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

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