b) A 'Zoretti curve' r is constru-cted so that r misses c O ^-^U lies in-. Int A, encloses K^ c/ K^, and bounds no disk in A . c) Some care needs to be taken to attach the ta i l s to r so that s 'misses one of the F^ . Construct a disk d C Int A with -centre on r, (see fig 12) so that d is big enough to hit U but small enough to miss one F^ . This is managed by a careful choice of the e assoc-iated with the Zoretti curve, d) There is an arc s near u K U' d which has the required properties. Details of proof. a) Let = c^ plus those, components of F^ which hit c^ . Let = plus those components of F^ which hit . We will show that U is a component of c V F^ U F^ . Let K be the component of c U F^ \J F 2 which contains the connected set U K^, and suppose that some point p e x i s t s i n K - (K^ u K^) . Then p l i e s i n one of F^, F^, say F^ . Since p ^ U K.^* no component of F^ meets both p and c f> F^ . By 1(9.3) of [10], there i s a separation of F^ i n t o compacta U^, containing c 0 F^ and p r e s p e c t i v e l y . Evidently U 2 misses not only but F 2 a n <* t n e whole of c ; thus u U c i s a compactum d i s j o i n t from U 2, and there i s therefore a separation of c U F^ U 7^ = u u c U F 2 into compacta containing c and p s e v e r a l l y . This denies the assumption that p l i e s with c i n a connected subset of c o F U F 2 . b) Since x 6 Int e and e misses c u F u F 2» a l l of U K 2 l i e s i n Int e by the usual argument. Since U i s a com-ponent of c u F^ U we can construct a Z o r e t t i curve r which misses c U F^ u F 2, encloses V K^, and l i e s withing a distance e of O K 2 . The following argument shows that r bounds no disk i n A : since c bounds no disk i n A, Int c, which i s a disk, must meet points of E - A . Since r encloses U K 2 D c, by (2.6) r encloses Int c . Hence the unique disk bounded by r meets points not i n A . To see that r C Int A : we 2 saw that c encloses points of E - A . These points cannot be i n Ext m by (2.6), since m encloses Int A o c . Hence Int c meets Int Z, and by a connectedness argument, since c misses Int Z, Int c >^ Int Z . Since r encloses Int c, r misses Int Z . Since r l i e s close to the s compactum U K 2 c Int m, r can be assumed to l i e i n Int m . Therefore r C Int m - Int|= Int A . c) We construct a (closed) disk of radius 2e with centre anywhere on r . C l e a r l y d w i l l h i t (J K^. We show that d c Int A by showing - 2 0 -that dc Int m and d misses Int £ . The distance e could have been chosen so that 4E (i.e. the diameter of d) is less than the distance separating c and Int £, 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 Int m). Since d hits K^U K^, d C Int m by the choice of e . If d hits Int £, then d must also hit c, since points of d n r l i e in the exterior of c which encloses Int £ as we saw. By the choice of e, d cannot meet both c and Int £; thus d misses "Int £ . We also assume that £ was chosen so, that for, i = 1, 2., 4e is less than the distance separating from F^ . The disk d must hit one of K^, K2> say (it does no harm i f d .hits both K^). Since d hits K^, d misses F^ since otherwise F^ would be closer to than the diameter of d . d) The continuum U d meets x and r and misses F^ . Using (2.5), let s be an arc in Int A which joins x and r and lies so near V d that s misses F^ . (note that although may not miss Bd A, (2.5) provides 6 that s misses Bd A). To see that Y = r O s has the required properties: r U s c Int A by construction; Y misses one of F^, F 2 because s misses one F^ and r misses both. The circle r bounds no disk in A as we saw in b); and finally we can assume that s has end points x and •• z 6 r with r n s - z = 0 by (2.1) . This completes"the construetion.of Y. assuming.that x 6 Int e . - 21 -Construction of c' . We maintain the assumption that x 6 Int e during t h i s construction. We w i l l f i r s t construct a continuum a c A which j o i n s Int £ to x so that a meets Y only at x . As suggested by f i g 13, the plane separation theorem can then be used to separate Y - x and Int Z u a - x . Construction of a : since r encloses c, x e Int r. Let Q be the open set Int r - s . " Q i s connected because s - r does not disconnect Int r ([10, VI(3.4)]). Evidently Bd Q = Y, and since each of r , s i s a l c continuum, so i s Y . Using (2.4), connect x to a point i n Int Z by an arc a which l i e s i n Q except f o r x . Since Y misses Int Z, , Y and Int Z U a are continua i n Int m which meet only at the point x . We now use (4.41) to separate Y - x and (Int Z U a) - .x by a c i r c l e c" which l i e s so close to Y that i t misses one of F^, F^ . Evidently c" must pass through x (since otherwise 2 (Int Z U a) - x and Y - x are subsets of a connected set i n E - c')« We know that c" C Int A because c" misses Int Z by construction and c' l i e s so near Y C Int m that c" C Int m. I t remains to show that c' bounds no disk i n A . To see t h i s ; we know that r encloses Z . This means that c^ cannot enclose r, since t h i s would imply that Int c' C r u Z (from (2.6)), whereas we know that c' separates r and Z - 22 r-Thus Ext c 1"'O r and Int c'D £ . The fact that Int c'^ £ implies that c* bounds no disk in A byxthe usual argument. The construction of C is now complete except that the restriction x £* Int e must be removed. Since the proof is easy i f x6 e, we only look at the case that xf Ext e . Since we know that A is homeomorphic to a nice annulus, i t is easy to construct a homeomorphism if of A onto itself which exchanges £ and m, i.e. <)>[£] = m, (j>[m] = £ . Then $[k n Ext e] = Int <f>[e] n A and f[A n Int e] F Ext <j>.[e] n A, using the fact that A n Ext e, A A Int e are connected to m, £ respectively in A - e . Then i f x 6 Ext e, apply earlier arguments to <j>[A], using the fact that <j>(x) £ Int <j>Ie],- etc Q, Theorem 7. Let A, F , F 2 be defined as in Th 5 and Th 6 except that F^ bridges A while F^ does not. Let c be a simple closed curve in Int A which bounds no disk in A and contains, the base point x . Then i f c has Property ^ P with respect to x, F^ F^, there exists a simple closed curve c" which meets x, ' lies in Int A, bounds no disk in A, and misses F^ . ».* Th 7 is proved in the same way as Th 6. At first glance one might think that one of Th 6, Th 7 is stronger than the other; but in fact this is not true. If F^, F^ f a i l to bridge A, one might wish to add pieces to, say, F^ so that the enlarged F^ would bridge A; this would obtain the conclusion of Th 7 which is stronger than that of Th 6 Csince i t predicts which F is hit by c'). However i t may not be possible to do this (F^U F 2 might be a number of circles concentric with m in A). Proof of Th 7. Use (5.1) with F taken to be F to construct - 23 -a simple closed curve e which lies in Int A, misses F2» and bounds no disk in A . If e meets x, then e is the required c'. If x e, assume that x £ Int e as befdre. Since e bounds no disk in A, e separates £ and m by (2.7); in particular e meets some component k of F^ which hits both £ and m (there must be at least one since F^ bridges A). For similar reasons, c meets the same component k . Let y' 6 k *~> e . Because c has Property ^  P, there is an arc b C c such that b misses one of F^, F^ and connects a point of k to x . Since k C F , evidently b u k misses F,, . Since b U k is a continuum which connects y' 6 e to x and misses F 2 (as does e), we can construct the lasso Y as in the proof of Th 6, reading b U k for c U d and e for r . In the proof of Th"6,. the curve r misses F^ \j F^> whereas here e misses just F2; however following the procedure of the proof of Th 6 will yield a lasso Y which misses F^ • The lasso Y is used to construct c" precisely as in the proof of Th 6, keeping in mind the fact that Y misses F^, so that the resulting c' also misses F^ . The assumption that x € Int c is removed just as in the proof of Th 6 • . CHAPTER TWO: BING'S DOGBONE'SPACE AND CURTIS' -CONJECTURE. 1. An upper semicontinuous decomposition G of E 3 into compact sets 3 (or simply 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 sets A of E such-that•the union of the elements of the decomposition 3 i s E , and each element A 6 G possesses a-system of open neighbourhoods which are unions of elements of G . The decomposition space G associated with G i s a t o p o l o g i c a l space i n which each point i s as element A 6 G, and the open sets are j u s t those subsets of G 3 the union of whose elements i s open when considered as subset of E Thus each point A of G has a system of neighbourhoods each of which 3 i s open 'both i n G and i n E ' . One can use th i s i n t u i t i v e idea to get a c e r t a i n geometric grasp of the topology of G simply by remembering that 'some points are sets' and keeping an eye on the neighbourhoods; for example one often does geometry on a torus or K l e i n b o t t l e by looking at the equivalent decomposition space of a rectangle 'with c e r t a i n sides 3 i d e n t i f i e d ' . I f an element A €: G contains more than one point of E , then A i s c a l l e d a b i g element of G . If A i s a si n g l e t o n , then A i s a small element of G . In G, the corresponding points are c a l l e d b i g and small points. The decompositions G i n which we w i l l be interested are a l l p o i n t l i k e t which i s to say that the complement of each A 6 G i s t o p o l o g i c a l l y equivalent to that of a point; i n p a r t i c u l a r , each A i s connected. We d e f i n i t e l y assume some acquaintance with these ideas and do not regard the present text as an adequate introduction. The c l a s s i c a l approach to decompositions and decomposition spaces may be found i n Ch VII of [10]. " Our approach w i l l be more along the l i n e s of [3,§6]. We w i l l use two main c l a s s i c a l r e s u l t s : i ) an upper semicontinuous decomposition space ( i . e . the decomposition space associated with an upper semicontinuous 3 decomposition) of E i s a separable metric space, i i ) there 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 0 of the quotient space carries each A € G onto the corresponding point <j>[A] in G . We will often write A* for 3 <p[A] i f A is a subset of E . In the sequel, 'decomposition space' will mean 'pointlike upper semi continuous decomposition space of E . An important question is: i f G is a decomposition space, 3 3 is G homeomorphic to E ? That G is homeomorphic to E is Wardwell's conjecture (in [8]) and is 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 decompo-sition space form a totally disconnected set. Detailed construction of the dogbone decomposition. We will describe an infinite sequence of compact sets whose elements intersect to form the set of big elements of the dogbone decom-position G . Our construction differs slightly from Bing's, but we assume an acquaintance with the original construction in [12] and will not prove, for example, that the various embeddings to be described can be assumed to be polyhedral. Dogbone space takes its name from the distinctive shape of the double handlecube A depicted in fig 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 fig 16a is called the upper eye of A . A path m <= jnt A which, makes one circuit of the curve marked m in the figure is called the lower eye 3 of A (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 £ and m laid out as nice circles plus a straight connecting arc a (with r of course, taken sufficiently small, say less than one-third of the common diameter of the nice circles £ and m) . We call £ v m u a the centre of A . The centre of a dogbone will not be impor-tant in this chapter (but will be needed in Chapters III, IV). The idea of A as an r-neighbourhood of its centre k is introduced mainly to pin down the embedding of k in A; we usually draw k and A as in 1 2 fig 16b. Fig 17 shows four short solid .'cylinders B , B , B , B2, which are subsets of A and cut into the eyes of A as the figure sug-1 2 gests. The removal of one of 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 its centre. Let A , . A , A^, A^ be four dogbones embedded as shown in fig 19 by embeddings h..: A -> A, j = 1, 2, 3, 4 so that the A_. = tu [A] are mutually disjoint and lie in Int A . In fig 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 A - B^  ^ B2 - B B . In the obvious way, the centre of A^  is called k.., j = 1, 2,3,4, with upper loop £_. and lower loop m... The £^ are placed so as to l i e as parallels on g^ . The connecting arcs a_. are laid out in a peculiar Facing page 27 (r,&) c/cftneS * point on the dish S. As f /ncreascs in 0< <p< 1TT/ (f SlVec/>s a toroid which is a. figure of revelation B.bout the. phTur ctrc/e. C-. A - 27 -way which i s c h a r a c t e r i s t i c of the dogbone construction. Using t o r o i d a l coordinates (which we- r e c a l l i n f i g 20) , we could define B^" and 3^ to be appropriate t r a n s l a t i o n s of the set r <_ 1, 9 = <f> and thus construct a band with an even double twist. However the bands i n the drawing are translations of the set r <_ 1, 6 = 0: IT/3 <_ <f> <_ 2TT r <_ 1, 6 = 6<fi: 0 <_ cf> <_ TT/3 This gives a ' f l a t t e r ' band and a better p i c t u r e . Another concession to art appreciation i s the plac i n g of 3^ and 3-j, so that t h e i r ' f l a t ' parts l i e on the plane of k . This necessitates a r i g h t angled bend i n the a^ near 3"^  and again near 3^ • The a d d i t i o n a l conditions are imposed on a. that a. misses 3^ U 3-, except at a. n JL. and 3 3 1 3 3 a. n m., and that the part of a. l y i n g within a distance e of 3 3 3 3^(3-^) consists of a si n g l e s t r a i g h t arc perpendicular to 3^(3^) • Note that the order of £ j ' s o n l s ^2* ^3' ^ 4 w n i l e the order of m.'s on 3, > due to the unusual embedding of the A., i s m. , m., m„, m..' j 1 j 1 3 2 4 The B. and B 1 locate the A. i n the following way (see f i g . 21): A l l i e s i n Int;(A - B2 - V A 2 l i e s i n Int(A - B2 - V A 3 l i e s i n Int(A - B1 - V A4 l i e s i n Int (A - B1 - V The closure of the component of A - B 1 -- B 2 -- B l - B B 1(3 1) i s c a l l e d K ( j y . F i n a l l y we l e t A U A 2 U A 3 U A^ - 28 -embed Now since each dogbone '.A is homeomorphic to A, we can four dogbones A.., A.„, A.