UBC Theses and Dissertations

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UBC Theses and Dissertations

Differentiable engulfing and coverings of manifolds MacLean, Douglas W. 1969

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DIFFERENTIABLE ENGULFING AND COVERINGS OF MANIFOLDS by DOUGLAS W. MACLEAN B.A., U n i v e r s i t y o f B r i t i s h Columbia,. 1 9 6 5  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 t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969  In  presenting  an  advanced  the I  Library  further  for  degree shall  agree  scholarly  by  his  of  this  written  thesis  in p a r t i a l  f u l f i l m e n t of  at  University  of  the  make that  i t freely  permission  purposes  may  representatives. thesis  for  be It  financial  for  8,  of  British  Canada  by  the  is understood gain  Columbia  for  extensive  granted  of  University  Vancouver  British  available  permission.  Department The  this  shall  reference  Head  be  requirements  Columbia,  copying  that  not  the  of  and  of my  I agree  for that  Study.  this  thesis  Department  copying  or  allowed  without  or  publication my  Supervisor:  Dr. E. L u f t -iiABSTRACT  There a r e now e n g u l f i n g theorems f o r t o p o l o g i c a l , p i e c e w i s e l i n e a r , and d i f f e r e n t i a b l e m a n i f o l d s . e n g u l f i n g so f a r was reduced t o p i e c e w i s e  Differentiable  linear engulfing  u s i n g the J . H. C. Whitehead t r i a n g u l a t i o n o f a d i f f e r e n t i a b l e m a n i f o l d and J . R. M u n k r e s ' t h e o r y o f o b s t r u c t i o n s t o t h e smoothing o f p i e c e w i s e - d i f f e r e n t i a b l e homeomorphisms.  I n the  f i r s t p a r t o f t h e t h e s i s we o b s e r v e t h a t ' the method o f p r o o f o f M. H. A. Newman's t o p o l o g i c a l e n g u l f i n g theorem a p p l i e s , up t o a l o c a l lemma, s i m u l t a n e o u s l y t o a l l t h r e e c a t e g o r i e s o f manifolds.  We p r o v e t h i s l o c a l lemma i n t h e d i f f e r e n t i a b l e case  and t h u s o b t a i n a d i f f e r e n t i a b l e e n g u l f i n g theorem w h i c h has a d i r e c t proof.  Then we s o l v e t h e p r o b l e m o f t h e e x i s t e n c e o f  a s t r e t c h i n g d i f f e o m o r p h i s m between complementary  subcomplexes  o f a s i m p l i c i a l complex i n E u c l i d e a n space w h i c h i s c r u c i a l f o r a l l a p p l i c a t i o n s o f engulfing,,  Next we p r o v e a theorem c o n c e r n -  i n g t h e u n i q u e n e s s o f open d i f f e r e n t i a b l e c y l i n d e r s w h i c h i s t h e d i f f e r e n t i a b l e analogue o f t h e u n i q u e n e s s theorem f o r open cones.  A consequence o f t h i s theorem i s t h a t i f M-^ and  a r e compact d i f f e r e n t i a b l e m a n i f o l d s w i t h i n t e r i o r s then  M-£ R  and  M^xR  diffeomorphic  a r e d i f f e o m o r p h i c , where  denotes t h e r e a l l i n e . • A n o t h e r consequence  (R  i s that i f a d i f f - .  e r e n t i a b l e m a n i f o l d i s t h e monotone u n i o n o f open d i f f e r e n t i a b l e c e l l s i t i s d i f f e o m o r p h i c t o E u c l i d e a n space. We p r e s e n t  •'  several applications of d i f f e r e n t i a b l e  e n g u l f i n g which a c t u a l l y hold i n a l l three categories o f manifolds.  - iiiOur methods are such t h a t they a p p l y a l s o t o noncompact manifolds. Theorem:  Let  and l e t U, =  u  U  i J  V.  (M - CIV,  #  #  be a d i f f e r e n t i a b l e ^  m  °P  e  e n  V..  . , - C l V. X,  .  .)  is  such t h a t M,  CIV.  1  ,  J > 1, and  + m >, n + 1,  m  onto i t s e l f such t h a t  C l V, -,, 1 < i < m, ^i "7  '  —  and  M =  M  is a  M  V. .,. X.J.  1  h^  i s the i d e n t i t y  on  m U h.(U.). i=l 1  1  For instance,  k-connected d i f f e r e n t i a b l e m a n i f o l d o f dimension  without boundary, then  =  t h e r e are d i f f eomorphisms  T h i s theorem has s e v e r a l c o r o l l a r i e s . if  '  m dM c U i  k^ + ... + k  , c V.  k.-connected, with.  J  x  1  M  i s open i n  i f k. > 0, .1 < i < m,  M  n-dimensional manifold  subsets o f  where each  1 , J ~rJ-  x  of  U  ., V.  k. < n - 3  h^  ' 5  ,,  1, J  Then, i f  M  may  k _< n - 3  be covered by  i f k>0, m<  and i f  m >  open d i f f e r e n t i a b l e  n-cells.  U s i n g t h i s r e s u l t , we g i v e a new  and d i r e c t p r o o f o f the  uniqueness o f the d i f f e r e n t i a b l e  structure of Euclidean  for  n > 5.  Theorem:  F i n a l l y , we prove a g e n e r a l  Let  M  2  w i t h two  connected boundary  such t h a t the i n c l u s i o n o f  e q u i v a l e n c e , i = 1,2. N-LX[0,OO)  onto  h-cobordism  n-space  theorem.  be a connected d i f f e r e n t i a b l e m a n i f o l d o f  dimension n, n > 5, N-], a n d ' N  n  M -  N  I  into  components M  i s a homotopy  Then t h e r e i s a diffeomorphism o f N . 2  - ivACKNOWLEDGEMENT I am g r e a t l y indebted t o P r o f e s s o r E. L u f t f o r s u g g e s t i n g the t o p i c o f t h i s t h e s i s , f o r a l l o w i n g me a generous amount o f h i s time and f o r h i s c o n s t r u c t i v e comments d u r i n g the p r e p a r a t i o n o f t h i s t h e s i s .  I a l s o wish t o thank  P r o f e s s o r J . V. Whittaker f o r h i s c r i t i c i s m o f the d r a f t  form  o f t h i s work, and Miss Doreen Mah f o r t y p i n g i t . The f i n a n c i a l support o f the N a t i o n a l Research C o u n c i l o f Canada and the U n i v e r s i t y o f B r i t i s h Columbia i s gratefully  acknowledged.  -  V -  TABLE OP CONTENTS Page 1  INTRODUCTION N o t a t i o n and Fundamental D e f i n i t i o n s  CHAPTER 1  Local  CHAPTER 2  The  CHAPTER 3  A  CHAPTER 4  Open C y l i n d e r s  48  CHAPTER 5  Coverings  55  BIBLIOGRAPHY  .  4  CHAPTER 0  c"- Engulf ing  7 15  C°°- E n g u l f i n g Theorem  C°°-Stretching Diffeomorphism  o f Manifolds  ...  35  65  1. INTRODUCTION There are now e n g u l f i n g theorems f o r t o p o l o g i c a l , p i e c e w i s e l i n e a r , and d i f f e r e n t i a b l e m a n i f o l d s .  Different-  i a b l e e n g u l f i n g so f a r was reduced ( i n [ 2 ] ) t o p i e c e w i s e l i n e a r e n g u l f i n g u s i n g the J.H.C. Whitehead t r i a n g u l a t i o n o f a d i f f e r e n t i a b l e manifold ([153)  and J . R . Munkres' t h e o r y o f  o b s t r u c t i o n s t o the smoothing o f p i e c e w i s e d i f f e r e n t i a b l e morphisms ( [ 9 ] ) .  homeo-  In the f i r s t p a r t o f t h i s t h e s i s we observe  t h a t the method o f p r o o f o f M.H.A. Newman's t o p o l o g i c a l i n g theorem ( [ 1 0 ] )  applies,  up t o a l o c a l lemma, s i m u l t a n e o u s l y  to a l l t h r e e c a t e g o r i e s o f m a n i f o l d s . lemma In the d i f f e r e n t i a b l e  engulf-  We prove t h i s l o c a l  case and thus o b t a i n a d i f f e r e n t -  i a b l e e n g u l f i n g theorem which has a d i r e c t After proving this d i f f e r e n t i a b l e  proof. e n g u l f i n g theorem,  we prove a theorem, c o n c e r n i n g the e x i s t e n c e o f a  stretching  d i f f e o m o r p h i s m between complementary subcomplexes o f a s i m p l i c i a l complex i n E u c l i d e a n space, which i s c r u c i a l f o r a l l applications  [12],  of engulfing.  T h i s s o l v e s a problem posed i n  p. 502. . Next we prove a theorem c o n c e r n i n g the uniqueness,  of open d i f f e r e n t i a b l e  cylinders  which i s the d i f f e r e n t i a b l e  analogue o f the uniqueness theorem f o r open ( t o p o l o g i c a l ) ([ are  5 ]).  A consequence o f t h i s theorem i s t h a t i f •  cones  and M  2  compact d i f f e r e n t i a b l e m a n i f o l d s w i t h d i f f e o m o r p h i c  interiors,  then  M-, y IR and M  ?  x R  a r e d i f f e o m o r p h i c , where  2. R  denotes the r e a l l i n e .  Another consequence  i s that i f a  d i f f e r e n t i a b l e . m a n i f o l d i s the monotone union o f open d i f f e r e n t i a b l e c e l l s i t i s d i f f e o m o r p h i c t o E u c l i d e a n space. We p r e s e n t s e v e r a l a p p l i c a t i o n s o f  differentiable  e n g u l f i n g which a c t u a l l y h o l d i n a l l t h r e e c a t e g o r i e s o f manifolds.  Our methods are such t h a t t h e y a p p l y a l s o t o noncompact  manifolds. Theorem 5»1«  Let  f o l d , and l e t  U  1-9  M  be a d i f f e r e n t i a b l e  • • -, U  m  n - d i m e n s i o n a l mani-  be open subsets o f  M  such t h a t  CO  U, = V  U V,  .,  (M-Cl V  i  k. < n-5 1  where each £>V±  i  j+i"  V. c l  v  *  i s open i n  i j)  i f k. > 0, 1 < i < m, ~  i s  j > 1  k  k^ + . „. + k  of  M  m  + m _> n+l,  and  M =  and M 0  c  m U V_. ', . i=l > ±  1^  M  is a  Then  ±  i s the i d e n t i t y on  h^  C l V\  ^»  m u h.(U,). i=l  T h i s theorem has s e v e r a l c o r o l l a r i e s . if  j c  t h e r e are diffeomorphisms  onto i t s e l f such t h a t  1 < i < m,  C l V.  i"connected, with  1  if  M,  k-connected d i f f e r e n t i a b l e  For i n s t a n c e ,  m a n i f o l d o f dimension n+l  n  without boundary,  then  M  may  k <: n-5  be covered by  m  i f  for  n > 5.  and i f  open d i f f e r e n t i a b l e  U s i n g t h i s r e s u l t , we g i v e a new ness o f the d i f f e r e n t i a b l e  k > 0,  m > n-cells.  and d i r e c t p r o o f o f the unique-  structure of Euclidean  F i n a l l y , we prove a g e n e r a l  n-space  h-cobordism  theorem.  Theorem 5«5« of dimension and Ng  Let  be a connected d i f f e r e n t i a b l e m a n i f o l d  n, n _> 5,  w i t h two connected boundary components  such t h a t t h e i n c l u s i o n o f  topy equivalence, of  M  i = 1,2.  N-, x [0,oo) onto  M-Np.  