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The fundamental surgery theorem and the classification of manifolds Cameron, Richard Bruce 1980

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THE FUNDAMENTAL. SURGERY THEOREM AND THE CLASSIFICATION OF MANIFOLDS by RICHARD BRUCE CAMERON B.Math., The U n i v e r s i t y  of Waterloo, 1978  A THESIS SUBMITTED IN PARTIAL  FULFILMENT  OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES (Department o f Mathematics)  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA ' June 1980 . ^ R i c h a r d Bruce Cameron, 1980  In  presenting this  thesis in partial  an advanced d e g r e e a t the L i b r a r y I further for  shall  the U n i v e r s i t y  make i t  agree that  freely  of  extensive  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  this  written  thesis for  It  f i n a n c i a l gain shall  Department of _ 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 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  I agree  r e f e r e n c e and copying of  this  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  i s understood that  permission.  the requirements  B r i t i s h Columbia,  available for  permission for  by h i s r e p r e s e n t a t i v e s . of  f u l f i l m e n t of  or  publication  be a l l o w e d w i t h o u t  my  Abstract The purpose of t h i s paper i s to p r e s e n t a survey of some important r e s u l t s i n the c l a s s i f i c a t i o n of 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 the Poincar£ c o n j e c t u r e and h-cobordism concerned  theorem. We  s u r g e r y problem developed i n the simply-connected  begin  i t s p a r t i a l s o l u t i o n u s i n g the  review next  w i t h the diffeomorphism  We  the work of K e r v a i r e and M i l n o r ,  c l a s s e s of homotopy spheres.  from t h e i r work, and we  case, by Browder.  The  present i t s s o l u t i o n  T h i s s o l u t i o n amounts to  the s u r g e r y i n v a r i a n t theorem, the fundamental s u r g e r y theorem and associated r e s u l t s . and  s e v e r a l important  and  Wall.  We  end our d i s c u s s i o n w i t h the plumbing  theorem,  c l a s s i f i c a t i o n theorems of Browder, Novikov  iii T a b l e of Contents  T i t l e page.  i  Abstract  i i  Chapter I .  The P o i n c a r e C o n j e c t u r e .  §1.  The P o i n c a r e C o n j e c t u r e and the h-cobordism  §2.  Exotic D i f f e r e n t i a l  §3.  Groups of Homotopy Spheres.  Chapter I I .  Structures  theorem.  on the 7-Sphere.  1 4 11  The Fundamental Surgery Theorem.  §4.  The Surgery Problem.  15  §5.  The Surgery I n v a r i a n t .  17  §6.  Surgery below the M i d d l e Dimension.  33  §7.  I n i t i a l R e s u l t s i n the M i d d l e Dimension.  45  §8.  The Proof of the Fundamental Theorem f o r m odd.  53  §9.  The Proof of the Fundamental Theorem f o r m even.  61  Chapter I I I .  Plumbing  and the C l a s s i f i c a t i o n  §10.  I n t e r s e c t i o n and Plumbing.  §11.  The Homotopy Types and C l a s s i f i c a t i o n  Bibliography  of M a n i f o l d s . 71  of Smooth M a n i f o l d s 76 80  1  Chapter §1.  I.  The P o i n c a r e C o n j e c t u r e .  The P o i n c a r e C o n j e c t u r e and the h-cobordism  The  Theorem.  