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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,. 1965 A THESIS "SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a Supervisor: Dr. E. L u f t - i i -ABSTRACT 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 , piecewise l i n e a r , and d i f f e r e n t i a b l e manifolds. D i f f e r e n t i a b l e e n g u l f i n g so f a r was reduced t o piecewise 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 of a d i f f e r e n t i a b l e manifold and J. R. Munkres'theory of 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 homeomorphisms. In the f i r s t p a r t of the t h e s i s we observe that' the method of proof of 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 to a l o c a l lemma, simultaneously to a l l three c a t e g o r i e s of mani-f o l d s . We prove t h i s l o c a l lemma i n the d i f f e r e n t i a b l e 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 proof. Then we solve the problem of the exi s t e n c e of a s t r e t c h i n g diffeomorphism between complementary subcomplexes of 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 a p p l i c a t i o n s of engulfing,, Next we prove a theorem concern-i n g the uniqueness of open d i f f e r e n t i a b l e c y l i n d e r s which i s the d i f f e r e n t i a b l e analogue of the uniqueness theorem f o r open cones. A consequence of t h i s theorem i s t h a t i f M-^  and are compact d i f f e r e n t i a b l e manifolds w i t h diffeomorphic i n t e r i o r s then M-£ R and M^xR are diffeomorphic, where (R denotes the r e a l l i n e . • Another consequence i s t h a t i f a d i f f - . e r e n t i a b l e manifold i s the monotone union of open d i f f e r e n t i a b l e c e l l s i t i s diffeomorphic t o E u c l i d e a n space. •' We present s e v e r a l a p p l i c a t i o n s 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 c a t e g o r i e s o f manifolds. - i i i -Our methods are such that they apply also to noncompact manifolds. Theorem: Let M be a d i f f e r e n t i a b l e n-dimensional manifold and l e t u i J # # ' 5 U m ^ e ° P e n subsets of M such that U, = U V. ,, where each V.. . i s open i n M, CIV. , c V. (M - CIV, ., V. . , - C l V. .) i s k.-connected, with. 1, J 1, J ~rJ- X , J 1 , ' m k. < n - 3 i f k. > 0, .1 < i < m, J > 1, and dM c U V. .,. x x i = 1 X . J . Then, i f k^ + ... + k m + m >, n + 1, there are d i f f eomorphisms h^ of M onto i t s e l f such that h^ i s the i d e n t i t y on m Cl V, -,, 1 < i < m, and M = U h.(U.). ^i1- — "7 ' i = l 1 1 This theorem has several c o r o l l a r i e s . For instance, i f M i s a k-connected d i f f e r e n t i a b l e manifold of dimension n without boundary, k _< n - 3 i f k > 0 , and i f m > then M may be covered by m< open d i f f e r e n t i a b l e n - c e l l s . Using t h i s r e s u l t , we give a new and d i r e c t proof of the uniqueness of the d i f f e r e n t i a b l e structure of Euclidean n-space for n > 5. F i n a l l y , we prove a general h-cobordism theorem. Theorem: Let M be a connected d i f f e r e n t i a b l e manifold of dimension n, n > 5, with two connected boundary components N-], and'N 2 such that the i n c l u s i o n of N I into M i s a homotopy equivalence, i = 1,2. Then there i s a diffeomorphism of N -LX[0,OO) onto M - N 2 . - i v -ACKNOWLEDGEMENT I am greatly indebted to Professor E. Luft for suggesting the topic of t h i s thesis, f o r allowing me a generous amount of his time and fo r his constructive comments during the preparation of t h i s t h e s i s . I also wish to thank Professor J . V. Whittaker f o r his c r i t i c i s m of the draft form of t h i s work, and Miss Doreen Mah for typing i t . The f i n a n c i a l support of the National Research Council of Canada and the Unive r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. - V -TABLE OP CONTENTS Page INTRODUCTION 1 CHAPTER 0 Notation and Fundamental De f i n i t i o n s . 4 CHAPTER 1 Local c"- Engulf ing 7 CHAPTER 2 The C°°- Engulf ing Theorem 15 CHAPTER 3 A C°°-Stretching Diffeomorphism ... 35 CHAPTER 4 Open Cylinders 48 CHAPTER 5 Coverings of Manifolds 55 BIBLIOGRAPHY 65 1. INTRODUCTION There are now engulfing theorems for topological, piecewise l i n e a r , and d i f f e r e n t i a b l e manifolds. Different-iable engulfing so f a r was reduced ( i n [2]) to piecewise l i n e a r engulfing using the J.H.C. Whitehead t r i a n g u l a t i o n of a d i f f e r e n t i a b l e manifold ([153) and J.R. Munkres' theory of obstructions to the smoothing of piecewise d i f f e r e n t i a b l e homeo-morphisms ( [ 9 ] ) . In the f i r s t part of t h i s thesis we observe that the method of proof of M.H.A. Newman's topolog i c a l engulf-ing theorem ( [ 1 0 ] ) applies, up to a l o c a l lemma, simultaneously to a l l three categories of manifolds. We prove t h i s l o c a l lemma In the d i f f e r e n t i a b l e case and thus obtain a d i f f e r e n t -iable engulfing theorem which has a d i r e c t proof. A f t e r proving t h i s d i f f e r e n t i a b l e engulfing theorem, we prove a theorem, concerning the existence of a stretching diffeomorphism between complementary subcomplexes of a simpli-c i a l complex i n Euclidean space, which i s c r u c i a l f o r a l l applications of engulfing. This solves a problem posed i n [ 1 2 ] , p. 502. . Next we prove a theorem concerning 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 of the uniqueness theorem for open (topological) cones ([ 5 ] ) . A consequence of t h i s theorem i s that i f • and M 2 are compact d i f f e r e n t i a b l e manifolds with diffeomorphic i n t e r i o r s , then M-, y IR and M ? x R are diffeomorphic, 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 . manifold i s the monotone union of open d i f f e r -entiable c e l l s i t i s diffeomorphic to Euclidean space. We present several applications of d i f f e r e n t i a b l e engulfing which a c t u a l l y hold i n a l l three categories of mani-f o l d s . Our methods are such that they apply also to noncompact manifolds. Theorem 5»1« Let M be a d i f f e r e n t i a b l e n-dimensional mani-f o l d , and l e t U 1 - 9 • • -, U m be open subsets of M such that CO U, = U V, ., where each V. * i s open i n M, C l V. j c V i (M-Cl V i £>V± j + i " c l v i j) i s ki"connected, with m k. < n-5 i f k. > 0, 1 < i < m, j > 1 and 0M c U V_. ', . Then 1 1 ~ i = l ±> ± i f k^ + . „. + k m + m _> n+l, there are diffeomorphisms h^ of M onto i t s e l f such that 1 ^ i s the i d e n t i t y on C l V\ ^ » m 1 < i < m, and M = u h.(U,). i = l This theorem has several c o r o l l a r i e s . For instance, i f M i s a k-connected d i f f e r e n t i a b l e manifold of dimension n+l n without boundary, k <: n-5 i f k > 0, and i f m > then M may be covered by m open d i f f e r e n t i a b l e n - c e l l s . Using t h i s r e s u l t , we give a new and d i r e c t proof of the unique-ness of the d i f f e r e n t i a b l e structure of Euclidean n-space for n > 5. F i n a l l y , we prove a general h-cobordism theorem. Theorem 5«5« Let M be a connected d i f f e r e n t i a b l e manifold of dimension n, n _> 5, w i t h two connected boundary components and Ng such that the i n c l u s i o n of Nj_ i n t o M i s a homo-topy equivalence, i = 1,2. Then there i s a diffeomorphism of N-, x [0,oo) onto M-Np. 4. CHAPTER 0 N o t a t i o n and Fundamental D e f i n i t i o n s In t h i s paper, R w i l l denote the s e t o f r e a l numbers, I w i l l denote the u n i t i n t e r v a l [ 0 , 1 ] , R n w i l l denote E u c l i d e a n n-space, H11 w i l l denote the h a l f - s p a c e [( x ^ , . . . , x ^ e R n: x n _> 0 ] , S n~^ w i l l denote the u n i t ( n - l ) - s p h e r e i n R n, and D n w i l l denote the c l o s e d u n i t n - b a l l i n R n. By the word map we s h a l l always mean a c o n t i n -uous map. , I f X i s a t o p o l o g i c a l space, id.^ w i l l denote the i d e n t i t y map of X. D e f i n i t i o n 0.1. I f X i s a t o p o l o g i c a l space and A c X i s a subset, we say th a t the p a i r (X,A) i s k-connected i f 7r n(X,A) = 0 f o r a l l n < k. I f A i s ( k - l ) - c o n n e c t e d and X i s k-connected, then (X,A) i s k-connected. D e f i n i t i o n 0.2. L e t Y be a m e t r i c space w i t h m e t r i c d. I f A and B are subsets o f Y, the d i s t a n c e , d i s t (A,B), between A .and B i s d e f i n e d to be i n f { d ( x , y ) : x e A, y e B}. I f X i s a t o p o l o g i c a l space, and f and g are maps o f X i n t o Y, the d i s t a n c e , d ( f , g ) , between f. and g i s d e f i n e d to be s u p { d ( f ( x ) , g ( x ) ) : x € X}. D e f i n i t i o n 0.5. I f K i s a s i m p l i c i a l complex,.the i - t h b a r y c e n t r i c s u b d i v i s i o n o f K w i l l be denoted by p 1 ( K ) , and the n - s k e l e t o n o f K w i l l be denoted by K ^ . I f S i s a subset o f | K | , the neighborhood o f S i n K i s d e f i n e d t o be t h e subcomplex N ( S , K ) = { A e K : A i s a f a c e o f J i n K and A fl S ^ 0). D e f i n i t i o n 0 . 