„, A.. in each. A. just as the A. j l j2V j 3 ' j4 1 3 are embedded in A . We could write A., = h.h, [A] . The union of the jk j k 16 A , j , k, chosen from 1, 2, 3, 4, is called 01 . The construction jk I proceeds as in [12] with the definition of 64 A^^ = h^h^h^IA] where h.h, h0 embeds A in A., just as A„ is embedded in A . The union j k -t jk t of the 64 A is called CL^ ; The construction proceeds in this way, defining at each m-th stage 4™ dogbones whose union is Ct^ . Let the intersection A n n $ 2 n ($^n ... = AQ . The components of AQ are compact and are defined to be the big elements of G while the re-3 maining points of E are the small elements. The dogbone space V is the associated decomposition space of G . Remark 1. In k^0 ... u k^, each upper (lower) eye fails to shrink to a point in the complement of any otheruupperflower)^eye. iThis is easily checked using, say, Ch XV of [6]. Remark 2. We are sure that the construction of V here is the same as that given by Bing in [12"]. In the Appendix we show a deformation of the upper part of k^U ... fJ \ to look like the upper part of Bing's con-struction. We think that the reader will see the plausibility, but we give no strict proof that our embedding of A^U ... V A^ is the same as the corresponding embedding in [12], and our attitude in this paper will 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 will usually apply results obtained for A to any A j K • • • r without further justification. Note that k_. has a property which is 1 not preserved by homeomorphism: a^ is perpendicular to g or g^ wherever i t lies near these sets. This property is lost after the first stage of the dogbone construction. This does not prevent the construction of V, but further comment will 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^ will not become apparent until Ch III (apart.from the fact that they cause the eyes to link to-gether) we will often use the picture in fig 22 to describe the embedding of k^ U ... V k^ in A . We will use pictures like fig 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_. is that no £^ (m ) will shrink to a point in the complement of any other Z.^fja..,) . Another 3 3 pictorial abbreviation shown in fig 22 is 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 fig 22, we will often show only the holes of A which will 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 is easy- to define a neighbourhood system for the - 3 0 r-big elements. Thus a lot can be. learned about the topology- of the decomr-position space by looking at elements of the associated decomposition. However i f we try to approach. V in this way, we find that the components of AQ, which constitute the big elements of G , are hard to see. To find a big element, note that each big element of G is the limit of a sequence of dogbones A, A., A Evidently each big element may be specified by an infinite sequence j , k, ... of integers chosen from 1 , 2 , 3 , 4 ; and the A, A., -A.,, ... constitute a neighbourhood 3 JK. system of this big element. Because A is compact, we know that i f A is a big element of G , then i f A lies in an open set V, some member A of the neighbourhood system lies in V.. (see I, 7 . 2 of [ 1 0 ] ) . It J.K. • • • IT is known that each big element of G is a tame arc (see [ 1 2 , § 2 ] ) . The canonical mapping $ is a local homeomorphism near small elements of G (because A Q is compact) but not of course in general. The fact that <f> is monotone means that cf> ^  preserves connectedness (VIII ( 2 . 2 ) of [ 1 0 ] ) . Simple connectivity properties are more complicated. As will appear later, any open set V*c V which lies in A* and contains a big point of V cannot be simply connected. We must expect a proof of this property to be delicate since i t is known that V is locally simply connected. ( [ 5 ] ) . (Roughly, what happens is that any mapping of into small neighbourhood V* of a big point of V will shrink to a point in the second smallest dogbone which, contains V* . Thus one can satisfy the definition of 'locally simply connected' by taking a smaller neigh-bourhood V* although V* itself will never be simply connected.) For the rest of this section we will prove a result which re-3 lates simple connectivity in V to the same property in E . A mapping - 31 -f of S"^  into a space X shrinks to a point in X i f f f is homo-topic in X to a constant mapping or represents the identity in TT^(X) for an appropriate base point. A third equivalent statement is : consider S"*" to be the boundary of a disk A : then frS"*" -> X shrinks to a point i f f f can be extended to a mapping f of A into X . (1.1). Let V* be an open,set in V . If f:S1 V* so that mg f -1 consists of small points, then <|> f will shrink to a point in V, where 3 <j> is the canonical mapping of E onto V . Corollary: i f V* is simply connected then so is V . •-'*"•". ' 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 its 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 will complicate the (pretty easy) proof slightly by introducing more generality in the method than is needed for the present argument. Outline of proof, a) Assume that f maps the boundary of a disk A into V* . Since f shrinks to a point, there is a mapping 7: A -* V* such that f = f . Recall that A* is the union of the I BdA 0 big points of V . The set f [^Aj5j] is compact. Let Q be a disk with holes such that Q c A, the outer boundary of Q is BdA, and the (open) holes of Q contain f "*/A*J . b) The mapping f maps Q into small points of. V; thus r <)> "^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 Bd u 'r 1 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 A into V . Details of proof, a) - -We know, that ' f :A + V*' 'so that" f = f. . Since A is compact, so is A* and ,'f~ 1[A*]. Note that f w l[A * l misses BdA o o o o because f[BdA] consists of small points, from the hypothesis. To obtain the disk with holes ' Q, we use the following result which will be needed several times in the sequel. 2 (1.2) Let A be a disk in E and S a compact set in A . Then there exist n disks W., ... W such that W A and I n - - - r - —-i ) W n W = 0, r j= s . r s ii) scw.u ... M . 1 n i i i ) Each point of Bd W^_ lies withing a positive distance e of S . iv) If S misses BdA, then S c Int W, u Int W„ a . . . u Int W , 1 2 n' and A - Int W. - ... - Int W is a disk with holes. If 1 n S hits BdA, then S misses Bd W - BdA for each r = 1, ... n . Proof of (1.2). We can assume that S ^ 0 and that A has the form of an equilateral triangle. Triangulate A 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 A each belong to one 2 - simplex only so that the three vertices of A cannot be cut points of any union of 2 - simplexes. The only properties of the 2 - simplexes which will 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 A which meet S . Evidently S is lc and each component of S is a lc continuum. For later reference, note that S 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 S, the entire star of v lies in A / S A S and v ^ Bd S . S cannot meet Bd S at the interior of an edge in A. ^ Int A by a similar argument. We alter S to a set :S which has no cut po —_ r> nts in this;way: a cut point of S cannot lie in the interior.of a 2r-simple in S, nor in the interior of an edge belonging to one "2-rSimplex, nor in th , A interior of an edge belonging to two 2-simplexes. Thus the cut points of S are a (finite) subset of the Vf.rtexes. Let the cut points be t, ,...t,, and cover each t g with a set b "s • = 1,.. .k, which is.a "disk of radius - e/6 and centre t i f t C Int A: and is a semi-disk of the same centre s s and radius i f t g lies on BdA and is not a vertex of the big triangle A (thus b is a 'disk relative to A'). We do hot define b for the s s three remaining points of A since these points are never cut points A A of S . Note that the \b are disjoint. Define S to be s A S u b, U . . . 1/ b, . It will turn out that the Bd W are some of the 1 k r A A - A A A components of Bd S , We know the following facts about S : the com-^  A A , ponents of S are lc continua and are consequently bounded apart. A A ' A A Every point of S (in particular every point of Bd S ) lies within a A distance 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. a-rigfc oc wny be rt(Air--}Xm-- 34 -times the number of subsets of 2 - simplexes). Two s t r a i g h t arcs i n Bd S meet as i n f i g 23; a s t r a i g h t arc meets a segment as i n f i g 24. Segments never meet because the b g are d i s j o i n t . Evidently S has no cut point on i t s boundary and hence no cut point at a l l . Let the components of S be ••• W^ j . These w i l l be reordered so that the f i r s t n components w i l l l i e i n the disks required i n (1.2). Since each W, a =.il, ... jn.> *-ts a l c continuum with no cut point, by IV (9.3) and VI (2.5) of 110],' the unbounded complementary domain of W i s bounded by a simple closed curve which w i l l be c a l l e d c . E v i d -a a ently c c W/ . Reorder the W^  and corresponding c so that a a a a c. , ... c are contained i n the i n t e r i o r of no other c i r c l e c , while I n a* each of c M . ... c , i f they e x i s t , i s contained i n the i n t e r i o r of n+1 m J > a c . Let W = Irit c , a = 1, ... m . We w i l l show that , ... W a a a I n s a t i s f y i ) , i i ) , i i i ) , i v ) of the statement of (1.2). Proof of i ) . I f r f s then neither of c , c i s contained r s i n the i n t e r i o r of the other. Then Int c O Int c = W O W F 0 D Y r s r s 1(1.6). Proof of i i ) . S c s = w;w ... ( y f . Each W c Int c , 1 m a a a = 1, ... m, since Ext c i s the unbounded complementary domain of W . Hence S c Wn u .. . u W . And i n f a c t S c W. u ... U W because a 1 m I n for n'+ 1 <^  a <_ m, c l i e s i n some Int c , r = 1, ... n, and by-1(1.6), Int c = W C W . ' a a r At Proof of i i t ) . Each Bd W i s a c C Bd S. and we saw e a r l i e r r r * ' . that a l l of (the closed set) S l i e s near S . Proof of iv). Take e less than the distance from (compact) S to BdA i f S misses BdA . The rest follows from the definition of a disk with holes, from 1), and from the fact that a point of Bd S and hence a point of Bd S misses- S wherever i t lies in Int A D. We now return to a) in the proof of CUI). Since f"1[A*] is compact, from (1.2) there are disks W^ , ... W^  which l i e in A and such that f_ 1[A*] C W. U ... U W and each Bd W lies within a L 0 1 n r distance e 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. Let u I n r be the 'holes' of q, r = 1, ... n, i.e. ur =  Int W^_. b) Since F"1[A*] lies in the holes u r of Q, flQ] consists only of small points. Thus <f) is a well defined mapping — -1— 3 when restricted to f[Q] and <}> f |Q m aps Q into V c E . We now find cubes in which to shrink <f> ^ f | B ( j > r = 1> ••• n • Since _ ' r f[A] O. A* is compact and the dogbones (considered as sets in V) evid-ently form a neighbourhood system of A*, there is a covering of f[A] n A* by a finite number of dogbones J*, ... J* each of which lies (J 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 ,, J J TJ . to be the four dogbones of the m + 1st stage si s2 s3 s4 lying in J . Note that since J C V, each J . lies in a cube M . ' ° s s sj S3 which is a subset of J . and hence of V (if J were the dogbone s s AC V then J , C M -, = A - B2 - B0, J _ lies in ti „ = A - B2 - B. s l s i 2 s2 s2 . • 1 etc.) Now in a) above, we could have chosen e so small that each f[Bd W'-] = flBd u ] lies in some J* Cfor V is a separable metric r r sj - 36 -space, and there is a minimum distance in the dogbone metric separating the compact set f{A]^ A* from the complement of the union of.the J* .). U S J Clearly <j> f[Bd u^] is defined and lies in the union of the J g ^ . We can assume the J . are dis-joint because we could have removed from the S J covering J^, ... any J g which was contained in any other member of the covering. Since (j)" ^ "f [Bd u j" is connected and the closed sets J . are separated, <j> ^ f[Bd u ] lies entirely in some one J . and S J I T L r J sj d> f i „ , shrinks to a point in M .c V . Thus there is an extension T Bd u r sj 1 r JY of d) "*"f i „, to a l l of u , i.e. y : u M . C V and 'r T I Bd u^ r r r sj Yr|Bd u = *"lf|Bd u * 1 r 1 r c) In view of the set-theoretic definition of function we can express the idea of 'mappings glued together' by unions of mappings. Consider the union ij) ^ f i . V y,y .. . U y . This is a well-defined |Q 1 n mapping of Q v dom y^v ... v dom Yn = Q u u-^  u . . . c u into V because each mapping in the union has its image in V. and because where the domains intersect the intersection is closed and the mappings agree on the intersection; in fact every point of domain intersection occurs on -1— a Bd u where'we know that y agrees with <f> f l g ^ by construction r r ' ur of . Finally we note that the new mapping dT^f | ^  U T±u • •• V Tn agrees with <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 will record the argument in this paragraph as a separate result. (1.3). Let A, W^ , ... Wn be .defined as in (1.2) including the fact that A - Int W. - ... - Int W is not neccessarily a disk with holes. 1 n J 3 Let g:A - Int W, - ... - Int W E . Then each gi ^ , IT is defined; 1 n Bd W r and i f g u , shrinks to a constant mapping in a space P , r = 1, ... n, Bd W r 1 r then there is a mapping of A into mg g u P^ u .. • " . In part-icular, § | B d A will shrink to a point in mg g 0 P^ o ... V P^ . Proof: The argument of c) in the proof of ( 1 . 1 ) is-3used;and is valid even i f A - Int W- - ... - Int W is not a disk with holes. It is 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 | M H is always defined Q, ' r Proof of the Corollary to ( 1 . 1 ) . Let IJKS 1 -»• V . If V* is simply connected, we can use (1.1)' to show that if) shrinks to a point in V only i f rng misses AQ . Evidently in order to apply ( 1 . 1 ) , i t is sufficient to show that i|> is homotopic in V to a mapping i^ rS"*" -> V - AQ .We use ' - ' to mean ?is homotopic in V to' . To construct TJJ': using an argument like that of b) in the proof of ( 1 . 1 ) , cover [^S"^ ] D AQ with dogbones J^, ... which are disjoint and lie in V . Dogbones J '., j =. 1, 2 , 3 , 4 , are defined just as in S J b) of ( 1 . 1 ) so that ^ J . covers T!»IS"*"] O An and each J . lies in r] rj r 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 is easier sj since we need not consider sets in V). We assume that some point z exists in tp[S'] n(E3;- - U Int J ), for otherwise, since S1 -is connected and the Int J . are separated, rng il; lies in one J . M «, and sj r ° sj s j ' the proof is concluded by shrinking i|> to a point in Mg.. V . Now choose § > 0 so that i f x and y are closer together on S"*" than the distance 6, then i|>(x) and i|>(y) a r e closer together than the - 38 -distance from Y[S''] P A . to E - .v< J . (remember that is a 0 sj sj sj sj neighbourhood of ip[S"] O A^ so that this distance is positive). If every 1 - 1 3 / / point of S lies closer than 6 to ip {E - \s J .],' then by the def-sj sj inition of 6 no point of S"*" maps under ip into A ^ , and we can let ip' = ip . If some point of fails to lie withing { of i|i "'"IE3 - ^  "'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 such that e' c e. c il "*TM, Int J .] . Then e, is a closed interval of 1 1 sj sj 1 length greater than 6 whose end points p^ and -q^ map into Bd j^ j J g ^ by the usual continuity argument. (Since S^" is a circle, we must make an easy allowance for the possibility that ip(p^). = ipCq-^ ) = z.) Because the J .. are separated and tple..] is connected, tple.l lies in the interior sj x x • • - - - i of just one J which we will call R , while Kp-,) and ipCq-',). lfe in sj ^ . J- • . 