o r i g i n a l form of the P o i n c a r e c o n j e c t u r e was the f o l l o w i n g :  1.1  I f M i s a c l o s e d 3 - m a n i f o l d such t h a t  (M) H ( S 3 ) , f i  A  " then'M=S3.  '  T h i s was shown t o be f a l s e , The b i n a r y i c o s a h e d r a l group I  through  the f o l l o w i n g  counter-example:  i s d e f i n e d by the g e n e r a t o r s A, B,  ft and C, and r e l a t i o n s A =B =C =ABC between them. I 3  a subgroup of S  3  2  i s p e r f e c t , and i s  5  . Define a closed 3-manifold M=S /I 3  ft ft . Then TTI ( M ) = I ,  ft and Hi  ( M ^ T T J ( M )  = 1  =1.'  By P o i n c a r g d u a l i t y ,  H  2  Thus,  ( M ) = 1 .  H J J . ( M ) = H ( S ) , b u t M i s n o t homeomorphic to S , because TTJ_ ( M ) = I , 3  3  A  whereas The 1.2  T T I ( S  3  ) = 1 .  f a i l u r e of the o r i g i n a l c o n j e c t u r e l e d to an amended f o r m u l a t i o n I f M i s a c l o s e d , simply-connected  Note t h a t , by t h e Hurewicz isomorphism duality  M-S3.  theorem, the P o i n c a r e  theorem, and the u n i v e r s a l c o e f f i c i e n t  t h a t M i s simply-connected  3 - m a n i f o l d , then  theorem, the h y p o t h e s i s  i m p l i e s t h a t i n f a c t -n ( M ) - T T ^ ( S ) , and 3  hence t h a t M - S . 3  Although  t h e r e have been p a r t i a l r e s u l t s c o n c e r n i n g t h i s c o n j e c t u r e ,  i t has n o t y e t been c o m p l e t e l y  settled.  The P o i n c a r e c o n j e c t u r e can be extended 1.3  t o dimensions  o t h e r than 3 :  I f M i s a c l o s e d n-manifold which i s h o m o t o p i c a l l y equivalent T h i s statement  to S  N  , i t i s homeomorphic t o S  has been proved  for  n ^ 3 , 4 .  N  In f a c t ,  . 1 . 3  s t a t e d i n an a p p a r e n t l y weaker form which i s , by the Hurewicz  can be isomorphism  theorem, a c t u a l l y e q u i v a l e n t t o 1 . 3 : 1.4  I f M i s a c l o s e d , simply-connected  n-manifold w i t h t h e  2 i n t e g r a l homology of S , then M i s homeomorphic n  to S . n  We w i l l prove the g e n e r a l i z e d P o i n c a r e c o n j e c t u r e i n dimensions g r e a t e r than 4 by means of the h-cobordism  theorem.  A smooth m a n i f o l d t r i a d i s d e f i n e d t o be a t r i p l e  (W;V,V')  S  where W i s a compact, smooth m a n i f o l d , and the boundary of W i s the d i s j o i n t union of two open and c l o s e d submanifolds V and V'. 1.5  Theorem  (h-cobordism  the p r o p e r t i e s :  (1) W,V,  theorem) : Suppose the t r i a d  (W;V,V) has  and V' a r e simply-connected,  (2) H (W,V)=0, A  (3) dim W=n>6. Then W i s d i f f e o m o r p h i c to V x [ 0 , l ] . The f o l l o w i n g p r o p o s i t i o n i s c e n t r a l to the p r o o f of the g e n e r a l i z e d conjecture: 1.6  P r o p o s i t i o n : Suppose W i s a compact simply-connected  n - m a n i f o l d , n>6, w i t h a simply-connected  boundary V.  smooth  Then the f o l l o w i n g  four a s s e r t i o n s are equivalent: (1) W i s d i f f e o m o r p h i c t o D . n  (2) W i s homeomorphic  to D . n  (3) W i s c o n t r a c t i b l e . (4) W has the i n t e g r a l homology of a p o i n t . Proof: I t i s c l e a r that  (l)->-(2)->-(3)->-(4) , so t h a t we need o n l y  (4)-Kl).  