4 . I f A c \Rn} and f : A - (Rm i s a map, we say t h a t f i s a C°°-map i f i t can be extended t o a C^-map o f a n e i g h b o r h o o d o f A i n t o R m. D e f i n i t i o n 0 . 5 . A C°°-n m a n i f o l d M i s a l o c a l l y E u c l i d e a n 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 C ^ - s t r u c t u r e J. J Is a. c o l l e c t i o n o f p a i r s (U,h) s a t i s f y i n g f o u r c o n d i t i o n s : (1) Each (U,h) e J c o n s i s t s o f an open s e t U c M t o g e t h e r w i t h a homeomorphism h w h i c h maps U onto an open sub-s e t o f H n. (2) The c o o r d i n a t e neighborhoods i n J c o v e r M. (3) I f (1^,1^), (U 2,h 2) e J, t h e n h ^ h g " 1 : h2(U1 n U 2) - H n i s a C°°-map :with nonzero J a c o b i a n . (4) J i s maximal w i t h r e s p e c t t o ( 3 ) . The boundary, gM, o f M i s d e f i n e d t o be t h e s e t o f p o i n t s o f M w h i c h do n o t have a n e i g h b o r h o o d w h i c h i s C65-d i f f e o m o r p h i c t o (Rn. D e f i n i t i o n 0 . 6 . I f M i s a C ^ - m a n i f o l d w i t h o u t boundary, a f a m i l y o f maps {h^: t € 1} ( u s u a l l y w r i t t e n h^) i s s a i d t o be C r o - i s o t o p y o f M i f each h^ i s a C r a - d i f f e o m o r p h i s m of M onto i t s e l f , and the map H: M x I - M d e f i n e d by H(m,t) = h t(m) • i s a C°°-map. 6. D e f i n i t i o n 0 . 7 . ' A t o p o l o g i c a l space X i s s a i d to be 1-connected at co i f f o r each compact set C c X there i s a compact set D r> C such that X-D i s simply connected. D e f i n i t i o n 0 . 8 . I f A, B c R n are j o i n a b l e subsets then A*B denotes the j o i n of A and B. 7. CHAPTER ONE Local C"-Engulf ing Lemma 1.1. Let T = {(a,B) e R : 0 < a < B < 1}. There i s a C^-map 9 : R x T - R such that i f 8?(x) = e(x,a,B) then e£(x) = x i f x fi (0,1), e£([0,a]) = [0,B] and ^ ( x ) > 0 for a l l x e R. .3/ Proof: We l e t 9(x,a,B) be of the form 6(x,a,B) = x + g(x,a, B). We construct a C^-map g: R x T - R such that i f g^(x) = g(x,a,B), then g£(x) =0 i f x i (0,1), gj^(a) = 0 - a, and 7J-~(X) > -1. See Figure 1. dg dx FIGURE 1 To construct such a map we use bui l d i n g block: l e t e: R x T «(x,a,p) = i [ X F * * ™ i t , J a x _< a, and e(x,a, P) = 1 i f the following C*-map as a R be defined by i f x e (ct,B), c(x,a,p) =0 i f x > B, where 8, 1 c = ~a^6 + t^j5 dt. See Figure 2. a + 3 a--» x FIGURE 2 To define g~(x) for x > a, we need a modified version of the map e« We define s ( x , a 3 a i f x < a + ^  , 6: RxT- R by fi(x,a,0) = < 1, i f a + izP< .x < 1 - 1-P l - e ( x , l - ^ , 1 ) , i f x > 1 - i ^ , <T1 See Figure 3< FIGURE 3 We define R x T - R by g(x,a,0) C = f 6 ( t , a , P ) d t > B-a. Now define J 0 1 C X 6(t,a,p)dt, where ( B-a) • e ( x , 0 , a ) , x _< a g(x,a,B) = ( B - a ) - [ l - e ( x , a , p ) ] , x > a. g i s a C" map, and, i f x _> a, d § a 1 ^ ( 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 . of [ 1 1 ] , p. 129-Lemma 1 . 2 . Let f - j_, • • • he n r e a l - v a l u e d d i f f e r e n t i a b l e f u n c t i o n s of n r e a l v a r i a b l e s . Necessary and s u f f i c i e n t c o n d i t i o n s that the mapping f: R n -> R n defined by f ( x ) = ( f (x),.. , , f (x)) be a diffeomorphism of 0Rn onto i t s e l f are: * f i (1) d e t ( — — ) never vanishes (2) l i m ||f(x)|| = oo. Ilxll— Theorem 1 . 1 . Let A M c R m c R n be an m-simplex, A M = v * A m - 1 , where A ™ " 1 C R " 1 " 1 c R m, b ^  = 0 , and v = ( 0 , . . . , 0 , 1 ) e R m A l i e s on the x - a x i s . Let p: R m - R m _ 1 be the orthogonal m c • ° • p r o j e c t i o n . Let A c Ara be a clo s e d subset such that A = P _ 1 ( p ( A ) ) fl A M . Let U be an open set i n R n such that A u v * 9 A m - 1 c U, and l e t F be a clo s e d subset of R n such t h a t F n A m c A u v * aA™"1. Then there i s a compact set 10. C c R n-F and a C^-isotopy h t : R n - R n such that h Q = i d R h t ( x ) = x i f x i C, and A M c h-^U). Proof: (1) Let c = ^ r d i s t ( R n - U , A U v * ^ A * 1 1 " 1 ) > 0. Let T$1 = {x G A M _ 1 : d i s t ( x , A \j v * S A " 1 " 1 ) > c}, and N 2 = (x G A M - 1 : d i s t ( x , A u v * o A m _ 1 ) > 2 c } . Then .- and N 2 are compact, and W 2 c N ^ . Further, p _ 1 ( A M _ 1 - N 2 ) n A M c U , s i n c e A m - 1 - N 2 c u , and i f -1 / m-1 ..T \ _ m o-^./ A m-l\ . x e p ( A -N 2) n A , then d i s t ( x , A u v * 3A ) < d i s t ( p ( x ) , A u v * 3 A m _ 1 ) < c. I f N 2 = 0, then A " 1 C U, so we may l e t C = 0 and h, = i d '. From now on, we z R n assume that N G ^ 0. Let d = i d i s t ( N - L , F ) > 0. ( 2 ) Let g: A M _ 1 -• R he the continuous f u n c t i o n defined by g(x) = ||s(x) - xjj, where s(x) i s the i n t e r s e c t i o n of the l i n e through x p a r a l l e l to the x m - a x i s w i t h v * 3 A m _ 1 . Let g: A M _ 1 -• R be a C*-^ -approximation A A, to g. I f x G N-p then g(x) _> c and hence g(x) - > g(x) - - 75 .> If"* For each x G ^ we define a " v e r t i c a l s t r e t c h i n g i n t e r v a l " . Let 11. v x(x) = x + (g(x) - -£)-v, v 2(x) = x + (g(x) - |).v v^(x) = x, and v^(x) = x - d«v. Then [v-^(x),Vg(x) ] c U. The stretching interval w i l l be [v-^(x),v^(x)] and by "stretching" we w i l l map [v^(x) 3v 2(x)] onto [v1(x),v-^(x) ]. The interval [v-^x), v^(x) ] has length Y(X) = g(x) x- -0- + d > 0. To apply Lemma 1.1, we map the interval [v 1(x),v^(x)] linearly onto [0,1] such that v 1(x) is mapped onto 0 and v^(x) is mapped onto 1. Then Vg(x) is mapped onto a(x) = 4 . x ^ and 0 onto B(x) = Y (y^~^« Note that a ( x ) < B ( x ) . See Figure 4. (3) Before constructing h^ . we must construct a 0°°-function W : R n -• I with proper compact support. Let v, : be a C -function such that v-j_(x) = 1 i f x e N 2 and Cl( v£ 1 ((0,l])) c Int N x = {xeA m - 1: dist(x,AUv*aA m _ 1)>c} which is open in R m - 1. Consider next the compact set C Q = {x = (x^...,x m) e Rm: p(x) € N 2 and -d < x m < g ( x ) - -J}. Then C Q n F = 0. Let r\ = dist(C ' p ) > o . Let 12. FIGURE 4 13 . v 2 : R - R b e a c " - f u n c t i o n such that 0 < v 2 ( t ) < 1, i f t < n , v o(0) = 1 and v ( t ) = 0 i f t > n. n rn Let 7r: R -» R be the orthogonal p r o j e c t i o n and l e t r = po i r . We defi n e li(x) = v 1(r(x)).v 2 ( 2||x - 7r(x)||) f o r x e R n. Notice t h a t l ^ - ( x ) = 0 . d Xm • Let C = {xeR n: Tr(x)eC 0 and Hx-7r(x) || < -3}. Then C i s compact and C 0. F = (2f. We note that c i ( ^ 1 ( ( o , i ] ) ) n 7r _ : L ( c o ) c C . We defi n e h m : R n - R as f o l l o w s . Let x e R n. I f r ( x ) e N-^ l e t h^(x) = x m + t « u ( x ) • [ { s t r e t c h i n g C^-diffeomorphism w i t h respect to [ v 1 ( x ) , v ^ ( x ) ] a p p l i e d to x m} - x m ] = ( l - t - u ( x ) )'.xm + t-u(x) x t-v(r(x)). 9^(^](- 7 r ?^ T T[xm-( S(r( X))-|)]) + g(x)-|]. I f r ( x ) £ ]$ l 9 l e t h£(x) = x m . We note 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 defined by h t ( x ) = ( x x , •••, x m_ 1,h£(x),x m + 1, ... 5x n) f o r x € R3 We compute 14. (x) - ( 1 - f ia(*)) + t . u ( x ) ^ ( ^ ) j ( - ^ r T [ x r a - ( g ( r ( x ) ) . | ) ] ) Therefore the rank of the Jacobian of h^ i s n. Obviously, l i m ||h. (x) || = 00. By Lemma 1 . 2 , h, i s a C^-diffeomorphism. By c o n s t r u c t i o n , h Q = i d , R h t ( x ) = x i f x fi C, and A M c h ^ U ) . C o r o l l a r y 1.1. Let A M = v * A M - 1 c R N be an a r b i t r a r y m-simplex, l e t T 1 1 1 " 1 be the hyperplane i n R N spanned by A M _ 1 , and if1 the hyperplane spanned by A M , l e t p: T m - T m - 1 be the p r o j e c t i o n such t h a t p(v) = b l e t A c A M be a closed set such that A = p - 1 ( p ( A ) ) H A M , l e t U be an open subset of R N such that A l j v * d A m _ 1 c U , and l e t P be a closed set i n R N such that P D A M c A U v*5A m - 1. Then there i s a compact set C c R n - F and a C r o-isotopy h^: R N - R N such that h = i d , h,(x) = x i f x ^ C. and A M c h-.(U). R . -15. CHAPTER TWO The C°-Engulfing Theorem I f a > 0, l e t c£ = { ( X l , . . . , x n ) e R n: \x±\ < a}, and l e t I n t C^ = [ ( x . ^ ...,x ) e R n: \x±\ < a}. D e f i n i t i o n 2 .1. I f M i s a C°°-n-manifold, a set X cr M i s s a i d t o be k-dominated i f there i s a system {cp_^} of n C -diffeomorphisms cp^ : c^ -» M such t h a t (1) X• c u cp i(lnt Cj) (2) For each i , cp~1( cp±( C^) n X) c P ±, where P ± i s a k-dimensional subpolyhedron of C^. The set {cpi3 i s c a l l e d a k-dominating system f o r X, and each cp^  i s c a l l e d k-dominating coordinate map f o r D e f i n i t i o n 2.2. I f M i s a C°°-n-manifold and K i s a f i n i t e s i m p l i c i a l complex, a map f : J K | -» M i s s a i d t o be l o c a l l y l i n e a r i z a b l e i f there i s a system {IJK} of C ^ - d i f f e -omorphisms IJK: C^ -* M such t h a t (1) f ( |K|) c u t ^ I n t Cj) (2) For each i , there i s a s u b d i v i s i o n CT^(K) of K such th a t f - 1 ( ^ ( c j ) ) = 1%!,' where H ± i s , a subcomplex of o \ ( K ) , and f± . f | | H > | : R± - c£ c R n i s l i n e a r . 