1 • . 1 Bd Rx. Define the mapping ip-rS + E so that = ^ o n s 1 - e while ^ i j ^ i s a'Pa t h i n (connected) Bd R± with end points. ip.^) and i p ^ ) (this is well defined because p1 ' and q^ map into Bd R under ip).. Both ip "and ip^  are paths in V and ip |— -.tp-^—. because they share end points and both map into the, same cube M . D L with M . <Z V. Evidently ip - ip . ' • - • S J 1 :"- S 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 AQ . We now look for an open interval in S e^ where e^  is of length greater than 6 and such that e£ maps into U i n t 3 . under ip, (and in fact under ip, since ip =. ip, on §^ ^ e_). sj sj 1 1 1 If , e^ does not exist, let ip^  =• ipf , If e^ exists, then there is an open interval e2 of maximal length; such that e^c e2-C S,^  - e ^ and C sj I n t Js j • The end points p2» q 2 of e2 map under ip1 into E3 - ^ / Int J ,, either because of the maximal!ty of e„ i f the sj sj l ^ 39 -end point Is in S1 - or, i f the., end point is in Bd (S1 - e1) = Bd e^ because if) [Bd e^ c Bd R^  . By a continuity argument, ^ [ e ^ lies in the interior of some one J called R2 and ^ 1(p 2) and ^Cq^) l i e 1 3 1 in Bd R2 . Define T\>2'.S -* E so that i ^ 2 agrees with lf^ on S - e2 (note that this means that ^ agrees with ' if) on S1 - e^ - e 2), and so that ^2\e ± s a P a t h i n B d R2 w i t n e n d P o i n t s ^(P^) a n d ^1^2^ ' By a previous argument, ip2 - ip^  - if) . Note that the fact that if*2 - ip on - e^ - e2 means that . ip2 = if) on the end points of both e^ and e2 . In general, suppose that mappings ifi^ * ... * ° f S1 into V, intervals e^, ... er_-^  and components R^ , ... Rr_-^ ° f Js j have been defined so that each e C-S''' - e_ - e s = 1, ... r - 1, s 1 . s-1 . . . . ' '' 1 . "• — .' ' ': '" " , 3 . and for each 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 length of e^ is greater than 6 and ij^ -^ te^ ] CL ^  Int J g . , or equiv-alently ib , [e 1 CL Int R , where R is a single J . (and hence J r - l r r r 0 sj lies in a cube M . C V). If there is no such interval, let i> , be sj . ' r - l ty' . If e^ exists, then let er be the largest open interval in S^" - e, - . .; - e , such that \b , [e ] C Int R . We know that 1 r - l Tr-1 r r 3 i ^ r _ ^ carries the end points p^, q^ of e^ into E - Int R^  by the maximality of e i f the %end point is in S''" - e ^, or, ,if the end point is in Bd(S^ - e, - ... - e .) C e, (j ... U e .. , 1 r - l 1 r - l because \p . ' carries e, U ,., l i e . into {J. Bd J . . (To see that r - l 1 r - l sj sj if) ,Ie ] lies in some Bd J .: for s = l , . . . r - l , il) [e ] C Bd R r ^ l s sj s s s by construction. i> ., agrees with if) on .S^" - e , - D e since J rs+l 0 s s+1 s e g + 1 lies in S1 - ex - ... - eg; i f ) g + 2 .agrees with ^ s + 1 °n S l " e a+2 Z ) S S i n C e 6s+2 C S1 - e ^ .. . - 7 - 7a+1; agrees with o n es» e t c > u n t i l ^ r _ i agrees.with $r_2 o n es • Since 'i> = > - . . . = if) , on e , if) . [e ] = if, [e ] C Bd R ) . • Since we know that the end. points "of e^ are mapped By if)^ ^  outside of Int R^ , and i|T •-rjf-r^]. Minis'JU, bv continuity, > - (p_). a f i d ~ i|> - lq ) l i e in Bd R . r - l r r r - - - i / ' / r - l r * r - l r , r 1 . ....... . - - . , - . -.. . Let \1>- = \b•, , on S - e - . and \b i — be a path in Bd R with end points r rr - l ~- r '•• - - -"re • . - - r • r" , ' r . . . . ^ r_l(P r) a n d ^ r - l ^ r ^ ' E v i d e n t ly * r | e '~.. * r _ i | e a n d "~ * r - l ' . - i r 1 r Since each er is of length greater than 6 and r6 must be less than the circumference of S"*", (it is easily checked that the e^ are disjoint) the sequence ij)^, . . . i f ^ , .... ends at if)^ . Let if)' be if), . We know that :S"^" -* V because each ijj does this; and if) - ip1 - . . . - if), = ip ' in V . Before we can show that mg if)' X AC misses A Q , we must show that if) = if)' = if^ on S"^  - e^ - ... - e^ . To see this: I(J£ = if) on e^ , 1 1 ^2 = ^1 o n S ~ e2 a n d *2 = ^ '~on"! ? ~ e l ~ e2 ' if)^ . = i f ^ - i on . S^" - e^ and if)^ = if; on S^ - e^ - ... - e^ . It is now easy to show that rng if)' = mg if; misses A Q , for I(J^ — 3 carries every eg into Bd RgC E - A Q by a previous result; and i f x £ S"*" - e- - ... - e , then x lies within a distance 6 of a point y such that if;, (y) € E^ - U Int J . . We can assume that J k sj sj y c - e^ - e2 - ... - because otherwise y £ e^u ... U and Facing page 41 Ay. may lc shrunk-, ^3L}AtjAf WW m/ss a.t least one Di. U \ yitist Jut loth. A1 lAtm*y. he ezch of 4j,A}. trusses one But 7)ew/Ij Vied-loth. - 41. -some point y" of Bd(e^O ... <J e^ ) S1 - e^ - ... - e^ lies closer to x than y does. Since ^ carries y* G Bd(e^V ... U e^ ) c e- U ...v e, into ^,Bd J . C E3 - 1/ Int J ., we could have 1 k sj sj sj s j ' originally chosen y" instead of y . But i f both x and y' l i e 1 1 in S - e^ - ... - e^, then since ip = ip^  on S - e^ - ... - e^ and ip, (y-*) ^  F? - Int J ., ip(x) = ip (x) lies so near to rk^ sj sj rk tyiy") = ^ (y-*) that ip(x) misses A^ by our definition of 6 P e 2. In his paper [12], Bing was concerned with an interesting property of G which we will make use of here and in Ch. IV. The formidable aspect of G lies in what might be called its 'topological idiom', as shown in fig 25: four double ended lassos strung in a special way inside a 2-holed torus. Bing's intent in using this idiom to con-struct G was to utilize this property: let D^ , D^, be the planar disks shown in fig 25. Then, no matter how the four lassos are deformed (provided that they remain linked and stay in the interior of the 2-holed torus), some one lasso will hit both and D2 • Figs 26, 27 show unsuccesful attempts by the lassos to avoid this necessity, and there is a proof of a very similar idea in §7 of [12]. Bing hoped to show that this property was induced through the construction of G in the following sense: assume that fig 25 shows D^ , in relation to the first stage of the construction of G, then, no matter how A^ ,. A 2 , A^, A^ were deformed, one of these, say A^, would hit 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 Q on the dogbones of the decomposition and show that A had this property. If a dogbone had property Q, this implied trivially that i t intersected both of D^ , B^; at the same time i t could be shown that i f a dogbone B had property 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 Q and the limit of the chain would be a big element of G which touched both and . We can express this idea in a slightly different way: (2.1) (Bing). Let be topological disks whose boundaries g^, .^^  l i e in A and link the upper and lower eyes respectively of A as shown in fig 28. Then either metts in A, or some big element of G meets both and T)^ • We will refer frequently to fig 28, which shows the relation-ship of C^, to A . Strictly speaking, we take c^, i = 1, 2, to 1 3 be an embedding of S in E ; however we frequently will confuse the embedding with the circle which is its 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 will be interested in this conjecture: 3 (2.2) . Let A be a 2 - simplex. For i = 1, 2, let f : A -> E so that f.i-r,,. = a \ and f „ i „ , A = C „ are paths whose ranges l i e in i|BdA 1 21 BdA 2 v & 3 E - A and which will not shrink to a point in the complements of the - 43. -upper and lower eyes respectively of A . Then either f^[A] and f^lA] intersect in A, or some big element of G meets both f^[A] and f 2 [A] . In (2.2), we replace the disks of (2.1) with singular disks f^[A] . The conjecture is plausible and lacks earthshaking sur-prise. It is interesting because i t leads directly to the following topological property of V: (2.3). If (2.2) is true, then V fails to possess arbitrarily small simply connected open neighbourhoods about any big point. The conclusion of (2.3) is 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) will be deferred to Chapters III and IV. Proof of (2.3). Supppose that A is a big element of G and that iii . V, A* = <|>[g] lies in a simply connected open neighbourhood V* such that A* c V* c A* • Clearly AC V c A, and V is open in 3 E .We could write A = A A (1 A j k ^ *** s o m e s e c l u e n c e ° f dog-bones A, A'., A By the Corollary to (1.1) (of lemma 1 of [2]), 3 Jk V . is simply.connected i f V* i s . Thus our assumption implies that V is simply connected. We will demonstrate that this is false by showing that AC VC A with V simply connected, implies that the upper eye t and the lower eye m of A shrink to a point in A . We define an upper (lower) principal path of A_. to be a mapping of S"*" into Int A^ which is homotopic in A. to the upper eye t. (the lower eye m.) of A. . Upper and lower principal paths of other dogbones, including - 44 -A, are defined analogously; t h i s a mapping of S^ " in t o Int A J K-which i s homotopic i n A to h., [&] i s an upper p r i n c i p a l path of Jk Jk A . As usual, we w i l l often confuse the mapping with i t s range. We 3 k know that A, A_. , ••• i s a neighbourhood system of A and, by a previous remark, that some member A., of the system l i e s j iC • • • I T S i n V . However th i s f a c t plus the following lemma leads to a contradiction. Lemma for (2.3). I f one of A^, j = 1, 2, 3, 4 contains an upper p r i n c i p a l path e^ and a lower p r i n c i p a l path which i n t e r s e c t and l i e i n V, then A contains upper and lower p r i n c i p a l paths which i n t e r s e c t and l i e i n V . In general, i f A. contains i n t e r s e c t -3 ... rs • ing 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 . .. r To apply the lemma we look at the neighbourhood A. 3 • • • r s which we know to be a neighbourhood of A i n V . Obviously V D A. contains i n t e r s e c t i n g upper and lower p r i n c i p a l paths 3 • • • r s of A. since any i n t e r s e c t i n g p r i n c i p a l paths w i l l q u a l i f y , j • • • rs The lemma implies that the dogbone A.. contains i n t e r s e c t i n g 13 ••• r 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 of the lemma leads to the conclusion that A contains an upper p r i n c i p a l path which l i e s , i n V . Since V c A, V i s simply connected, and the upper p r i n c i p a l paths of A are a l l homotopic to t i n A , therefore Z must shrink to a point i n A . This i s c l e a r l y f a l s e from f i g 16a. Thus the proof of (2.3) w i l l be complete when we have proved the lemma. x Proof of the lemma for (2.3): S i m p l i f i e d version. Suppose that e^ and l i e i n A^ . • The following 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 of the proof. Although the 'proof we give now i s Facing page 45(1) / / / Facing page 45 'di) A nox-pl«n*r dL Tndy Tffect A3 [t is neitf fivsilh Cd ConsTriut px so £t*>? pL t/ees -n*t y. v / 4 l „ u i r t/>e U/fzr A»Je <?/ 4 is ?e qui re J. Fig 31b. curve- pj way iS censtruotcU 31*.. und f* l'°f> once aiout - 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. Suppose that by good fortune the paths e^, &2 take the form of the double ended lasso J in fig 29. J consists of circles C^, and connecting arc a as shown. The circle will not shrink to a point in the complement of the upper eye of A^. Similarly will not shrink to a point in. the complement of the lower eye of A^ . We also pretend that J lies in V and that disjoint planar disks d^ bounded by C^, also l i e in V . By (2.1) some big element g in A^ meets both d^ and d^l and g lies in V 3 d^ since V contains every element of ' G that i t intersects (remember that V is the pre-image of an open set in V). We can now construct the upper principal path p^ shown in fig 30 from parts of V lying in a, d^, d^> g« A similar procedure using A^ instead of A^ will yield the lower principal path p2 . The paths p^, p2 intersect in A so that p^ U P 2 is the set required by the conclusion of the lemma. The above 'proof is far to easy and will f a i l i f we allow d^, d2 to be non-planar, for then p^ may not be a principal path as figs 31a, b show-. We ensure that p^ makes one .circuit about the upper hole of A 2 2 by trapping p^ C\ a in the cube A - B - B (which is easy) and Pl ° ^dl ° g ^ d2^ = q i n t l i e C U b e A ~ b 1 ~ B2 (s e e f i§ 3 2) • 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 p^ by constructing certain arcs in Facing page 46 A - 46 -cubes only works i f the l i e in A - B1 - TS^ . But i f we use the obvious candidate for J, viz. C. = e. with arc a degenerate, 1 1 then fig 33 shows that this may not happen, and in fact J cannot usually be e^U • However we show that, provided that intersecting principal paths exist in A^ n V, there is a double ended lasso (perhaps with singularities) in AO V which has just the properties which we assigned to J . We will 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. We first give the proof assuming that. &2<~ A l ' t n e n indicate alterations in the case that el ^ e2 -*--*-es ^ n ^2' A3 ° r A4 " a^ L e t e l * &2 ^ e uPPer a n d lower principal paths of A^ which l i e in V and intersect at least at p . We follow the sketch of the 'proof already given, but as previously explained, we cannot use e^ U e2 for J in fig 3j2.. We construct J = U <J a so as to satisfy five properties i ) , ... v) . Sometimes we will regard as a mapping (not necessarily an embedding) of S^  and sometimes as the range of this mapping. The set J must satisfy the following properties: for i = 1, 2, i) C±[BdA] c V r> Int (A - B1 - B2 i i ) mg misses A^ . 3 3 i i i ) (C2) fails to shrink to a point in E - £^(E - m^ ) iv) There is a point y^ € rag ry mg e^ n and a point y2 in rng C2 O mg e 2/l A - B - B2 - ^ (recall that is the topological cube which is the closure of the upper 1 2 component of A - B - B , see fig 34) . Facing page 47 - 47 -v) the arc a C V ^  and the points -y , y2 of iv) are the end points of a . The idea of i i ) and i i i ) is that we want the Gi to act like the circles C^, C 2 in fig 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 in 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 will join y^ and y2 by a path like q ^ A - B^ - B2 in fig 35. b) For i = 1, 2, let f. :A -»- V so that f. . ' = C. . x l|BdA I Using (1.2) and (1.3), obtain a new mapping which agrees with ."f on A - Int - ... - Int W , where Int W . r = 1, ... n, are holes 1 n r ' 7 in A; in particular f± = f^ in BdA . The fi[Wr] may leave V (!) but this does not harm the proof, c) • By (2.2), f f A ] and f"2[A] either intersect in A^ or hit the same big element g in A^ ,. There is a path q from a n to a n which resembles q in fig 35 and lies in V and in Int (A - B1 - B 2). The path q travels to A 3 in f 1 [ A 3 , passes from f^A] to f2[A] in A 3 either at the inter-section of f^[A] and f2[A] or using the element g, and then proceeds to a n C2 by. means of f 2 tA] • d) T n e path which begins at an C^, travels to a A G 2 on q and returns to aa on a, is an upper princ-ipal path of A which lies in V , e) The lower principal path of A in V may be constructed as in a), b), c), d) above, using A 2 and 2 1 A - B1 - B instead of A 3 and A - B - B2 . f) If k = 2, 3, 4 the lemma remains true. Details of Proof. Suppose that e.