If D  satisfies  the c o n d i t i o n s o f the h-cobordism  by e x c i s i o n  0  i s a smooth n - d i s c imbedded i n intW, then theorem.  prove  (W\intD ,3D ,V) 0  0  In p a r t i c u l a r ,  (W\intD , 3D )=H^. (W,D )=0. 0  S i n c e the cobordism a product cobordism  0  0  (W;c(>,V) i s the c o m p o s i t i o n of (Dg ; (J>, 9r/) 0  (W\intD ;9Dg,V), W i s homeomorphic 0  to Dg.  of M i l n o r shows t h a t the c o m p o s i t i o n p r e s e r v e s d i f f e r e n t i a b l e so t h a t W i s i n f a c t d i f f e o m o r p h i c to Dg.  QED  with  A theorem structures,  3 We  are now  Proof of 1.4: satisfies H  i  ready  to prove  Case 1: n>5.  If  the g e n e r a l i z e d c o n j e c t u r e .  the h y p o t h e s i s of 1.6\  ( M \ i n t D ) = H N _ : L ( M \ i n t D ,3D ) 0  0  - H  In p a r t i c u l a r ,  by P o i n c a r e  0  ( M , D Q )  N  i s a smooth n - d i s c , then M \ i n t D g  D Q £ M  duality  by  excision  by  the exact cohomology sequence.  ' 0 i f i>0  Hi i f i=0 M=(M\intDg)UDQ  Consequently, D^,  D ^ of the n - d i s c w i t h the boundaries Such a m a n i f o l d i s c a l l e d  I K S D ^ - ^ D ^ .  completed  to S .  by  showing t h a t any  Let S i ^ D ^ S  n  of S £ & n  n + 1  .  be w r i t t e n  1 1  S ,  f o r case  1.  n  and  Case 2: n=5. 1.7  ||x|=l,  x n + 1  -°}-)  n + 1  , where  use  E a c t l  proof i s  P°int of fi^ may  by g ( u ) = g ( u ) f o r  n  1  e  = n  +  1  ueD^,  ( ° »• • • , 0 ' , l ) ' e f f , n+1  Then g i s a w e l l - d e f i n e d i n f e c t i v e  i s hence a homeomorphism.  We  The  diffeomorphism  the s o u t h e r n hemisphere  Define g : M + S  (v))+cos(^|)e  f o r a l l p o i n t s tveD^'. onto  a t w i s t e d sphere.  be an embedding onto  t v , 0<t<l, veSD^. 1  i d e n t i f i e d under a  copies  t w i s t e d sphere M = D ^ u , D^J i s homeomorphic 1 h 2  ( I . e . the s e t {x|  g(tv)= s i n C ^ g ^ ^ C h  map  i s d i f f e o m o r p h i c to a u n i o n of two  continuous  T h i s completes the p r o o f  here:  Theorem: Suppose  i s a c l o s e d , simply-connected  w i t h the homology of S . n  smooth m a n i f o l d  Then i f n=4,5, or 6, M bounds a smooth,  compact, c o n t r a c t i b l e m a n i f o l d . Thus, 1.7  and  to D 6 , so t h a t M Remark: The The  5  1.6  S  i s homeomorphic to  S . 5  g e n e r a l i z e d c o n j e c t u r e h o l d s i n dimensions  proof i s t r i v i a l ,  1- and  imply "tha't"M ""bb'unds a m a n i f o l d homeomorphic  2-manifolds.  1 and  2 as w e l l .  because of the well-known c l a s s i f i c a t i o n  of  4 By u s i n g 1.7  and 1.6  homology n-sphere has proved devoted  § 2.  one  can show t h a t i n f a c t a simply-connected  i s d i f f e o m o r p h i c to S , n  t h a t t h i s i s not t r u e f o r n=7.  to an e x a m i n a t i o n of t h i s  f o r n=5,6.  However, M i l n o r  The next s e c t i o n w i l l  be  result.  E x o t i c D i f f e r e n t i a l ' S t r u c t u r e s on the 7-Sphere.  The  invariant  For  every c l o s e d o r i e n t e d smooth 7-manifold s a t i s f y i n g  2.1  X(M ) 7  the h y p o t h e s i s  H (M)=H (M)=0 3  we w i l l  tt  d e f i n e a r e s i d u e c l a s s X(M)  modulo 7.  