16. The set • i s c a l l e d a l i n e a r i z i n g system f o r f . 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 ( | K | ) i s k-dominated. D e f i n i t i o n 2.3« I f K i s a s i m p l i c i a l complex, Y i s a t o p o l o g i c a l space, S <z Y i s a subset, and f , g: | K | -» Y are maps, we say that f and g agree on S i f there i s a s u b d i v i s i o n CJ(K) of K such t h a t N ( f - 1 ( S ) , o - ( K ) ) = N ( g _ 1 ( S ) , cr(K)) = N, and f | N = g | r Ip i s w e l l known that a t o p o l o g i c a l manifold i s an absolute neighborhood r e t r a c t , see, f o r i n s t a n c e , [ 4 ] , p. 9 8 . In p r o v i n g the 0 ° ° -engulfing theorem, we s h a l l need the f o l l o w -i n g r e s u l t from homotopy theory? Lemma 2 . 1 . Let Y be a m e t r i z a b l e absolute neighborhood r e t r a c t w i t h metric d, and l e t e > 0 . Then there i s 6 > 0 such t h a t f o r every c l o s e d subset A of a metric space X and f o r a l l maps f - ^ f ^ 2 A -» Y w i t h d ( f 1 , f g ) < 6 , i f f1 has A A an extension r ^ : X -» Y, then f ^ has an extension f ^ : X - Y such that d ( ^ 1 , ^ 2 ) < e. Proof: This i s Theorem V . 3 . 1 of [ l ] , p. 103. Theorem 2 . 1 . Let M be a C^-n-manifold without boundary, V an open subset of M such that (M,V) i s k-connected, X c M a closed and k-dominated subset such t h a t X-V i s compact, k <: n - 3 . Let K be a f i n i t e s i m p l i c i a l k-complex, 17. f : | K | -» M continuous, L c K a subcomplex such t h a t f | j L j 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 system 2 = {iL-3 such that each A. i s a l s o a k-dominating coordinate J J map f o r X. Let e > 0 . Then there i s a map g: | K | -» M, a compact set C c M, and a C^-isotopy h^: M -» M such t h a t : (1) h Q = i d M , h t ( x ) = x i f x £ C, and h-^V) ^ X U g ( | K | ) . (2) g|, L| = f | , L , (3) d ( f , g) < e f o r some f i x e d m e t r i c d on M. C o r o l l a r y 2 . 1 0 (C™-Engulfing Theorem) I f M i s a C°°-n-manifold without boundary, V i s an open subset of M such t h a t (M,V) i s k-connected, X c M i s closed and k-dominated, X-V i s compact and k _< n - 3 , then there i s a compact set C cz M and a C^-isotopy h^: M -» M such t h a t h Q = i d M , h t ( x ) = x i f x j . C, and h^V) => X. Proof: Let K = 0 i n Theorem 2 . 1 . Proof of Theorem 2 . 1 . We f o l l o w Newman's proof of the t o p o l o g i c a l e n g u l f i n g theorem, [ 1 0 ] , a l l o w i n g f o r d i f f e r e n t i -a b i l i t y and u s i n g the us u a l method of s i m p l i c i a l c o l l a p s i n g , i n s t e a d of 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 three steps. For each x e M, we choose a C°°-coordinate map u x : C^ - M such t h a t x e n x ( M c £ ) , and 1 8 . ( 1 ) i f x e f ( | L | ) , \IX e S (2) i f x € X - f ( | L | ) , (j. i s a k-dominating coordinate map f o r . X such that M x (cJ) fl f ( | L | ) = 0 (3) i f x /•/ X U f ( then u x( c j ) fl (X U f ( | L |)) = 0. Step I r Reduction t o the case X c V Let A(m) denote the theorem w i t h the added hypothesisr X - V c u x ( I n t c £ ) U ... U ^ ( I n t c £ ) f o r some x ^ . . . 5 x m e M. (a) A ( l ) i m p l i e s A(m). Proof: Let u. = u } 1 < i < m. We use i n d u c t i o n on m m > 2. Let Xffi = X - u m ( l n t c £ ). Then X m - V c u-^Int C^1) U . . . u ^ ^ ( I n t s o t h e hypotheses of A(m-l) are s a t i s f i e d . Thus there i s a map g m r |K| _> M5 a co rn compact set' C m c M, and a C ' - i s o t o p y h^r M -* M such that ( 1 ) h™ « i d M , h£(x) = x i f x i C f f i 5 and hJ(V) = X m U g m( |K| ) ( 2 ) Sml|L| 8 8 f l | L | (3) d C ^ f ) < e/2 - -Now l e t f = g m , V = h*(V). Then X - V l c u m( Int c £ ) , so A ( l ) may be appl i e d r there i s . a map gr | K| -» M5 a compact set C c M, and a C r a-isotopy h i : M -» M such t h a t : • 19. (1) = i d M , h£(x) = x i f x i C«, and h'(V') 3 X U g ( |K|). & g ' | L | * f , l | L | ( 3 ) d ( g , f ) < e/2. Let C = C U C , and l e t h t = h^oh™. Then (1) h Q = i d M , h t ( x ) = x i f xdC, and h ^ V ) => X u g ( | K | ) . ( 2 ) S I | L | = f , i | L | = gml|L| = f'|L|-( 3 ) d(g,f)"< d ( g j g m ) + d(g r a,f) < | + | = e. (h) A ( 0 ) im p l i e s A ( l ) . Proof: Let n = u where X-V a u ( I n t C ? ) . Since u —————— 1 1 i s a k-dominating coordinate map f o r X, (-f^X fl u (C^)) c P, where P i s a k-dimensional subpolyhedron of C^. I f x1 £ f ( | L | ) , l e t f = f U y | : |K| U P - M. Then there i s a s u b d i v i s i o n a^(K) of K and a s u b d i v i s i o n crg(C^) °^ w i t h a subcomplex H c o-g(C^) such that |H| = P, and u _ 1 o f | ^ ( L ) : a-j_(L) -* CgCC^) ^ s a s i m P l i c i a l imbedding. I f A l e a l ( L ^ 3 1 1 ( 1 Ag e H are such that f(A-^) = u^Ag), i d e n t i f y A-^  and Ag, and l e t K* be the s i m p l i c i a l complex obtained from Og(K) U H by t h i s i d e n t i f i c a t i o n . Let p: o-^KjlIEHK* be the p r o j e c t i o n . I f x^ i f ( | L | ) , l e t H be a s i m p l i c i a l 20, complex i n C 1 such that |H| = P, and l e t K* = K U H. Let p: K U H _ K * he the i d e n t i t y , and l e t f = f U |i| p. Let f * : |K*I — M be defined by f = f*«p, and l e t L* = p ( c f 1 ( L ) U H), X* = X-u(Int c£). Then X* c V, and f * | i s a l o c a l l y l i n e a r i z a b l e imbedding, so we may apply A ( 0 ) : There i s a map g*: J K * | - » M , a compact set C c M , and a C^-isotopy h, : M - » M such t h a t : (1) h Q = i d M , h t ( x ) = x i f x £ C, and h x(V) r> X* u g*( |K*|). (2) g * | | L # j = f * | j L # j . (3) d(f*,g*) < e. Let g = g*op|j K|. Then: (1) h]_(V) z, X* U g*( |K*|) = X - u ( l n t C*) U. g( | K | ) U u(P) O X U g( | K | ) . (2) g| | L| = f | | L| : i f x e |L|, g(x) = g*.p(x) = f*.p(x) = f ( x ) = f ( x ) . (3) d(g,f) < s. Step I I : Reduction to-the case X c V, and f ~ 1 ( V ) c Int A*', f o r some A^ e K Let B ( i ) denote the theorem w i t h the added hypotheses: X c V and dim N(|K| - f _ 1 ( V ) , K ) < I, i . e . i f 21. A e K and f ( A ) n (M-V) £ 03 then dim A cr Let B(£,m) denote B(^) w i t h the added hypothesis: f ( | K ^ / _ 1 ^ | ) c V, and there are at most m £-simplices e K such that f ( A ^ ) $zf V, 1 < i _< m, and, f o r each {.-simplex t\l e K , i f f( Ll) O.X ^ 0, then f ( A*) c V. Note that A £ , . . . , A ^ must be p r i n c i p a l i n K . (a) B(i3 1) i m p l i e s B(i3m) f o r a l l m. Proof: We use i n d u c t i o n on m. Suppose B ( l 3 l ) and B(£,m-l) are t r u e , and the hypotheses of B(i,m) h o l d , f o r some m _> 2. Without l o s s of g e n e r a l i t y , we assume that e i s so small that f o r any e-approximation g: | K | -» M t o f , i f A ^ s . K i s an {.-simplex such that g( A ^ ) fl X ^ 03 then g( A ^ ) cr V. ( i ) Let K' = K - { A ^ } , L» = L - £ A £ ] , f ! ' ' = f | | K i | . By Lemma 2.1, , there i s e' > 0 such that i f g J : S A ^ - M i s an €'-approximation to f k A ' then there i s an extension m A t , A r g: L\t -* M of gt such that d ( f | A ^ g ) < Now m m • . d dim N( |K« I - (f«)" 1(V ) , K T ) < l3 f ( | ( K I ; ) ( H ^ |) c V 3 and there at most (m-l) -t-simplices A £ , . . . , A ^ ^ e K 1 such that f ' ( A ^ ) <zf V, !_<!_< m-l, so the hypotheses of B(-t,m-l) are s a t i s f i e d . Thus there i s a map g 1 : | K ' | -» M, a compact set C c M, and a C^-isotopy h£: M -* M - such t h a t : 22. (1) bJ = i d M , h£(x) = x i f x e C , and hj_(V) 3 X U gt( |K « I) . ( 2 ) S ' I | L ' | = f , | ! L « | " (3) d(f',g«) < e«. Let f: [Kj -» M be defined as f o l l o w s : f j j ^ j j = g 1 ; i f A^ G L, f | A-t = f | n i f ^ L. • l e t f | .1 be an 4 _ m > A m Affi m ^ 5 . 1 A m 2 approximation to f|A^ which extends g ' l ^ - t . ( i i ) Let V-•= h £ ( V ) . Now dim N ( | K | - f _ 1 ( V ) , K ) < i, • f ( |K^ _ 1^|) C V, and there i s only one t-s implex, A^ } i n K such t h a t f( A^) <£ V . Thus the hypotheses of B(|,,l) are s a t i s f i e d , so there i s a map g: |KJ M, a compact set C c M, and a C - i s o t o p y h^: M -* M such t h a t : (1) h Q = i d M , h t ( x ) = x i f x £ C, and h-^V) = X U g( |K|).. (2) g|, L, = f | | L | . (3) d(f,g) < |. Let C = C» u C, h t = htoh£.