^ , e2 are upper and lower - 48 -principal paths respectively of which l i e in V and, intersect at p . For i = 1, 2, since e. shtinks to a point in V, there are mappings e^ :A -> V such that e^ | = e^ • We'do not claim that mg e^ lies in A^ or even A . We use (1.2), taking f, S, to be e,. e.^IA - B"*" - B„] . Thus• there are disjoint disks Wn, ... W in 1 1 z 1 m A such that e l1 [A - B1 - B„]c:W- u W„ U ... u W , each point x of 1 z 1 z m Bd W^  l i e s near e " 1 ^ - B 1 - B 2J , and x6 Bd misses e1"L[A - B 1 - B'2] i f x^BdA . Those W which hit BdA are called W. , ... W ; those r r 1' n W which miss BdA are W ,,, ... W (with obvious adjustments of one r n+1* m J or the other class does not exist). We now apply (1.3) with g taken to be the restriction of e, to A - Int W. - ... - Int W . For 1 1 m r = n+1, ... m, ei|gd W = ^Ifid W faaPs in t o Ext(A - B^ -•B^ ) • because —"-1 1 Bd W misses BdA for r > n and hence misses e, [A - B - B_] . r 1 2 Thus for r = n+1, ... m, e. |„, TT shrinks to a point in Ext(A B''" - B_) 1 Bd W z 1 r 3 1 which is the exterior of a cube in E ; and we can let Ext (A -• B - B2) be P ,-=...= P in the hypothesis of (1.3). There is no chance that n+1 m Jf v el|Bd W' Ci ^o r r > n' s i n c e -^3 misses Ext(A - B^ B2) = . We suspect that is an e^lgd y ^o r r — n • Assume that every r 3 e i l t j j TT = §IT>J TT will shrink to a point in E - -£„ . Use (1.3) again, J. bd w Bd W i ....... ' 1 r 1 r 3 1 letting P = P = ... = P be E - I. . Then with Ext (A - B - B j ± z n J z taken to be P ... = ...= P , gi„,, = e, will shrink to a point in n+1 m' °IBdA 1r mg g U P. U ... U P u E u ... u P . Each P misses either 1 n n+1 . m r 3 by-de^tnit-ipri of -because\'P misses A - B"^  ~ B 2 • And rng g misses Facing page 49 - 49 -l3 as well; for g = e ig _ _ and from (1.2) i i , i v , 1 1 m the only points of A - Int W^  - ... - Int W^  which can map into A - B"*" - are those in BdA . Such points are in dom e^ and map into 1 3 A . Hence rng g C A U Ext (A - B - <=• E - £ . • Therefore 3 8|T>JA = e i shrinks to a point in rng g U P, y ...UP C E — £_, which I BdA 1 1 m j contradicts the fact the e^ is an upper, principal path. Thus i t is — 3 3 false that every 6J. B ( J ^ , r <_ n, shrinks to a"point-in E - £^ . Let • - r 3 C be one of the e i , which f a i l s to shrink to a point in E - £„ . 1 1 Bd W 3 1 r _ As regards C^: the above argument plus the fact that rng C rng e^c V shows that i i i ) is true; i i ) is true because from. (1.2) i v , every point 3 1 x in Bd W^_ is either in Int A, in which case C^(x) € E - (A - B - B^) 3 3 C E - A 3, or x 6 BdA, when (^(x) = e^x) 6 A± C E - A 3 . In general,i) i s not true because some candidates for C^(x) l i e outside of A - B^ - B 2 as we have just seen. However we,can assume that C^[Bd W ] lies in Int (A - B"*" - B^) by the following argument: By (1.2), we assume that dom C. (which is one of the Bd W ) l i e s so near 1 r ej 1[A - B 1 - B 2] that, rng lie s within £ of A - B 1 - B 2 (remember that = g ^ e^ on dom C^ ) . In this paragraph a) so far, we could have replaced A - B 1 - B 2 by a cube K => A^ such that an e-neighbourhood of K l i e s in Int (A - B"*" - B 2) . Such a cube is shown in f i g 36. If this had been done, we would have rng in the e-neighbourhood of K, i.e. rng Int (A - B 1 - B 2) . We assume that this was done and that rng C± c Int (A - B 1 - B 2) . Proof of i v ) : In (1.2) i v , each Bd W misses S (in (1.2)) except where Bd W hits BdA . In the r r - 50 -r. present context, with e^fK] for S (i.e. continuing to. use K for A - B1 - B 2), the domain of is a Bd Wr and C^Bd Wr - BdA] misses K . To show that there is a y^ 6 mg \ mg e^ -• C,[Bd W ]/) e, [BdA] n K. " C. [Bd W ft BdA] f) K_ , assume that 1 r 1 1 1 r 1 C [BdA n Bd W ] Z).^ = 0 . In fig 36, note the two cubes K , K2 which are placed so- that £_ C K c K n K. and m„ C K„ c K <0 K„ . J 1 1 5 1 2 . Then . C^Bd Wr f) BdA] .0^ = 0 because ^ D K ; and C.jBd Wr - BdA] /) fC^  = 0 because C^[Bd Wr - BdA] misses K as we just saw. This means that a l l 3 3 3 of . mg <- E - and shrinks to a point in E - E -which contradicts the choice of . We repeat the entire procedure of this paragraph a) taking e2, e2, m1, m^, K2> A - B1 - B2 - K^, for e±, e±; l±, . K±i ¥L± . This is just the preceding argument 'upside down' and constructs the path C2 3 y2 as required. The only unexpected thing is the use of the cube A - B"*" - B2 - for the original cube K^ ; this reflects the fact that y^ should be found in and y2 not necessarily ihr\ K2 but merely 'in A^ and below B^ 1. We now have y^ and y2 as required by iv). To construct a, join y^ to p by a path a^ lying in mg e^ C A^ O V; and y2 to p by a path a 2 in mg e^c A^H V . Let .a = a (j a^ . b) We can assume that dom = BdA, i = 1, 2 . Since mg C C V, C. shrinks to a point in V and there is a mapping i • i f^:A -> V such that ^ j e d A = C i * * t ^ aPP e n s t o ^ e true that (2.2) gives us a big element g' in V/1 which hits both f^A] and f2[A] - 51 -(unless they i n t e r s e c t ) , but we are not sure that there i s a connected set i n f ± £ A] that w i l l j o i n %' and, y± and stay i n A -, B - T>2 > so that i t i s not yet possible to b u i l d qC'V H (A - B^ B 2) T as i n f i g 35. By (1.2), taking S to be Ext(A - B-1 - B 2) , there ... are disks WI\ ^ i n A such that f~ [Ext(A - B - B„)]CH U-..-. \J W . 1 n l 2 1' n Since BdA. misses f ~ 1[Ext(A - B 1 - B2') ] (because fj.[BdA] C Int (A - B 1 - B 2 ) l , i i A - Int Wn - ... - Int W i s a disk with holes. We assume that each 1 n Bd W1 l i e s so near fT 1[Ext(A - B 1 - B j ] that f.[Bd W1] l i e s close r l 2 l . r 1 1 to Ext(A - B - B 2) . Since we know exactly what A - B - B 2 looks l i k e , we- can construct an E-neighbourhodd N of Ext (A - B"*" - B 2) so that N i s simply connected. We can assume that each f.[Bd W 1 ] C N; l r then f^[Bd W*] shrinks to a point i n N; and by (1.3), taking N = P = P = . .. = P , and g = f . , A T fc I T i T fc „i , .there, 1 2 n' i A - I n t W. - ... - Int W 1 1 n i s a mapping f.:A -> rng g WP. y ... U'.P = f . [ A - Int W?" - ... - Int W^UN n" such that f. = f. • on A - Int W, r- ... - Tnt W1 . In p a r t i c u l a r i i 1 n r f,, = f. on BdA . I t i s important that f.[A - Int W* - ... - Int W1] i l l 1 n = f.[A - Int - ... - Int W1] C V . l I n c) Since --rng C.^  = f [BdA] = f^tBdA] misses A3 and f a i l s to shrink to a point i n the absence of the appropriate eyes of A^f by (2.2), f^[A] and f 2 [ A ] eit h e r i n t e r s e c t i n A^ or h i t the same b i g element X i n A^ . We can combine these ideas by saying 'f^[A] and f 2 [ A ] meet the same element A i n A^' and allowing A to be e i t h e r a b i g element or a small element. Since f o r a small e, N misses A^ , A O rng 1. must l i e i n rng f. - N c f.[A - Int - ... - Int W1] c V,, I i i 1 n 7 and f ^ [Aj c: A - Int W* - ... - Int W^ , which we saw was a disk with Facing page 52 V7' - 52 -holes and which contains c^^y^)  c d o m c x ^ B d A • Since X n f^lA] and lie- in the image under f\ of a disk with holes which, maps into V,. there.is a path ^ which, joins y^ and A in V . Futher-more V'^ C A - B"*" - B2, 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 J r 1 \>/> C v H (A - B - B2) • Let q be a path joing y^ and ; y^ in Hi U A U v . . clearly q c y ^  (A - B1 - B^ . d) We will show that the path which travels from y^ to y2 in q and returns to y in -a is an upper principal path of A in V by showing that c A <"> V and that 5 ^  is homotopic to £ in A . Let £ be decomposed into two paths Z' • and £" such that £" c B 1 and Z' E3 - B1 . We assume that £ pierces Bd B1 in just two points z^, z 2 as shown in fig 37 . We can do this because Bd is horizontal near £ and because £ can be a nice circle. 1 2 Construct arcs z^ y^ and z 2 y 2 in the cubes and A - B—B2 - B -respectively. (The idea here is that both z^ y^ will l i e in A - B^ - B2> 2 the cube which locates A^ Dq, and in A - B - B2> the cube which locates A^ Z3 a). The path which begins at z^ and travels to z 2 2 through z^ y^, a, and z 2 y2 is homotopic in the cube A - B - B2 to £" . The path which begins at z 2 and travels to z^ via z2^2' ^' and z^ y 2 is homotopic in the.cube A - - B2 to Z" .' Combining homotopies, the path ^ which.begins at z^, travels to z 2 in z^ y^ u lit- z 2 y2 and returns to z^ in z 2 y2 y a u z^ y^ is Facing page 53 - 53 -Nomotopic in A to I . The path is evidently homotopic to ^ . Note that ^ passes through the point p £ a . Eventually p will be the 'official' intersection point of the principal paths ^ and ?2 of A . e) There is no difficulty in altering, the argument to construct a lower principal path 5'2 ^ o n e keeps in mind the fact that 'the pictures are different' and that everything in the construction of ^ t be repeated. We cannot, for example, use the from a) because mus they were defined with respect to A - B1 - B2 and we must replace A - B1 - B2 (the cube which located A3 and 'shaped' the right side 2 of ?1) with A - B-B1 which locates A2 . The idea is to start with e^  and e2 as before, but to use A2 rather than A^ as suggested in fig 38 which, in a sense, is a replacement for fig 35. The new £ 2 turns out to contain p 6 e^ n e2 just as 5^ does; this establishes that C ^ ^ ^ 2 ^ 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 7 l 3' I 2 point y^ such that! y£ 6 mg C£ n mg e^ f> (A - B - B^  - K2) and y2 £ mg C2'^i mg e2 n K2 . The arc a' lies in V .0 A^ and has end points y^ ,' y2 . 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 , B1, B2> A - B1 - B2 - K±, K2> and priming every new construction. It will be found that the arc a' con-tains p just as the original a does. For the construction of K', K£, , replace fig 36 by fig 39. It is quite easy to adapt b) and Facing page 54 - 54 c) by keeping In mind that the important cube is A - B - B^ which replaces A - B^ - in the construction of £^ . (The point is that in b), c), one must use a cube whose boundary encloses the 'important' dogbone A2> see fig 38).: Finally we construct a path q' which joins y£, in A - B - B 1 . This plus a' C A - B - B 2 can be combined into the path -£> which can be shown to be homotopic to m by adapting d) above, decomposing m into paths m"C B^ and vx" C. E - B^ etc. The path ^ lies in V by an argument which should appear naturally from the adaptation of a), b), c) to construct ?2; and ^ 1 S clearly in A . Since the point p lies in both a and a' and hence in both %^ and ?2 ; , therefore 5: fi K 2 ^ 0. f) The proof is now complete i f e^ V lies in A^, i.e. i f 'j••-= 1 . The is no difficulty in constructing a proof for the lemma when j = 4 in view of the symmetry of the construction of the A.. . We will give only an outline of the proof for'- j =v2 (and by symmetry for j = 3) for these reasons: 1) the details can be filled in along the lines of a) ... e) above, and, 2) the argument in a), ... e) is 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 A H A . A.. A... ... where i , j , k, ... are chosen l i j xjk from 1 or 4. • To construct the principal path 5 ^  when j = 2, use the paths e^, e2> which we now assume to l i e in A2 f\ V, and the cube A - B 1 - B^ (see fig 40) which acts toward A2 just as A - B 1 - B 2 acts toward A^ in a) (i.e. A - B^ - B^ separates A2 just under the Facing page 55 upper eye while A - B1 - B2 does the same for A^. Using the argument of a), construct a lasso J = o a u which is related to V, el ^ e2' A4' a n d A ~ B1 - B ^ just as J in a) was related to V, el u e2' A3' A ~ B l ~ B2 ' F i s ^  s n o w s the n e w J • When Cj. and G2 shrink to a point in V they hit the same element \ of A^ (A may be a big element or a point). The path q joining the end points of a in V ^  (A - B"*" - B^ ) may be constructed by adapting the argument of b), c), and ^ = a V q may be shown to be an upper principal path of A by an argument like that of d). Just as in the case j = 1, the arc a contains a point p e n e2 . Thus p € ^ . . To construct the lower principal path 5 when j = 2, we start as before with 2 e1 u e2 c A2> but we use the cube A - B - B2 'O A^ and construct J ' = C^a'UC^' so that J ' is related to V, e1 v; e2, A . 2 A - B - B2, just as J ' in e) is related to V, e^'U e2', A2> 2A - B - B , see f l g 0 41 . Fig^ 41 also shows q'~ which ts used with a' to form £'2 . The arc a" and hence the path K2 turns out to contain p ; hence £ n £ 2 ± 0 as before, 0. - 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 Q such that A had Property Q, and i f a dogbone Ajk ... r h a d P r° Pe r ty Q. o n e o f Aj k ... ri> Aj k ... r2> \. », A , had Property Q . This meant that there J K ... r j j K ... r^ f was a chain A 3 A^ . Aj^^3 ... of dogbones with Property Q . Since the possession of Property Q implied intersection with both disks B.. in II (2.1), the limit of the chain was a big element A which hit both and (see the discussion in I I § 2 ) . We follow Bing's proof closely (in spite of the fact that we alter Property Q to a property which has to be applied to a whole 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) in [12], i t is evident that the crucial part of the argument is the proof of [12, Th 10], where i t is 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 also fails to have Property P, (The precise definition of Property P is unimportant until Ch IV). In Ch IV we will prove just this result with the disks D. in [12§7] replaced by the singular disks f [A] in II (2.2). Our l i proof will differ from the proof of [12, Th 10] in that whereas in [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 is 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 (in fig 19)'misses one of f^[A], f 2[A], then the new f^[A] may be constructed so that not only does each k. miss one . f7['A], but both f1[A] miss each of the J j x x arcs ^ and shown in fig 42. The lie in g and g^ and tie the upper and lower loops of the k_. together as shown in the figure. If we can obtain such singular disks f^[A], the reward is considerable, for then parts of the k^  can be erased as shown in fig 43, leaving the set .b^  U b 2 u b^ \j b^ u K^U ?2 shown in this figure. Since each K C k_., each b^ misses one ~f^[A] while £^ 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 ?2 instead of to O'pq.r.s in [12, fig 2], This can be done with very l i t t l e change in the argument of [12] and results in 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 ZC E and mg g^ D rng g2 n Z = 0, i.e. i f f .the ranges are disjoint at least in Z . Lemma One. Consider A, A_., g \ k. as defined in Ch II (see fig 19). 