Nj_  into  M  i s a homo-  Then t h e r e i s a d i f f e o m o r p h i s m  4. CHAPTER 0 Notation  and F u n d a m e n t a l  In t h i s paper, I  will  denote  Euclidean  n-space, e R :  (n-l)-sphere  in  in  R . X  D e f i n i t i o n 0.1. a subset,  7r (X,A) = 0 n  X  is  between If  X  and  n  S ~^  D  will  n  be  denote  denote  the unit  the closed unit  always  mean a c o n t i n -  id.^w i l l  denote the  X. If  X  i s a t o p o l o g i c a l s p a c e and  then  Let  Y  If  (X,A)  (X,A) A  is  is  of  Y,  is  k-connected i f  ( k - l ) - c o n n e c t e d and  space w i t h  the distance,  t o be  d(f,g),  A c X  k-connected.  be a m e t r i c  i s defined  the distance,  metric dist  d.  (A,B),  i n f { d ( x , y ) : x e A, y e B}. f and g  a r e maps o f  b e t w e e n f. and g  X  i s defined  s u p { d ( f ( x ) , g ( x ) ) : x € X}.  barycentric  of  If  K  i s a simplicial  subdivision of  n-skeleton  subset  will  n  denote  i s a t o p o l o g i c a l space,  are subsets  A .and B  Y,  R  numbers,  the half-space  will  n  f o r a l l n < k.  D e f i n i t i o n 0.5.  the  _> 0 ] ,  denote  i s a t o p o l o g i c a l s p a c e , and  into to  R ,  k-connected,  A and B  n  will  11  the set of r e a l  [0,1],  we s a y t h a t t h e p a i r  D e f i n i t i o n 0.2. If  denote  By t h e word map we s h a l l  n  i d e n t i t y map o f  is  H x  n  uous map. , I f  will  the unit i n t e r v a l  [(x^, . . . , x ^  n-ball  R  Definitions  of  K  will  K  will  be d e n o t e d  be d e n o t e d b y  | K | , the neighborhood  complex,.the  of  S  K  ^  in  K  by  i-th  p (K), 1  . If  S  and is a  i s defined to  be t h e s u b c o m p l e x N(S,K)  { A e K : A i s a f a c e o f J i n K a n d A fl S ^ 0).  =  D e f i n i t i o n 0.4. say t h a t  f  I f A c \R  and  n  isa  C°°-map  of a neighborhood o f Definition 0.5.  A  A  }  f : A - (R  i f i t c a n be e x t e n d e d t o a into  m  C°°-n m a n i f o l d M  Is a. c o l l e c t i o n o f p a i r s  (1)  Each  (U,h) e J  i s a locally  set o f  (U,h)  h  satisfying four  w h i c h maps  (3)  If  J  (1^,1^), ( U , h ) e J, 2  2  C°°-map : w i t h n o n z e r o  The b o u n d a r y , M  D e f i n i t i o n 0.6.  If M  a f a m i l y o f maps  { h ^ : t € 1}  onto  cover  M.  h ^ h g " : h (U 1  2  1  n U)  r o  itself,  - H  2  n  (3). i s d e f i n e d t o be t h e s e t o f C 65  n  C -isotopy  of  isa  M  C^-manifold w i t h o u t boundary, (usually written  i f each  a n d t h e map  H(m,t) = h ( m ) • i s a t  o n t o an open sub-  w h i c h do n o t h a v e a n e i g h b o r h o o d w h i c h i s (R .  M  together  Jacobian.  gM, o f M  diffeomorphic to  t o be  then  i s maximal w i t h r e s p e c t t o  points of  conditions:  n  The c o o r d i n a t e n e i g h b o r h o o d s i n J  (4)  U  U c M  J.  H .  (2)  is a  Euclidean  C^-structure  c o n s i s t s o f an open s e t  w i t h a homeomorphism  C^-map  R .  H a u s d o r f f space w i t h a c o u n t a b l e b a s i s and a J  i s a map, we  m  C°°-map.  h^  isa  H: M x I - M  h^)  i s said  C -diffeomorphism of r a  d e f i n e d by  6. D e f i n i t i o n 0.7.' 1-connected a t compact s e t  A t o p o l o g i c a l space  i s s a i d t o be  co i f f o r each compact s e t  D r> C  Definition 0.8.  X  such t h a t  I f A, B c R  denotes the j o i n o f  A and B.  X-D n  C  c  X  there i s a  i s s i m p l y connected.  are j o i n a b l e subsets then  A*B  7. CHAPTER ONE Local  Lemma 1.1. a  e  L e t T = {(a,B) e R : 0 < a < B < 1}. 9: R x T - R  C^-map  £(x)  = x  C"-Engulf i n g  i f x fi (0,1),  such t h a t i f  There i s  8?(x) = e(x,a,B)  .3/ e£([0,a]) = [0,B]  and ^ ( x )  then  > 0  f o r a l l x e R. Proof:  We l e t  x + g(x,a, B). if  9(x,a,B)  be o f the form  We c o n s t r u c t a  g^(x) = g(x,a,B),  then  C^-map  g£(x) = 0  dg and 7J-~(X) > -1. dx  gj^(a) = 0 - a,  6(x,a,B) =  g: R x T - R  such that  i f x i (0,1),  See F i g u r e 1.  FIGURE 1  To c o n s t r u c t such a map we use the f o l l o w i n g l e t e: R x T  b u i l d i n g block:  «(x,a,p) = i [ J  x _< a,  and  e  X  R  F * * ™ i t , if  as a  be d e f i n e d by  x e (ct,B), ( x , a , p ) = 0 c  a  ( x , a , P) = 1 i f  C*-map  x > B,  where  i f  8, 1  ~a^6  c=  +  t^j5 dt.  See Figure 2.  a + 3  a-» x FIGURE 2 To define  g~(x) f o r x > a, e«  the map  We define s  6: RxT- R by  we need a modified version of  fi(x,a,0)  = <  (x,a a  i f x < a + ^  3  1,  ,  i f a + izP<.x < 1 - 1-P  l-e(x,l- ^ , 1 ) , i f x > 1 -  i^,  See Figure 3<  <T1  FIGURE 3 We define C = f 0 J  R x T - R  6(t,a,P)dt  > B-a.  by  g(x,a,0)  Now define  1 C  X  6(t,a,p)dt,  where  (B-a) • ( x , 0 , a ) ,  x _< a  e  g(x,a,B) = (B-a)-[l- (x,a,p)], e  g  isa  C" map,  and, i f x _> a,  1  §a  d  x > a.  ^ ( x )  = ( B - a ) . [- -±6(x,a,B)] > - 1 .  Q.E.D.  The f o l l o w i n g lemma i s C o r o l l a r y 4 . 3 . o f [ 1 1 ] , p. 129Lemma 1 . 2 .  L e t f-j_, • • •  functions of n  he  n  real variables.  c o n d i t i o n s t h a t t h e mapping  f ( x ) = ( f (x),.. , , f (x))  real-valued  differentiable  N e c e s s a r y and s u f f i c i e n t  f:R  n  -> R  n  defined by  be a d i f f e o m o r p h i s m o f 0R  onto  n  i t s e l f are: * i det(——) f  (1) (2)  never vanishes  l i m ||f(x)|| = oo.  Ilxll—  Theorem 1 . 1 . where  A™"  Let A CR" " 1  1  M  c R  c R , m  1  c R  m  be an  n  b ^  = 0,  m-simplex,  A = v*A M  and v = ( 0 , . . . , 0 , 1 )  m - 1  e R  m  A  l i e s on t h e x - a x i s . m projection.  L e t p: R  Let A c A  c  ra  A u v * 9A that  F n A  m  c U,  m _ 1  be t h e o r t h o g o n a l • ° •  be an open s e t i n R  M  m - 1  - R  be a c l o s e d s u b s e t such t h a t  A = P ( p ( A ) ) fl A . L e t U _ 1  m  and l e t F  c A u v * aA™" . 1  n  such t h a t  be a c l o s e d s u b s e t o f R  n  Then t h e r e i s a compact s e t  such  ,  10.  C c R -F  and a  n  h (x)  = x  t  h : R  C^-isotopy  i f x i C,  - R  n  t  A  and  such t h a t  n  h  Q  =id R  h-^U).  M c  Proof: (1)  c = ^rdist(R -U, A U v *  Let T$ N  1  2  n  = {x  G  A  M  _  1  :  d i s t ( x , A \j v *  = (x  G  A  M  -  1  :  dist(x, A u v  and N  .p  _  1  ^A*  ( A  x e p  M  _  1  -1/  - N  2  n  )  A  M  c  m-1 .. \ _ m (A -N ) n A , 2  dist(p(x),A u v * 3A  m _ 1  C = 0  so we may l e t  )  A  m  _  m - 1  N  (2)  Let  g: A  M  _  1  G  ^ 0.  -• R  v * 3A to  A g.  g(x) -  m _ 1  .  Let  x  >  Then  2c}.  Further,  2  u,  c  A  If N  2  v  = 0,  h, = i d '. R  and i f * 3A  m-l\ . ) < A " C U,  then  1  From now on, we  n  d = idist(N- ,F) L  > 0.  f u n c t i o n d e f i n e d by  s ( x ) i s the i n t e r s e c t i o n o f  p a r a l l e l to the x - a x i s w i t h m  g: A  I f x G N-p > g(x) -  where  )  o-^./  he t h e c o n t i n u o u s  g(x) = ||s(x) - x j j , the l i n e t h r o u g h  Let  > c } , and  1  -N  z  assume t h a t  1  Let  dist(x,A u  < c.  and  1  2  then  T  0.  >  1  SA" " )  *oA  since  U,  " )  W c N^.  a r e compact, and  2  1 1  M  _  1  then  -• R  be a  C*-^  A, g ( x ) _> c  - 75 .> If"*  and hence  F o r each  a "vertical stretching interval".  -approximation  Let  x G ^  we d e f i n e  11. v ( x ) = x + (g(x) - -£)-v, x  v ( x ) = x + (g(x) - | ) . v 2  v^(x) = x,  and  v^(x) = x - d«v. [v-^(x),Vg(x) ] c U.  Then be  [v-^(x),v^(x)]  onto  and by "stretching" we w i l l map  [v (x),v-^(x) ].  The i n t e r v a l  1  Y ( X ) = g ( x ) - -- + d > 0. x  mapped onto  0  mapped onto  a(x) = 4 . ^  that (3)  and  v^(x)  -• I  n  Let v-j_(x) = 1  0  and  3  2  has length  h^.  v (x)  is  1  Then  Vg(x)  B(x) = (y^~^«  onto  Y  is  Note  4.  See Figure  Before constructing  such that  i s mapped onto 1.  x  a(x) < B(x).  W : R  [0,1]  l i n e a r l y onto  1  [v-^x), v^(x) ]  [v^(x) v (x)]  To apply Lemma 1.1, we map the i n t e r v a l  0  [v (x),v^(x)]  The stretching i n t e r v a l w i l l  we must construct a  0°°-function  with proper compact support.  v, :  be a  if x e N  Cl( v £ ( ( 0 , l ] ) ) c Int N  x  which i s open i n  .  1  m-1  such that  and  2  R  C -function  = {xeA  m-1  : dist(x,AUv*aA )>c} m_1  Consider next the compact set C  = {x = ( x ^ . . . , x ) e R : m  Q  Then  m  C  Q  n F = 0.  Let  p(x)  €  r\ = dist(C  N  2  and -d < x ' p) > o .  m  < g ( x ) - -J}.  Let  12.  FIGURE 4  13.  bea  v : R -R 2  if  t < n,  v (0) = 1  let  and v ( t ) = 0  o  n Let  2  i f t > n.  rn  7r: R  r = poir.  0 < v ( t ) < 1,  c " - f u n c t i o n such t h a t  -» R  be t h e o r t h o g o n a l p r o j e c t i o n and  We d e f i n e  l i ( x ) = v ( r ( x ) ) . v ( 2 | | x - 7r(x)||)  forx eR . n  2  1  l^-(x) = 0 . m •  Notice that  dX  C = {xeR : Tr(x)eC  Let  and  n  0  Hx-7r(x) || < -3}. Then C  i s compact and C 0. F = (2f. We n o t e t h a t  c i ( ^ ( ( o , i ] ) ) n 7r (c ) c C . 1  _:L  o  We d e f i n e  h :R m  r ( x ) e N-^  n  -R  as f o l l o w s .  l e t h^(x) = x  m  Let x e R . I f n  + t«u(x)•[{stretching  C^-diffeomorphism w i t h respect t o [ v ( x ) , v ^ ( x ) ] a p p l i e d 1  to  x } - x ] = ( l - t - u ( x ) )'.x + t - u ( x ) x m  m  m  t-v(r(x)). ^(^](9  If  r ( x ) £ ]$  7r?  ^ [x -( (r( ))-|)]) (x)-|]. TT  m  S  l e t h£(x) = x .  l9  m  X  +g  We n o t e t h a t  h£ i s a  C°°-map. F i n a l l y , l e t h^: R  n  -R  n  be d e f i n e d b y  h ( x ) = ( x , •••, _ ,h£(x),x x  t  We compute  x  m  1  m+1  , ... x ) 5  n  forx  R  3  €  14.  (x) - ( 1 - f ia(*)) +  t.u(x)^(^)j(-^  r T  [x  T h e r e f o r e t h e rank o f the J a c o b i a n o f Obviously,  h^  is  n.  By Lemma 1 . 2 ,  l i m ||h. ( x ) || = 00.  C^-diffeomorphism.  -(g(r(x)).|)])  r a  By c o n s t r u c t i o n ,  h  h,  = id  Q  is a  ,  R  i f x fi C,  h (x) = x t  C o r o l l a r y 1.1.  Let  m-simplex, l e t T A  by p: T  M  _  1  - T  m  ,  let  U  "  M  =  M  P  n  = id  R  c  h^U).  M  -  c  1  R  be an a r b i t r a r y  N  R  be the h y p e r p l a n e i n  1  M  A ,  R  be a c l o s e d s e t i n m - 1  R  M  - 1  Aljv*dA  such t h a t  N  . Then t h e r e i s a compact s e t  C -isotopy ro  , h,(x) = x  h^: R  i f x ^ C.  N  - R  and  let  p(v) = b  such t h a t  N  M  A = p (p(A))HA ,  be a c l o s e d s e t such t h a t  and a  spanned  N  the h y p e r p l a n e spanned by  1  c A U v*5A  C c R -F  M  v * A  be an open s u b s e t o f  and l e t  h  1 1 1  A  A  be the p r o j e c t i o n such t h a t  m - 1  A c A  P D A  if  and  let  and  such t h a t  N  A  M  c h-.