• Then (1) h Q = i d M , h t ( x ) = x i f x £ C, and h x(V) 3 X U g(|K|). W g | | L | = f l | L | * ' ' . (3) d(f,g) < e. 23« (b) B(i-l) and B({,,m), f o r a l l m, imply B ( { ) . Proof: Suppose B(i-l) and. B( {,m), f o r a l l m, are t r u e , and the hypotheses of B(l) are s a t i s f i e d . Let o(K) be a s u b d i v i s i o n of K so f i n e that I f A*' e a(K) i s an {-simplex such that f ( A1') fl X ^ 0, then f ( C V. Let K Q = [A e o-(K): f ( A) c V ] , = K Q U ( a ( K ) ) ( ^ _ l ) , = a ( L ) n^, f t = fljj^J. Then dimNCJK^I - (f!)"V),^) < t-1. L e t N e ! > 0 be such that any e'-approximation g': |K^| U | L q | M to ±*I ]Ki | y j L | c a n be extended to an -|-approximation g: | K | -• M to f. The hypotheses of B ( i - l ) are s a t i s f i e d , so' there i s a map g": I K ^ I -• M, a compact set C c M, and a C^-isotopy h£: M -» M such t h a t : ( 1 ) h^ = i d M , h£(x) = x i f x fi C J, and hj_(V) 3 X lj g«( | K ^ j ) . ( 2 ) g«| | L,, - f ' l | L , , . ( 3 ) d(f«,g") < e ' . • Let g»: | K ^ | U | L | - M be defined by g* I | K, | = g" 3 s '1|L| = f'|L| j' 3 2 1 ( 1 l e t ^ : ' K' ~* M h e a x x extension of g ! such that d ( f , f ) < -|, and I f A s a(K)-K^, then f ( A ) n X = 0. Let V = h£(V). 24. For some m, the hypotheses of B(t,m) are s a t i s f i e d , so there i s a map g: |K| -* M, a compact set C c M, and a C - i s o t o p y h^: M -* M such that (1) h Q = i d M , \(x) = x i f x £ C, and S 1(v) 3 x u g(|K|)., (2) g | | L , = f | | L | . (3) d(g,f) < f . Let C = C U C J, h t = h t«h^o Then (1) h Q = i d M , h t ( x ) = x i f x £ C, and h x(V) o X U g(|K|). ' (2) g | | L , = f | | L | . (3) d(g,f) < d(g,f) + d ( f , f ) < | + | = e. Step I I I : Proof of B ( t , l ) In view of Step I I , we need only show that B ( t - l ) i m p l i e s B ( t , l ) , i < k, since t h i s proves B ( k ) , and hence the theorem. Thus we may assume: (1) X c V. • (2) there i s an t-simplex A^ e K such t h a t | K | - f - 1 ( V ) c Int( kl) . (3) f ( A*) n X = 0 . (.'4) B(-t-l) i s t r u e . Let G = K U v * ^ , where v*A't = i s an ( t + 1 ) - s i m p l e x not i n K. Since (M,V) i s k-connected, there 25< A A i s a map f: |GI -* M such t h a t f | = f, and f(v*b&1) c V. Let H = (K-{A^}) U v^"3A't. Then . A A G = H U v*A% X U f ( |H|) c V, and X n f( |L n v*A^|) = 0. There are p o i n t s x.p...,x^ e M such that f ( A ^ ) c ( I n t LJ ... U W^( Int C^). Let u ± = u x_, 1 < i < N. Case A Suppose f ( A ) c u ( I n t C, ). Let u = u . 1 x x l There i s a number a such that 0 < a < 1 and f ( A W ) c u ( l n t eg). ( 1 ) X fl li(C^) c u ( P ) 5 where P. i s a k-dimensional sub-polyhedron of C^. There i s a s u b d i v i s i o n a Q(G) of G and a s i m p l i c i a l complex R l y i n g i n C^ such that |R| = P, ( ^ I J L J ) """(^(R)) i s a subcomplex of oQ(&) a n d t-l~1°f l(f | j L | ) - 1 ( ^ R ) ) : ( f I | L| ) " 1 ( l - L ( R ) ) - R i s s i m p l i c i a l . I f A 1 e aQ(L)3 Ag € R are such that f ( A-^ ) = u(Ag),_ i d e n t i f y A-^  and Ag and l e t G* be the s i m p l i c i a l complex obtained from a (G) U R by t h i s i d e n t i f i c a t i o n * Let p: cr Q(G) U R -i C-* be the p r o j e c t i o n , l e t L* = p ( a Q ( L ) U R), and l e t f * : |G*| - M be define d by f*«p = f u u| p. ( I f f ( l a Q ( L ) | ) n u(P) = 0 then 2 6 S G* = a Q(G) U R). ( i i ) By Theorem 4 of {10], there i s a map f * * : JG*| -> M such t h a t : (1) f * * and f * agree on M-u(lnt c " ) . ( 2 ) n _ 1 o f * * : |G*| -» agrees w i t h a PL-optimal map i n C^. (3 ) f * * l | L * | = f * l , L * | . ( 4 ) f**op(|H|) c V, and d ( f * f**) <-|-Let f« = f * * o P | | G | : | G | - M. Then: A N (1) f and f agree on M-ia(lnt C^). ( 2 ) l i ' ^ e f 1 agrees w i t h a PL-optimal map i n which i s " i n general p o s i t i o n " w i t h respect to P 0 C . (5) f'l, L| - f|| L, - f | | L ) . ( 4 ) f»(|H|)cV, and d(f«,f)<-§. ( i i i ) Let a]_(^) ^e a s u b d i v i s i o n of o-Q(G) such t h a t ^ • ^ ( f . - ^ O ) , *,(<*)) 1 N(f,_1(^cS))^ CTI(G)) -cl -1 ^. I , . ,T,„.-1, i s optimal and " i n general p o s i t i o n " w i t h respect to P fl c£ In Int C^, and such that c-L(G) ^ ^ ( H ) : there i s a sequence { E ^ + 1 } s of (t + l)-complexes l , i i=0 n such t h a t E1*1 = 0, E l + 1 = CTl(At+1)-, E^"1"1 = E ^ + 1 U A, X , 1,0 l , s . l , i + l l , i 2 7 . n n . - l -. n n . - l A ± = x ± * A i x 3 and (a-^H) lj E"^_) n A ± 1 = x±**&± 1 3 1, i 1 < 1 < s - 1 » ( i v ) I nduction Hypothesis: There i s a map g^: |G| - a compact set C. cz M, and a C°°-isotopy h. , : M -» M such that h i , 0 = ldW h i , t ^ x ^ = x i f x ^ C i > a n d h ± a l ( V ) o X U g^laxCH) U (E^V 0!). < 2 ) g i l | a 1 ( L ) U N ( f ' - 1 ( n ( C ^ ) ) , a 1 ( G ) ) | = = f ' l i a 1 ( L ) U N C f - ^ ^ C ^ ) ) , a x ( G ) ) | . (3) d(g.,f) < ICl-2"1). This i s c l e a r l y true i f i = 0 . (v) Induction Step: We have < l + 1 f o r i _< s-1, u ' ^ A . 1 ) n u(P)) U u ' ^ f U A / ) n f«( |a 1(H ) l j(a 1( | )) H " 1 ^ ' ( A ^ 1 n ( 0 l ( H ) U ( o 1 ( ^ + 1 ) ) ( ^ ) ) U Q±5 where . Q± i s 1 n.. a subpolyhedron of u~ < > f l ! ( A i x ) such that dim Q± _< (t+1) + k-n < t-2. n. , n. n . - l , n i - 1 Consider now A = u~ ° f ! ( A i ), A = ^ o f ! ( A ± ) , Thus ^ 1 ( f « ( A . n i ) n (X u f ' ( | a 1 ( H ) U ( a-, ( A ^ 1 ) ) ^ | ) ) ) c 28. n . - l and v = u ° f ! ( x i ) , l e t T 1 be the plane i n R n n . - l n. . determined by A^  "L , l e t T 1 be the plane i n R n.. n. n i ~ x determined by A "% and l e t 7r: T 1 -* T be the projection n. with TT(V) = b n 1 . Let = A 1 fl-/r C',nR(QI)), a subpolyhedron A 1 _ n. of A 1 of dimension _< {-1. Let P ± = ( f M ' ^ ^ A . ) ) = g 1 _ 1 ( u ( A 1 ) ) , D ± = | C T l(H) U ( E t + 1 ) ^ } ! U P ±, ~s i and l e t D. a = D. fl ( f ' ) ~\ u( C*J)). Then ' i s a polyhedral subset of |G|. (v i ) We now show that there i s a continuous map §j_+2_: IG-1 -* Ms a compact set C* c M5 and a 0°°- isotopy h|: M - M such that (1) h* = id M,h*(x) = x i f x i C*, and h ^ h . ^ V ) ) o X U g 1 + 1 ( D ± ) . ^ S i + l ! | L ! u | N ( f - 1 ( t i ( C ^ ) ) , a 1 ( G ) ) | = : = f , 1 | L | U |N(f«- 1 (^cS)) ,a 1(0))|-(3) d ( g i 3 g i + 1 ) < - I| 2. Proof: There i s a subdivision cr 2(G) of CT]_(G) such that CT2(Di) and a2^±^ a r e subcomplexes of r j 2(G), and 29. f i a 2 ( D ^ a ) - 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 complex i n C^. I d e n t i f y A-^  and A 2 i f A-^ , A g e P ( a 2(D i a)) and f ' ( A 2 ) = f ' ( A 2 ) , and l e t K* be the s i m p l i c i a l complex obtained from P(a 2(D i))- by t h i s i d e n t i f i c a t i o n . Let p: p ( a 2 ( D i ) ) - K* be the p r o j e c t i o n , l e t f * : |K*| - M be defined by f*»p = g^, and l e t L* = p(B(o- 2(L) n a 2 ( D i ) U a 2(D°))). Then f * l | L * | i s a l o c a l l y l i n e a r i z a b l e imbedding. Let V* = h. The hypotheses of B ( i - l ) are s a t i s f i e d by K*, L*, f * , X and'V* si n c e N(|K*| - ( f * ) _ 1 ( V * ) , K * ) c P ±, and dim P i < l-l. Thus there i s a map g*: |K*| -» M, a compact set C* c M, and a C^-isotopy h£: M -• M such t h a t : (1) h£ = i d M , h*(x) = x i f x £ C*, and h*(V*) 3 X U g*(|K*|). ( 2) «*«|L*| = f * ! | L * | ' (3) d(g*,f*) < Let g i + 1: IGI — M be defined by g i + 1 l |D.j = §*°P> and g i + 1 | At+1 = g ± |^Vrl = f• | A t + 1 . Then S l + l ' l L l u " |N( f'- 1(y ( cS)),a 1 (G) | = = f ' l | L | U | N ( f 1 ( u ( C ^ ) ) , a 1 ( G ) ) h i ^ 30. X U S i + l ( D i ) = X U 6*(I K *D c h j ( h i , l ( V ) ) -( v i i ) Let U = n" 1(h|oh j. X(V) n u( Int C*)), and l e t P = A. U M _ 1(X u f ' ( | a 0 ( H ) U (El+1)^1^ \)). Then d l , i n. n , - l n , - l P 0 A x = A ± U v*3A 1 and A ± \J v * 9 i 1 c U, so we may apply C o r o l l a r y 1.1. There i s a compact set C c R n - P and a c"-isotopy h£: Rn - R n such that h^ = i d n , h£(x) = x i f x / C , and h£(U) ^ A n. i R' Let C i + 1 = u(C^) U C . U C*, and l e t h t : M - M he defined by h t | ^  ^ = M.h£ and h j ^ c g ) * ldM. :-n(cg)- L e t h i + i , t = ht°( h? o hi,t)- T h e n (1) h i + 1 ^ 0 = i d M , h 1 + 1 ^ t ( x ) = x i f x £ C i + 1 , and h i + l j l ( V ) U g i + 1 ( | a 2 ( D . ) | U A."1) ^ X U S i + i d a ^ H ) U ( E ^ + 1 ) d ) | ) . l , i + l -<2> «i+il| L|U | N ( f t - l ( u ( c n ) h ffi(G))| = ^ f , | | L | u | N ( f ^ ( ^ C g ) ) , a ^ G ) ) ] ' (3) d ( g i + 1 , f ) < d ( g i + 1 , g i ) + d ( g . , f ) <-tp2 + |(1 - 2"1) = |(1 - 2 - ( i + 1 ) ) . 51. Thus we have completed the induction step, ( v i i i ) Let g = gsI|K|: |K| - M, h t = h t,. C = C g. Then (1) h Q = i d M , h t(x) = x i f x £ C, and h l ( V ) " h s , l ( y ) => X ' U MM 1*) U ( E ^ ) U ) I ) D X U g(|K|) & g i | L | " g s l | L | = f , ' | L | = . f ' l | L | -(3) d(g,f) < d ( g , f ) + d ( f ' , f ) < | ( 1 - 2 _ s ) + | < e . Thus Case A i s proved. Case B A ,1+1 For some N, £( b l ) c u-^Int c£) U...U u^ C Int c£) ( i ) There i s a number a such that 0 < a < 1 and f ( b l + 1 ) c u-^Int c£) u ... U M j j C l n t cjj). Let a 1(G) be a subdivision of G such that f o r each A € a 1(A*' + 1), there i s an integer j( A) such that f ( A) C ^ - ( A ) ( I n t C^), and a 1(G) ^ C T l ( H ) : there i s a sequence {E } of (t+1)-complexes such that l , i i=0 E l + 1 = 0, E ^ + 1 = o ^ A ^ 1 ) , E ^ + 1 = E ^ + 1 U A * 1 , where 1,0 l , s ± l , i + l l , i 1 n. n . - l , n. \ n . - l L± 1 = x i*A i 1 , and (a x(H) u E^ h ±) fl A± 1 = x ± * 3 A 1 1, l 1 < i < s - 1 . 