3 3 Let ZO A and let C.. :BdA E - Z . Let g. :A E be Z-disioint 1 x mappings such that S^EdA = Ci " L e t ^ ^e t'i e sPn e r e shown in fig 44 consisting of the cylindrical annulus Q with disks d^, d2 for end caps. Each pierces each d^ exactly once and g "^ misses £2 . Let S C Int A and let N be an n - neighbourhood of S such that Nd"Int A . The arc x.^ shown in fig 42 lies in 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 gi - Int S) u N , i i i ) If £. U Q misses rng g., then L misses rng g. . 3 ~^ J Corollary. Let be the cube defined in Ch II (see'fig 21Y. Then ii ) and i i i ) in Lemma One may be replaced by ii ) " , g^ 'A -* rng S± V &x > i i i ) ' If k. V fi misses rng g., then k. misses rng g. . j i j • a i The proof of Lemma One is delayed to §2, which may be read after Ch IV i f desired. We give a second lemma which is intended to repair a gap-which would otherwise appear in the proof in Ch IV. This lemma is quite special-ized, but appears here because its proof is just a variation of the proof of Lemma One. As before, the proof is delayed to §2 and may be omitted on a first reading. Lemma Two. Consider A, Z, k_., a.., g \ as defined in fig 42. Let E be the sphere shown in fig 45. The sphere E together with an n - neighbourhood W of £ lies in Int A; g"*" Cl Int E - n, and each 3 of a^, a^y a^ pierces E as shown. Let mappings g_^:A -> E be Z-disjoint with g . = C. , where c. is defined as in Lemma One. i|BdA i i Both rng g^ miss the set ^ u k^ \j y k3 . Let U ^ J be arcs 1 1 1 1 1 in g which join a^ rt g and rt g , a^ O g and a^ rt g respectively and miss rng g^ . Let v^2' V13 loe a r c s i n ^ which 1 1 1 1 join a^ rt g and rt 3 , a 2 Cl g and a^ rt g respectively and miss rng g2 • The arcs u^2» U13' V12' V13 a r e n o t necessarily Facing page 59 - 59 -d i s j o i n t . Then there e x i s t Z - d i s j o i n t mappings g7:A -*• (rng g.. - IntE) U W such that S^'" C-^ o n BdA, and one of rng g^, rng g^ misses k i u V V C o r o l l a r y . One of rng g^, rng g^ misses b^u b^U b^ y £2> and g^:A rng g i U . Although u^2 l i e s i n an annulus and i s joined to E by the orderly arcs a^ rt Int E, a 2 O Int E, the arc (a^ y u ^ 2 w a^) rt Int E may be knotted i n Int E, as a few moments experiment w i l l show (an arc ab i n a cube K with ab c Bd-D = a U b i s knotted i f there i s no disk D c K with ab c ;BdQDer ;Bd<CKU ab) . To be knotted the arc must make more than one c i r c u i t on the twisted annulus. A s i m i l a r comment applies to U13' V12' V13 * 2. Proof of Lemma One. 3 3 (2.1). As a preliminary, we describe an untwisting function y:E -> E which i s onto and one-to-one and which unwinds the twist i n g\ i . e . (/(g^ ) i s the planar annulus shown i n f i g 46. For w e l l known reasons y cannot be a mapping» but we ensure that y w i l l be discontinuous only on the curved c y l i n d r i c a l surface z shown i n f i g 47. In f i g 47, the end caps of z are c a l l e d M^, M 2, and the cube Int ( z U U M2) i s ca l l e d K . Eventually y w i l l be composed with a mapping whose range misses z . Thus the E e s u l t of the composition w i l l be a mapping. The 3 function y i s defined to be the i d e n t i t y on E - K and on both . To define y i n Int K: Imagine K to be cut free of the space by means of a cut on Z and on M 2, remaining attached only on . Faing page 60 - 60 -K may--.be • thought of as a stack of circular disks of infinitesmal thickness. These disks span the cylinder z and each meets in a straight arc. 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' in 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) in place. When this happens, the disk M^ , which is attached to the space, necessarily remains fixed and does not rotate. Each disk intermediate between and M2 rotates through an angle which is close to zero for disks close to and approaches 2TT for disks whose location approaches that of . The rotations of the various disks in the stack can be contrived so that B ^ H K is carried onto the plane which contains B " * " - K, and so that the final result is homeomorphic to K . In fig 48, the 'typical disk', which is located half-way between and M2 will rotate through an angle of TT . This carries its intersection with B " * " on to the desired plane. Since M2 has returned to its original position, we can restore the cut at M2 Evidently y is 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 in the definition of y as- a discontinuity on z . Clearly y carries B ^ into the plane containing ft'*' - K . (2.2). We will prove a simpler version of Lemma One to show the general approach. 3 (2.21). Let S be a sphere in E having a simply connected neigh-3 bourhood 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 in 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 in A such -I -1 that g [Int S] c W1u ... U . Since g [Int S] misses BdA, -1, A - Int W., - ... - Int W is a disk with holes and g lint S] c 1 n Int W^  ... Int . If e in 11(1.2) is sufficiently small, then g carries each Bd W into N, for g[Bd W ] lies close to, but not in r r Int S and hence close to S . In 11(1.3), take (simply connected) N to be P = P2 = ... = Pn to obtain the mapping I = g|A _ I n t „ _ _ _ I n t w 1 1 n (j J Y 2 o ... J Yn;A rng g V N . Since each point x in A lies either in A - Int W^  - ... - Int Wn in which case g(x) £ E _ i n t s, or in some W^_, in which case g(x) £ N, rng g misses Int S - N . Thus rag g lies in (rng g - Int S)U N . Finally g = g on BdA because the two mappings differ only in u W2o ...tfW^ , which misses BdA . (2.3). We will now give a formal proof of Lemma One. Case one: neither 8-^ ]^ meets S . Let g^ = g^ • Since g^[A] meets Ext S, 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. Facing page 62 - 62 -Case two: exactly one rng g^ meets S . The rng gi which meets S also meets Q . Assume that rng g^ meets Q <= S . Let j*2 = • Evidently -i)'., i i ) , i i i ) of Lemma One are true of g^ . Apply the argument of 2.2, taking g in 2.2 to be g^, and construct a mapping g^ :A -* (rng g^ - Int S) U N which agrees with on BdA . With regard to g^, 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 g2-. Thus 0 = Z n rng g1 n rng g2 = Z r> Crng g1 U N) r\ rng g2 D Z rt rng g1 n rng g2 . Case three: both rng g^ meet S . One rng g^, say rng g^, meets Q; the other (rng g2) does not. The aim of the proof will be k k to construct an intermediate pair of mappings g such that rng g2 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, a) Choose a component of rng g 2D S . It is important that, since rng g2 misses Q, rng g2 n S c Int d^ u Int d2 . Using this fact, we construct a circle c^ c Int d^ o Int d2 - rng g^ - rng g2 which encloses (on. one20f '.the .;. d^ ) points of exactly one of rng g^ n S, rng g2 n S . Although we choose z^ c rng g2 n S, c^ may turn out to enclose points of rng g / i . S • b) Assume that Int c^ C Int d^ . Construct a sphere w u 6^ v <52 in the shape of a pill-box as shown in fig 49 so that c^ is the equator of to u <5^  v &^ • c) An argument like that of case two but using co V S^u 6~2 instead of S yields a pair of Z-disjoint mappings 1 1 g.:A rng g. u N such that g. R C... on BdA, Int c. misses 1 1 1 rng g.^  y rng g^; and i f rng gi misses Q u , . then so does rng g^ . The argument of case two is used virtually as is i f c^ encloses points of rng g^n S . If c^ encloses points of rng g2 f\ S the argument must be modified somewhat, since the method of case two would not ordin-arily ensure that rng g^ would miss a l l the t. that rng g2 misses. d) It turns out that rng g^ misses 0. . 1 If some component cr rng g2 n S exists in, 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 3 4 r Si ' gi ' '*" ' first 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 Q, o Z^, then so do rng 2 r r+1- T T t . ^ ^ r „ r+1 rng g^, ... rng g^ rng gi .We show that rng n S O rng g^ . - (\ S and.that the sequence of mappings ends at a pair g^ for which k k rng g2 H S - 0 . although 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 is satisfied. Details of proof, a) If we assume that rng g2 misses Q but hits S, then there is a component of rng g 2n S lying in Int d^ or Irit d 2, say Int d^ . By the Zoretti Theorem, there is a circle in Int d^ which misses rng g 2, encloses z^, and lies so near to z^ that i t misses rng g^ (by 'encloses' we mean 'encloses relative to d^'). If encloses points of just one of rng g^ n S, rng g2 r\ S, -64 -then let ^e c i • We must expect that x-^  will enclose .points of both rng g^ r> S and rng g2 n S (z^ could be itself a circle enclosing points of rng g^ n S). In this case use the Plane Separation Theorem in Int x-^  C Int d^ to construct a circle x 2 <- I n t X-j_ which misses (g^ I'A]'C A]) <n S and separates the component of g2IA] ^ s from a component z' of g^A] n S in Int X-j_ .• Since may be in Int - Int x2» w e cannot predict that Int x 2 contains points of rng g2 n S; but by the Plane Separation Theorem we know that X2 en-closes points of (rng g^ u rng g2> n S . If x 2 encloses points of both rng g^ n S and rng g2 n S, then separate Int x 2 ^ (rng g1 u rng g2> s t i l l further by means of another application of the Plane Separation Theorem. We repeat this procedure, defining circles x^> X^ > ••• > Xr being constructed in Int Xr_-^ whenever Int Xr_^ contains points of r - l both rng g_^  f\ S . The following argument shows that the sequence X, , Xo> ••• ' must be finite:, each annulus Int x ~ * n t X >i contains l. z. r r+l points of (rng g^ *J rng g2) o S . Without loss of generality in the construction of the xr> we could have replaced the sets rng g^n S with 'thickened' sets obtained by covering the )^ rng g^ r> S by small disks of area a (from compactness, the thickened rng g^ n S can be assumed to remain disjoint from the thickened rmg g2 r» S) . But since each of the disjoint open annuli must therefore have area at least a, the number of- x r must be finite.and the sequence ends at some x t • Since Xt+1 could be defined i f x f c encloses points of both rng g^ n S and rng g2 n S, x t must enclose points of only one of rng g^ n S, rng g2 n S . Let x f c be c^ . We repeat that we do not know which of rng g^ n S, rng g2 n S is intersected by Int x t = ^ n t . 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 vert-ical cylinder OK and end caps <5 ^ , <$ 2 , which are parallel to d^ (see fig 49). The cylinder co intersects d^ only at c^ and extends equal distances above and below d . Thus Int c^ (considered as a subset of d^ ) lies in Int(co o u <$2) • Since c^ misses rng g^ u rng g^ we build to so near c^ that co misses rng g^ u rng g2 . Fig 50 shows <fru 8^ <J <$2 and part of d^ . We assume that c^ has been moved slightly, i f necessary so as to miss (JJ^ u #2'c/ %^) n d^ . We also assume that co misses every although this may necessitate making co smaller or even curving co slightly to follow the curve of . The sphere toi/6-^ u <$2 is constructed as in fig 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 fig 51. We construct v . so that 6. - - I- - H- - I, is a l x 1 2 3 4 deformation retract of v . - J l . - and so that v . misses x 1 2 3 4 x S => Int c^ and any & which misses Int c^, i.e. any which misses co u S^u 6 2 • We assume that cot/ u 6 2 u u v 2 has been constructed so that co u v §2 u u v 2 lies in N, and (since c^ c: Int d2) so that Int(co u U § 2 ) c/ u v 2 misses o d2 . c) Assume for the moment that Int c^ contains points of rng g^ n S . Because rng g^ hits 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 co u 6^  u 6'2 u u v 2 lies- so near Int c^ that co u u <$2 u u v 2 misses rng g2 . Since - 66 -co also misses rng g^ , rng g^ meets to U 6^  o fi^ only in 6^  or 62- Apply an argument like that of 2.2 to corru 6 w 6^  , i . e . use 11(1.2), taking S in 11(1.2) to be & u 8^ a n d N t o b e v-|_ 17 v2 ' t o obtain disks W.^, ... Wn in A such that g1 [Int (co ^  '5 ^ &2)] lies in Int W-.f ... U Int W . Since gg,[Bd W ] lies near 1 n • 1 r rng g^ rt (co iy <5^  u 6,,) as we saw in 2.2, and since co misses rng g^, therefore g-^ lBd W^ ] lies near 6^  U <$2, i.e. in U v2 . Evidently ;. each g,[Bd W ] lies in one v. . Since each g,[Bd W ] lies in a 1 r l 1 r simply connected subset of U v 2, we can construct the mappings in 11(1.3) and, following the argument of 2.2, define a mapping 1 1 g^ :A -* [rng g1 - Int(co c ^  u 6 2)] u u such that g^ = on BdA 1 1 1 Let g2 = g2 . Then i t is true of both g^ that g^A -*• rng g^ U N, gj = on BdA, and i f rng g^ misses Q, O then so does rng g^ (this last property is only true because rng g^ hits 0, and because g„, which can miss some QU t., is identical to g^). Additionally Int c^ on d^ misses rng g2 because g2 = g2 and misses rng g^ because Int c, misses v. U v„ . The %~. are Z-disioint because 1 1 2 ° i J rng g2 = rng g 2, and rng g^ exceeds rng g^ only in U which misses rng g2 . (By 'rng g^ exceeds rng g^ only in N', we mean that rng g1 u N .3 rng g* .) Suppose that instead of rng g^rt S, Int c^ encloses points of: rng g2 rt S . Let g^ = a n d construct co u v <-> u as in b) , this time so that co u <5^  u <$2 U u v2 misses rng g^ and co misses rng g2 . Constructing g2 is harder than constructing g^ as we did in the last paragraph for we must ensure that rng g2 misses any set V £j that rng g2 misses. We must take careful account of the various £^ . Some are not missed by rng g2 and can be ignored. 67 -Some £^ are missed by rng g2 but do not meet Int c^; we note that co o 6^  v &2 a y has been constructed so that any £^ which mis isses Int c^ also misses co u <5^  u 62 U y v 2 • With this pre-caution, i t is safe to ignore those Z^ which miss rng g2 and also miss Int c^ . In the remainder of this paragraph we will assume that £ and £ are those £. which miss rng g_ and hit Int c .We think that the procedure in the general case that some subset £. , Jl £j , ... £„ of £^, £ 2» £^, £^ misses rng g2 and hits Int c^ 2 s -• will be evident. We proceed to define g2 using 11(1.2) and 11(1.3) as before. The only difficulty occurs when we wish to shrink 82|Bd W r to a point so as to define yr • It was easy to shrink S^ig^ w t o ' r a point in one component, say , of v v 2 in the course of defining g^ . But in this case we must shrink 8213^ y t o a point in - £^ ~ £ 2» 1 r otherwise rng Yr a nd hence rng g2 will hit £^ v Z^ • The reason that g o i ^ j T7 will shrink to a point on v- - Z. - £» is that g0[W ] 1 r misses £^ U Z^ (because rng g2 does) and can be assumed to miss z in fig 47 without loss of generality. There is a retract R (though v»". 