A c c o r d i n g to Thorn every  c l o s e d smooth 7-manifold M i s the boundary of a smooth The  i n v a r i a n t X(M)  8-manifold,B.  w i l l be d e f i n e d as a f u n c t i o n of the index T and  the P o n t r j a g i n c l a s s p.]_ o f B . 8  I f yeHy(M ) i s the d i s t i n g u i s h e d 7  v e H ( B , M ) i s determined 8  7  8  by  g e n e r a t o r , then an  the r e l a t i o n s h i p  8v=u.  orientation  Define a quadratic  form over the group H ^ ( B , M ) / t o r s i o n by the formula a-><v,a >. 8  Let  T ( B ) be 8  7  2  the index of t h i s form  (the number of p o s i t i v e  terms  minus the number of n e g a t i v e terms when the form i s d i a g o n a l i z e d over R ) . Let of  p eH^CB ) be the f i r s t 8  B .  (For the d e f i n i t i o n of P o n t r j a g i n c l a s s e s , see  8  The h y p o t h e s i s 2.1 pair  8  [ M i l n o r 1974].)  ( t o g e t h e r w i t h the l o n g cohomology sequence of the  7  q(B )=<v,(x 2.2  bundle  ( B , M ) ) i m p l i e s t h a t the i n c l u s i o n homomorphism i i H ^ ( B j M ) - ^ ( B ) 8  i s an isomorphism. 8  P o n t r j a g i n c l a s s of the tangent  _ 1  T h e r e f o r e , we  7  can d e f i n e a ' P o n t r j a g i n number'  ) >. 2  P l  Theorem; The r e s i d u e c l a s s o f 2 q ( B ) — r ( B ) modulo 7 does not  depend on the c h o i c e of the m a n i f o l d  8  B . 8  8  8  D e f i n e A(M ) as t h i s r e s i d u e  class.  7  As an immediate consequence,  we have: 2.3  i s n o t the boundary of an 8 - m a n i f o l d  C o r o l l a r y : I f A(M )*0 then M 7  w i t h f o u r t h B e t t i number Proof of T h e o r e m ' 2 . 2 :  zero.  L e t Bf , B | be m a n i f o l d s b o t h h a v i n g boundary  (We may assume they a r e d i s j o i n t . ) Then C =B^U 7B 8  8  M  i s a closed  M. 7  8-manifo  which p o s s e s s e s a d i f f e r e n t i a b l e s t r u c t u r e compatible w i t h t h a t o f Bf and B . 8  Choose t h a t o r i e n t a t i o n v f o r 6 . w h i c h i s c o n s i s t e n t 8  the o r i e n t a t i o n vj_ o f B  8  (and t h e r e f o r e c o n s i s t e n t w i t h - V 2 ) .  L e t q ( C ) denote the P o n t r j a g i n number  <v,p (C )>.  8  According  2  8  to [Thorn 1954] we have  T(C )=<v,^-(7p (C )-p (C )>, 8  8  2  8  2  and  therefore 45T(C )+q(C )=7<v,p (C )>=0 8  8  (mod 7)  8  2  This (1) 2.4  implies 2q(C )-T(C8)E0  (mod 7)  8  Lemma: Under the above c o n d i t i o n s we have  (2)  T ( C ) = T ( B ) - T ( B ) , and  (3)  q(C )=q(B )-q(B ).  8  8  8  8  8  8  ( 1 ) , ( 2 ) , and ( 3 ) c l e a r l y  Formulae  imply  2q(B )-T(B8)E2q(B )-T(B ), 8  which i s j u s t  8  that  (mod 7)  8  the statement o f the theorem.  P r o o f of Lemma 2 . 4 : C o n s i d e r the diagram: H ( B , M ) e H ( B , M ) «—|L_ H (C,M) n  n  1  n  2  ± ®± l  H (B )®H (B ) < n  n  1  j  2  2  H (C) n  Note t h a t f o r n=4 these homomorphisms a r e a l l isomorphisms.  with  6 I f a=jh (4)  1  (cqSK^XeH * (C) , then 1  <v,a >=<v,jh ( a e a ) > = < v ® ( - v ) , a ® a | > = < v , a > - < v , a > 2  1  2  2  2  1  2  2  2  1  2  Thus the q u a d r a t i c form of C i s the ' d i r e c t sum' of the q u a d r a t i c form:j of B\ and the n e g a t i v e o f the q u a d r a t i c form o f B . 