32. ( i i ) Let e' > 0 be so small that i f g: | G | M i s an A e'-approximation to f, then g( A) c yx^ ^(int C^), for a l l A € a 1 ( A / f / + 1 ) , and, for any subcomplex Q of A*' + 1), IQl A any e'-approximation g: |Q| -• M to f l i^i may be e A extended to an —--approximation to f. 2 s ( i i i ) Induction Hypothesis: There i s a map g i : | G | -• M, a compact set c: M, and a C°°-isotopy h^ ^: M -» M such that: x h i , 0 = i dM» h i , t ^ x ) = x i f x ^ Ci> ^ h. (V) 3 X U g j M H ) U x> x 1 1 l , i <2) «il|L| = f | | L | * A (3) d(g.,f) < e ( l - 2 _ 1 ) . ( i v ) Induction Step: We reduce Case B to Case A: Let V = h. ,(V), H' = a.(H) U ( E l + 1 ) ( l \ x>x x 191 K' = H« U A ^ 1 \ G' = K' U A ^ 1 , f = g i l | G i j , n. n . - l L' = CTl(L) 0 G'. Then H' fl A ± = x i*SA j L 1 X u f'( |H' j) c V , and n. x n f'( | L « n A i x \ ) c x n f'( | L n A * - | ) = so the hypotheses of Case A are s a t i s f i e d . Thus there i s a map 33. g': |G'| -• M, a compact set C c M, and a C^-isotopy h£: M - M such that h^ =• i d M , h£(x) = x i f x fi 0Xt and X U g ' ( l K ' l ) c hJ(V' ) . g''|L'| = f , 1 | L ' | * d C f ' j g 1 ) < e«. Let g ± + 1 : |G| - M be defined by g 1 + 1 l | G i j = g 1 , g i + l l | L | ° f ' | L h a n d d ( S i + l ^ ) < < l - 2 - ( I + L ) ) . L e t h i + l , t = ht° hi,t> Ci+1 = C ' U C i ' T h e n : h i + l , 0 = idM> h i + l , t ( X ^ = x i f x * Ci+1> ^ . X U gi+xCla^H) U ( E ^ + 1 c X u g ' ( | K ' | ) c h 1 + 1 X ( V ) . 1,i+1 * S i + l ' l L l = f l | L | ' d ( g i + 1 , f ) < e ( l - 2 - < I + 1 ) ) . This completes the induction. Let g = g s, C = C s, h t = h g t . Then h Q = i d M , n-fc( x) = x i f x £ C, and X u g(|K|) c X U g s ( l o i ( H ) U ( E * + 1 ) ^ | ) c h l ( V ) . 1, s S||L| = f l | L | ' • d(g,f) < c. This proves Case B, and hence B(t3l). Q.E.D. 34, We note that C o r o l l a r y 2.1 holds f o r k = 0 w i t h no r e s t r i c t i o n on n. 5 5 . CHAPTER THREE A C"-Stretching Diffeomorphism Let A m = A k*A*- c R M c R N , where m = k + -t + l < n . We let El = { ( 0 , . . . , 0 , x k + 1 , . . . , , x k _ K r l ) e R M } , and let L c R M denote the x m-axis. We assume that A m is situated in R M such that A k c R K , A^ c E l , b , = 0 , and b . = ( 0 , . . . , 0 , l ) e L . A A Let ir: R N - R M , p: R M - R " 1 " 1 , and q: R M - L he the ortho-gonal projections, and let r = po-jr and s = q»7r. For \ e R , let H Q m( \) = {(x^ .. . , x j e R M : x m >-\), and l e t H m(\) = {(x.^ ... ,x m) e R M : x m > \ ) . Lemma 5 . 1 . Let U c R N be an open set and let e > 0 be such that o A m c U U H Q m ( l - 2 e ) . Let F be a closed set in R N , such that F 'rt A m c d A m . Then there i s . a compact set C c R N - F an, a C-- i soto P y h^: R n -» R n S U c h that: ( 1 ) h = id , h.(x) = x i f x 4 C, and K z A m c h^U) U H Q m ( l - 2 e ) . (2) If T is any linear subspace of R N which contains A m , then h.(T) = T for a l l t e l . 36. Proof: Step A: We f i r s t construct a "horizontal C*-stretching d i f f eomorphism" h •: R n - R n. If m = 1, let h' = id . We z z R n assume that m _> 2 for the rest of Step A and further that 1 - 2e > 0. (1) Let e Q > 0 be such that Am n (R m - K™(2eQ)) c U. If 6 > 0, let Bg" 1 = {x e R111*"1: ||X|| < 6}. We choose a fixed 6 > 0 such that i f DJJ = p'^B™" 1) n (H m( e Q)-Hj!J(l- c)), then c Int Am. Let D m - p " 1 ^ 1 ) n (H m(2e Q) -H*(l-2e)) See Figure 5. Finally, l e t FIGURE 5 37. C = ^min{dist(oA m - H™(l-2e),,Rn - U), dist(D m,, 9 A m)} > 0. We wish to construct a C^-isotopy h£: R n R n and a compact set C c R n - P such that: (1) h^ = i d n , h£(x) = x i f x e C, . and R (A m-D m) U C l ( a D r a - H m ( l - 2 e ) ) c h£(U) U H m ( l - 2 c ) . (2) h|(x) - x € R m ~ x for a l l x e R n _ 1 ' and a l l t e l . (2) Consider the continuous map g: 7r" 1(H^(o) - H m ( l ) - L) -• R A defined hy g(x) = ||f(x) - s(x) ||, where f(x) i s the point of 9A m l y i n g on the ray from s(x) through T T ( X ) . A A Note that g(x) = g(s(x) + \»r(x)) for a l l \ > 0 and a l l x e 7r" 1(H m(0) - H m ( l ) - L) . We construct a c"-c-approxi-A mation to g| n w with the same TT" (H^( e Q) - H ^ l - e ) - L) property as follows: Consider S f f l - 2 X [ e 0 , l - c ] = p - 1 ( { x £ R r a " 1 : ||x|| = l } ) ^ c c ) - H ^ l - c ) and g| o : S m " 2 x [ e _ l - e ] - . R. Let S m ~ ^ x [ e 0 , l - e ] ° g: S x [ e Q , l - e ] -• R be a C^-c-approximation to ^ s m - s x [ e o , i - c ] - L e t s ( x ) = * ( s ( x ) f o r a i 1 x e i r \ ^ t o ) - HjU-e) - L ) . 38. Now we may construct a "horizontal stretching interval" for each x e ir" 1(H m( e Q) - H™(1 - e) - L). Let U ] L(x) = s(x) + (g(x) - c)« [^x|n , u 2(x) = s(x) + (g(x) - 2 c ) - 1 ^ | | , u 5(x) = s(x) + U 4 ( X ) = B ( X ) + • The stretching interval w i l l he [u 1(x),u^(x)] and by "stretching" we w i l l map [u-^x),u 2(x) ] onto [u 1(x),u^(x) ] 6 6 The length of [u 1(x),u ] +(x) ] is y(x) = g_(x)-c-^g(x)-2'c-g > (4c+6) - 2c - | = 2c + - | > 0. Notice that [ ^ ( x ^ u ^ x ) ] c u i f x e Hm(2cQ) - K™(l-2e) - L, since d i s t ( u 2 ( x ) , 0 A m - H™(l-2e)) < |!f(x) - u 2(x)H < 3 c To apply Lemma 1.1, we map the interval [u 1(x),u 1^(x)] linearly onto the interval [0,1] by a transformation such that U-j^x) is mapped onto 0 and u^(x) is mapped . ||u (x)-u (-x)|| onto 1. Then u 2(x) is mapped onto a(x) = | | u ^ x ) . u ^ x j \\ c •  • , , , s M x ) ^ i ( x ) H B ' y ( x l ; a a d u 3(x) is mapped onto B(x) - ^ ( x j - u ^ x ) \  = Y(x)-| Y(x) • o f course a(x) < B(x). 5 9 . . Before defining h"., we must construct a C°°-function cp: R n -» R with the proper support. . Let p 1: R -• R he a C 0 0-function such that 0 < PiC*) < 1 f o r a 1 1 t € R, p 2(t) =1 i f 2 c o < t < 1 ~ 2 e» a n d P 2( t) = 0 i f * < e0> o r t > x " e Let C Q = { x e H m ( c o ) - H ^ ( l - e ) - L : ||p(x)H < g(x)-c} U . (L n ( H m ( c Q ) - H ^ ( l - e ) ) ) . Then C Q c Int Am. Let n = dist(C Q,P U SA m). Let p 2: R -» R he a C™-function such that 0 < p 2(t) < 1 for a l l t e R, p 2(t) = 1 If t < 0, and p 2(t) = 0 i f t > n. Let cp(x) = P l(s (x)).p 2 (2 | l x-7r (x) ||) for x e Rn. Let C = Cl(cp" 1 ( (0,l])) 0 7r" 1(Gl(C 0)). Then C is compact, C Q c C, and C D P = 0, Note that cp(x) does not depend on ||r (x ) | | . If x e T r " 1 ( H m ( e 0 ) - H m ( l - e ) - L), we let h £ ( x ) = x + t«cp(x). [{stretching diffeomorphism with respect to the interval [u 1 ( x),u i + ( x ) ] applied to x ) - x ] = fl/„, ||p(u1(x))'||-||r(x).|| x + t-q>(x)[8 P x (" ^ ' ) K ( x ) - u 1 ( x ) ) + ( 7 r ( x ) - u ] L ( x ) ; a ( x ) "v ' • • otherwise, we let h £ ( x ) = x . 40. Obviously, lim ||hjL(x) || = 00, so to apply Lemma 1.2, i|xII— • we need only show that the Jacobian matrix of h£ is always non-singular. By the definition of h|, i f ||r(x) || _< -|, then h£(x) = x, that i s , h£ is the identity on a neighborhood of ir~ 1(L). Thus we need only show that hi I , has a non-* R ^ i r " 1 ^ ) singular Jacobian matrix. We perform a coordinate transformation. We define a 0°°-diff eomorphism e: R n - T T - ^ L ) - R +xS r n- 2x(R m" 1) 4-' o where R + = {t e R: t > 0), S m" 2 = (x e R111"1: ||x|| = 1) , and j_ denotes the orthogonal complement, as follows: i f x e Rn-nr""'"(L), le t e(x) = (Hr(x)||, ^[Xj || , x - r(x)). Consider H+ = e..(h. I „ -, ) o e - 1 : R ^ x S ^ ^ R ™ " 1 ^ R ^ x S ^ x C R^ V . * ^ R -7T (L) + Let f t ^ y ) : R + - R + for (u,y) e Sm*"2x(Rm"1)J-, and t e I, be defined by ft,(u,y)M = Wh^ e'^ w^ u^ y)) )Jj . Then Ht(w,u,y) = ( f t . (u^y;).(^),u,y), and i f e-:L(w,u,y) € i r " 1 ( H m ( e 0 ) - H^(l-c) - L), then f ^ K y ) ( w ) = a(e"x(w,u,y)) Y(e (w,u,y)) 41. g(e - 1(w,u,y)) + c + w], otherwise f t ^ u y ) ( w ) = w. We must show that the differential of is every-where nonsingular. To prove this we have only to show that df. / 0 n —'kw > 0 f o r a 1 1 (w,u,y) € RfxSm"'^x(Rm*"J-)-J-, and a l l ' t e l . If e'-^w^y) £ r" 1(H m(e 0) - H^(l-e) - L), then d f t , d w l Y ) ( W ) = L S u P P ° s e e-^w^u^y) e>- 1(H j n(c 0)-H m(l- e)-L). F i r s t we note that g(e"1(w,u;,y)) does not depend on w by the construction of g. Hence a(e~'J"(w,u,y)), ' P(e-1(w,u,y) ), and Y( e~x(w,u, y)) do not depend on w. Further d ^ q ( z ^ > 0 for a l l z e R. We.differentiate: d f M U^ y ) ( W ) - l-t.cp(e-1(w,u,y))[-a;^(<1(w^u^))(^e"Viuiy))-c-w) d W o ^ K ^ y ) ) v ^ V ^ y ) ) ' + 1 ] > 0. Thus the rank of the Jacobian matrix of h£ i s n, so by"Lemma 1.2, h£ is a C^-diffeomorphism. It satisfies the required properties ( 1 ) and ( 2 ) . Step B; Next we construct a "vertical C*-stretching diffeo-morphism" h^: R n - R n such that-( 1 ) h^ = id n , h£(x) - x i f x £ C, and R 42. Am c hJ(hj_(U)) U 1^(1-26). h[l(x) - x e L for a l l x e R n and a l l t e l . Since Cl( S.Dm-1^(1-25)) c hj(U) 3 we may let d = dist(R n - h£(U), Cl( SDm - 1^(1-26))) > 0. (If m = 1, let D 1 = 3D1 = L n (H 1(2e Q) - H^(l-2e))) and D 1 = L n (H 1 ( e ) - H Q ( 1 - C 0 ) ) . If m > 2, we assume that d < 6. We notice that Am - H m(2e Q) c h£(U) hy the construction of h£. Let v = (0,...,0,l) e L. For each x e R n we define a "vertical stretching interval". Let ^ /-v x(x) = r(x) + eQ«v, v 2(x) = r(x) + 2eQ.v, v 5(x) = r(x) + (l-2c)-v, v^(x) = r(x) + (l-e).v. The "stretching interval" w i l l be [v 1(x),v^(x)], and by "stretching" we w i l l map [v-^(x),v 2(x)] onto [v 1(x),v^(x)]. The interval [v 1(x) Jv^(x)] has length Y = l-e~e 0. To apply Lemma 1.1, we map the interval [v 1(x),v^(x) ] linearly onto [0,1] such that v-^x) is mapped onto 0 and v^(x) is mapped onto 1 . Then v 2(x) e is mapped onto a = and v-^(x) is mapped onto B = We note that a < B. See Figure 6. 