3 not a deformation retract) of E - z - £^ - £ 2 onto 6^  - £^ - £ 2 . Additionally i t turns out that R restricted to - £^ - £ 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 Vi ~ ^ 1 ~ ^ 2' a n d s i n c e • ' r ' r R g2 [ Bd W shrinks to a point in Rg^W^] c - £^ - £ 2 > therefore ' r ^2[Bd Wr shrinks to a point in - £^ - £ 2 as required. We delay the description of the retract R and the proof that g2[Bd Wr] can miss z until the end of this proof. Except for the use of R to make ^2lBd W shrink to a point, the construction of g2 is like that of g^, ' r - 68 -and we have g2:A •> [rng g2 - Int (to v $^ ^  <$2)] v (v^ o - t^ - £2) . 1 The g^ are Z-disjoint by an argument like that in the previous para-1 1 graph, and g^ :A -*- rng g^ o N, g_. = c\ on BdA as before. We know 1 1 that rng g2 misses SI because rng g2 exceeds rng g2 only in u v 2 which is remote from Si . If £ misses rng g 2, then either £j meets Int c^, in which case t^ misses rng g2 because £^ is one of t- , £„ above; or else t. misses Int c, , in which case t. 1 2 j 1' j misses rng g2 because £^ is remote from v v 2 . Note that the g^ satisfy the hypothesis of Lemma One. The important difference between g^ and g^ is that Int c^ n rng g^ = 0 whereas Int c^ hits one of rng g^, rng g2 . Since rng g"|" a rng g^ \j v 1 V v 2, rng g* n S <=• rng g± n S (because v u x>2 misses S) . Evidently we can write (rng g^ v rng g2) n S C (rng g^ rng g2)O S where the inclusion is proper. d) Since the g^ satisfy the hypothesis of Lemma One, we look for a component of rng g2 n S in Int d^ U Int d2 and repeat a), b), c) to obtain a circle c 2 c Int d^ \J Int d2 2 1 2 and Z-disjoint mappings g^ :A ->• rng g^ U N with g. = on BdA, and 1 2 if rng g^ misses SI U t^, then rng g_^  and rng g,. also miss Si U t^ . 2 2 1 1 Furthermore (rng g1 J rng g2) n S c (rng g1 u rng g2) ,o S C (rng g u rng g2) f) S, both inclusions being proper. We can continue in this way, defining map-pings g^, g^, ... and components z^ c rng g2 D (Int d^O Int d2> , z4 c rng g3 o (Int d± u Int d 2), ... so that g*;A -> rng g^"1 N, g^ = on BdA, and i f rng g^ misses SI U £^, then so does rng g_^  . - 69 -v TC ir—1 ir~l Furthermore (rng g^ u rng g2) H S c (rng g^ C rng g2 ) O S where the inclusion is proper. An argument from compactness like that used in 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 g^ . Since g^+"*" k k could be defined i f z-^j_ existed in rng g2 n S, therefore rng g^ must miss S . r r - l S ince : rng 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. 1 x x x i x on BdA, while i f ft V t. misses some rng g., then ft f £ misses 3 ^ j k k rng g^ . Since rng g2 misses S, the argument reduces to either Case one or Case two. In the course of the argument of Case one or two, g 2, whose range already misses S, will be set equal to g^ . This means — k that g2 has the properties of g2; thus i ) , i i ) , i i i ) of Lemma One are true for g2 . The argument of either Case one or of Case two will — k — now construct a new g^ :A -* rng g^ u N C rng g^ \j N with g^ = on BdA, so that g^, g2 are Z-disjoint. This proves i ) , i i ) of Lemma One for 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 will 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 is rng g2 . When we arrive at the point in Case three where g^ is defined, we can k — let 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 <5'2 in Case three. Thus a quick proof is possible by adapting Case two. Case five: Both rng g^ hit 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 k g 2). When the g^ are defined, the argument reduces to Case four or Case one. Case six: Both rng g^ hit S; both rng g^ hit fi. This case is not used in the applications of Lemma One, which always require every set 9, U 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 will show how 3 to define R: E - z - - •+ & - I - I ; the definition of R when 6^ is replaced by <52 is similar. Strictly speaking, the proof of Lemma One requires a retraction onto d, - £. - ... - Z , where 1 Jl Js Z. Z. are some subset of Z Z 2 > -£o> Z,; however we continue Jl Js the assumption in 2.3 Case three c) that rng g2 misses Z^ V Z^ . Assume that the unique boundary component of (3^  which is a planar circle lies on the Y - Z plane and that the centre of this circle is 1 the origin. The idea is that i f we untwist g by means of y, a l l the circles y[ZV\ will be nice circles on the Y - Z plane with centre the origin. We assume further that 6^  lies on the left-hand X - Y half-plane. We describe R in terms of several mappings which are app-3 lied in sequence to E • z - Z^ - Z^ . Each mapping will leave Facing page 71 - 71 -61 " l l ~ ^ 2 f i x e d » a n d t t i e l a s t W i l 1 b e ° n t ° 51 ~ ^ 1 " ^ 2 * First: untwist g1 by applying the mapping £/jE3 _ z _ £ „ £^ Ct/ becomes a mapping by restricting i t so that the domain misses the 'bad' set z). Each y[Z^] is a plane circle with centre the origin. 3 Second: using the symmetry of E - y(Z^ U Z^ across the X - Z plane, reflect the right-hand half-space minus y(Z^O Z^) onto the left-hand 3 * half-space minus y(l U Z^ . This reflection carries E - z - yiZ^ U 3 into those points in E with non-positive coordinates which do not lie on Z^U Z^ • Third: retract the left-hand half-space minus y(Z^ (J Z^) (which is the same as the left-hand half-space minus £^ 0 Z^) onto the left-hand X - Y half-plane minus £^ U Z^ . This is easy be-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 X - Y half-plane minus Z^U Z^ . Finally retract the left-hand X - Y half-plane minus £^ y Z^ onto <S^  - £^ - Z^ • The four successive mappings define R . Note that R acts on as a deformation retract (this was used in §2.3 Case three c) ) . Finally we will show that g2lWr] in 2.3 Case three c) can be assumed to miss the curved cylinder 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 in fig 51a and which agrees - 72 -with g2 on every point of A which maps under g2 outside of a small nelghr-bourhobd of Z:. In fig 51a, z. is constructed so as to contain Q •*>• X and to J S £ ^ U t^. Thus rng g2 can be assumed to miss .-£•]_" ^  ^2* ^ e s e t s v i 1 mxs£v2 (see 2.3 Case Three b)) miss ft and could have been constructed so as to miss a l l of z.; Hence we assume that g2 = g2 °n Bd W^  since §2^^ r^""' C V] A, by the deflation of W^  . We do not intend that g2 should replace g2 since g^, g2 may not be Z-disjoint; but i f g2[W ] hits z, we can apply the retract R not to S2IW ^ U t t 0 g2|w a n d U S e t ^ i e * - a c t t n a t 82|Bd W s h r i n k s t o a P °i n t i n vl " £1 " l 2 l f f ^2lBd W d 0 e S * ' r 1 r Essentially the same argument applies to the construction of the other mappings in the sequence g2> g ^ , g 2 ^ , . . . D . 3. Proof of Lemma Two. We will 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 fig 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 in fig 49, where c^ is planar and the 6^  can be considered to be horizontal while co is vertical. It will 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 is any circle on Z and Int c is defined, then a sphere co u §^ u <$2 can be constructed together with neighbourhoods v^, so that co o v 62 f u v2 behaves like the corresponding set In §2, i.e. 3 co o Z. fr c, <S^ , 5 2 are disks in E - Z which meet co only at its Facing page 73 0 - 73 -two boundary components, while the are simply connected neighbour-hoods of the S. which miss E . Furthermore, i f a. hits Int c, i j then a. pierces each Int <S. just once and misses to; while i f a. 3 i J misses Int c, then a_. misses Int(co u 6^  \j S^) u u • We require that when Int c and hence 6_. hits just one a ^ , then - a^ is homeomorphic to the structure shown in fig 52. This require-1 ment is 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 is easy to make a_. n W a straight arc perpendicular to E). The hard problem is that in Lemma Two we cannot use the retract R, which was crucial to the proof of i i i ) of Lemma One. The reason is that we permit the arcs (a^ V L/ n Int E, etc. to be knotted, and in general deviate from the specialized geometry in fig 19. Recall that R was used to show that certain mappings g. \T>, TT will shrink to a 1 r point in - - ~ £3 _ ^ • 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 in fig 52. Let f :BdA -> - a^ . Let F:A -> E so that f = F on BdA and rng F misses a simple closed curve L such that L 3 . The curve L may be knotted. Then f shrinks to a point in vj ai • A similar result is true i f a^ or a^ replaces al * Proof. Let u be the small circle shown:in fig 52. Then u can be considered to represent the sole generator y of - a^, - 74 -and also (by consulting, say, the definition in [6 Ch VI]) a generator 3 of the Wirtinger presentation of E - L (we specify the particular presentation only to be sure that u does not represent a trivial 3 generator,). If i:ir^(v^ - a^ ) TT^(E - L) is the inclusion homomorph.ism, then, with a change of basepoint, f G y m for some integer m, and l(y) is an element of T T ^ ( E ^ - L) . Then f £ i(ym) = (i(y))m, which 3 is the identity of TT^(E - L) because f shrinks to a point in 3 E - L . Since i(y) is a non-trivial element (in fact a Wirtinger 3 generator) of ir^(E - L), either m = 0 or i(y) is an element of finite order. 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 y^ in ~ aj) ^  ° To prove Lemma Two, we will apply arguments like those of §2 to a disk D rather than the sphere S . We will first define some simple closed curves to play the part of L in (3.1): Let ' a i 3 be the unique simple closed curves which are subsets of ?2 y al u U12 U a2' ^2  U a l U U'13 U a3' r e sP e c t l v e xy' L e t ^12' u i 3 ^ e identical to ai 2 ' U13 r e s P e c t i v e x y except that is replaced by and u ^ by . From the hypothesis of Lemma Two, i t is clear that u.^ and LL miss rng g^ while v-_2> y'^3 miss rng g^ . Now i f D is a disk which is a subset of Zh and Bd D misses rng g^ U rng g^ u a^ua^o a^, then i f rng g^ U rng g^ meets 'D, we can define a circle 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 g^ :A •*• rng g^ u u v^ , where the are the usual neighbour^ hoods of the ^ and the gl have properties like the g"!" in §2, Case 1 1 1 three. If Int c^ and hence co U <5 2 m eets just one of a^, a 2, - 75 -say a^ , then we use (3.3) instead of the retract R to shrink the various mappings g . i _ , TT to a point in v. u v 0 as r 1 was done in §2, Case three. Thus i f rng g^ hits Int c^, rng g^ misses a^ because Int c^ meets only one of a^, a.^ a^; while i f rng g_£ misses Int c^, rng g} misses a^ because g^ - g., . And by the usual arguments, and a^, which are remote from to U u continue to miss both rng g^ . In applying (3.1), we let L be O R UT.2 ( a s s u m i n g that Int c^ hits a^ ) depending as rng g^ or rng hits Int c^ . If Int c^ hits a^ only, let L be o r aga i n» i f Int c^ hits a^ only, let L be or 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 fails (because the argument with (3.1) is weaker than the original argument in §2 which used the retract R), and gj cannot be constructed so that 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 3 the above ideas to further pairs 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 D C. £ , and let Bd D miss rng g^ u rng g2 u a^ u u Then there exist circles c^, C2> ••• c m i n l n t D and Z-disjoint mappings 1 2 m , J_T_ i_ r * r^ -1 , , _N 1 2 m g^ .. . g±, ••• g± such that g^A -*• rng g^ U In - £;) , . 8 ± H. g ± <" ••• ~ %± = c± r-1 on BdA, c^_ encloses (relative to D) points of just one of rng g_. n E } rng g2 O £, and rng g^ misses D . If, additionally, each c r can be constructed so that c^ encloses just one of a^ n E, a2n E, - 76 -A E, then rng g™ can be constructed so as to miss a^u o a^ . Proof. (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 g2 evidently takes the place of the condition in Lemma One that ft misses rng g^ \J rng g2 . Clearly we cannot have rng g™ miss Int E in this version of the argument because D 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 r r—1 rng g^ c rng g^ 0 (n - E). The reason that rng g exceeds rng g_^  only in n - E i s , as usual, that u is remote from E . (3.3). Corollary. If, additionally to the hypothesis of (3.2), D misses rng g^, then rng g™ misses a^ u U a^ regardless of the number of a^, a2» a3 which are hit by the Int c r . Similarly, i f ^ m rng g2 misses D, then rng g2 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 first glance: Do there exist Z-disjoint mappings f^ :A -*- (rng g_^  u vi) - a^ - a^ - a3 with f± ~ on BdA, and a decomposition of E into disks D-^» D2 so-that D^U D2 F E and D1 n D2 =. Bd D^  - Bd D2, and so that Int D2 misses one rng f^, say rng f^, and hits a 2 and a^; while Int D^ . hits - 77 -Case one: the mappings f exist as described above. Look at the decomposition E = D^u T>^ •• Apply (3.2) to to convert the fj. to Z-disjoint mappings f^:A rng u (n - E) with = on BdA. and such that rng f^ misses a^ and . In (3.2), the condi-tion that Int c hit at most one of a., a_, a_ is satisfied because r 1 2 3 a 2 > a^, miss 3 Int c^_ . Furthermore rng f misses a^ and a^ by the usual argument. Now apply (3.3) to D2, to replace the f^ with Z-disjoint mappings g':A -> rng f. O (n - E) with g'|_JA '= C . . We know i x i|BdA i that rng f^ misses since rng f^ does, and f^ evidently satisfies the hypothesis of (3.2) and (3.3). By (3.3), rng g^ misses a^ u a 2 i> a^, although rng g2 probably does not. Since rng g_T now misses a l l of E, rng g^ misses Int E except perhaps in 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 in the construction of Z-disjoint mappings 8i = 8i: A r n g 8i U ( n " E ) W i t h gi = Ci °n B d A and'~<such t h a t. r n g gi misses D and a 2 u a^ . We know that .every c^ encloses at most one of a^ n E, a 2o E, a^rt E, as required by (3.2): for otherwise r-1 g^ , Int c r, E - Int c r satisfy the definition of f^, D^, X>2 r-1 given above, which means that (since _g exists) c^ contradicts the Case two assumption. Evidently rng g£ misses not only D v a^ U a^, but also d u a^, so that 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 will use Lemma One, Lemma Two and I§5 to prove 11(2.2). The organization of the proof is 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 first give a (somewhat altered) definition of Property Q, then induce Property Q through'.the steps of the dogbone construction. This argument occupies most of the length of this chapter. 