(U). .  -  m _ 1  cU,  15.  CHAPTER TWO The If  and l e t  C°-Engulfing Theorem  a > 0,  l e t c£ = { (  I n t C^ = [ ( x . ^ . . . , x ) e R : n  D e f i n i t i o n 2.1. i s s a i d t o be  If M  i sa  k-dominated  (1)  X• c u c p ( l n t C j )  (2)  F o r each  \x \  < a},  ±  n  < a}.  ±  C°°-n-manifold, a s e t X cr M  i f t h e r e i s a system  n -» M cp^: c^  C -diffeomorphisms  \x \  ,...,x ) e R : n  X l  {cp_^} o f  such t h a t  i  i , cp~ ( cp ( C^)  n X) c P ,  1  ±  where  ±  P  ±  is a  k - d i m e n s i o n a l s u b p o l y h e d r o n o f C^. {cp3  The s e t X,  and each  i scalled a  i  cp^ i s c a l l e d  D e f i n i t i o n 2.2.  If M  k - d o m i n a t i n g system f o r  k - d o m i n a t i n g c o o r d i n a t e map f o r  isa  C°°-n-manifold and K  f i n i t e s i m p l i c i a l complex, a map  f : J K | -» M  l o c a l l y l i n e a r i z a b l e i f t h e r e i s a system omorphisms  I J K : C^ -* M  f ( |K|) c u t ^ I n t C j )  (2)  F o r each  o\(K),  f  {IJK} o f  CT^(K)  i , there i s a subdivision  - 1  ( ^(cj))  and  f  ±  i s s a i d t o be C^-diffe-  such t h a t  (1)  that  isa  = 1%!,' .f||  H >  |: R  where  ±  H  such  i s , a subcomplex o f  ±  - c£ c R  of K  n  i s linear.  16. The s e t •  i s c a l l e d a l i n e a r i z i n g system f o r  Note t h a t i f f : | K | - M  i s l i n e a r i z a b l e , then  f.  is  f(|K|)  k-dominated. D e f i n i t i o n 2.3«  If  S <z Y  t o p o l o g i c a l space, are  K i s a s i m p l i c i a l complex,  maps, we say t h a t  subdivision  CJ(K)  of  N(f (S),o-(K)) = N(g - 1  Ip  K  _ 1  agree on  is a  f , g: | K | -» Y  i s a s u b s e t , and  f and g  Y  S  i f there i s a  such t h a t  (S),  = N,  cr(K))  and  f|  N  =  g|  r  i s w e l l known t h a t a t o p o l o g i c a l m a n i f o l d i s an  a b s o l u t e neighborhood r e t r a c t , see, f o r i n s t a n c e , [ 4 ] ,  0 ° ° - e n g u l f i n g theorem, we s h a l l need t h e f o l l o w -  I n p r o v i n g the ing  r e s u l t f r o m homotopy t h e o r y ?  Lemma 2 . 1 .  Let  Y  be a m e t r i z a b l e a b s o l u t e neighborhood  retract with metric  d,  e > 0.  and l e t  such t h a t f o r e v e r y c l o s e d s u b s e t for  a l l maps  f-^f^  A -» Y  2  an e x t e n s i o n rA ^ : X -» Y, such t h a t  1  , f g )  <  i f  6,  and has  f  1  A  has an e x t e n s i o n  f^: X - Y  2  Let  an open s u b s e t o f a c l o s e d and  compact,  f^  1  X  d ( ^ , ^ ) < e.  Theorem 2 . 1 .  X c M  then  d ( f  6 > 0  Then t h e r e i s  A o f a m e t r i c space  with  T h i s i s Theorem V . 3 . 1  Proof:  V  98.  p.  k <: n - 3 .  M  be a M  o f [ l ] , p.  C ^ - n - m a n i f o l d w i t h o u t boundary,  such t h a t  k-dominated Let  K  103.  (M,V)  is  k-connected,  s u b s e t such t h a t  be a f i n i t e s i m p l i c i a l  X-V  is  k-complex,  17.  f:  | K | -» M  L c K  continuous,  a subcomplex  such t h a t  i s a l o c a l l y l i n e a r i z a b l e imbedding w i t h l i n e a r i z i n g 2 = {iL-3 such t h a t each J map f o r X.  e > 0.  Let  A. J  and a  (1)  h  =  (2)  g|, | =  (3)  d ( f , g) < e  =  i d  M  ,  h (x) t  x  C^-isotopy i f  x  (M,V)  Proof:  h-^V)  d  V  on  X  U  g ( | K | ) .  i s compact and  k _< n - 3 ,  X-V  and a i f x  Let K = 0  C^-isotopy C,  j.  M. isa  i s an open s u b s e t o f X c M  t  and  i s c l o s e d and then there i s a  h^: M -» M  such t h a t  h ^ V ) => X.  We f o l l o w Newman's p r o o f o f t h e  t o p o l o g i c a l e n g u l f i n g theorem,  [10],  allowing f o r differenti-  a b i l i t y and u s i n g t h e u s u a l method o f s i m p l i c i a l  collapsing,  instead o f c o l l a p s i n g through p r i n c i p a l s i m p l i c e s .  We d i v i d e  the p r o o f i n t o t h r e e s t e p s . F o r each x  M  i n Theorem 2 . 1 .  P r o o f o f Theorem 2 . 1 .  u : C^ - M  a  such t h a t : ^  If M  i s k-connected,  = id , h (x)= x M  and  (C™-Engulfing Theorem)  0  compact s e t C cz M  Q  h^: M -» M  f o r some f i x e d m e t r i c  C°°-n-manifold w i t h o u t boundary,  h  g: | K | -» M,  L  Corollary 2.1  k-dominated,  C,  system  f|, ,  L  such t h a t  £  L  k-dominating coordinate  Then t h e r e i s a map  compact s e t C c M,  Q  i s also a  f|j j  x e M,  such t h a t  x e n  we choose a x  ( M  c£),  C°°-coordinate map and  1 8 .  ( 1 )  i f x e f ( |L|),  (2)  i f x € X - f ( | L | ) , (j. for. X  e S  \I  X  isa  k-dominating  M ( c J ) fl f ( | L | ) = 0  such t h a t  x  then  u (cj)  Step I r R e d u c t i o n t o t h e case  X c V  (3)  i f x /•/ X U f (  Let X-V c u (a)  A(m)  x  fl (X U f ( | L |)) = 0.  denote t h e theorem w i t h t h e added h y p o t h e s i s r  ( I n t c £ ) U ... U ^  x  ( I n t c£)  f o r some  L e t u. = u  m > 2.  Let X  ffi  1 < i < m.  }  = X-u (lnt c£). m  e M.  m  Then  u ^^(Int  A(m-l)  Thus t h e r e i s a map  1  m  are s a t i s f i e d .  compact s e t ' C  m  c M,  , h£(x)  h™ « i d  ( )  Sml|L|  (3)  d C ^ f ) < e/2  88  M  f  s  co  ( 1 )  and a = x  i f  c M,  h  e  hypotheses o f  g r | K | _> M m  C  ffi5  h^r M -* M hJ(V)  and  a  5  such t h a t = X  U g ( |K| )  m  m  f  = g ,V m  = h*(V).  Then  A ( l ) may be a p p l i e d r t h e r e i s . a map C  t  rn  C'-isotopy x i  o  l|L|  Now l e t  set  5  We use i n d u c t i o n on m  X - V c u - ^ I n t C^ ) U . . .  so  x ^ ... x  A ( l ) i m p l i e s A(m).  Proof:  2  c o o r d i n a t e map  and a  C -isotopy ra  h i : M -» M  -  X-V c u ( Int c £ ) , l  m  gr | K | -» M  5  such t h a t :  a compact  • (1)  = i d , h£(x) = x  i f x i  M  19.  C«, and  h'(V') 3 X U g ( |K|). &  '|L|*  g  d(g,f)  (3)  l|L|  f ,  < e/2.  Let  (1)  h  ( ) 2  S  C = C  U C ,  = i d , h ( x ) = x i f xdC,  Q  M  I|L|=  f , i  |L|  =  g  ml|L|  d(g,f)"< d (  (h)  A ( 0 ) implies A ( l ) .  Proof:  g j  i sa  £ f(|L|),  1  m  where  A  _  l  o f | ^  1  e  a  y  a^(K) o f K  l (  (L)  L  ^  :  a-j_(L) -* CgCC^)  3 1 1 ( 1  Og(K) U H  :  Ag e H  u  ^  s a  s  such t h a t i  m  P l i i l  a r e such t h a t  c  a  C^. I f  Then t h e r e i s  and a s u b d i v i s i o n  crg(C^)  °^  |H| = P, and  imbedding.  f(A-^) = u^Ag),  I f  identify  be t h e s i m p l i c i a l complex o b t a i n e d  by t h i s i d e n t i f i c a t i o n .  be t h e p r o j e c t i o n .  Since  (-f^X fl u ( C ^ ) ) c P,  |K| U P - M.  H c o-g(C^)  A-^ and Ag, and l e t K* from  (Int C?).  k-dimensional subpolyhedron o f  w i t h a subcomplex u  X-V a u  1  let f = f U |  a subdivision  g( | K | ) .  r a  1  P  u  '|L|-  k - d o m i n a t i n g c o o r d i n a t e map f o r X,  where x  f  Then  g ) + d ( g , f ) < | + | = e.  Let n = u  ——————  =  = h^oh™.  t  and h ^ V ) => X  t  (3)  is a  and l e t h  I f x^ i f ( | L | ) ,  L e t p: o-^KjlIEHK*  letH  be a s i m p l i c i a l  20, complex i n C  such t h a t  1  |H| = P,  and l e t  Let  p: K U H _ K * he t h e i d e n t i t y , and l e t  Let  f * : |K*I  L* = p (  c f 1  f*|  — M  be d e f i n e d by  X* = X - u ( I n t c£).  ( L ) U H),  i s a locally linearizable  A(0):  There i s a map  and a  C^-isotopy  (1)  h  Q  g*: J K * |  h, : M - » M  M  t  p  and l e t X* c V,  - »M , such  and  so we may a p p l y  a compact s e t C c M , that: and  r> X* u g*( |K*|).  x  (2)  g*|| j  (3)  d ( f * , g * ) < e.  L #  Let  =  f*|j  g=  L #  j.  g*op|j |. K  Then:  h _(V) z, X* U g*( |K*|) = ]  X - u ( l n t C*) U. g( | K | ) (2)  Then  f = f U |i| .  imbedding,  i f x £ C,  = i d , h (x)= x  h (V)  (1)  f = f*«p,  K* = K U H.  U u ( P ) O X U g( | K | ) .  g| | | = f | | | : i f x e |L|, g ( x ) = g*.p(x) = f * . p ( x ) L  L  = f(x) = f ( x ) . (3)  d ( g , f ) < s.  Step I I : R e d u c t i o n t o - t h e case f o r some Let hypotheses:  X c V,  and  f ~ ( V ) c I n t A*', 1  A^ e K  B ( i ) denote t h e theorem w i t h t h e added X c V  and  dim N(|K|  - f  _ 1  (V),K)  < I,  i.e.i f  21. A e K  f ( A ) n (M-V) £ 0  and  Let f ( | K ^  /  _  1  B(£,m)  c V,  ^ | )  e K  that (a)  A £ , . . . , A^  3  Proof:  B ( ^ ) w i t h t h e added m  i f f( L)  O.X ^ 0,  l  then  must be p r i n c i p a l i n  implies  B(i m)  We use i n d u c t i o n on  m.  B(i,m)  B(l l) 3  an (i)  g: | K | -» M  e-approximation  L e t K'  L» =  =K-{A^},  f  k A' m  and  t h e r e a t most  such t h a t B(-t,m-l)  f'(A^)  that:  3  f  <zf V, !_<!_< m - l ,  c M,  • .  (|(K  (m-l) -t-simplices  are s a t i s f i e d .  a compact s e t C  < l  <  A  m T  r  A  d(f| ^g)  m  1  I ;  Now  d  )( ^ H  |) c V  ^ e  A£, . . . , A ^  3  K  1  so t h e hypotheses o f  Thus t h e r e i s a map and a  i s an  t h e n t h e r e i s an e x t e n s i o n  such t h a t  dim N( |K« I - ( f « ) " ( V ) , K )  that  By Lemma 2.1, ,  K  ,  gt  m _> 2.  g( A ^ ) cr V.  then  3  !  t  of  B(£,m-l)  i f A^ s .K i s  f,  f ''=f|| i|.  L-£A£],  and  i s so s m a l l  J  €'-approximation t o g: L\t -* M  e  such t h a t i f g : S A ^ - M  t h e r e i s e' > 0  A  to  g( A ^ ) fl X ^ 0  {.-simplex such t h a t  Note  h o l d , f o r some  W i t h o u t l o s s o f g e n e r a l i t y , we assume t h a t f o r any  f ( A*) c V.  m.  Suppose  a r e t r u e , and t h e hypotheses o f  and, f o r each  K.  for a l l  3  hypothesis:  £-simplices  such t h a t f ( A ^ ) $zf V, 1 < i _< m,  l  B(i 1)  denote  and t h e r e a r e a t most  t\ e K ,  {.-simplex  dim A cr  then  3  C^-isotopy  g : | K ' | -» M, 1  h£: M -* M - such  22. (1)  bJ = i d , h£(x) = x M  hj_(V)  S'I|L'|  (3)  d(f',g«) < e«.  =  Let if  A^ m  !L«|"  f , |  f : [Kj -» M f | -t = f | n A  L, >  G  A  m  be d e f i n e d as f o l l o w s : i f  m ^  Affi  approximation  to  ( i i ) L e t V-•=  h£(V).  Now  dim  5  N( |K|  - f (V),K) < _ 1  f ( A^) <£ V .  a r e s a t i s f i e d , so t h e r e i s a map and a  h  i f x £ C,  = i d , h (x)= x M  t  i, •  t-s implex,  A^  C -isotopy  g: |KJ h^: M -* M  M,  g|, , = f |  (3)  d ( f , g ) < |.  