1 f' / FIGURE 6 ( 2 ) Before defining h£ we must construct a 0°°-function ijc R n R with the proper support. If m = 1 , let 7] = d i s t ( D ^ F ) . If m > 2 , note that DJJ C C and hence dist(D^F) < dist(C 0,F) = TI. Let X1: R -» R be a C*0-function such that 0 <: X^t) < 1 for a l l t e R, \{t) = 1 i f t < 0 and = 0 i f 4 4 . t > 1> If m = 1, let * ( x) = *i(2||x-ir(x) | | ) , and let C = 7r _ 1(D 0) n C l ( r 1 ( ( 0 , l ] ) ) o If m > 2, let XgJ R R be a C w rfunetion such that 0 < X 2(t) < 1 for a l l t e R, X 2(t) = 1 i f t < 6-d, and X 2(t) =0 i f t > 6. Let ty(x) = X 1(2 |1X-TT(X) ||)»X2( ||r(x) ||) for a l l x e R n„ Note that |i-(x) = 0, and m c iCT^CO,! ] ) ) n 7r-1(Hm(co) - H£(1-C)) cc, Let x € Rn. Similarly as in Step A, we define h^ m(x) = x m + t.*(x)[Y-e£(^S-!2) + e Q - x j , - and then hj(x) = ( x i ^ - ^ x m _ i ^ h t m ( ' x ) i x m + i > « « ^ x n ) ' » W e compute oh" r a oh" m(x) Q x — s zsr-W'- —Ix— = (i-t - t(x)) + t . t ( x ) e ' ^ ) > o. m m r Hence the rank of the Jacohian matrix of h£ is n, and again lim ||h£(x) || = oo . By Lemma 1.2, h£: R n - R n is a C°°-diff eomorphism onto R n which satisfies properties (1) and (2) by construction. Combining Step A and Step B, we let h t = h£oh£. Then h t(x) - x € R m for a l l x e Rn, so ht(-T) = T i f T is a linear subspace of R n which contains Am» Q.E.D. 45. I f A m = t^*^ c R n i s an a r b i t r a r y ra-simplex i n R n, l e t E(A m) be the m-dimensional plane determined by A m, l e t E (A k, L l ) be an (m-1)-dimensional plane i n E(A m) p a r a l l e l to the planes E(A k) and E(A^) determined by the s i m p l i c e s A and A r e s p e c t i v e l y , and w i t h E(A , A ) D I n t A m £ 0. Let H^( A m, A*') be the component of E( A m) - E( A k, A^ which contains A C o r o l l a r y 5.1. I f A m = t^*h*' i s an a r b i t r a r y m-simplex i n R n, U i s an open set i n R n such t h a t S i 1 " c U U H^( A m, A* and P i s a c l o s e d subset of R n such that P. n Ara c SA m, then there i s a compact set C c R n - F, and a C^-isotopy h t : R n - R n such t h a t : (1) h = i d , h,(x) = x i f x £ C, and R A m c h^U) U H m( Am,A^-). (2) I f T c R n i s a hyperplane c o n t a i n i n g A m, then h t(T) = T f o r a l l t e l . Theorem 5.1. Let K be a s i m p l i c i a l complex i n R n, L a f u l l f i n i t e subcomplex of K, and L c = {AeK: A 0 L = 0} the subcomplex complementary to L. Let U and V be open sets i n R n such that |L| C'U and | L C | C V. Let F c R n be a cl o s e d set such t h a t F 0 |K| C |L| U |L°|. Then there i s a 46. compact set C c R n - F and a C°°-isotopy h^: R n -» R n such t h a t : (1) h Q = i d n , h t ( x ) = x i f x i C, and |K| C h-^U) U V. R ( 2 ) h t ( A) = A f o r a l l A e K and t e l . Proof: I f A e K - (L U L C ) , then A = ^*t\l where A k e L and A^ e L°. For each p r i n c i p l a l simplex A e K - ( L U L ° ) l e t H™( A, A^) be chosen so th a t A 0 H J ( A, A 1) c V. „if A e K - (L\U L°) i s not a p r i n c i p a l simplex, then l e t E£(A,A* ) .= „ n H £ ( A , ^ ) A i s a p r i n c i p a l simplex i n K-(LUL ) w i t h A<A . Let F« = F U |L| U |L°|. Induction Hypothesis: There i s a 0 ° ° -isotopy h^""1": R n R n and a compact set C m _ i c R n - p» such t h a t (1) h ™ _ 1 = i d n , h^" 1(x) = x i f x i c and f o r a l l R A e K ^ " 1 ) - (L U L C ) , A c h ^ - ^ U ) U H™( A, A^). ( 2 ) h£ - 1(A) = A f o r a l l A e K and t e l . This i s c l e a r l y t r u e f o r m = l(h° = i d f o r a l l t e I ) . • z R n Induction Step: Let there be k m m-simplices A^,...,A^e _ ( L u L c ) . F i r s t note t h a t i f A' i s a face of A, then 47. lA A', A'l) c H £ ( A, A*). Hence d A^ c h ^ U ) U l£( A*; A*) f o r 1 < J < k . Let P. = P' U {AeK: A 0 Int A™ = 1 < J < km- T h e n P i H A™ = 3A™. For each j.= 1 , k , we apply C o r o l l a r y 3.1 wi t h A"1 = Ak*A*', where A k e L, A^ e L°, F. i s the closed subset, h^" (U) i s the open subset, and w i t h respect to H ^ ( A ^ A J ) . There are i s o t o p i e s h 3 ^ : R N - R N and compact subsets C . c R N - F. such t h a t : W ho'3 = l d Rn' h?' J' ( x ) = X " X * ^ A»? C h ^ h ^ U ) ) U 4*), i = 1 . . . . . V (2) I f T c R N i s a hyperplane c o n t a i n i n g A™, then h^ J'(T) = T f o r a l l t e I . We conclude that A) = A f o r a l l A e K and t e I. Let h£ = h^N.-....^-1.^1 and C m = C ^ U C % 1 U 3 m (1) h™ = i d n , h m ( x ) = x i f x i C m, and f o r a l l R A 6 K ( m ) - (L U L C ) , A c h ^ ( U ) U H™(A,A^). (2) h^(A) = A i f A 6 K and t e l . I f dim K = k, l e t = h k and C = C^. Q.E.D. 48. CHAPTER POUR Open C y l i n d e r s Theorem 4.1. Let M-^  and Mg be compact connected C™-manifolds and l e t f : M-^x R-* MgX R be a C 0 0 - d i f f eomorphism such that MgX{0} c f(M-xR). Then, f o r any number p > 0, there i s a A C 0 0 - d i f f eomorphism f of M-* R onto MgXR such t h a t *\x[-p,p] " f ' M l x [ - p , p ] - ^ t h e r , i f f(V(-»*P)) => A M 2x(-oo,0], we may r e q u i r e that f l ^ ^ . ^ ^ j = f ^ ( - 0 0 , p]' Proof: Our proof i s s i m i l a r t o t h a t used by K. W. Kwun i n [ 5 ] . (1) There are p o s i t i v e numbers a and b, w i t h a > p, such that M 2x[-b,b] c f ( M - ^ - a , a ) ) . Without l o s s o f g e n e r a l i t y , we may assume th a t f (M-Lx( a+1, oo )) D MgX(-oo,b) = 0 (otherwise f i s re p l a c e d by i t s r e f l e c t i o n ) . Let g Q be a C°°-diff eomorphism of M-^ IR onto i t s e l f such that g o l M 1x((R-(-a,a+l)) = i d M 1 x ( R - ( - a , a + l ) ) 3 ^ d f »g0(M1x(-oo,a]) c M 2x(-oo,b). Let - fogQ. Then f- L(M 1x(-co , a]) c M 2x(-co,b) and M 2x[-b,b] c f 1 ( M 1 x ( - o o , a + l ) ) . g Q and a l l other C^-diffeomorphisms used i n t h i s proof may be constructed by u s i n g Lemma 1.1. 49. Suppose we have constructed a sequence f ^ , ... f o f C^-diffeomorphisms of M^xe i n t o MgXR such that f i ( M 1 x ( - o o j , a + i - l ) ) c M 2 x ( - o o , h + i - l ) a M 2 x [ - b 3 b + i - l ] c f . ( M 1 x ( - o o 3 a + i ) ) , and f±\u1x(-CD,a.+l-2] = f i - l ' M 1 x ( - c o , a + i - 2 ] ^ 1 ^ 2 ' L e t h, be a C * - d i f f eomorphism of M J R onto i t s e l f such that \ l M 2 X ( _ 0 0 a b + k . 1 ] = i dM 2x(-oo,b+k-lp 8 1 1 ( 1 h^of^(M1x(-co,a+k)) o MgX[-b,b+k]. Let g^ be a 0°°-diff eomorphism of M-^ x IR onto i t s e l f such t h a t ^ l M B - ( ^ 1 . ^ l , , - ^ ( ^ l . ^ ) ) . -nKe f k 6 M M l x ^ a + k ^ c M 2x(b+k-l,b+k). Let - f k + 1 = h k°f k 9S k< Let f = l i m f±. Then MgX(-b,,oo) c f ( M 1 x e ) . Note t h a t j_-»eo f'M 1x(-co,a+l] = hl° so 8SllM 1x(-oo,a+l]' H e n c e . - 1 - 1 h l ° f° go lg1»g0(M1x(-oo,a+l]) = f'g 1<»g 0(M 1x(-co Ja+l])-Note that g- L< ,g 0(M 1x(-oo,a+l)) r> M-Lx(-co, a ] . Let f * = h - 1 . ? ^ - 1 ^ ; 1 . Then f l ^ x ( . p > p y - f l V ( - p , p ) » and M 2x(-b,oo) c f * ( M 1 X R ) . I f f(M 1x(-co/p)) 3 MgX(-co .,0 ], 50. l e t A f ( x ) « < f ( x ) , xeM 1x(-oo,p) f * ( x ) , xeM x(-p,oo'). I f f(M 1 x(-oo, p)) | M 2 X ( - O O , 0 ] v f * may be extended i n a manner symmetrical to the methods of (2) to o b t a i n the A r e q u i r e d f . Q.E.D. Lemma 4.1. There i s a 0*°-diff eomorphism h of [ 0 , l ] x ( 0 , l ) onto ( - l , l ] x [ 0 , l ) - [ 0 , l ] x { 0 ] which leaves a neighborhood of ( l } x ( 0 , l ) f i x e d . Proof: Let f : [-1,0] - [0,1] be a C 0 0-function such that f l ( - l , 0 ) i s a 0"-imbedding, f ( - l ) = 1 , f ( 0 ) = 0 , | | ( x ) < 0 and f ( x ) < -x f o r -1 < x < 0. See Figure 7. We move (0}x(0,l) onto the graph of o) t y m e a n s 0 l* a h o r i z o n t a l s t r e t c h i n g diffeomorphism h 1 . The obvious l i n e a r t r a n s f o r m a t i o n which c a r r i e s -| onto 0 and -1 onto 1 c a r r i e s 0 onto f _ 1 ( y ) onto B(y) = ^ ^ ( y ) f o r a l l a = and c a r r i e s y e (0,1). I f ( x , y ) e [ 0 , l ] x ( 0 , l ) , l e t h x ( x , y ) = ( | - | e ^ y ) ( - | ( x - | ) ) , y ) . Then h 1 ( [ 0 , l ] x ( 0 , l ) ) = {( x , y ) : r ^ y ) < x < l , 