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 will not present a formal induction, but will show that i f A has Property Q, then so does A^\j A^ U A^U A^ (Bing proves that one A^ . has Property Q; our version of Property Q is only useful when applied to a^, &2, ^ » > ° f Ex 2 in §2). The proof of this is divided into Part I and Part II as in [12 §7]. In Part I, we look at the set ^'u b± u b2 u b3 U b^ U r,^ (see fig 43) which serves a purpose like that of the set M pq^r^s in 112, fig.2]. We show that the f^ in 11(2.2) can be replaced by mappings :>'g^  such that each rng g". misses one b. and both . We call the set t,^ U b^ U b2 U b^ U b^U t,^ t a e cradle of A, and later represent i t as in fig 53, which preserves the embedding of t,^U b^ V b^ U b^ U b^v l in A . In 112], L/ pq..^s 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 will define a Property P on double ended lassos Zu a U m with respect to closed sets Y^, Y2.. The lasso Z u a o m consists of circles Z and m connected by an arc a . In Ch II we often specified constructions only up to homotopy (e.g. the intersecting prin-cipal paths of Ch II). The consequence was that we ignored singularities in .these constructions. In this chapter, this practice is emphatically not allowed; in particular, in the lasso Z V a u m, the circles Z and m 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 in terms of their negatives, which we write Property 'VP and Property Q^. A double ended lasso Z U a V m 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 Y2 Cor both), fVP(b).; Z ^ a V m meets both Y '' and Y2 . The set a {J m misses Y^  U 1 Z contains a point y f Z a such that of the two distinct arcs in Z with end points y and Z n a, one. misses 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^  and the point y may be in m - a . Regard-less of whether Z U a <j m has Property 'VP (a) or Property 'VP(b), each of Z and m has Property "VP as defined in I §5 for circles with base point (the base points here are taken to be Z r> a, m n a) . This statement, which is important, is 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 in 112]. It is 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 x^ 6 m and Z u a u m has Property 'VP (by our definition) , then there is an arc in Z U a U m with end points x^, x^ which misses one of Y^, • We will neither use nor prove the complete equivalence of the two definitions here, although a proof will be found to be straightforward. Property Q« is defined on dogbones. If a dogbone X "> *~2_' *"2 has Property Q^^  , this means roughly that the centre of X has A' 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 X has Property ~Q • i f f there exist Z-disjoint mappings ,g1, g„ 1' 2 i l 3 such that • g. :A 'E , g. R C , on BdA, and the centre of X has Property'VP with respect to rng g^, rng • We also say 'X has Property -x-Q- with respect to g.,, g ' with the obvious- meaning. z.,c.^ , x £ We define X to have Property Q ' tff X fails to have Property (i.e. with respect to every qualified pair of map--"»'-•^ ' 2 pings g.) . Note that a statement like 'X has Property Q„ X Z , , C-2 - 81 -with respect to ^2 m e a n s very l i t t l e . Example 1). Suppose Z = X = A and c^, C2 are the two circles shown in fig 28. Then A has Property Q . For i f c. (d'--). L , c.^ , c 2 y --shrinks to a point, i t must hit the upper (lower) eye £ (m) of A ., Thus i f f is an extension ot c\ to a l l of A, then rng f^ hits £ and rng f 2 hits 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 f.[Al miss both £ and m or that one of £ or l m miss both f^[A] . Example 2). Let Z = A^; c^, c 2 as in Ex. 1). Then A^ has Property ; for the c. can shrink to a point so as to , ^  A-^  > 2^ 1 miss Z and A . We emphasize that 'X has Property Q ' 1 (~2 does not imply that C^, link the eyes of X . Evidently i f X has Property Q and f , f are "» <~-^ » "^2 any Z-disjoint mappings of A into E"^  with f^ = on BdA, then the centre of X fails to have Property 'VP (a) with respect to f^fA], f-Xh], and consequently both f^[A] and f2[A] meet (the centre of), X . This suggests that the obvious way to attack the proof of 11(2.2) is to let Z = A and let C^, C2 be the in I1C2.2), and. we will eventually do this. But i t turns out that in this case there is no sequence A ^  A., o A^D ... such that each, of A, A ^, A , ... has Property Q. , with, the c - defined" as-in H:C2.2); jK. " » C--^  > c- 2 J-in fact every dogbone X f, A has Property • We overcome c2 this difficulty with the next definition. - 82 T-A set {X , ... X } of dogbones has Property <\,Q i f f each X , r - 1, 2,'. ... m has Property with respect to r L , , *~2 the same pair of mappings f^, a n d t n e same triple Z, c.^ , . If {X-, ... X } fails to have Property HL , then we will say J. m z, Cj., ^ 2 that {X , ... X } has Property . I f the set of components of 1 m AyC-, G „ i z . 3 some 0. has Property Q„ and i f g : A -> E is an extension of o z , , i _ ^ , i_2 i C\, i = 1, 2, and the g^ are Z-disjoint; then some component X of a fails to have Property with respect to the g. . As we s z,c^, l saw earlier, this, means that both g^[A] meet X . We will say that a has Property Q i f f the set of components of a has s L*, , C-2 s Property Q /,--•' Eventually we will show that each of a., a_, <X_, .. z,, , •-- 1 2 J has Property Q_ . Z,'G1» c2 3. We now give our version of [12, Th 10]. 3 (3.1). Let Z 3 A and C^, by any circles whatever in E - Z . In particular, the c do not necessarily link the eyes of A . Then i f {A , A , A_, A.} has Property 'vQ , so has A . -L Z j H Z , C ^ ' , Ll^ We remark that in [12], the proof of Th 10 does not use the fact that Bd Dx, Bd T>2 (in fig 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 is that in later applications of the argument of the proof of [12, Th 10] (which is a disguised induction step) to, say, A1 and , A^ 2> ^13* ^14' t I i e B d D i d o n p t i n fact link the eyes, of A^ . For a similar reason we state (3.1) for very- general circles c. rather than the c... in fig 28. We assume that - 83 -Z, c^, have been chosen once and for a l l before the proof of (3.1) begins, and will now write Property 'v-Q for Property We will not refer to Bing's Property Q again in this paper. We will 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 (A^, A^, A^, A^} has Property with respect to mappings g^, g^ and show that the 3 g^ can be replaced by Z-disjoint mappings g7:A -* E with g^ = on BdA and with the property that in the cradle £^ u b^ o b^o b^ j ^ (see fig 43), each b^ misses one rng g^ while t,^ u t,^ misses both. By the definition of Property ^Q, each k has Property M? with respect to the rng g . Look at g1U g^, and recall the definition of bridging in I §5. The construction of the g^ divides into three cases depending on the way that the sets rng g n ( g 1 U g^) bridge g^ and g 1 . If rng g^n g"*" or rng g^ Ct g"*" bridges g \ but not both, then we say that g^ " is bridged once by rng g^ or rng g^ 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 g , g^ is bridged twice. Case three. One of g \ g^ is bridged twice. 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^ is bridged twice, the other cannot be bridged even once). Case one. Since each k. misses one rng g.., this case sug-gests an immediate application of Lemma One. 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 g^ by means of the following argument: assume that k^ misses rng g^ and k^ misses rng g2 (if another pair of k. miss the rng g^ or i f a l l four miss the same rng g^, the method is similar or easier). Since £^ misses 3 1 rng g^ > there is a circle £ C E — 3 which lies near £^ and approximates i t so that £ misses rng g^ . We imagine £ sliding on the surface of the twisted band 3^ and eventually coming to rest directly over Z^ . Although we use the term 'slide', we intend that Z stays 1 1v close to but does not touch g . By sliding Z on the side of 3 which is free of the arcs a., we are assured that Z can move without 3 3 touching the a. . This shows that there is a homeomorphism M of E 3 1 onto itself such that M is fixed on E - K.. and on 3 u a, u a„ v a_ u a, ; 1 1 2 3 4 and carries Z to a position directly over M[£^] = £^ . Clearly M[£] misses rng Mg^, Z^ misses rng Mg2 . Construct a small annulus a so that its boundary components are M[£] and Z^ . This can be done so that a misses k'1? k ^ k 3 and k^ - £^ . By Th 5 (in Ch I), Int a contains a simple closed curve £ which bounds no disk in ct and which misses both rng Mgi . Figs 54a, . ,.,'d show how £ may be moved to 1 location of an equator of S without hitting 3 u o k2 U k3 v k^ the - 85 -3 This shows that there is a homeomorphism H of E onto itself which 1 3 A fixes 3 , every k ^ , E - K^, and carries t onto the location A W[l] shown in fig 54d. Evidently misses both M'Mg^lA]; and in fact we can assume that a l l of 0, misses both 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^  continue to be Z-disjoint and M'Mg.j. on BdA, while each kj has Property "vP'(a) with respect to the WMg^ f'A] because both M' and M are fixed on each k.. . We can now apply Lemma One to construct Z-disjoint mappings g^ with g^ = on BdA, such that rng g^ C~ rng M'Mg^  U and (since JJ misses both rng M^ Mg^ ) rng g_^  misses every k^  that rng g^ misses. Since Int S - N, both rng g^ miss £^ . Since M and M' are fixed outside of K^, rng g_^  c rng g^ u . Now apply a result like Lemma One to 'the 3^ end' of {J kj to construct Z-disjoint mappings g^ such that g^ = on BdA, rng g^ G rng g± V K^ ,, and rng g^ misses ?2 as well as any k^  that rng g^ misses. It may be necessary to alter the g\ with homeomorphisms which act like M, M' above, in order to make rng g^ miss those k. which rng g. misses. Evidently rng g7 misses both J 1 X ?. and every k. that rng g. misses. Since each k, misses one l J i j rng g^ » the cradle of A has the required property. Case two. In this case we allow some of the k. to have Property-VP (b) with respect to the rng g± . We reduce this case to Case one by converting the . k.. with Property ^ p(h) to Property 'vPGa) or, more accurately, we will define mappings = g^, 7 G ^ > e t 2 ' ^t3' G^ with the usual properties such, that k^  has Property 'VP(a) with respect to rng G^ ., rng G2^, and in fact each k_. misses one rng G^ , - 86 -Th_e argument then reduces to Case one. We w i l l show how G . , i s constructed and i n d i c a t e the construc-xl t i o n of the other G . . . I f k.. has Property M?(a). w i t h respect to xj 1 the rng g^,- then l e t g± *= G.Q - G ^ . I f k 1 has Property 'vP-Cb) w i t h respect to rng G^ D » r ng > we assume that a ^u m^ misses rng G. v rng G 0 and that has Property 'VP(b) (as defined i n I §5 Io /.o I f o r a c i r c l e s w i t h basepoint £ ^ a ^ ) , s i n c e otherwise we simply 'turn the p i c t u r e upsidedown'. Now by Th 6 or Th 7 ( i n I § 5 ) , s i n c e at most one of rng G^q f) g 1 , rng G 2 q n g 1 , bridges g 1 . there i s a c i r c l e £ / C I n t which bounds no d i s k i n g \ contains the base p o i n t r\ a^ and misses one of the rng say, rng G ^ q . We now have a centre (or at l e a s t a double ended lasso ) w i t h Property 'vP(a) s i n c e £/ u U m^ misses rng G ^ Q v but £ ' i s l i k e l y to be a ver d i s o r d e r l y c i r c l e and among other d e l i n q u e n c i e s , probably h i t s U u k^ (which means that i." U a^U m^  can't be used i n Lemma One (the c o n s t r u c t i o n of 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 get 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 the f o l l o w i n g procedure which r e c a l l s the manipulation of t i n Case one. L et X be a simple closed curve which l i e s near and approximates £ ' but misses g"^  . A short s t r a i g h t arc a connects (\ a^ to a base p o i n t on X so that a meets X at only one p o i n t . We can assume that A u a misses rng and we now regard A u a u a^ c m^ as a double ended l a s s o which misses rng G = rng g^ . Now s l i d e A over g \ keeping the base p o i n t f i x e d , so that the f i n a l p o s i t i o n of A i s d i r e c t l y over £^ . As b e f o r e , we choose the ' r i g h t ' s i d e of g^\ to s l i d e A on so that 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 upper loop rides near but not on g , and i t remains only to telescope a o X so that . a collapses and A moves to the location of i.^ . We 3 conclude that there is a homeomorphism M" of E onto Itself which 3 fixes E - K^, k^, k^, k^, and carries A u a v a^o m^  onto k^ . Evidently M"G^ g has the required properties of G ^.. Clearly k^ misses G-Q » a n d since rng G^ n k2 = rng G±Q n k2» = k2 continues to. have Property 'VP with respect to the rng G _ ^ , and a similar argu-ment applies to k^, k^ . Since M is not fixed on g \ we must ask how the rng G^ bridge g"*" . It is clear that g"^  is bridged at-most the once by the rng G ^ , since which misses rng separates boundary components of g"*"; and of course g^ is bridged by the rng G_^ just as i t was bridged by the rng G ^ . Thus neither of g \ g^ is bridged twice by the rng G ^ . If k2 misses one of rng G-Q». rng G , , ^ , then let G ^ = G ^ 2 , . Otherwise k3 has Property ^P(b) with respect to the rng G,_, and we construct G.„ so that i l i2 rng G I 2 n (k^ \J k^ u k^) = rng G ^ n (k^ U k^ u k^) , and k2 misses one of the rng G^2 (note that we may have to work at the lower end of the figure; the fact that neither band, g^ or g^ is bridged twice by the rng G ^ is used in the second application of Th 6 or Th 7). Evidently k^ misses one of the rng G^2 . Proceeding in the same way-, we define G„_ so that k0 misses one rng G.„, and since G..- can be constructed i i 3 i3' i3 so that rng G I 3 n (^ u k^) = rng G ^ n (k^ u k^) each, of k^, k2 misses one rng G I 3 . Finally define G ^ so that each k^  misses one rng G ^ . Evidently the G ^ can be used in the argument of Case one to construct the 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 Z . Similarly each G.. = a.. on BdA . 1 Case three. In this case we know only- that one of g , 8'^ , 1 1 say- g is bridged twice. It is easy to see that i f g is bridged twice, then no k^  can have Property 'vPGa) . For this would mean 1 that some t. misses, say, rng g^; then by 1(1.7), rng g^ n g ° 1 cannot bridge g , so that the number of bridges is at most one. But if each k. has Property 'VP(b), then in every case, m. must miss rng g^ o. rng g^ and £. must have Property 'vPCb) . For evidently i f any _3 £j,c: E: - . 