2  This c l e a r l y  implies formula ( 2 ) . Define a j = i i ( B ^ ) 1  and a = i 2  1 2  p (B ). 1  2  Then t h e r e l a t i o n  k(Pl (C))=Pi (B^ )©Pi ( B ) 2  implies that j h ( a ^ $ a ) = P i ( C ) .  The computation  1  2  (4) now shows t h a t  <v,p (C)>=<v ,a >-<v ,a >, 2  2  2  1  which i s j u s t f o r m u l a of  2  (3).  T h i s completes  the p r o o f o f the lemma and  the theorem. The f o l l o w i n g p r o p e r t y of the i n v a r i a n t A i s c l e a r :  2.5  Lemma: I f the o r i e n t a t i o n of M i s r e v e r s e d , then A(M) i s m u l t i p l i e d  by -1. As a consequence we have: 2.6  C o r o l l a r y : I f A(M )*0 then M 7  d i f f e o m o r p h i s m onto  7  possesses no o r i e n t a t i o n - r e v e r s i n g  itself.  A p a r t i a l c h a r a c t e r i s a t i o n o f the n-sphere C o n s i d e r the f o l l o w i n g h y p o t h e s i s c o n c e r n i n g a c l o s e d mani f o l d 2.7  M : 11  There e x i s t s a d i f f e r e n t i a b l e f u n c t i o n f:M>R h a v i n g o n l y two  c r i t i c a l points X  Q  .  Furthermore,  these  critical  p o i n t s a r e non-degenerate. (That i s , i f u j , . . . , u of 2.8  x  a r e l o c a l c o o r d i n a t e s i n a neighbourhood  (or X } ) then the m a t r i x (3 f/9u^8u^.) 2  0  Theorem: I f M  11  i s n o n s i n g u l a r a t X Q (or x^)) .  sa t i s f i e s h y p o t h e s i s 2.7 then t h e r e e x i s t s a  homeomorphism of M onto S  n  which i s a d i f f e o m o r p h i s m except p o s s i b l y  7 at  a single point.  Proof: This r e s u l t The  i s e n t i r e l y due t o [Reeb 1952].  p r o o f w i l l be based on the o r t h o g o n a l  t r a j e c t o r i e s o f the  the f u n c t i o n so t h a t f ( X Q ) = 0 , f ( x ^ ) = 1 .  manifolds  f=constant.  Normalise  According  to [Morse 1925, Lemma 4] t h e r e e x i s t  local  coordinates  v i ,. . . ,v i n a neighbourhood V/"of xn so t h a t f ( x ) = v i + . . . + v f o r xeV. n n (Morse assumes t h a t f i s o f c l a s s C , and c o n s t r u c t s c o o r d i n a t e s o f 3  c l a s s C , b u t the same p r o o f works i n the C ds =dv? + ..,+dv i n 2  2  case.) The e x p r e s s i o n  d e f i n e s a Riemannian m e t r i c i n the neighbourhood V.  Choose a d i f f e r e n t i a b l e Riemannian m e t r i c f o r M* which c o i n c i d e s w i t h t h i s one i n some neighbourhood V' o f x g . by  [Steenrod  (This i s p o s s i b l e  1951, 6.7 and 12.2].) Now the g r a d i e n t of f can be  c o n s i d e r e d as a c o n t r a v a r i a n t v e c t o r  field.  F o l l o w i n g Morse we c o n s i d e r the d i f f e r e n t i a l •^r=  grad  equation  f/|grad f | . 2  In the neighbourhood V' t h i s e q u a t i o n has s o l u t i o n s (vi ( t ) , . . . , v where a=(a^,..., a ^ e H  1 1  n  ( t ) ) = (ai St,...,a  n  St)  f o r 0<t<e ,  i s any n - t u p l e w i t h Z a = l . 2  extended u n i q u e l y to s o l u t i o n s x ( t ) f o r O ^ t ^ l .  These can be Note t h a t  these  3.  solutions satisfy  the i d e n t i t y  f ( x (t))=t.  Map the i n t e r i o r o f t h e u n i t sphere o f i ? i n t o M* by the map n  St)—>x (t)  ( a - i / t , . . . ,a  n It  i s easily verified  open n - c e l l onto M \ { x i ) . Given  1  .  