L  and  h  Q  = i d , h (x)= x M  x  | L |  t  t  = h oh£.• Then t  i f x £ C,  and  3 X U g(|K|).  h (V) W  g  (3)  d ( f , g ) < e.  |  .  C = C» u C, h  Let (1)  | L |  =  f  l|L|*  }  ''  a  such t h a t :  h-^V) = X U g( |K|).. (2)  _  Thus t h e hypotheses o f  compact s e t C c M, Q  4  g'l^-t.  in  B(|,,l)  1  m  and t h e r e i s o n l y one  such t h a t  = g ;  1  f ( | K ^ ^ | ) C V, K  fjj^jj  ^ L. • l e t f | .1 be an . A 2  f | A ^ w h i c h extends  _ 1  (1)  and  X U g t ( | K « I).  3  ( ) 2  i f x e C,  .  23«  (b)  and B({,,m),  B(i-l)  Proof:  Suppose  a subdivision of  K  B(l)  a(L)  |K^| U | L |  M  N  q  -|-approximation  fljj^J.  =  > 0  Let  e  ft  Then  K  g: | K | -• M  to  C  c M,  and a  C^-isotopy  h^ = i d , h£(x) = x  (1)  M  - (f )"V),^)  c  a  The hypotheses o f  h£: M -» M J  !  be extended t o an  n  g": I K ^ I -• M,  i f x fi C ,  ,  e'-approximation |  L  f.  are s a t i s f i e d , so' t h e r e i s a map  _ l )  dimNCJK^I  ±*I ] i | y j  be  C V. L e t  (  be such t h a t any to  o(K)  f(  U (a(K)) ^  Q  are true,  Let  then  1  = K  m,  A*' e a(K) i s an  f ( A ') fl X ^ 0,  !  < t-1. g':  n^,  for a l l  are s a t i s f i e d .  = [A e o-(K): f ( A) c V ] , =  imply B ( { ) .  so f i n e t h a t I f  {-simplex such t h a t Q  m,  and. B( {,m),  B(i-l)  and t h e hypotheses o f  K  for all  B(i-l)  a compact s e t  such t h a t : and  hj_(V) 3 X lj g«( | K ^ j ) .  (2)  g«| ,,  (3)  d(f«,g") < e ' . •  - f ' l |  | L  Let  g* | , | = g" I  K  extension of then  L  , , .  g»: | K ^ | U | L | - M s  3  g  '1|L| !  =  f  '|L| '  such t h a t  f ( A ) n X = 0.  j  3 2 1 ( 1  be d e f i n e d by l  e  t  ^  :  d ( f , f ) < -|,  L e t V = h£(V).  ' ' ~* K  M  and I f  h  e  a x x  A s a(K)-K^,  24. F o r some  m,  t h e hypotheses o f B(t,m) a r e  s a t i s f i e d , so t h e r e i s a map C c M, (1)  h  and a  C -isotopy  = i d , \(x)  Q  M  S (v) (2)  g|  (3)  d(g,f) < f .  | L  ,= f |  Let h  Q  = x  such t h a t  i f x £ C, and  | L |  .  C = C U C ,  h  J  = i d , h (x) = x M  h (V)  (2) g | , = f | | L  | L |  t  = h «h^o  Then  t  i f x £ C, and  t  o X U g(|K|).  x  (3)  h^: M -* M  a compact s e t  3 x u g(|K|).,  1  (1)  g: |K| -* M,  '  .  d ( g , f ) < d ( g , f ) + d ( f , f ) < | + | = e.  Step I I I : P r o o f o f  B(t,l)  I n v i e w o f Step implies  B(t, l ) ,  t h e theorem.  s i n c e t h i s proves  B(t-l)  B ( k ) , and hence  Thus we may assume:  (1)  X  (2)  t h e r e i s an  c V.  i < k,  I I , we need o n l y show t h a t  • t-simplex  A^ e K  such t h a t  |K|-f (V) c Int( k ) . - 1  l  (3)  f ( A*) n  X  (.'4)  B(-t-l)  i s true.  Let  =  0.  G = K U v*^,  where  ( t + 1 ) - s i m p l e x n o t i n K. S i n c e  v*A' = t  (M,V) i s  i s an k-connected, there  25< A  i s a map  A  f : |GI -* M c V.  f(v*b& ) 1  .  c  f(A^) 1  = f,  t  A  X U f ( |H|)  There a r e p o i n t s  f|  H = (K-{A^}) U v^"3A' .  Let  G = H U v*A%  such t h a t  c V,  LJ  (Int  Then 0.  X n f( |L n v*A^|) =  and  x.p...,x^ e M  A  and  such t h a t  ... U W^( I n t C^).  Let  u  =  ±  u _, x  < i < N.  Case A Suppose  f(A  ) c u  ( I n t C, ).  There i s a number f ( A (1)  W  )  a  polyhedron of  where  5  C^.  |R| = P,  If  A  1  (^IJLJ)  |j | )  f  - 1  L  e a (L) Q  identify  and  :  o b t a i n e d from  Ag  L* = p ( a ( L ) U R ) , f*«p = f u u | . p  i s  C^  Q  1  L  R  R  a r e such t h a t G*  of  G  such t h a t  subcomplex o f  a  L  and l e t  a (G)  i s  o (&)  a n d  Q  simplicial.  f ( A-^) = u ( A g ) , _  be the s i m p l i c i a l  by t h i s i d e n t i f i c a t i o n *  complex Let  be the p r o j e c t i o n , l e t  Q  Q  lying i n  f  a (G) U R  p: cr (G) U R -i C-*  k - d i m e n s i o n a l sub-  ( I | | )" (l- ( )) -  Ag € R  A-^ and  R  """(^(R))  (^R))  3  P. i s a  There i s a s u b d i v i s i o n  and a s i m p l i c i a l complex  1  . l  x  0 < a < 1  such t h a t  X fl l i ( C ^ ) c u ( P )  l  u = u  eg).  c u(lnt  t- ~ ° l(f  Let  x  1  and l e t  (If  f * : |G*|  f(la (L)|) Q  - M  n u(P) = 0  be d e f i n e d then  by  26  S  G* = a ( G ) U R ) . Q  By Theorem 4 o f { 1 0 ] , t h e r e i s a map  (ii)  f * * : JG*| -> M  such  that: (1)  f * * and f *  (2)  n  (3)  f * * l |  (4)  f**op(|H|) c V,  _  1  |G*| -»  o f * * :  * |  L  agree on M - u ( l n t c " ) .  =  Let  f * l ,  f« =  agrees w i t h a * | .  L  and d ( f * f * * ) <-|f * * o  P  | |  G  f  (2)  li'^ef  Then:  N  and f  agree on M-ia(lnt C ^ ) .  agrees w i t h a  1  - M.  |G|  | :  A  (1)  P L - o p t i m a l map i n C^.  P L - o p t i m a l map i n  which i s  " i n general p o s i t i o n " with respect t o P 0 C . (5) (4) (iii)  f'l, | - f|| ,  - f| and d(f«,f)<-§.  L  L  f»(|H|)cV, L e t ]_(^)  ^e  a  a  .  s u b d i v i s i o n o f o- (G)  such t h a t  Q  ^•^(f.-^O), -1 ^. I  | L )  ,  *,(<*))  . , ,„.-1, N  1  ( (^ S))^ T  f,_1  c  CT  I( )) G  -l c  i s o p t i m a l and " i n g e n e r a l p o s i t i o n " w i t h r e s p e c t t o P fl c £  I n I n t C^,  and such t h a t  t h e r e i s a sequence such t h a t  E* 1  1  1,0  =  {E^ } l , i i=0 + 1  0, E  = l,s  (A )-, t+1  CTl  .  L  o f (t + l ) - c o m p l e x e s  s  l + 1  c- (G) ^ ^ ( H ) :  n E^" " = E ^ U A, , l,i+l l , i 1  1  + 1  X  27.  n A  =  ±  <  1  x *A ±  <  1  (iv)  s  -  n.-l -. and (a-^H) lj E"^_) n 1, i  n  x  A  3  i  n.-l  x **&  =  1 ±  ±  1 ±  3  »  1  Induction Hypothesis: C. cz M,  compact s e t  There i s a map and a  C°°-isotopy  g^:  |G| -  a  h. , : M -» M  such  that h  i,0  h  <  2 )  g  ± a l  W  =  ld  i,t^ ^  h  x  =  x  i  f  ^ i>  x  C  a  n  d  (E^V !). 0  ( V ) o X U g^laxCH) U  i | a ( L ) U N(f'- (n(C^)),a (G))| = l  1  1  1  =  a (G))|.  ' ia (L) U NCf-^^C^)),  f  l  x  1  (3)  d(g.,f) <  ICl-2" ). 1  i = 0.  This i s c l e a r l y true i f (v)  Induction Step: Thus u  '  H "  ^  1  ^  A  .  ( f « ( A .  1  1  ^ ' ( A ^  n  i  )  1  n (  0  l  1  U (o ( ^  (H)  Consider  + 1  1  1  ±  n  ' ^ f U A / )  a s u b p o l y h e d r o n o f u~ dim Q _< (t+1)  f o r i _< s-1,  n ( X u f ' ( | a ( H ) U ( a-, (  n u(P)) U u  )  < l+ 1  We have  <>f  l  !  (A  f«(  1  ) )  |))) c  ^  |a (H)lj(a (  ))(^)) U  n.. )  A ^  1  | ))  1  Q  ±5  where .  Q  ±  is  such t h a t  x  i  + k-n < t-2. now  n. A  , = u~ °  f  !  (  A  i  n. n.-l ), A =  , ^  i  n  o f  !  ( A  -  ±  1  ) ,  28. n.-l and  v = u°f (x ),  let  !  i  determined by  n.-l A^ " ,  determined by  n.. A "%  T  be the p l a n e i n  1  n. let  L  T  and l e t  be the p l a n e i n  1  7r: T  n.  i ~ -* T  n  1  R  n  . R  x  be the p r o j e c t i o n  n. with  TT(V) = b  n  1  A n. A  of P  1  .  Let  C' n (Q )),  fl-/r  1  ,  a subpolyhedron  R  I  1 _  o f dimension _< {-1.  = (fM'^^A.)) = g  ±  = A  _ 1 1  Let  (u(A )),  D  1  =  ±  |  CTl  (H) U ( E  t  +  1  )^ !  U  }  P , ±  ~s i  and l e t  D.  = D.  a  fl ( f ' ) ~\ u( C*J)).  subset o f  |G|.  ( v i ) We  show t h a t there  now  a compact s e t  Then '  is a  i s a continuous map  C* c M  and a  = x  x i C*,  5  polyhedral  §j_ 2_  :  +  0°°- i s o t o p y  IG-1  h|: M - M  -* M  s  such  that (1)  h* = i d , h * ( x ) M  h^h.^V)) o X U ^  S  i+l |L!u  (3)  d(g  Proof:  CT2  (D ) i  i 3  g  i + 1  g  1 + 1  ±  1  t  f , 1  1  1  <- | . I  2  There i s a s u b d i v i s i o n and  a  2^±^  =  1  |N(f«- (^cS)),a (0))|-  |L|U  )  and  (D ).  |N(f- ( i(C^)),a (G))|  !  : =  if  a  r  e  cr (G) 2  subcomplexes  of  of  ]_( ) G  CT  rj (G), 2  such t h a t and  29. f i a  ( D ^  2  )  a  -  complex i n C^. and  i s a s i m p l i c i a l map onto a s i m p l i c i a l  Identify  f'(A ) = f'(A ), 2  and l e t K*  2  P(a (D ))-  o b t a i n e d from  2  p: p ( a ( D ) ) - K* 2  i f A-^, A  2  by t h i s i d e n t i f i c a t i o n .  i  f*»p = g^, 2  2  i  K*, L*, f * , X and'V* < l-l.  ( ) 2  (3)  a r e s a t i s f i e d by ±  C^-isotopy  i f x £ C*,  g*: |K*| -» M, h£: M -• M  a  such t h a t :  and  X U g*(|K*|).  3  =  f  * |L*|' !  d(g*,f*) < Let  and  B(i-l)  _ 1  and a  M  «*«|L*|  L e t V* = h.  N(|K*| - ( f * ) ( V * ) , K * ) c P ,  since  h£ = i d , h * ( x ) = x h*(V*)  be  isa  L  Thus t h e r e i s a map  compact s e t C* c M, (1)  Let  f*l| *|  Then  2  The hypotheses o f  i  i  and l e t  l o c a l l y l i n e a r i z a b l e imbedding.  dim P  a  2  be t h e s i m p l i c i a l complex  L* = p(B(o- (L) n a ( D ) U a (D°))).  and  e P(a (D ))  g  be t h e p r o j e c t i o n , l e t f * : |K*| - M  i  d e f i n e d by  A-^ and A  g  i + 1  g : IGI  —  i+1  | t + 1 = g |^Vrl A  M  be d e f i n e d by  = f•| t+1.  ±  1  i+1  Then  A  S l + l ' l L l u " |N(f'- (y(cS)),a (G)|  g  =  1  = 'l|L| U |N(f (u(C^)),a (G))h f  1  1  i  ^  l| .j D  = §*°P>  30. X  U  S  i+l  (vii)  (  D  i  )  =  X  6*(I *D K  U  c  h  j( i,l( ))h  V  L e t U = n" (h|oh . ( V ) n u( I n t C * ) ) , and l e t 1  j  X  P = A. U M ( X u f ' ( | a ( H ) U (E )^ ^ l , i l+1  _ 1  \)).  1  0  Then  d  n. P 0 A = A x  ±  U v*3A  n,-l and  1  A  we may a p p l y C o r o l l a r y 1.1. C c R  - P  n  h^ = i d , R' C  he d e f i n e d by ld  M.:-n(cg)-  (1)  h  i + 1  ^  h  i + l j l  L  e  h |^ ^ t  h  = id ,h  0  M  (V)  1 + 1  U g  oh  ^ (x) = x t  n  - R  h£(U) ^ A  T  h j ^ c g ) *  h  e  n  i f x £ C  i + 1  ,  ( | a ( D . ) | U A." ) ^ 1  i + 1  2  X U Si+ida^H) U ( E ^ )d)|). l,i+l+ 1  <> 2  «i+il| |U L  ^ (3)  d(g  i + 1  |N(ft-l  f , |  |L|u  ,f) < d(g  ( u ( c  n  ) h  ffi(G))|  |N(f^(^Cg)), i + 1  ,  g i  =  a^G))]'  ) + d ( g . , f ) <-tp2  + |(1 - 2" ) = |(1 - 2 - ( 1  such t h a t  n  t  i + i , t = t°( ? i,t)h  so  n. i  and l e t h : M - M  and  M  h  1  and  C*,  = .h£  t  h£: R  i f x/C,  = u(C^) U C . U  i + 1  n,-l c U,  There i s a compact s e t  c"-isotopy  h£(x) = x  n  Let  and a  \J v * 9 i  ±  i + 1  )).  and  51. Thus we (viii)  (1)  & (3)  g = g I| |: |K| - M,  Let  h  s  = id ,  Q  have completed the i n d u c t i o n s t e p ,  h (x)  M  l( )  " s,l( )  g i  |L|  "  V  = x  t  h  h  y  g  sl|L|  =  h  K  => ' X  if  1  U  t  ,.  C = C.  Then  g  and  (E^)  I)  U )  U g(|K|)  D X  = . 