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 - rng g^ - rng g^ • Except for the fact that m ay n o t miss rng g^v rng g^, the picture begins to resemble fig 45,- (though we s t i l l must construct the arcs U-^v etc.). We first alter the g^ so that 2^ misses rng g^ v rng g^ •• This is done just as tn Case one. Fig 55 shows the k_. and a sphere S placed tn the usual way wtth, respect to ?2 a n d mj • *n ^8 55, the 'k do not have Property^ >P(a), so that we use Lemma One.itself and not the corollary. Using the method of Case one, construct Z-dtsjoint mappings g. ;A.-*-(rng gv o K„) i -X^ X, 4. c. such that rng g^  misses every nu that rng g^ mtsses-. Thts stmply means that rng g.^  _j rng g^ misses each, m^  . A n examination of the ... method of Case one shows that i f rng misses aj> so does rng g^; « 89 thus rng g. rng g„ misses a l l four xn. u a . We also know that each point of which misses rng g^ also misses rng g^; this means that t. has Property r"P(b) with respect to rng g., rng g ... Therefore the four k^  have Property- 'VP with respect to the rrng "g. On the other hand, the fact that the inclusion rng g^c^rng g^ u J^)" C2 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 g2 A 8^ is zero (because of the presence of, say-, 3 - -m^cE _ r ng g^ „ r ng 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 nor 3 1 is bridged twice. However in the,'worst case', 3 ^ continues to be bridged twice. 1 — i If 3 is bridged twice by the rng g^ n 3 , then we use Lemma Two. The hypothesis of Lemma Two is satisfied except that we must — 1 1 construct u 1 2, u13> vi2» V13 ' S i nce rng g^ n B bridges f? , — i there is a component Q of rng g^ o B 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 and a continuum e2 in £ 2 such that e^  contains a^ O and misses one of rng g^, rng g2 . Since e^ hits Q C rng g 2, e^ must miss rng g^ . Since the whole continuum e^u Q U misses rng g 2, we use 1(2.5) to construct an arc u^2 which, joins n a^ and n a^ in B^ and misses rng g1 . The constructions of u13> v^2' V13 a r e simi la r« - 90 -Now by-Lemma Two there are Z-disjoint mappings g,T:A -^(rng g^ - Int z)u n (where Z, n are the sets described in Lemma Two) with g_T = = on BdA, and such that one rng g^ , say rng g^ , •' misses b^ u b^ U b^ while both rng g^ miss ^ u ?2 • Evidently rng g^ c rng g^ y . In the argument of Case three we did not suceed in constructing the g^ so that ?T. ^  ^ 2 m i s s e s rn8. g i ^  r n g g2 a n d e a c ^ ^-j m i s s e s one rng g^; instead t,^ u £ 2 misses both rng g^ and three b^ miss the same rng g^ . In Part II of the proof of (3.1), i t turns out that it is sufficient to define the gl so that, three "b. miss the same r ng (the same thing happens in 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 g^ so that each 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 r ng g^ C rng g^ u U K2 . Thus we can write rng g^ c rng g.D A . This will be important when we apply the argument of (3.1) to the com-ponents of OL^y etc. To summarize the situation: i f {A^, A2, A^, A^ } has Property with respect to g^, g 2, then there exist Z-disjoint 3 mappings g_T:A -> E such that gi F Ci °n B d A~ ' rng g£ c mg g^U k , i f U X^y b^u b2 u b^ U b^ is the cradle of A, then both rng g£ miss Z^O X2 a n d either each b^ misses one rng g' or three 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 II. 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 is 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 will 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 b ^ f ... V U ^ by the figure iy^pq.r.s shown in fig 56. This can be done so that either each arc J J 1 pq^ .r^ s misses one rng g_T or three pq^r^s miss the same rng g^ . Our terminology is now like that of [12] except that rng g.T replaces in [12]. We follow the division into cases found, in [12], We will not prove that the three cases given in [12] exhaust the possibilities, but remark for plausibility that the case division ... 1) Three pq^r^s miss one rng g^, 2) P I - J ^ T 8 P lu s Pq 2r2s m i s s e s rn8 P ^1^ plus pq^r^s misses rng g^, 3) P^^s Plus pq^r^s misses rng g^ ; P.q-2r2S Pl u s P^3r3S m i s s e s r ng §2 " ' s e e m s a t first glance to ignore the possibility: pq^r^s plus pq3r3S misses rng g£, pq2r2s plus P.^ '4r4s hisses rng g^ • However this last variation is just Case Two with the diagram inverted. We will now describe how Bing's Part II can 3 be. altered to show that there exist Z-disjoint mappings F_.:A E such that F^ = on BdA and the centre of A has Property 'VP with respect to rng F^, rng F^ . Facing page 92 - 92 -Case One: any three of pq.r.s (j - 1> 2, 3, 4) miss the same rng gC • If pq?r.s is ah arc which fails to miss rng gl , 1 mC mC • 1 — o . . o then the structure shown in fig 57 lies near pq-rs V pq^s-U pq,r,s -and 1 1 3 3 4 4 this ses rng gf . The structure in fig 57 can-be moved $6% fcKe°j5osition of the xo 3 centre k of A by a homeomorphism M,. which fixes E - A . , Evidently M^ g^ , M^g2 are the required F^, F 2 . If pq^ r^ -s is aim arc which fails to miss rng gl , then one uses the structure in fig 58 which lies o near pq^r^s u pq2r2s u pq^r^s and misses rng g«, . If pq^r^s or o-pq^r^s f a i l to miss rng g^ , the method is like one of those already o given. If a l l four ^q.r-ss miss rng g' , then 'forget' one of them. J J J Xo Case Two. pq^r^s plus pq2r2s misses rng g^, pq^r^s plus pq^r^s misses rng g2 • We replace Li pq^r^s with the more compli-cated construction in fig 59. In fig 59, s has been replaced by s,, s„, s„, s. which li e near s so that the s. and arcs s.s„, 1 2' 3 4 j 1 3 s^^, S2S4. mis s'arng g^ U rng g2 . Abusing the notation slightly, 3 we have arcs p qjrjSj w i t n P ^ irisi ^ pq2r2S2 C E ~ r n g Sl * 3 pq^r^s^ U pq^r^s^ c E - rng . We build two new arcs: p'q^r^s^ , which lies near pq^r^s^ and misses rng g2> and p'q^r^s2 which lies near pq^r^s^ and also misses rng g£ . Apply a move which carries s^^q^p'q^r^s^ to the location shown in fig 60 and fixes pq^r^s^., s^s3 » S2S4' a n d S1S4 * I j O O K a t a disk in A bounded by the circle pq1r1s1s3M6CrpM6(qpM6(p^)M6CqpM6Cr^)s2r2q2p . We will call this disk T and assume that i t is just the obvious disk suggested by the'figure. Thus T misses a l l but the end points of pq^r^,s^s2 . Later we will need the fact that T can be constructed so as also to miss- a l l but the end points of pq^r^s^ (in Case 3). There is an arc Facing page 93 - 93 -A C T with end points and s^ which misses both rng MgS^ because arc s^gCrpMgCqpMgCp^MgCqpMgCr^^ misses rng M gg 2 and arc s 3 s i r i 9 i P 9 2 r 2 S 2 m ^ s s e s r n S ^gS^ • (We w i l l now begin to abbreviate our arc nomenclature). Define a move which moves A to the position of arc s^MgCp'is^ and fixes each pq^r^sy and s 3 s ^ s ^ s 2 . Although we do not know the location of A in T, this can be done by means of the A - move defined in I §3. Evidently rng M^ Mgg^  misses pq^r^s^, rng M^Mgg2 misses p q ^ r ^ , and both rng M^ Mgg^  miss the circl e s^s 3Mg(p')s 2s^s^ . Fig 61 shows P.9]_r-j_sj_" <J P.9^r4s^ u s i s 3 M 6 ^ P ^ ^ s 2 s 4 s l r e p i a c e d by a s e t U a' O m' which li e s very near the f i r s t set so that m"v a' misses both rng M^ Mgg^ , and t' has Property ^P(b) with respect to the rng M^ Mgg^  (i t is easy to give V this property since much of V can coincide with s^r^q^pq^r^s^). Evidently f U a'u m' can be moved to the position of the centre £ O a u m of A . If this is accomplished by a move Mg, then the centre of A has Property ^P(b) with respect to the rng MgM^ M^ g^ , which we define to be the required F^ . Case three, pq^r^s plus pq^r^s misses rng.gj, pq 2r 2s plus pq^r^s misses rng g 2 . The mechanism of this case resembles that of Case two. We repeat the construction i n f i g 59 and define Mg precisely as in Case two, so that we arrive once more at f i g 60. However, since the rng g^ are related differently to the various parts 3 of the figure, we have this time: s^^a^s,, c E - rng l ^ g ^ ~ rng Mgg2 as usual, but p q ^ r ^ w pq^v^s^u M, (p')Mg Cr^s^ c E 3 - rng M ^ , 3 p q 2 r 2 s 2 U p q 3 r 3 s 3 t / MgCp'iMgCrps^ E - rng M &g 2 . In this case we Facing page 94 e - 94 -must use a fact that we stated but did not completely use in Case 2, viz. that Mg fixes a l l four pq^r^s . We assume tht the disk T is placed so as to miss pq^r^s^ . We use Th 4 from I §4 at-this point; at the analogous place in [12], Th 7 of [12] is used. By Th 4, since s3M6(qpMg(p') and pq 2r 2s 2 miss rng Mgg2 and M6(p')M6(q^)s2 and S3SiriqlP m i s s r n g M6gl ' t n e r e i s a n a r c A C T with end points s^, s2' such that X misses either rng Mgg^  or rng Mgg2 . Apply a move M', similar to M.,, to move X to s„M/.(p')s_ . This can be / / 3 o 2 done by an A - move as before; but some care should be taken so that fixes every pq.r.s. (as well as, of course .'s„s,.s.s„) : - the 7 J ] j 3 1 4 27' reader might first prefer to move pq^r^s^ to a new location where i t cannot interfere with the collar of T used in the A - move. The proof is now completed along the lines of the previous cases,: using the fact that i f s^MgCp^s,, misses rng M^ M^ g^ , then the set shown! in fig 62 lying near s ^ M C p ^ s ^ ^ v s ^ q ^ q ^ r ^ misses rng M^ Mgg^ ; while i f s3Mg(p'')s2 misses rng M7Mgg2> then the set in fig 63 lying near s^s3Mg(p')s2s^s^V s3r.jq3pq2r2s2 misses rng M^ Mgg,, . This com-pletes 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 g , g2, then A has Property with respect to mappings F^, F2 such that rng F c. r ng g. , Proof. We know that 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 {A^, ... A^} has Property then A has Property .. Our argument now diverges somewhat from 3 Bing's in [12]. Suppose that f.:A-> E such that f. i„ J A = C. and r r i i|BdA i the f^ are Z-disjoint (we continue to take Z, c^, c 2 to be assigned arbi t r a r i l y according to the remark at the beginning of §3). We w i l l show that i f A has Property Q, then each of a^, a^, cx^ , ... has Property Q; the proof of 11(2.2) follows directly from this fact. If A has Property Q, then by (3.1), {A^, ... A^ > has Property Q. i.e. ct^  has Property Q . (3.1) does not imply that some A^  has Property Q for the luminous reason that each A^ has Property ^Q, as the argument in §1 Ex 2 shows. However we can show that has Property Q by adapting the argument of the proof of (3.1) to show that for each A^, i f ••• A j 4 ^ n a s P r°perty then so does A^  . This i s easy to do since the proof is simply restated i n terms of images under the embedding h_. of various subsets of A . Occasionally in the proof of 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, C]_> ' c2 of course, since we intend to show that the same Property Z ' C i ' ' C 2 is possessed by each of a^, a2> P~y ... • We originally defined Z to contain A so that we have Z D i ^ as required. We intend of course to l e t Z F A eventually. To show that CL^ has Property Q, assume that - 96 -the set of components of &2 has Property with respect to qualified mappings 8^' § 2 " ^PP-^ a r e s u x t l i k e the corollary of (3.1) to {A^^, ... A^} to obtain Z-disjoint mapping F^:A -> rng g^ U A^ such. that F ^ - on BdA, and A^ has Property with respect to the F ^ . We can see that since rng F.^  does not exceed rng f.. in 3 E ~ A^, the dogbones A^^' ••• A24' A31' **' A34' A41' A44 c o n ~ tinue to have Property • M}- with respect to the E^^, for as we saw earlier, possession of Property 'VP depends on the fact that rng g^ misses certain continua in various dogbones, and this property is inherited by rng F.. 3 at least for dogbones i n E - A^. Construct Z-disjoint mappings; F i 2 : A r n g F i l U A2 S U c h t h a t F i 2 = C i o n B d A a n d A2 ^ a s P r ° P e r t y with respecl "3 to the F ± 2 . Once again, dogbones in' E - A 2 . ^ which. have Propertiy/v,Q with respect to the F^ . v.-, continue-to have .Property,.with respect to the '.F.0 . This means that not only 'A2> but A^ 'A'31, ... A^,- A41, ... A ^ have Property 'vQ with respect to the F^ 2 . Evidently we can continue i n this 3 way and f i n a l l y derive Z-disjoint mappings F^:A -> E which agree with c\ on BdA and with respect to which, a l l of A^, ... A^ have Property M}. Assume that the set of components of have Property . Then an argument.like that of (3.1) Corollary can be applied to each A., in (perhaps lexicographic) order to show eventually that . has Property . If A has Property Q, then by induction, a^, have Property Q and must also have Property Q . We think that i t Is now evident hpw^  to proceed in 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 A , then 11(2.2) ts true; thus we consider only the case that rng f^ f\ rng f 2 n A = 0, 97 -i . e . the case that the f are A - d i s j o i n t . In the preceding argument we showed that f o r a f i x e d choice of Z, G^, c^, i f A has Property Q„ , then so does each a . I f Z = A and C, ,. c „ are the ^Z,a^, G 2 m 1' 2 c\ l i n 11(2.2) then A c Z c: E3 - rng - rng G 2 as r e q u i r e d , and A has Property Q by an argument l i k e that of §1 Ex 1. By the i n d u c t i o n argument, every CL has Property Q„ . As we saw e a r l i e r , t h i s m z , c ^ , G 2 means that both rng f . h i t some component of a f o r m however l a r g e . l m F i n a l l y we w i l l show that both rng f ^ must h i t a b i g element A. A of the dogbone decomposition G . Let G be the s e t of a l l elements of the dogbone c o n s t r u c t i o n ( i . e . a l l components of <X^ , a^, ••• ) which meet both rng f ^  and rng f 2 . E v i d e n t l y G i s i n f i n i t e , f o r by the arguments^of t h i s rtchaptef, each a must c o n t a i n an element of m /\ r, G . C l e a r l y one of A^, ... A^ , must c o n t a i n an i n f i n i t e subset of G, f o r the four A. c o n t a i n a l l of G . I f A. contains an i n f i n i t e 3 3 subset of G, then one of A.,, ... A.., say A., , contains an i n f i n i t e j l j 4 ' jk' subset of G . There i s a sequence A 3 3 A ^ r> ^ j j ^ ••• each of which contains i n f i n i t e l y many dogbones which meet both rng f . Obviously each member of the sequence meets both rng f , and the i n t e r -s e c t i o n An A. n A n A m e e t s both rng f . . One can a l s o use 3 3^- 3 i the dogbone m e t r i c to show that i f the images of the rng are d i s j o i n t i n V, then there i s a neighbourhood system of the p o i n t s of V c o n s i s t i n g of 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 of dogbones about the b i g p o i n t s such that no neighbourhood of diameter s m a l l e r than e Ctn the dogbone me t r i c ) meets both images of the rng . This i m p l i e s that some a.^ has Property ^Q, cf>;proof of Th 12 of [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 from E-= , Notices Amer. Math. Soc. 12 (1965) p. 363. 3 [12] R. H. Bing, A decomposition of E into points and tame arcs such 3 that the decomposition space istopologicallydifferent 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 -- 1 0 1 -- 102 -

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