a that t h i s defines a diffeomorphism  of the  The a s s e r t i o n of the theorem now f o l l o w s .  any d i f f e o m o r p h i s m  g : S ^—>-S \ n  n  an n - m a n i f o l d  can be  o b t a i n e d as f o l l o w s . 2.9  C o n s t r u c t i o n : L e t M ^ g ) be the m a n i f o l d o b t a i n e d from two c o p i e s  of  n  i ? by matching the s u b s e t s i? \{0} under the d i f f e o m o r p h i s m n  (Such a m a n i f o l d i s c l e a r l y homeomorphic to S .  I f g i s the  n  i d e n t i t y map, 2.10 if  then M^g)  i s d i f f e o m o r p h i c to S  C o r o l l a r y : A manifold M and o n l y i f i t s a t i s f i e s  P r o o f : I f M (g) F  (  x  >  =  .  w i l l satisfy  l  (1+  n x ;  "  l  ,  v  can be o b t a i n e d by  the h y p o t h e s i s  i s o b t a i n e d by  n  .  n  z  1  (1+  v  the h y p o t h e s i s 2.7.  ... z  Consider group S0(4) bundles with  .  The  converse  over  as s t r u c t u r a l group.  The  i be  sphere bundle 2.11  be d e f i n e d by  (by  i s understood  corresponding  Lemma: The  ca ) hj  (see  to  For each odd where h and m a n i f o l d has  1951]) i s 7  by  later.  isomorphism  .  (Quaternion  equation.) L e t E, . be  1  the  ) equals ±2(h-j)i. One  can show t h a t the  e q u a l to  characteristic  (h )x) + J  the t o t a l space of the bundle  the e q u a t i o n s h+j=l, h-j=k.  a n a t u r a l d i f f e r e n t i a b l e s t r u c t u r e and  w i l l be d e s c r i b e d  §18])  [f^.. ] £ ^ 3 (SO (4)) .  i n t e g e r k l e t M^ be  j a r e determined  such  For each (h,j)eZ©Z, l e t  for ^ ( S * ) .  P o n t r j a g i n c l a s s p^(E  [Steenrod  A specific  on the r i g h t of the  the s t a n d a r d g e n e r a t o r  rotation  [Steenrod, 1951,  (u) • v=u^« v u ^ , f o r veR  (The p r o o f w i l l be g i v e n l a t e r . class  2.8.  e q u i v a l e n c e c l a s s e s of  between the groups i s o b t a i n e d as f o l l o w s .  Let  e s t a b l i s h e d by  the 4-sphere, w i t h the  a r e i n one-to-one correspondence  multiplication  can be  7-manifolds  3-sphere bundles  S0(4)  then the f u n c t i o n  )  the elements of the group ^ 3 ( S 0 ( 4 ) ) = Z ® Z .  f^:S—y  2.9  2.7.  a s l i g h t m o d i f i c a t i o n o f the p r o o f of theorem  Examples of  the c o n s t r u c t i o n  the c o n s t r u c t i o n 2.9,  -  u )  .)  E^j >  This  o r i e n t a t i o n , which  9 2.12  Lemma: The i n v a r i a n t X (M ^)  2.13  Lemma: The m a n i f o l d s a t i s f i e s  7  Combining 2.14  i s the r e s i d u e c l a s s modulo 7 o f k - l . 2  the h y p o t h e s i s 2.7.  these we have:  Theorem: F o r k ^ l mod 7 the m a n i f o l d M ' i s homeomorphic, b u t 2  7  not d i f f e o m o r p h i c , to S . 7  (For k=±l t h e m a n i f o l d M  i s diffeomorphic to S , but i t i s not  7  7  known whether t h i s i s t r u e f o r any o t h e r k w i t h k = l mod 7.) 2  C l e a r l y any d i f f e r e n t i a b l e s t r u c t u r e on S throughout i? \{0}. 8  2.15  7  can be extended  However:  C o r o l l a r y : There e x i s t s a d i f f e r e n t i a b l e s t r u c t u r e on  which  g cannot be extended throughout 'R . T h i s f o l l o w s immediately from the p r e c e d i n g a s s e r t i o n s , t o g e t h e r w i t h c o r o l l a r y 2.3. Proof o f Lemma 2.11: I t i s c l e a r t h a t t h e P o n t r j a g i n c l a s s i s a l i n e a r f u n c t i o n o f h and j .  