'l|L|-  '|L|  f ,  = h  x £ C,  MM *)  U  t  f  2 )  + |  ,1+1 £( b ) c u-^Int c£)  U...U  d(g,f) < d ( g , f ) + d ( f ' , f ) < |(1  -  _s  <  e.  Thus Case A i s proved. Case B F o r some (i)  N,  A  l  There i s a number f(b Let  l + 1  )  a (G)  a  A € a (A*' ),  f ( A) C ^ - ( ) ( I n t C^),  and  A  {E  } l,i  l + 1  1,0  L  ±  = 0,  1  of  + 1  1 < i <  i  n.-l , 1  i  s-1.  1  cjj).  such t h a t f o r each j ( A)  ^  C T l  such t h a t  (H):  there i s a  (t+1)-complexes such t h a t  1  ±  = x *A  a (G)  and  i=0  E^ = o^A^ ), l,s  n.  G  there i s an i n t e g e r  + 1  1  E  u ... U M j j C l n t  c u-^Int c£)  be a s u b d i v i s i o n of  1  sequence  0 < a < 1  such t h a t  c£)  u^C Int  E^ l,i+l  = E^ U A* , l , i  + 1  + 1  1  , and  (a (H) u E^ ) h ±  x  where  1  1, l  fl A  1 ±  n. \ n.-l = x *3A 1  ±  32. (ii)  Let  e' > 0  g: | G |  be so s m a l l that i f  M  i s an  A  e'-approximation t o all  A € a (A  ),  / f / + 1  1  f,  g( A) c yx^ ^(int  then  C^),  and, f o r any subcomplex  Q  of  for A*' ), +1  A  any  e'-approximation  |Q|  g:  -• M  f l i ^IQl i  to  may  be  A  extended  e  t o an  —--approximation to 2  f.  s  (iii)  Induction Hypothesis: compact that: h  i,0  h. x  >  x  set  and a  | G | -• M,  g : i  C°°-isotopy  h^ ^: M -» M  a such  x  =  i d  M»  h  i,t^ ) x  =  x  i  f  (V) 3 X U g j M H ) U 1  <)  «il|L|  (3)  d(g.,f) <  2  c: M,  There i s a map  =  f |  1  ^ i>  ^  C  x  l , i  |L|*  A  (iv)  e(l-2  _ 1  ).  I n d u c t i o n Step: Let  V  =  We reduce Case B t o Case A:  h. x  K'  = H«  U  A^  ,(V),  =  a.(H)  >  ( E  U  G'  = K'  U  A^ ,  f  1  =  n. L' =  CTl  (L)  0 G'.  l  +  1  ) (  l  \  11  x  9  \  1  H'  x  Then  X u f ' ( |H' j) c V ,  H' fl A  ±  gil| ij, G  n.-l =  x i  *SA  1 j L  and  n.  x n f'( |L« n hypotheses  A  x  i  \)  c  x n f'( | L n  A*-|)  o f Case A are s a t i s f i e d .  =  so the  Thus t h e r e i s a map  33. g':  |G'| -• M,  a compact s e t C  h£: M - M  such t h a t  h^ =• i d ,  h£(x) = x  M  ''|L'|  =  and a  C^-isotopy  and  X t  c hJ(V').  XUg'(lK'l) g  i f x fi 0  c M,  |L'|*  f , 1  d C f ' j g ) < e«. 1  Let  g  i+ll|L| L  h  g  e  °  t  h  ±  +  : |G| - M  1  ' | L h  f  i+l,t  i + l , 0 = M> id  h  =  a  n  d  d  be d e f i n e d by (Si l^)  <  +  h  t° i,t> h  i+l,t( ^  =  X  X U gi+xCla^H) U ( E ^  C  x  C  f  U  C  i '  +  T  * i+1>  x  l| ij =g , 1  1  < l - 2 - (  i+1 = ' i  g  G  1  I  +  h  L  e  ^  C  )  n  ) .  :  .  cXug'(|K'|) c h  + 1  1  1,i+1 Si+l'lLl d(g  i + 1  =  f l  ,f) <  ( l - 2 - <  I  +  1  ) ) .  T h i s completes the i n d u c t i o n . Let h  g = g , s  s  = i d , -fc( ) = n  Q  C = C , x  x  M  h  t  = h  g t  .  Then  i f x £ C, and  X u g(|K|) c X U g ( l o i ( H ) U ( E *  + 1  s  )^|) c  h l  (V).  1, s S||L| = l | L | ' f  1 X  *  |L|' e  +  •  d ( g , f ) < c. T h i s proves Case B, and hence  B(t l). 3  Q.E.D.  (V).  34, We  n o t e t h a t C o r o l l a r y 2.1  no r e s t r i c t i o n on  n.  holds f o r  k = 0  with  55.  CHAPTER THREE A  C"-Stretching Diffeomorphism  Let We  E  let  A  =  l  denote the such that  =  m  k  {(0,...,0,x  x -axis. k  c  R  K  k + 1  A^  ,  R  c  M  R  N  ,...,,x _ k  where  ,  K r  l)  We assume that  m  A  c  A *A*-  c  l  e R },  and  M  A  b , =  E ,  m = k + -t + l < n .  ir: R  - R ,  N  p: R  M  1  \ e R,  U c R  o A c U U H (l-2e).  such that  F 'rt A  (1)  m  i s  h  = id  ,  K  A (2)  m  c h^U)  If T then  c dA . m  h^: R  P y  :  x  F  -» R  h.(x) = x  n S U  be  be a closed set i n R  Then there i s . a compact set  n  >-\),  m  > \).  m  Let  m  Q  C-- oto  M  be an open set and l e t e > 0  N  such that  an, a  he the ortho-  M  R  M  s = q»7r.  m  m  m  - L  Q  m  Let  and  M  l e t H ( \) = { ( x ^ . . . , x j e R : x  and l e t H (\) = {(x.^ ... ,x ) e Lemma 5 . 1 .  q: R  1  gonal projections, and l e t r = po-jr For  R  C c R  N  c h that:  i f x 4 C,  and  z  U H (l-2e). m  Q  i s any l i n e a r subspace of h.(T) = T  for a l l  M  A  - R " " , and  M  R  b . = (0,...,0,l)eL.  A  Let  L c  i s situated i n  m  and  0,  let  R  t e l .  N  which contains  A , m  N  ,  - F  36. Proof:  Step A:  d i f f eomorphism"  We f i r s t construct a "horizontal h •: R  n  - R . n  I f m = 1,  l e t h' = i d . R  z  assume that  m _> 2  C*-stretching  z  f o r the rest of Step A  We  n  and further that  1 - 2e > 0.  (1)  Let e > 0 Q  6 > 0, fixed then  be such that  A  n ( R - K™(2e )) c U. I f  m  m  Q  l e t Bg" = {x e R *": ||X|| < 6}. 1  6 > 0  111  We choose a  such that i f DJJ = p ' ^ B ™ " ) n (H ( e )-Hj!J(l- c ) ) , 1  m  Q  c Int A .  See Figure 5.  1  m  Let D  m  Finally, l e t  FIGURE 5  - p "  1  ^  1  )  n (H (2e ) m  Q  -H*(l-2e))  37. C = ^min{dist(oA  - H™(l-2e),,R  m  - U), dist(D ,, 9 A )}  n  m  We wish t o c o n s t r u c t a a compact s e t C c R (1)  n  - P  C^-isotopy  > 0.  m  h£: R  R  n  n  and  such t h a t :  h^ = i d , h£(x) = x  i f x e C, . and  n  R  (A -D ) m  U C l ( a D - H ( l - 2 e ) ) c h£(U)  m  r a  (2)  h|(x) - x € R ~  (2)  Consider the continuous map  m  U H (l-2c).  m  for a l l x e R  x  n _ 1  m  '  and a l l t e l .  g: 7 r " ( H ^ ( o ) - H ( l ) - L) -• R 1  m  A  d e f i n e d hy point o f  g(x) = ||f(x) - s ( x ) ||,  9A  l y i n g on the r a y from  m  A  Note t h a t  where  f ( x ) i s the  s ( x ) through  A  g(x) = g ( s ( x ) + \»r(x)) f o r a l l  x e 7r" (H (0) - H ( l ) - L). 1  TT(X).  m  \ > 0  c"-c-approxi-  We c o n s t r u c t a  m  and a l l  A  mation t o  g| w i t h the same TT" ( H ^ ( e ) - H ^ l - e ) - L) n  w  Q  p r o p e r t y as f o l l o w s : Consider S  f f l  -  2  and  [e ,l-c]=p- ({x R 1  X  0  r a  £  " :  c )-H^l-c )  ||x|| = l } ) ^  1  c  g| o : S " x [ e _ l - e ] - . R. L e t S ~^x[e ,l-e] ° m  2  m  0  g: S  x [ e , l - e ] -• R Q  ^s - x[ m  x e  s  e o  ,i-  i r \ ^  t  o  c ]  )  -  L  e  t  be a  s  (  x  )  =  C^-c-approximation t o  *  (  s  - HjU-e) - L ) .  (  x  )  f  o  r  a  i  1  38. Now we may construct a "horizontal stretching i n t e r v a l " for each  U]L  x e ir" (H ( e ) - H™(1 - e) - L ) . Let 1  m  Q  ( x ) = s(x) + (g(x) -  c)«^[x|n ,  u (x) = s(x) + (g(x) - 2 c ) - ^ | | , 1  2  u (x)  = s(x) +  U (X)  =  5  4  B(X)  +  •  The stretching i n t e r v a l w i l l he "stretching" we w i l l map  > (4c+6) - 2c - |  dist(u (x), A 2  m  0  onto  2  Notice that  B  c  'y(xl  Then  Q  <  |!f(x) - u (x)H < 3 c 2  Y  i s mapped onto  2  , ,  u ( x ) i s mapped onto  ; a a d  Y(x)-| (x) •  [0,1]  3  o  f  course  a(x) < B(x).  [u (x),u ^(x)] 1  1  by a transformation 0  u ( x ) i s mapped onto  •  ••  since  m  l i n e a r l y onto the i n t e r v a l  onto 1.  6  i f x e H (2c ) - K™(l-2e) - L,  - H™(l-2e))  U-j^x)  1  6  To apply Lemma 1.1, we map the i n t e r v a l  such that  [u (x),u^(x) ]  i s y(x) = g_(x)-c-^g(x)-2'c-g  ]+  = 2c + - | > 0.  [^(x^u^x)] c u  and by  1  [u-^x),u (x) ]  The length of [u (x),u (x) ] 1  [u (x),u^(x)]  and u^(x) i s mapped . ||u (x)-u (-x)|| a(x) = | | ^ ) . ^ j \\ u  , s  x  u  x  M )^i( )H x  x  B(x) - ^ ( x j - u ^ x ) \\  =  .  5 9 .  Before defining cp: R  -» R  n  p : R -• R 1  o  <  t  <  he a  1  < PiC*) <  2 c  we must construct a  ~  1  Let  C°°-function  with the proper support.  . Let 0  h".,  C  f o r  2 e  a  n  00  t € R,  a 1 1  »  C -function such that  P ( )  d  t  =  p (t) =1 i f 2  0  i  * < >  f  e  2  m  Q  > "  t  x  e  ||p(x)H < g(x)-c} U .  - H^(l-e))).  c Int A .  o  (L n ( H ( c )  r  = {xeH (c )-H^(l-e)-L: m  Q  o  0  Then  C  m  Q  Let  n = d i s t ( C , P U SA ). m  Q  Let  p : R -» R 2  0 < p (t) < 1  n  2  Pl  1  1  0  Q  c C,  t < 0,  cp(x) = (s(x)).p (2|lx-7r(x) ||) f o r x e R .  C = Cl(cp" ((0,l])) 0 7r" (Gl(C )). C  If  2  and  i f t > n.  2  Let  C™-function such that  f o r a l l t e R, p ( t ) = 1  2  p (t) = 0  he a  and  C D P = 0,  on  ||r(x)||.  If  x e Tr" (H (e ) 1  m  0  x + t«cp(x).  Note that  - H ( l - e ) - L), m  Then  C  Let  i s compact,  cp(x)  does not depend  we l e t h £ ( x ) =  [{stretching diffeomorphism with respect to [ u ( x ) , u ( x ) ] applied to x ) - x ] =  the i n t e r v a l  1  i +  /„, ||p(u (x))'||-||r(x).|| x + t-q>(x)[8 (" ^ ' )K(x)-u (x)) ( r(x)-u (x); a(x) "v ' • • fl P  1  x  1  otherwise, we l e t h £ ( x ) = x .  +  7  ] L  40. l i m ||hjL(x) || = 00, i|xII— •  Obviously,  so to apply Lemma  we need only show that the Jacobian matrix of non-singular. h£(x) = x, ir~ (L).  By the d e f i n i t i o n of  that i s , h£  h|,  if  i s always  ||r(x) || _< -|,  then  i s the i d e n t i t y on a neighborhood of  Thus we need only show that  1  h£  1.2,  hi I , * R^ir" ^)  has a non-  1  singular Jacobian matrix. We perform a coordinate transformation.  We define a  0°°-diff eomorphism e: R  n  -  - R x S - x ( R " ) 4-' rn  TT-^L)  2  m  1  +  o  where  R  = {t e R: t > 0), S " m  +  2  = (x e R " : ||x|| = 1), 111  denotes the orthogonal complement, as follows: e(x) = (Hr(x)||, ^[ j  let  || , x - r ( x ) ) .  X  H+ = e..(h. I „ * ^ R -7T Let and  t e I,  Then  -, f  t  ^  +  y  )  : R  be defined by  +  - R  +  for  (u,y) e  t  u  m  0  m  2  m  1  t  and i f  y  then  a(e" (w,u,y)) x  S *" x(R " )J-,  f,(u,y)M = Wh^e'^w^u^y)) )Jj.  H (w,u,y) = ( f . ( ^ ;).(^),u,y), t  n  R^xS^xCR^V.  (L)  i r " ( H ( e ) - H^(l-c) - L), 1  1  1  i f x e R -nr""'"(L),  Consider  R^xS^^R™" ^  )oe- :  and j_  1  f^  K  Y(e  y  )  e (w,u,y) € -:L  (w) =  (w,u,y))  41. g(e (w,u,y)) + c + w],  otherwise  -1  f  t  ^  ) ( ) = w. w  u y  We must show that the d i f f e r e n t i a l of where nonsingular. df. / —'kw  >  'tel.  d f t ,  dw  To prove this we have only to show that 0  0  f  o  r a  1  m  1  l Y ) ( W )  =  L  PP°  S u  and a l l  J  then  m  e-^w^u^y)  s e  g(e" (w,u ,y)) 1  ;  x  for a l l z e R.  U  J  0  Y( e~ (w,u, y))  M ^  m  I f e'-^w^y) £ r " ( H ( e ) - H^(l-e) - L),  the construction o f g. Hence  d f  n  (w,u,y) € R xS "'^x(R *" -)- -,  1  f  F i r s t we note that  and  i s every-  y ) ( W )  -  d W  e>- (H (c )-H (l- )-L). 1  j n  m  0  e  does not depend on w by  a(e~' "(w,u,y)), ' P(e (w,u,y) ), J  -1  do not depend on w.  