0 < y < l } . FIGURE 7 51. Next we construct a v e r t i c a l C09- s t r e t c h i n g d i f f eo-morphism hg which c a r r i e s h ^ ( [ 0 , l ] x ( 0 , l ) ) onto ( - l , l ] x [ 0 , l ) - [ 0,l]x ( 0 ) by moving the graph of f q N onto (-1,0)x{0}. The obvious l i n e a r t r a n s f o r m a t i o n which c a r r i e s -x onto 0 and - i onto 1 c a r r i e s f ( x ) onto a(x) = f ( x } + x and c a r r i e s 0 onto p(x) = ~ r , f o r x < 0. x x- g v x- 3 C l e a r l y , a(x) < B(x) , since f ( x ) > 0. I f (x, y ) e h 1 ( [ 0 , 1 ]x ( 0 ,l)), and x < 0, l e t hg(x,y) = ( x, ( x - i ) 0^ x j (g|)-x). I f ( x , y ) e [ 0 , l ] x ( 0 , l ) , l e t hg(x,y) = ( x , y ) . Then hg'h-^ [ 0 , l ] x ( 0 , l ) ) ± S C*, and i s a d i f f eomorphism si n c e |^((x-|) B^||j ( ^ ^ ) - x ) = (x - | ) 9 ' ^ | x j ( x 5|) x^x > 0 f o r a l l x < 0 and a l l y e R. Further, h g o h 1 ( [ 0,l]x ( 0,l)) = ( - l , l ] x [ 0 , l ) - [0,l]x{0} . Let h = hgoh^. Q o E . D . C o r o l l a r y 4.1. Let M-^  and Mg be compact C^-manifolds such t h a t I n t M-^  and I n t Mg are C^-diffeomorphic. Then M-^x R and MgXR are C^-diffeomorphic. Proof: Let f±: S M ^ t - l , oo) - M i be a C°°-collaring of bM± i n M ± (see [8], p. 56), and l e t M[ = M ±-f ±( S M ^ t - ^ O ) ) , M i = M i - f i ( S M i x t - 1 j 1 ) )> 1 = 132' W e construct a C°°-diff eomorphism h i o f M ^ C 0 * 1 ) o n t o I n t M ix [ 0,l) - Mj_x{0}. Let h ± be the c"- d i f f eomorphism of (M^-Int MV)x (0,l) onto aM ix|-l,l) x ( 0,l) 52. defined by h^(m, t ) = ( f T 1 ( m ) , t ) . Let h be as i n Lemma 4.1, A and l e t h ± : B M ^ O , ^ ^ , ! ) bM ±x( ( ~ l , l ] x [ 0 , l ) - [ 0 , l ] x { 0 } ) A A be defined by h ^ m ^ y ) = (m,h(x,y)). leaves a neighborhood of 9 M i x { l } x ( 0 , l ) f i x e d . Let h£: Mj_x ( 0,l)-IntM ix [ 0,l)-Mj_x{ 0 3 be defined by 1 A ^ M - I n t M " ^ , ! ) "  hl #V hi» ^ hi>Mi;x ( 0,l) = i dMi;x (o,i)-Let g be a C a-diffeomorphism of Int onto Int Mg. Let D = ( h j _ ) " 1 . ( g x i d [ o ^ 1 ] ) " 1 o h 2 ( M 2 x ( 0 , l ) ) c M£ X(0,1). Since M-^  and M 2 are compact, there i s a number a e (0,1) such t h a t A 1 M£x(a,l) c=D, and i f we l e t f = (hg) o ( g x i d j - Q 1 ) ) * h j _ l D , then A f o r some b e ( 0,1), f ( M j x ( a , l ) ) r5 MgX{b}. Then Theorem 4.1 im p l i e s that M'x(a,l) i s C°°-diffeomorphic to Mg x R, and th e r e f o r e that l^xflR i s C°°-diffeomorphic t o MgX R. Q.E.D. C o r o l l a r y 4.2. I f and M 2 are compact C*-manifolds such t h a t I n t i s C w-diffeomorphic t o Int Mg, ' then BM-^R i s C^-diffeomorphic t o SMgX R •. Theorem 4.2. Let M be a C^-n-manifold such that 00 M = U 0 . , where O1? i s an open C ^ - n - c e l l i n M w i t h i = l 1 1 °i C °1+1 > f o r a 1 1 1 - l m T h e n M i s c°°-diffeomorphic to R n. 55. Proof: Let f ^ : R n -* M be a C w - d i f f eomorphism such that f i ( R n ) = 0*? , i > 1. We may assume that f ^ O ) = p e M, i > 1. Since M i s the union of countably many compact s e t s , we may 00 f u r t h e r assume that M = _U f ±(D*?) (where D n = {xeR n: ||x||<i3). We construct a sequence of C^-diffeomorphisms g i of R n i n t o M such t h a t g± = f±, S i + 1 l D n = S±\r^9 g±(&n) = o£, and S i + 1 ( D i + 1 ) ^ f i + 1 ( D ^ + 1 ) . 1 > !• Suppose that g ^ . . . ^ are c onstructed. Define S k +]_ a s f o l l o w s : consider fk+l° sk ' e n -{0} *  R n " ^ 0 ^ - R n - { 0 } . By Theorem 4 .1 , there i s a C™-diffeomorphism h k + 1 : R n-{0} such that V l ^ - C O } = f k i l e g k l ^ - { 0 } J W^" W > = ^ " CO}, and W^+i* 3 D£+r L e t W x> = g k ( x ) i f X 6 ^ and g k + 1 ( x ) = f k + l o h k + i ( x ) i f x e R n - {0}. Then g = l i m f i i s a C^-diffeomorphism of R n onto M. Q.E.D„ i - * o o The f o l l o w i n g theorem may be proved i n a s i m i l a r ~ manner: Theorem 4 . 5 . Let M be a C "-manifold w i t h compact connected boundary 3M. I f there are C ^ - c o l l a r i n g s f^: 3Mx[0,oo) -M CO such that M = U f.(SMx[0,oo)), and i = l 54. f ±(aMx[0,oo)) c f i + 1(5Mx[0,oo)) , i > 1, then M i s C* 0-diffeomorphic to SMx[0,oo). 55. CHAPTER FIVE Coverings o f M a n i f o l d s The f o l l o w i n g lemma i s a consequence o f C o r o l l a r y 2.1: Lemma 5.1. L e t M be a C™-manifold, and l e t g: R n -» M be a C™-diffeomorphism. L e t P be a k-d i m e n s i o n a l sub-polyhedron o f R n, not n e c e s s a r i l y compact, such t h a t g(P) i s c l o s e d , and l e t U c M be an open s e t such t h a t g(P)-U i s compact. L e t E D SM be a c l o s e d s e t such t h a t E c u , and (M-E,U-E) i s k-connected. I f k _< n - 3, t h e r e i s a compact s e t C c M-E, and a C°°-diff eomorphism h: M-*M . such t h a t h(U) g(P), and h(x) = x i f x £ C. Note t h a t h | E = i d E , and, i n p a r t i c u l a r , t h a t h the i d e n t i t y on a neighborhood o f dM. Lemma 5.2. L e t M be a C^-n-manifold, l e t U^ ,...,U^ , V-^ , ...,V m be open subsets o f M such t h a t C l V i c and ( M - C l V j_,U i-Cl V ± ) i s k - c o n n e c t e d , i f k ± > 0, l e t k^ < n - 5, 1'< i < m. L e t E^, . . . , E m be c l o s e d subsets o f m M such t h a t E. c V-, 1 < i < m, and SM c U E.. L e t 1 1=1 1 g: C^ - M be a C™-diff eomorphism and l e t 0 < a < 1. I f k n + ... + k + m > h + 1, there are compact s e t s C-,,...,C 1 m — 3 l - 5 m, i n M such t h a t • C i 0 ( E . U SM) = ^ 1 < i < m, and C^-diffeomorphisms h^ o f M onto i t s e l f such t h a t h i ( x ) = n m i f x £ C., 1 < i < m, and g (c" ) c U h.(U.). 5 6 . Proof: L e t G be the s i m p l i c i a l complex determined by a s i m p l i c i a l subdivision of C n such that C^ i s the set of points of a subcomplex K of G, • JN(K,G)| C I n t C^9 and for any simplex A e G such that g( A) fl E . ^  0} we have g ( A ) c V i , i . e . : g( |N( g~X( E ^ G ) |) c V±9 1 < i < m. Let L Q = K. We construct i n d u c t i v e l y two sequences L L -i and K, ,...,K n of s i m p l i c i a l complexes as o' 3 m-1 1' 3 m-1 (k.) follows: suppose Lj__]_ i s defined. Let = 1 and l e t L i be the complementary complex of i n P C L ^ - ^ ) , 1 < i < m - 1. Then dim L. = n-i-(k,+.. .+k.). Thus — — I V 1 i d i m Lm-1 = n - m + 1 " ( k l + " - + k m - l ) < V L e t Km 8 8 Lm-1' We"now apply Lemma 5»1 with respect to each K^. Let P ± = g" 1(g(|K i l ) - C l V i ) . Then P i i s a k^dimensional polyhedron i n Int C^ - g ' ^ C l V^), g(P i) i s closed i n M - Cl \V^9. and g(Pj_) - i s compact, so there are C -diffeomorphisms h|: M - M and compact sets C^9 1 < l < m,• such that h|(x) = x i f x £ Cf±9 , and g(P i) c h j ( u \ ) a Let W = g ' ^ h ^ ^ ) ) , 1 < i < m. Then \K±\ C W±. The barycentric subdivisions used"in the d e f i n i t i o n s of and L i imply that K i and are f u l l subcomplexes of 57 o 6(L^ ^ ) , 1 _< i _< m - 1. Applying Theorem 3°13 we construct in d u c t i v e l y a sequence of C^-diffeomorphisms s m _ i ' -* such that S .^ i s the i d e n t i t y on - |N(K,G)| U |N(g" 1(E i) 5G) I, 1 < i < m - 1, and l Lm - 2 l c S m - l ( W m - l ) U Wm * K-?\ c Sm- 2( Wm- 2) U Sm-l(Vl) U V | K | = i L j c S ^ ) U . . . U S ^ W ^ ) U V For example, we construct ^. In the notation of Theorem 3.1, l e t U = W ^ V = W , L = K m_ 1 U O ( A ) : A € L m _ 2 and g( A) n ^ ^ 0} L C = {A G P ( L m _ 2 ) : A fl L = 0}s F = (R n - |H(K,G)|)'U g'^ Vl) Note that |L| C W m , L i s f u l l i n P ( L m _ 2 ) , and F 0 | P ( L m _ 2 ) | c |L| U |L C|. Let be the h± obtained i n Theorem 3.1. We l i f t the C 0 0 - d i f f eomorphisms onto M: l e t A A -1 ' n S ±: M - M be defined by S ±(p) = g,°S ± a g (p), i f p e g(Cj), A and S±(p) = p I f pj^ g( (N(K,G) |), 1 < i < m - 1. Note • 5 8 . A that S ±| E_ = i d E _ . It follows that g(C^) = g(|K|) c S ^ h ' ^ ) U ... U S f f l . r h ; - l ( \ - l ) U h m ( V ' L e t A h± = S ±«h|, 1 < i < m - 1, and l e t h m = h^. Q.E.D. Theorem 5 . 1 . Let M 'be a C°°-n-manifold, and l e t 09 u i * » * * * u m b e ° P e n subsets of M such that U. = U V. where V ^ i s open, C l V ^ j c V ^ J + 1 , (M- CIV. ,,V. , , - C1-. v . .) i s k.-connected, k. < n - 3 5 m i f k. > 0 , \ j > 1, 1 < i < m, and 3M c 0 V. r Then, i — 1 i f k^ + ... + k^ + m _> n + 1, there are C°°-diffeomorphisms h^: M -• M such that m h i ' CIV = i d C l V 3 1 — 1 — m> 3 X 1 ( 1 M = U hi( Ui.)° i , 1 i , 1 ' i = l Proof: Let g .: C^ -» M, j = 1, 2,... be a sequence of C O co ' n. C -diffeomorphisms such that Int M = U g . ( C i ) . Suppose we 0=1 J *• have constructed m sequences {f^ Q , . . . , f ^ 3 i = l,...,m, of c " -diffeomorphisms of M onto i t s e l f such that k m f i.J l vi Jaj-2"' f l-J- l l Ti,a^2' r ± 3 S * > w h e r e 59. We apply Lemma 5.2 w i t h E. = G l V. 1 i,2k> v i i,2k+l = v\ 2k+2 8 1 1 ( 1 g ~ g k + l *° ^ e z 0°°-diff eomorphisms " ^ i k+1 3 — — m j °^ ^ °nto i t s e l f such t h a t k+1 m Let h.