Further  d  ^q( ^ > 0 z  We.differentiate:  l-t.cp(e- (w,u,y))[-a;^(< ( ^ ^))(^ "Vi iy))- - ) 1  1  o ^ K ^ y ) )  +  1] >  w  u  e  v ^ V ^ y ) ) '  0.  Thus the rank of the Jacobian matrix of h£ so by"Lemma 1 . 2 ,  u  h£  is a  C^-diffeomorphism.  i s n,  It satisfies  the required properties ( 1 ) and ( 2 ) . Step B; morphism" (1)  Next we construct a " v e r t i c a l h^: R  n  -R  n  such that-  h^ = i d , h£(x) - x R n  C*-stretching  i f x £ C, and  diffeo-  c  w  42. A  c hJ(hj_(U)) U 1^(1-26).  m  h[l(x) - x e L for a l l Since  D  m = 1, l e t D  that  1  = 3D  1  - 1^(1-26))) > 0.  m  = L n (H (2e ) - H ^ ( l - 2 e ) ) ) Q  If  0  We notice that  construction of For each  h£.  x e R  Let  and  1  ( e ) - HQ(1-C )).  1  d < 6.  interval".  3  - h£(U), Cl( SD  n  tel.  we may l e t  m  = L n (H  1  and a l l  n  Cl( S.D -1^(1-25)) c h j ( U ) d = dist(R  (If  x eR  A  m > 2,  we assume  - H ( 2 e ) c h£(U)  m  m  Q  hy the  v = (0,...,0,l) e L.  we define a " v e r t i c a l stretching  n  Let  ^  /-  v ( x ) = r(x) + e «v, x  Q  v ( x ) = r(x) + 2e .v, 2  Q  v ( x ) = r(x) + ( l - 2 c ) - v , 5  v^(x) = r(x) + ( l - e ) . v . The "stretching i n t e r v a l " w i l l be and by "stretching" we w i l l map [v (x),v^(x)]. 1  Y = l-e~e . 0  The i n t e r v a l  1  [v-^(x),v (x)] 2  [v (x) v^(x)] 1  J  onto  has length  To apply Lemma 1.1, we map the i n t e r v a l  [v (x),v^(x) ] l i n e a r l y onto 1  [v (x),v^(x)],  [0,1]  such that  v-^x)  i s mapped onto  0  and  v^(x) i s mapped onto 1 .  Then v ( x ) 2  e i s mapped onto  a =  We note that  a < B.  and  v-^(x)  i s mapped onto  B =  See Figure 6 . 1  f'  /  FIGURE 6  (2)  Before defining ijc R  R  n  If that Let  we must construct a  0°°-function  with the proper support.  m = 1,  DJJ C C X : R -» R 1  for a l l  h£  t e R,  l e t 7] = d i s t ( D ^ F ) . and hence be a \{t)  If m > 2 ,  d i s t ( D ^ F ) < d i s t ( C , F ) = TI. 0  C* -function such that 0  =  note  1 if t < 0  and  0 <: X ^ t ) < 1  =  0  if  44.  t > 1>  l e t * ( ) = *i(2||x-ir(x) ||),  I f m = 1,  C = 7r (D ) n C l ( r ( ( 0 , l ] ) ) _1  1  0  If that  and l e t  x  m > 2,  l e t XgJ R  0 < X (t) < 1 and  R  be a  f o r a l l t e R,  2  t < 6-d,  o  X (t) =0  C f u n e t i o n such w  r  X (t) = 1 i f 2  i f t > 6. Letty(x)=  2  X (2|1X-TT(X) ||)»X ( ||r(x) ||) f o r a l l x e R „  Note that  n  1  2  |i-(x) = 0, and m  ciCT^CO,!])) Let  x € R.  m  m  -  x  _i^ t (' )i h  m  m  x  x m +  i>««^  x n  )'»  m  zsr-W'-  — x — I  W  compute  e  Qx —s  oh" (x)  ra  cc,  H£(1-C))  Q  x  m  o  + t.*(x)[Y-e£(^S-!2) + e - x j , - and then  hj(x) = ( i ^ - ^ oh"  m  1  S i m i l a r l y as i n Step A, we define  n  h^ (x) = x  n 7r- (H (c )  = (i-t-t(x)) + t . t ( x ) e ' ^ )  m  > o.  r  Hence the rank of the Jacohian matrix of h£ i s n, and again l i m ||h£(x) || = oo . By Lemma 1.2, h£: R - R i s n  a  C°°-diff eomorphism onto  R  n  which s a t i s f i e s properties  n  (1) and (2) by construction. Combining Step A and Step B, h (x) t  - x € R  m  for a l l  a l i n e a r subspace of R  x e R, n  n  we l e t h so  t  = h£oh£.  h (-T) = T t  which contains  A» m  if T  Q.E.D.  Then is  45. If  A  R,  l e t E(A )  let  E(A ,L )  n  m  = t^*^  c R  be t h e  m-dimensional p l a n e determined by  m  be an  l  k  i s an a r b i t r a r y  n  (m-1)-dimensional p l a n e i n  p a r a l l e l t o the p l a n e s  E(A )  simplices  r e s p e c t i v e l y , and w i t h  IntA  A  £ 0.  m  and  Let  which contains  R,  and  P  A  n  (1)  h  (2)  m  If  If  E( A ) - E( A , A^ m  k  A  = t^*h*'  m  i s an a r b i t r a r y  m-simplex  S i " c U U H^( A , A*  i s a c l o s e d subset o f  R  n  such t h a t  P. n A  n  T c R  - F,  n  and a  m  c  ra  SA , m  C^-isotopy  i f x £ C,  and  U H ( A ,A^-). m  m  i s a hyperplane c o n t a i n i n g  n  for a l l Let  K  f u l l f i n i t e subcomplex subcomplex  C c R  1  such t h a t :  c h^U)  Theorem 5.1.  n  E(A , A ) D  such t h a t  t  R  d e t e r m i n e d by the  n  h (T) = T  in  m  A  = i d , h,(x) = x R  A  m  R  - R  t  A,  E(A )  be t h e component o f  m  t h e n t h e r e i s a compact s e t h : R  E(A^)  in  i s an open s e t i n  U  n  and  k  H^( A , A*')  C o r o l l a r y 5.1. in  ra-simplex  t e l .  of  K, to  | L | C'U  c l o s e d s e t such t h a t  then  m  be a s i m p l i c i a l complex i n  complementary  such t h a t  A,  and L.  and  L  Let  = {AeK:  c  U  and V  | L | C V. C  Let  R,  L  n  a  A 0 L = 0}  the  be open s e t s F c R  n  be a  F 0 |K| C |L| U | L ° | . Then t h e r e i s a  46. compact s e t C c R  n  - F  and a  C°°-isotopy  h^: R  -» R  n  such  n  that: (1)  h  (2)  h ( A) = A  Q  = i d , h (x)= x R n  If  C,  i  for a l l A e K  t  Proof:  i f x  t  and  A = ^*t\  then  C  and  A^ e L ° .  F o r each p r i n c i p l a l s i m p l e x  let  H™( A, A^)  be chosen so t h a t  E£(A,A*).= „  C  h-^U) U V.  and t e l .  A e K - (L U L ) ,  A e K - ( L \ U L°)  |K|  l  where  A  k  e L  AeK-(LUL°)  A 0 H J ( A, A ) c V. „if 1  i s not a p r i n c i p a l simplex, then l e t n  H£(A,^)  A i s a p r i n c i p a l s i m p l e x i n K-(LUL ) w i t h A<A . Let  F« = F U |L| U | L ° | .  Induction Hypothesis:  There i s a  and a compact s e t C _ i c R  n  m  (1)  h™  _ 1  - p»  = i d , h^" (x) = x R 1  n  0°°-isotopy  h^"" ": R 1  R  n  n  such t h a t  i f x i  c  and f o r a l l  A e K ^ " ) - ( L U L ) , A c h ^ - ^ U ) U H™( A, A^). 1  (2)  h£ (A) = A -1  C  for a l l A e K  and t e l .  T h i s i s c l e a r l y t r u e f o r m = l(h° = i d • I n d u c t i o n Step: _ (L u L ). c  L e t t h e r e be  k  m  z  m-simplices  F i r s t note t h a t i f  A'  R  for a l lt e I ) . n  A^,...,A^e  i s a face o f  A,  then  47. lA A', A' ) c H £ ( A, A*).  Hence  l  for 1  1 < J < k .  < J < mk  T  h  e  L e t P. = P' U {AeK: A 0 I n t A™ =  n  i H A™ = 3A™.  P  F o r each with  j.= 1  A" = A *A*', 1  subset,  h^" (U)  subsets  where  k  N  - F.  W o' = n' ?' ' ld  3  h  J  (x)  (2)  h ^ h ^ U ) )  C  If T c R  we a p p l y C o r o l l a r y 3.1  ,  e L, A^ e L°, F.  i s the closed  h ^: 3  R  N  -  =  X  "  X  *  h^ '(T) = T  and compact  N  ^  4*), i =  U  1.....V  i s a hyperplane c o n t a i n i n g  N  R  such t h a t :  R  A»?  k  k  i s t h e open s u b s e t , and w i t h r e s p e c t t o  .c R  C  h  A  ,  There a r e i s o t o p i e s  H^(A^AJ).  U l£( A*; A*)  d A^ c h ^ U )  A™,  then  for a l l t e I.  J  We c o n c l u d e t h a t  A) = A  for a l l A e K  and  t e I. Let h£ = h^N.-....^- .^ and C = C ^ U C 1  1  m  m  3  (1)  h™ = i d , h ( x ) = m  n  R  A  (2)  6  K  % 1  (  m  )  and f o r a l l  m  - ( L U L ) , A c h ^ ( U ) U H™(A,A^). C  h^(A) = A If  x i f x iC,  i f A 6 K  d i m K = k,  and let  t e l . = h  k  and  C = C^.  Q.E.D.  U  48. CHAPTER POUR Open C y l i n d e r s Theorem 4.1. and l e t  Let  M-^ and Mg  be compact connected C™-manifolds  f : M-^x R-* MgX R be a  MgX{0} c f ( M - x R ) .  C - d i f f eomorphism such t h a t 0 0  p > 0,  Then, f o r any number  there i s a  A  C - d i f f eomorphism  f  0 0  *\x[-p,p]  of  M-* R  " ' x[-p,p]-  onto  ^ther,  f  M l  MgXR  such t h a t  f(V(-»*P))  i f  =>  A  M x(-oo,0],  we may r e q u i r e t h a t  2  Proof: (1)  =  f ^ ( - 0 0 , p]' [5].  Our p r o o f i s s i m i l a r t o t h a t used by K. W. Kwun i n  There a r e p o s i t i v e numbers that  a and b,  M x[-b,b] c f( M - ^ - a , a ) ) . assume t h a t  (otherwise be a  f  =  i d  1  M-^IR  1  1  2  Let  f- (M x(-co , a ] ) c M x ( - c o , b )  and  M x[-b,b] c f ( M x ( - o o , a + l ) ) .  g  2  2  1  1  Q  3  ^  g  Q  -  d  fog. Q  Then  and a l l o t h e r  C ^ - d i f f e o m o r p h i s m s used i n t h i s p r o o f may be by u s i n g Lemma 1.1.  Let  onto i t s e l f such t h a t  M x(R-(-a, a+l))  f »g (M x(-oo,a]) c M x ( - o o , b ) . 1  such  L  i s r e p l a c e d by i t s r e f l e c t i o n ) .  olM x((R-(-a,a+l))  L  a > p,  f (M- x( a+1, oo )) D MgX(-oo,b) = 0  C°°-diff eomorphism o f  0  with  Without l o s s o f g e n e r a l i t y ,  2  we may  g  f l ^ ^ . ^ ^ j  constructed  49. Suppose we have c o n s t r u c t e d a sequence C^-diffeomorphisms o f M^xe  into  MgXR  f (M x(-oo ,a+i-l)) c M x(-oo,h+i-l) i  1  j  f ^ , ... f  2  of  such t h a t  a  M x [ - b b + i - l ] c f . ( M x ( - o o a + i ) ) , and 2  3  1  ±\u x(-CD,a.+l-2]  f  =  f  1  h,  \l  i-l'M x(-co,a+i-2]^  M 2 X  ^  1  1  C * - d i f f eomorphism o f M J R  be a  that  3  (_  0 0 a b + k  . ] 1  '  2  L  e  t  onto i t s e l f  = i d M 2 x(-oo,b+k-lp  such  8 1 1 ( 1  h^of^(M x(-co,a+k)) o MgX[-b,b+k]. L e t g^ be a 1  0 ° ° - d i f f eomorphism o f M-^x IR  ^ l n  K  e f  B - ( ^  M  k M l ^ M  6  x  . ^ l , , - ^ ( ^ l . ^ ) ) .  1  a + k  ^  M x(b+k-l,b+k).  c  f = lim f.  Let  onto i t s e l f such t h a t  Then  ±  Let - f  2  -  = h °f S < 9  k + 1  k  MgX(-b,,oo) c f ( M x e ) .  Note  1  k  that  j_-»eo f  'M x(-co,a+l]  =  1  -1 h  l  h  l° o SllM x(-oo,a+l]' s  8  e  n  c  .  e  -1  ° ° o lg »g (M x(-oo,a+l]) f  g  1  Note t h a t  0  1  1  =  f  'g <»g (M x(-co a+l])1  0  1  J  g- < g (M x(-oo,a+l)) r> M- x(-co, a ] .  Let  ,  L  0  1  f* = h - . ? ^ - ^ ; . and 1  1  L  Then  M x(-b,oo) c f * ( M X R ) . 2  H  1  1  f l^ . x (  p > p  y -  I f f(M x(-co/p)) 1  f l  3  V  k  ( - p , p ) »  MgX(-co .,0 ],  50. f ( x ) , xeM x(-oo,p) 1  A  let  f(x)  «<  f * ( x ) , xeM x(-p,oo').  f ( M x ( - o o , p)) | M X ( - O O , 0 ]  If  1  2  f * may be extended i n a  v  manner s y m m e t r i c a l t o t h e methods o f ( 2 ) t o o b t a i n t h e A  required  f . Q.E.D.  Lemma 4.1. onto  0*°-diff eomorphism  ( - l , l ] x [ 0 , l ) - [0,l]x{0]  of  (l}x(0,l)  Proof: f  There i s a  and  i s  a  f ( x ) < -x  0"-imbedding,  f o r -1 < x < 0.  stretching  diffeomorphism -| onto f  and c a r r i e s  y e (0,1).  be a  _ 1  0 (y)  00  See F i g u r e 7. o)  t  y  m  e  a  n  s  ||(x) < 0  We move 0l  *  horizontal  a  h .  The o b v i o u s l i n e a r t r a n s f o r m a t i o n  and  -1  1  onto  onto  1  carries  B(y) = ^ ^ ( y )  0  onto  for a l l  I f ( x , y ) e [0,l]x(0,l), l e t  h (x,y) = (| - | e ^ ) ( - | ( x - | ) ) , y ) . y  x  h ([0,l]x(0,l)) 1  C - f u n c t i o n such t h a t  f ( - l ) =1, f(0) =0,  onto t h e graph o f  a =  [0,l]x(0,l)  fixed.  (0}x(0,l)  which c a r r i e s  of  w h i c h l e a v e s a neighborhood  L e t f : [-1,0] - [0,1]  l(-l,0)  h  = {(x,y): r ^ y )  Then <x<l,  0<y<l}.  FIGURE 7  51. Next we c o n s t r u c t a v e r t i c a l morphism  (-1,0)x{0}.  onto  carries f  x  +  Clearly,  by moving t h e graph o f  0  i onto 1  and 0  and c a r r i e s  x  x  onto f  q  The o b v i o u s l i n e a r t r a n s f o r m a t i o n  - x onto  a(x) = ( } x- g  09  h^([0,l]x(0,l))  hg which c a r r i e s  ( - l , l ] x [ 0 , l ) - [0,l]x(0)  C - s t r e t c h i n g d i f f eo-  a(x) < B(x),  since  onto  v  3  x < 0.  for 1  x  l e t hg(x,y) = ( x , y ) .  onto  I f (x,y)eh ([0,1]x(0,l)),  0^ j(g|)-x).  x  which  f(x)  p(x) = ~ r , x-  f ( x ) > 0.  and x < 0, l e t h g ( x , y ) = ( , ( x - i ) (x,y)e[0,l]x(0,l),  carries  N  If  Then hg'h-^ [ 0 , l ] x ( 0 , l ) )  C*, and i s a d i f f eomorphism s i n c e |^((x-|) B^||j ( ^ ^ ) - x ) =  (x-|)9'^| j( 5|) ^x x  x  >0  x  hgoh ([0,l]x(0,l))  QoE.D.  h = hgoh^.  Let  C o r o l l a r y 4.1.  M-^x R  and MgXR  bM±  h  i  M i  o  f  -  S M ^ t - l , oo) - M  ±  in M  =  f i  be compact  C^-manifolds  (  S  Then  are C^-diffeomorphic.  Let f:  Proof:  i  L e t M-^ and Mg  I n t M-^ and I n t Mg a r e C ^ - d i f f e o m o r p h i c .  such t h a t  Further,  = ( - l , l ] x [ 0 , l ) - [0,l]x{0}.  1  M  f o r a l l x < 0 and a l l y e R.  i  be a  C°°-collaring o f  ( s e e [ 8 ] , p. 56), and l e t M[ = M - f ( S M ^ t - ^ O ) ) ,  ±  ±  x t - j ) )> 1  M i  M^C * ) 0  1  1  o  n  t  o  c"- d i f f eomorphism o f  = 3' 1  1  I  n  t  2  W  e  construct a  M x [ 0 , l ) - Mj_x{0}. i  (M^-Int MV)x(0,l)  onto  ±  C°°-diff eomorphism Let h  ±  be t h e  aM x|-l,l) x(0,l) i  ±  S  52. d e f i n e d by  h^(m, t ) = ( f T ( m ) , t ) .  A  ±  ±  A  be d e f i n e d by  A  h ^ m ^ y ) = (m,h(x,y)).  9M x{l}x(0,l)  x  l e a v e s a neighborhood  L e t h£: M j _ x ( 0 , l ) - I n t M x [ 0 , l ) - M j _ x { 0 3  fixed.  i  be as i n Lemma 4.1,  bM x( ( ~ l , l ] [ 0 , l ) - [ 0 , l ] x { 0 } )  and l e t h : B M ^ O , ^ ^ , ! )  of  Let h  1  i  be d e f i n e d by 1 ^ M - I n t M " ^ , ! ) Let  g  be a  A  l V i»  h  "  #  ^  h  C -diffeomorphism o f  h  i>Mi;x(0,l)  Int  a  onto  =  id  I n t Mg. L e t  D = ( h j _ ) " . ( g x i d ^ ) " o h ( M x ( 0 , l ) ) c M£ (0,1). 1  1  [ o  M-^ and M  1 ]  2  2  X  A  o(  A  b e (0,1),  implies that therefore that  l^xflR  C o r o l l a r y 4.2.  If  such t h a t BM-^R  Int  Q  1  ))*hj_l , D  x  i s C°°-diffeomorphic t o MgX R. and M  a r e compact  2  i s C -diffeomorphic t o w  Let M  be a  and Q.E.D.  C*-manifolds I n t Mg, ' t h e n  SMgX R •.  C ^ - n - m a n i f o l d such t h a t  00  M =  U 0. , i=l  where  O? 1  1  °i to  C  °1+1 > R . n  i s an open  C^-n-cell i n M  with  1  f  o  r  a  1  1  1  -  l m  T  h  e  n  M  i  s  °°-diffeomorphic  c  then  Then Theorem 4.1  i s C°°-diffeomorphic t o Mg R,  i s C^-diffeomorphic t o  Theorem 4.2.  gxidj-  f ( M j x ( a , l ) ) r5 MgX{b}.  M'x(a,l)  such t h a t  1  M£x(a,l) c=D, and i f we l e t f = (hg)  f o r some  Since  a e (0,1)  a r e compact, t h e r e i s a number  2  Mi;x(o,i)-  55. Proof:  Let f^: R  -* M  n  f ( R ) = 0*? , i > 1. M  C - d i f f eomorphism such t h a t w  f ^ O ) = p e M, i > 1.  We may assume t h a t  n  i  Since  be a  i s t h e u n i o n o f c o u n t a b l y many compact s e t s , we may 00  f u r t h e r assume t h a t  M = _U f (D*?)  We c o n s t r u c t a sequence o f into and are f  M S  such t h a t ( i  ) ^  D  i + 1  + 1  constructed. n  s  R n  = kil f  W^+i* £+r 3D  and  g  g = lim f  isa  i  +  1  S  ±  ( ^  ).  D  i  0  +  1  i  +  S ]_  a s  k +  e g  : R -{0}  kl^-{0} J  L e t  o h  W > x  l  = S \r^  n D  ±  = {xeR : ||x||<i3). n  of  i  g (& ) n  9  ±  follows:  R  n  = o£,  g^...^  consider there i s a  such t h a t W> = ^  W^" g  i f  g  Suppose t h a t  = k( )  x  k+i( )  n  By Theorem 4 . 1 ,  n  k + 1  1  > !•  1  n  h  (x) = k+l f  k + 1  f  = f,  ±  D  C^-diffeomorphisms  "^ ^ -R -{0}.  C™-diffeomorphism Vl^-CO}  g  Define  k + l ° k ' e - { 0 } **  (where  ±  x  x e R  n  i f  X 6  " CO}, and  ^  - {0}. Then  C^-diffeomorphism o f R  n  onto  M.  Q.E.D„  i-*oo  The f o l l o w i n g theorem may be p r o v e d i n a s i m i l a r  ~  manner: Theorem 4 . 5 .  Let M  boundary  I f there are C^-collarings  3M.  be a  C " - m a n i f o l d w i t h compact connected  CO  such t h a t  M =  U f.(SMx[0,oo)), i=l  and  f ^ : 3Mx[0,oo)  -M  54. f (aMx[0,oo)) c f ±  i + 1  ( 5 M x [ 0 , o o ) ) , i > 1,  C* -diffeomorphic t o 0  SMx[0,oo).  then  M  is  55. CHAPTER F I V E Coverings o f Manifolds The  f o l l o w i n g lemma i s a c o n s e q u e n c e  Lemma 5.1. be a  Let  M  be a  is  closed,  is  compact.  and  R ,  and l e t Let  set  such that  U c M  is  k-connected.  V-^, ...,V  g ( P ) , and h|  Let M  M  i  < n - 5, 1'< i < m.  k  n  1  in  + ... + k M  m  Let  M  such that  E^, . . . , E  k  ClV  such t h a t • C  C^-diffeomorphisms  i  and  h^  of  M  onto  SM c  n  if  x £ C., 1 < i < m,  and g ( c " ) c  and  subsets o f  1  0 < a < 1. sets  1 < i < m, itself  c  m U E.. L e t  t h e r e a r e compact  0 ( E . U SM) = ^  i  be c l o s e d  m  C ™ - d i f f e o m o r p h i s m and l e t  3  h  > 0, l e t  ±  1=1  + m > h + 1, —  that  C ^ - n - m a n i f o l d , l e t U^,...,U^,  E . c V-, 1 < i < m,  be a  h : M-*M .  dM.  1  g: C^ - M  E c u ,  i f x £ C.  i s k-connected,i f  ±  such that  of  g(P)  g(P)-U  and, i n p a r t i c u l a r ,  E  be a  (M-Cl V _,U -Cl V ) j  -» M  there i s  C°°-diff e o m o r p h i s m  = id ,  E  k _< n - 3,  h(x) = x  be open s u b s e t s o f  m  n  k - d i m e n s i o n a l sub-  s e t such t h a t  If  and a  i d e n t i t y on a n e i g h b o r h o o d  Lemma 5.2.  be a  be a c l o s e d  C c M-E,  h(U)  P  g: R  be a n o p e n s e t s u c h t h a t  E D SM  Note t h a t  k^  Let  and l e t  n o t n e c e s s a r i l y compact, s u c h t h a t  n  (M-E,U-E)  a compact  the  C™-manifold,  C™-diffeomorphism.  polyhedron of  o f C o r o l l a r y 2.1:  C-,,...,C l m, 5  and  such that  m  U h.(U.).  If  h (x) = i  56. Let  Proof:  G  be the s i m p l i c i a l complex determined  simplicial subdivision of  C  p o i n t s o f a subcomplex  K  of  f o r any simplex  such t h a t  g(A)cV ,  L  -i and K, ,...,K 1' m-1  m  m-1  =  n  -  Lj__]_  Then  —  L  i n d u c t i v e l y two sequences  i  defined.  s  Let  m +  " ( l "-  1  k  +  + k  m-l) < V  L  e  t  K  m  = g" (g(|K l) - C lV ) . i  i  Then  P  i  isa  PCL^-^),  in  V  1  ±  1  dim L. = n - i - ( k , + . . .+k.). I 1 i 88  We"now a p p l y Lemma 5»1 w i t h r e s p e c t P  (k.)  =  be the complementary complex o f  i  1 < i < m - 1. i  1 < i < m.  3  suppose  and l e t L  we have  }  o f s i m p l i c i a l complexes as  n  follows:  d  g( A) fl E . ^ 0  We c o n s t r u c t  m-1  3  —  = K.  Q  and  9  ±9  L  i s the set o f  G, • JN(K,G)| C I n t C^  X  Let  o'  C^  i . e . : g( |N( g~ ( E ^ G ) |) c V  i  L  A e G  such t h a t  n  by a  L  Thus  m-1'  t o each  K^. L e t  k^dimensional  polyhedron i n I n t C^ - g ' ^ C l V ^ ) , g ( P ) i s c l o s e d i n i  M - C l \V^ . 9  and  g(Pj_) -  C -diffeomorphisms such t h a t  h|: M - M  h|(x) = x Let  W  i s compact, so t h e r e a r e and compact s e t s  i f x £ C  = g'^h^^)),  f ±9  , and  g(P )  1 < i < m.  i  C^  9  1 < l < m,•  chj(u\)a  Then  \K \ C ±  The b a r y c e n t r i c s u b d i v i s i o n s u s e d " i n the d e f i n i t i o n s o f and  L  i  imply t h a t  K  i  and  a r e f u l l subcomplexes o f  W. ±  57 o 6(L^ ^ ) , 1 _< i _< m - 1. inductively such t h a t  Applying  a sequence o f S  ^.  i  l m- l  c S  m-l( m-l)  K-?\  c  S  2  W  m- ( m- ) W  2  U  U  2  cS^)  |K| = i L j  1 < i < m - 1,  5  W  m  U S ^ W ^ )  U ...  L  C  m  = {A G P ( L _ ) : m  F = (R  n  -  |L| C W , m  2  0}  g( A) n ^  the  s  L  is full  in  be the  C - d i f f eomorphisms  A  P(L _ ), m  h  and  2  obtained  ±  onto  0 0  A  M - M  ^ 0}  3.1.  We l i f t ±  V  U  Let  C  S :  and  g'^Vl)  |H(K,G)|)'U  F 0 | P ( L _ ) | c |L| U |L |. i n Theorem  -*  ^. In the n o t a t i o n o f  and  2  A fl L =  2  Note t h a t m  '  i  V = W ,  ^  U O ( A ) :A € L _  1  _  V  U  m-l(Vl)  S  Theorem 3.1, l e t U = W  m  m  *  For example, we c o n s t r u c t  L = K _  s  construct  i s the i d e n t i t y on  1  L  we  3  C^-diffeomorphisms  |N(K,G)| U | N ( g " ( E ) G ) I,  -  Theorem 3°1  be d e f i n e d by  S (p) = g,°S  = p  (N(K,G) | ) ,  ±  a ±  -1  g  (p),  M:  let  '  i f p e  A  and  S(p) ±  If  pj^g(  1 < i < m - 1.  Note  n  g(Cj),  •58. A  that  S | _ = id _. ±  E  I t follows that  E  U ... U S f f l . r ; - l ( \ - l ) h  S^h'^)  g(C^) = g(|K|) c L e t  m(V'  U h  A  h  1 < i < m - 1,  = S «h|,  ±  ±  Theorem 5 . 1 .  and l e t h  L e t M 'be a  = h^.  m  Q.E.D.  C°°-n-manifold, and l e t 09  u  i*»*** m u  where  b  e  V ^  °P  e n  subsets o f  i s open,  ClV^j  (M-  CIV. ,,V. , , - C1-. v .  if  k. > 0 , \  if  k^ + ... + k^ + m _> n + 1,  h^:  M -• M  h  i ' CIV i, 1  Proof:  =  M  such t h a t  c V^  J + 1  there a r e  C°°-diffeomorphisms  such t h a t i  d  ClV  3  1  i, 1  —  1  g .: C^ -» M,  have c o n s t r u c t e d  m  — > m  3 X 1 ( 1  M  =  U  j = 1, 2,...  sequences  c"-diffeomorphisms o f  h  m i( i.)° ' i=l U  be a sequence o f CO  n. U g.(Ci).  0=1  *•  J  {f^ , . . . , f ^ Q  3  .J i aj-2"' -J- i,a^2' fl  J  llT  Suppose we i = l,...,m,  M onto i t s e l f such t h a t  m  lv  5  r  ' C -diffeomorphisms such t h a t Int M =  fi  ,  i s k.-connected, k. < n - 3 m 1 < i < m, and 3M c 0 V. Then, i—1  j > 1,  Let  k  U V.  .)  co  of  U. =  r  ±  3  S  * >  w  h  e  r  e  59.  We a p p l y Lemma 5.2 = v\ 2k+2 " ^ i k+1  8 1 1 ( 1  —  3  —  ~ k+l  g  g  °^  m j  ^  with *° ^  E. = G l V. 1 i,2k>  v  i  i,2k+l 3  0°°-diff eomorphisms  e z  °nto i t s e l f such t h a t  m  k+1  ,2k  Let  h.