(x) = l i m f± k ( x ) f o r a l l x e M. Q.E.D. k-»oo * C o r o l l a r y 5»1« Let M be a k-connected C^-n-manifold without boundary, w i t h k < n - 3 i f k > 0 . Then, i f m > , M may be covered w i t h m open C°-n-cells. Proof: Let U^, ...,Un be open C w - n - c e l l s i n M. Then (M,UN) i s k-connected, so i f we l e t k i = k, 1 _< i _< m, we have k^+ ... + k + m = mk + m_>n + l . C o r o l l a r y 5.2. Let M be a k-connected C^-n-manifold ( w i t h k _< n-3 i f k > 0 ) w i t h I boundary components N1,...,N^, and l e t f±t N ix[0,oo) -» M be C w - c o l l a r i n g s , n- Jt +l oo 1 S. 1 S l t I f m 2 k ^ 2 3 there are l C -diffeomorphisms h^ of M onto i t s e l f such that each h i i s the i d e n t i t y on a neighborhood of and m,.C°°-diff eomorphisms g^ ": iRn -• M 3 ,2k such that I m M= U h. of ( N , x[0,oo))u U g,(R ). i = l 1 \ • i = l 1 6o. P r o o f ; (M,N^) i s a t l e a s t 0-connected, and t + mk + ra > n + l . C o r o l l a r y 5»5° L e t M be a c o n n e c t e d C ^ - n - m a n i f o l d , n > 5, ' w i t h two con n e c t e d boundary components and such t h a t t h e i n c l u s i o n o f i n t o M i s a homotopy e q u i v a l e n c e , i = 1, 2. Then t h e r e a r e C69- d i f f eomorphisms h^: N^x[0,oo) -» M such t h a t h i ( x , 0 ) = x f o r a l l x e N^, i = 1,2, and M = h ^ x l O , ® ) ) U h 2 ( N 2 x [ 0 , c o ) ) . C o r o l l a r y 5.k. L e t M be a c o n t r a c t i b l e C™-n-manifold w i t h o u t boundary, n > 5» Then M can be c o v e r e d w i t h two open C°-n- c e l l s . Theorem 5.2. L e t M be a c o n t r a c t i b l e C ^ n - m a n i f o l d w i t h o u t boundary, n > 5, w h i c h i s 1-connected a t co. Then M i s C ^ - d i f f e o m o r p h i c t o R n. P r o o f : By Theorem +.2, we need o n l y show that i f C c M i s compact, t h e r e i s a C w - d i f f eomorphism f : R n -• M such t h a t C c f ( ! R n ) . L e t f 1 5 f 2 : ^ n ~* M b e C * - d i f f eomorphisms such t h a t M = f-^(R n) U f 2 ( R n ) < , S i n c e M i s a no r m a l space, t h e r e a r e c l o s e d s e t s A 1, A g c M w i t h c f ^ ( R n ) , A g c f 2 ( R n ) and M = A^ U Ag. 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 s u b d i v i s i o n o f R n i n t o a s i m p l i c i a l complex K such t h a t ( a ) i s t h e s e t o f p o i n t s o f a subcomplex o f K , i _> 1. 61. . (b) I f A € K and f ^ A ) fl Ag ^  0, then f j A ) c f g ( R n ) . (c) I f A c C^ + 1 - Int C n, then diam f-^( A) < j , i > 1. Let L = N ( f ^ 1 ( A 1 ) , K ) , and, f o r a l l i > 1, l e t K i = ^ l ^ l ) n c i ^ K ) ' E a c h K i i s a subcomplex of L, and CO L = U K . . Let D n C be a compact set such that M. - D i s i = l 1 simply connected. Then (M, M-D) i s 2-connected, x*-^ ( (H,^  ^ ^ I) i s c l o s e d (because of c o n d i t i o n ( c ) ) and 2-dominated, and f-j_( | L^ 2)|) fl D i s compact, so by Lemma 5 .1 , there i s a compact set C-^  c M and a G 0 0 - d i f f eomorphism h^: M -+ M w i t h f x ( 2^ | ) c h^(M-D), and h-^x) = x i f x £ C±a Since U D i s compact, there i s an i n t e g e r i > 1 w i t h C1 U D c f ^ l K j ) U f 2 ( C n ) n Ag. Let H n" 5 be the subcomplex of P(K^) complementary t o 8(K^^ ^ ) ) o ^e have f x ( l ^ ^ 2 ^ |) c h^M-D). Let E = f 1 ( |N(f^1(f2(cJ) n A 2 ) , K ) | ) . By c o n d i t i o n ( b ) , E c f 2 ( R n ) . By c o n d i t i o n ( c ) , i f A c c " + 1 - I n t C^, f o r i > 1, then diam(f 1 ( A ) U f 2 ( C ? ) n Ag) < diam f 1 ( A) + diam(fg(C*?) n Ag) < ~ + diam (fg(c£) n Ag). Therefore diam E _< 2 + diam(fg(C^) fl Ag). Since E i s bounded, there 62* i s an i n t e g e r j > i such that f p ( l n t C1?) r> E, that i s , i f f ^ ( A ) 0 f ^ C ^ ) n A 2 ^ 0, then f-^( A ) c f 2 ( Int C*?). Let M 2 = M - fg(C*j), and l e t Vg = f 2 ( l n t C^- j -C 3 ?) . Then (Mg,Vg) i s (n-2)-connected, ^ ( I H 1 1 " 5 ! ) n M g i s closed and (n-3)-dominated i n Mg, and f ^ l H 3 1 " 5 ! ) 0 (Mg-Vg) i s compact. By Lemma 5.1, there i s a compact set Cg c Mg and a C 0 5 - d i f f eomorphism hg: Mg - Mg w i t h hg(Vg) ^ ^ ( I H 3 1 " 5 ! ) n Mg and h 2 ( x ) = x i f x £ C 2 . We may extend h 2 to a l l of M by l e t t i n g h 2 ( x ) « x i f x e f2(cj). Then I H 1 1 " 3 ! )'c h 2of 2(Int C ^ + 1 ) . Next we consider two open subsets o f R n , U = f^ 1 o h 2 o f g ( R n ) ; ( and V = f^ 1 o h 1 ( M - D ) 0 We apply Theorem 3.1 w i t h L = H 3 1 - 5 U O ( A ) : A e and A c U] and L C = { A e & ( K ± ) A n L = 0} a B ( K [ 2 ^ ) . We l e t P = |L U L C U NCf^CfgOtfJ) n A 2 ) , K ) | U ( R n - Int C ^ + 1 ) and ob t a i n a compact set C c R n - P and a C™- d i f f eomorphism s: R n - R n such that s(x) = x i f x £ C, s ( A ) = A f o r a l l A e P ( K ± ) , and s(U) U V => I K J . Let s: M M he defined s(p) = f1»Sof"1(p) i f p e ZA??1), and 's(p) =.p otherwise. 63. Then f ^ l K j ) U f 2 ( c £ ) n A 2 c h i ( M ~ D ) u S o h 2 o f 2 ( R n ) . ( s i n c e s o h g l g = l d E ) . Consequently C 1 UD.ch^M-D) U S o h 2»f 2 ( l R n ) . Since M - (C^ U D) c M - C 1 c h (M-D), we have M = h^(M-D) U S o h g o f ^ R 3 1 ) ^ or M = (M-D) U h ^ o S o h g o f ^ R 3 1 ) . Let f .= h ' ^ s . h g o f g . Then f ( R n ) r> D D C. Q.E.D,, We can strengthen 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 C^-n-manif o l d , L n _> 5 5 w i t h two boundary components and N 2 such t h a t the i n c l u s i o n of N ^ i n t o M i s a homo top y equivalence, . - i = 1,2. Then • there i s a C 6 0 - d i f f eomorphism of N ^ x ^ c o ) onto M - N g . Proof: Let g j : - M, j = 1,23 ... be a sequence of C™-diff eomorphisms such t h a t Int M = U g - j ( C i ) . Let f 0=1 J s ° _ be the C*- d i f f eomorphism h-^  of C o r o l l a r y 5«3» We construct i n d u c t i v e l y a sequence f Q , f g , . . . of C 0 8 - d i f f eomorphisms of 'N-^xCOjOo) i n t o M such t h a t - f o r each j > 1, j ) i g i ( c | ) c f ^ J + i ) ) , and f j l ^ o ^ , = t > ± | [(Jj},. 64, Let h: N,~x[0,oo) -» M be a C 0 5-collaring such that h(N 2x[0,co)) n ( g j + 1 ( C x ) U f j(N 1x[0,j+2 ])) = 0. Let M - = M - f . ( N , x[0,j+l)). By Theorem 5 . 1 , there are C^-diffeomorphisms r , and r of M . onto i t s e l f which are t— j the identity on a neighborhood of the boundary of M . such that M j c J ( N - L X [ j + l , j+2))) Ur 2(h(N 2x [ 0,oo)). L e t fj+l'N 1 x [0 , j+l] = ^ ' N ^ O , j + l j ^ j + l ^ ^ x t j+l ,oo ) = = r 2 1 o r 1 o f J | N i X [ j + ^ o o r Then M = f J. + 1(N 1 x [0, j+2)) U h(N 2 x[0, oo)). Since S j + l ( c | ) n h(N 2x[0,oo)) = 09 we have j+l _ U S i ( C i ) c f , (N, x[0, 3+2)). Let f = lim f . . Then i = l 1 2 J - 3 M - N 2 = fC^xtO^oo)). Q.E.D.* Corollary 5«5. If M is a C^-n-manifold, n > 5, with two boundary components N-^  and Ng whose inclusions into M are homotopy equivalences, then N-jX R ;N,.x fR^ and Int M are C°°-diffeomorphic. If M is compact, then MXR is C°°-diffeomorphic to N^tOjlJxR,' i = 1,2. 65 . BIBLIOGRAPHY 1. K. Borsuk, Theory of R e t r a c t s , P o l s k a Akademia Nauk Monografie 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 n, Annals of Mathematics, v. 80, (1964) 5 pp. 94 -103 . 3. E. H. Conn e l l , A T o p o l o g i c a l h-cobordism theorem f o r n _> 5, I l l i n o i s J o u r n a l of Mathematics, v. 11,. (1967) , pp. 300-309. 4 . S. T. Hu, Theory of R e t r a c t s , Wayne State U n i v e r s i t y Press, 1965. 5. K. W. Kwun, Uniqueness of the Open Cone Neighborhood, Proceedings of the American Mathematical S o c i e t y , v. 15, (1964) , pp. 476-479. 6. E. L u f t , Coverings of Manifolds w i t h Open C e l l s , I l l i n o i s J o u r n a l of Mathematics, ( t o appear) 7. E. L u f t , C o n t r a c t i b l e Open Man i f o l d s , Inventiones Math., v. 4, (1967) , pp. 192-201. 8. J . R. Munkres, Elementary D i f f e r e n t i a l Topology, Annals of Mathematical S t u d i e s , v. 54, rev. ed. (1965) . 9 . J . R. Munkres, Higher Obstructions t o Smoothing, Topology, v. 4, (1965) , PP. 27-45. 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 , Annals of Mathematics, v. 84 (1966) , pp. 555-572. 6 6 . 1 1 . R. P a l a i s , N a t u r a l Operations on D i f f e r e n t i a l Forms, Transactions of the American Mathematical S o c i e t y , v. 9 2 , ( 1 9 5 9 ) , PP. 125-141. 1 2 . J . S t a l l i n g s , On T o p o l o g i c a l l y Unknotted Spheres, Annals of Mathematics, v. 77, (1963), pp. 4 9 0 - 5 0 3 . 1 3 . J . S t a l l i n g s , The Piecewise-Linear S t r u c t u r e of Euc l i d e a n Space, Proceedings of the Cambridge P h i l o s o p h i c a l S o c i e t y , v. 58 , ( 1 9 6 2 ) , pp. 481-488. 14. J . S t a l l i n g s , On I n f i n i t e Processes Leading t o D i f f e r e n t -i a b i l i t y i n the Complement of a P o i n t , D i f f e r e n t i a l and Combinatorial Topology, P r i n c e t o n Mathematical S e r i e s , v. 27, ( 1 9 6 4 ) , pp. 245 -254. 1 1 5 . J . H, C. Whitehead, On C -complexes, Annals of Mathematics, v. 41, ( 1 9 4 1 ) , pp. 8 0 9 - 8 2 4 . 

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