(x) = l i m f  k-»oo  C o r o l l a r y 5»1«  ±  *  k  (x)  Let M  for a l l  be a  k < n - 3  M  m  Proof:  L e t U^, ...,U  (M,UN) have  k ^ + ... + k + m  ( w i t h k _< n-3 N ,...,N^,  h^  S.  1  S  C - n - c e l l s i n M. w  l  t  of M  I  f  be a  i f k>0)  m  2  k  i  = k, 1 _< i _< m,  with  k-connected I  C^-n-manifold  be  i  C -collarings, w  n- Jt + l oo ^ 2 3 there are l C -diffeomorphisms  a neighborhood o f  h  i  i s t h e i d e n t i t y on  and m,.C°°-diff eomorphisms  such t h a t I  U h. of ( N , x[0,oo))u  i=l  1  we  boundary components  onto i t s e l f such t h a t each  M=  Then  = mk + m_>n + l .  Let M  ±  Then, i f m >  C°-n-cells.  and l e t f t N x [ 0 , o o ) -» M  1  C^-n-manifold  i f k>0.  i s k - c o n n e c t e d , so i f we l e t k  C o r o l l a r y 5.2.  1  open  be open  n  Q.E.D.  k-connected  w i t h o u t boundary, w i t h may be c o v e r e d w i t h  x e M.  \  •  m  U g,(R ) .  i=l  1  g^": iR -• M n  ,  6o. Proof;  (M,N^)  i s at least  5»5°  Corollary  Let  M  0-connected,  be a c o n n e c t e d  w i t h two c o n n e c t e d b o u n d a r y  components  the  M  inclusion of  i = 1, 2.  into  Then t h e r e a r e  such t h a t  h (x,0) = x  5.k.  Let  Let  h ^ : N^x[0,oo) -» M  69  i = 1,2,  2  M  be a c o n t r a c t i b l e  and  Then  M  C™-n-manifold  c a n be c o v e r e d w i t h t w o  be a c o n t r a c t i b l e  which i s  C^-diffeomorphic t o Proof:  n  that  Let  f  1  w  f  ^  : 2  n  M = f-^(R ) U f ( R ) < , n  n  2  closed sets  and  A , A  M = A^ U A g . R  n  1  co.  we n e e d o n l y show t h a t  C - d i f f eomorphism 5  at  Then  M  is  i f C c M  i s  n  B y T h e o r e m +.2,  C c f(!R ).  1-connected  C^n-manifold without  R .  compact, t h e r e i s a  (a)  that  i s a homotopy e q u i v a l e n c e ,  2  M  n > 5,  boundary,  of  such  C°-n- c e l l s .  T h e o r e m 5.2.  are  and  C - d i f f eomorphisms  n > 5»  w i t h o u t boundary, open  n > 5, '  Uh (N x[0,co)).  M = h^xlO,®)) Corollary  t + mk + ra > n + l .  C^-n-manifold,  f o r a l l x e N^,  i  and  g  c M  ~*  M  b  e  Since  f: R  n  -• M  C * - d i f f eomorphisms M  with  i s a normal c f^(R ), n  We c o n s i d e r a f i x e d s i m p l i c i a l  i n t o a s i m p l i c i a l complex  such  K  such  that  such  space, t h e r e A  c f (R ) n  g  2  subdivision  that  i s t h e s e t o f p o i n t s o f a subcomplex  of  K , i _> 1.  61. .  (b)  If A € K  (c)  I f A c C^ Let  K  i  ^ l ^ l )  =  f ^ A ) fl Ag ^ 0,  and  - Int C ,  L = N(f^ (A ),K), 1  c  i^ )' K  E  a  c  h  i  K  n  i s  a  i > 1.  i > 1, l e t  and, f o r a l l  1  n  f j A ) cfg(R ).  diam f-^( A) < j ,  then  n  + 1  then  subcomplex o f L, and  CO  UK.. i=l  L =  Let D n C  be a compact s e t such t h a t  M. - D i s  1  simply connected.  Then  (M, M-D)  i s 2-connected,  i s c l o s e d (because o f c o n d i t i o n ( c ) ) and f-j_( | L ^ ) | ) fl D C-^ c M  f (  U D C  1  and a  P(K^)  f ( l^^ ^ 2  x  U f ( C ) n Ag.  i f x £ C i > 1  Let H "  n  n  2  with  5  ^  o  e  Since  ±a  with  be t h e subcomplex have  |) c h ^ M - D ) . E = f ( 1  E c f (R ). n  2  i > 1,  then  diam(fg(C*?)  n Ag)  for  and h-^x) = x  complementary t o 8(K^^ ^ ) )  Let (b),  h^: M -+ M  0 0  i s compact, t h e r e i s an i n t e g e r  UD c f^lKj)  of  G - d i f f eomorphism  ^ | ) c h^(M-D),  2  x  2-dominated, and  i s compact, so by Lemma 5 . 1 , t h e r e i s a compact  2  set  x*-^( (H,^ ^^ I)  |N(f^ (f (cJ) 1  2  n A ),K)|).  By c o n d i t i o n ( c ) , i f A c c " diam(f (A) 1  < ~ + diam  By c o n d i t i o n  2  + 1  - I n t C^,  U f ( C ? ) n Ag) < diam f ( A) + 2  1  (fg(c£) n A g ) .  diam E _< 2 + d i a m ( f g ( C ^ ) fl A g ) .  Since  E  Therefore i s bounded, t h e r e  62* i s an i n t e g e r  j > i  such t h a t  f ^( A ) 0 f ^ C ^ ) n A  if  (Mg,Vg)  i s (n-2)-connected,  By Lemma 5.1,  compact.  C - d i f f eomorphism  by l e t t i n g  h (x) « x 2  h of (Int 2  1  "  1  f^lH  3 1  1  n  2  L = H  =  { A  e  3  1  & ( K  P = |L U L  f(cj).  i f x e  2  C  -  5  ±  ; (  and  V = f^ oh (M-D) 1  U O(A): Ae  - R  n  A e P(K ), ±  s(p) =  n  ) A  L =  0}  U NCf^CfgOtfJ)  such t h a t and  a  B(K[ ^). 2  n A ),K)|  0 (Mg-Vg)  5  ^ ^ ( I H  h  3  1  "  5  is  and a  ! )  n Mg  to a l l of M  2  I H " !)'c 1 1  3  1  p  and a  s(x) = x  i f p  ZA?? ), 1  e  R ,  of  n  We a p p l y Theorem 3.1  0  We l e t  U (R  2  s ( U ) U V => I K J .  f »Sof" ( ) 1  i s closed  g  and A c U] and  n  n  " !)  Then  1  o b t a i n a compact s e t C c R - P s: R  nM  ! )  + 1  U = f^ oh ofg(R )  C  5  hg(Vg)  with  Next we c o n s i d e r two open s u b s e t s  L  3  C^ ).  2  with  C^-j-C ?).  2  We may e x t e n d  2  2  Vg = f ( l n t  t h e r e i s a compact s e t Cg c Mg  i f x £ C .  h (x) = x  ^ ( I H  and  hg: Mg - Mg  0 5  and  2  and l e t  (n-3)-dominated i n Mg,  and  f-^( A ) c f ( I n t C*?).  then  2  that i s ,  1  p  M = M - fg(C*j),  Let Then  ^ 0,  2  C ?) r> E,  f (lnt  n  - IntC^ ) + 1  and  C™- d i f f eomorphism  i f x £ C, L e t s: M  s ( A ) = A for a l l M  and 's(p) =.p  he d e f i n e d otherwise.  63.  Then (since  Since  =  sohglg  C  f (c£)  U  f ^ l K j )  n  2  l  d E  ) .  A  2  c  h  i( ~ ) M  D  n  2  2  Consequently  UD.ch^M-D) U Soh »f (lR ). n  1  2  2  M - ( C ^ U D) c M - C  1  c h (M-D),  M = h^(M-D) U S o h g o f ^ R ) ^ M =  (M-D)  h^oSohgof^R  U  f .= h ' ^ s . h g o f g .  Then  3 1  we have  or  3 1  Let  Soh of (R ).  u  ).  f ( R ) r> D n  C.  D  Q.E.D,,  We c a n s t r e n g t h e n C o r o l l a r y 5 „ 3 as f o l l o w s : Theorem 5 . 3 °  Let  M  be a connected and N  w i t h two boundary components N ^ into  of  there i s a  Proof:  M  gj:  j = 1,2  - M,  3  ...  Int M =  inductively of j)  'N-^xCOjOo)  i g i  (c|)  a sequence into  M  h-^ f , Q  g  U g-j(Ci). J  of  and  Let  f °  s  We  5«3»  _  construct  C - d i f f eomorphisms  such t h a t - f o r each  c f ^ J + i ) ) ,  M -Ng.  onto  of Corollary f ,...  Then •  be a sequence o f  0=1  C*- d i f f eomorphism  5  such t h a t t h e i n c l u s i o n  2  N^x^co)  6 0  C™-diff eomorphisms such t h a t be t h e  n _> 5  L  i s a homo top y e q u i v a l e n c e , . - i = 1 , 2 .  C - d i f f eomorphism o f  Let  C^-n-manif o l d ,  f j l ^ o ^ ,  0 8  j > 1, = t >  ±  |  ,.  [(Jj}  64,  h: N,~x[0,oo) -» M be a  Let  h(N x[0,co))  n (g  2  j + 1  C - c o l l a r i n g such that 05  j  C^-diffeomorphisms  1  By Theorem 5 . 1 , there are  M - = M - f . ( N , x[0,j+l)).  of M . onto i t s e l f which are  r , and r  j  t—  i d e n t i t y on a neighborhood of the boundary o f M . such that  the  M  L  e  t  f  j  J ( N - L X [ j + l , j+2))) U r ( h ( N x [ 0 , o o ) ) .  c  2  j + l ' N x [ 0 , j+l]  =  1  ^ ' N ^ O ,  =  r  1 2  J  or of | 1  J  N  i  X  [  j  +  ^  j+l _ U Si(Ci) c f 2  1  M - N  2  , (N, x[0, 3+2)). J  = fC^xtO^oo)).  Corollary 5«5.  9  If M  two boundary components  -  o  r  we have Let f = l i m f . . 3  Then  Q.E.D.* is a  C^-n-manifold, n > 5,  with  N-^ and Ng whose inclusions into  are homotopy equivalences, then C°°-diffeomorphic.  o  =  Since  2  2  +  are  j + l j ^ j + l ^ ^ x t j+l,oo )  1  +1  S j l ( c | ) n h(N x[0,oo)) = 0  i = l  2  M = f . ( N x [ 0 , j+2)) U h(N x[0, oo)).  Then  M  ])) = 0. Let  ( C x ) U f (N x[0,j+2  N-jX R N,.x fR^ and Int M ;  I f M i s compact, then  C°°-diffeomorphic to N ^ t O j l J x R , '  i = 1,2.  MXR  is  65. BIBLIOGRAPHY 1.  K. Borsuk,  Theory o f R e t r a c t s , P o l s k a Akademia Nauk  M o n o g r a f i e Matematyczne, ~ t . 44, 1967. 2.  E, Ro C o n n e l l , D. Montgomery, and C. T. Yang,  Compact  Groups i n E , A n n a l s o f M a t h e m a t i c s , v . 8 0 , ( 1 9 6 4 ) n  3.  E. H. C o n n e l l , n _> 5,  A Topological  5  pp. 9 4 - 1 0 3 .  h-cobordism theorem f o r  I l l i n o i s J o u r n a l o f M a t h e m a t i c s , v . 11,. ( 1 9 6 7 ) ,  pp. 300-309. 4.  S. T. Hu,  Theory o f R e t r a c t s , Wayne S t a t e U n i v e r s i t y P r e s s ,  1965. 5.  K. W. Kwun,  Uniqueness o f the Open Cone Neighborhood,  P r o c e e d i n g s o f t h e American M a t h e m a t i c a l S o c i e t y , v . 1 5 , ( 1 9 6 4 ) , pp. 4 7 6 - 4 7 9 . 6.  E. L u f t ,  C o v e r i n g s o f M a n i f o l d s w i t h Open C e l l s ,  Illinois  J o u r n a l o f M a t h e m a t i c s , ( t o appear) 7.  E. L u f t ,  C o n t r a c t i b l e Open M a n i f o l d s ,  Inventiones  Math.,  v. 4, ( 1 9 6 7 ) , pp. 1 9 2 - 2 0 1 . 8.  J . R. Munkres,  E l e m e n t a r y D i f f e r e n t i a l Topology, A n n a l s  o f M a t h e m a t i c a l S t u d i e s , v . 54, r e v . ed. ( 1 9 6 5 ) . 9.  J . R. Munkres,  Higher Obstructions  t o Smoothing,  Topology,  v. 4, ( 1 9 6 5 ) , PP. 2 7 - 4 5 . 10.  M. H. A. Newman,  The E n g u l f i n g Theorem f o r T o p o l o g i c a l  M a n i f o l d s , A n n a l s o f M a t h e m a t i c s , v . 84 ( 1 9 6 6 ) , p p . 555-572.  66. 11.  R. P a l a i s ,  N a t u r a l O p e r a t i o n s on D i f f e r e n t i a l Forms,  T r a n s a c t i o n s o f t h e American M a t h e m a t i c a l S o c i e t y , v . 9 2 , (1959), 12.  PP. 125-141.  J . Stallings,  On T o p o l o g i c a l l y U n k n o t t e d Spheres, A n n a l s  o f Mathematics, v . 77, (1963), pp. 4 9 0 - 5 0 3 . 13.  J . Stallings,  The P i e c e w i s e - L i n e a r S t r u c t u r e o f E u c l i d e a n  Space, P r o c e e d i n g s o f t h e Cambridge P h i l o s o p h i c a l S o c i e t y , v. 5 8 , ( 1 9 6 2 ) , pp. 481-488. 14.  J . Stallings,  On I n f i n i t e P r o c e s s e s L e a d i n g t o D i f f e r e n t -  i a b i l i t y i n t h e Complement o f a P o i n t , D i f f e r e n t i a l and C o m b i n a t o r i a l Topology, P r i n c e t o n M a t h e m a t i c a l S e r i e s , v . 27, ( 1 9 6 4 ) , pp. 2 4 5 - 2 5 4 . 1 15.  J . H, C. Whitehead,  On  C -complexes, A n n a l s o f Mathematics,  v. 41, ( 1 9 4 1 ) , pp. 8 0 9 - 8 2 4 .  

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