Furthermore i t i s known t o be  independent o f the o r i e n t a t i o n o f the f i b r e . of S  3  Pl^^j)  But i f the o r i e n t a t i o n  i s r e v e r s e d , then E, . i s r e p l a c e d by £ . , . T h i s shows t h a t hj * ' -j,-h  P l ( 5 ^ j ) i s g i v e n by an e x p r e s s i o n o f the form c ( h - j ) i .  Here c i s a  c o n s t a n t which w i l l be e v a l u a t e d l a t e r . Proof o f Lemma 2.12: A s s o c i a t e d w i t h each 3-sphere bundle t h e r e i s a 4 - c e l l bundle p, :B —>S . k k  8  bundle i s a d i f f e r e n t i a b l e m a n i f o l d w i t h boundary M . group H^CB ) i s generated by the element ct=p, ( i ) . 8  y,v f o r M  7  and B  8  so t h a t < v , ( i a ) > = + l . g _ 1  1  The t o t a l space B o f t h i s k  h  8  —vS *  2  The cohomology  Choose o r i e n t a t i o n s  Then the index T ( B ) w i l l be 1.  The tangent bundle o f B^ i s the Whitney  8  sum o f (1) t h e bundle o f  v e c t o r s tangent t o the f i b r e , and (2) t h e bundle o f v e c t o r s normal t o the f i b r e .  The f i r s t bundle  (1) i s induced (under p^) from the bundle  10 E^_. > is  a  n  i t h e r e f o r e has P o n t r j a g i n c l a s s p^=p^(c (h-j ) i )=cka.  {  induced  from the tangent  Pontrjagin c l a s s zero.  The second  bundle of S , and t h e r e f o r e has f i r s t 4  Now by the Whitney p r o d u c t  theorem:  p (B )=cka+0. 8  x  For the s p e c i a l case k = l i t i s e a s i l y v e r i f i e d  that B  8  i s the  2 q u a t e r n i o n p r o j e c t i v e plane QP  w i t h an 8 - c e l l removed.  But t h e  P o n t r j a g i n c l a s s p^(QP ) i s known t o be twice a generator 2  of H^CQP ). 2  T h e r e f o r e the c o n s t a n t c must be ± 2 , which completes the proof of 2.11. Now q ( B ) = < v , ( i ( ± 2 k a ) ) > = 4 k , and 2 q - x = 8 k - l = k - l 8  _ 1  2  2  2  (mod 7 ) .  2  K.  T h i s completes the p r o o f o f Lemma 2.12. Proof o f Lemma 2.13: As c o o r d i n a t e neighbourhoods i n the base space take t h e complement of the n o r t h p o l e , and the complement o f the south pole.  These can be i d e n t i f i e d w i t h the E u c l i d e a n space B.^ under  stereographic p r o j e c t i o n .  Then a p o i n t which corresponds  under one p r o j e c t i o n w i l l correspond The  t o t a l space M  7  to u'  =—p|^-pr  t o ueff.^  under t h e o t h e r .  can now be o b t a i n e d as f o l l o w s ( c f . [Steenrod 1951  h J u u vu §18]). two c o p i e(u,v)—>-(u',v') s of R xS and the d i f f Take eomorphism = i d e n t i vf y t h e subsets , ( u(ff' s i n g \{0})xS q u a t e r nby ion k  multiplication). Replace Consider where  r^F'~nrr  +  T h i s makes the d i f f e r e n t i a b l e s t r u c t u r e of M  the c o o r d i n a t e s  7  3  precise.  (u',v') by ( u " , v " ) , where u"=u'(v')  the f u n c t i o n f : M 7 — « d e f i n e d by  f( x ) = —  R  R(v) denotes t h e r e a l p a r t of the q u a t e r n i o n v .  verified and  3  t h a t f has o n l y two c r i t i c a l p o i n t s (namely  t h a t these a r e non-degenerate.  ^  U  —  ,  I t i s easily  (u,v)=(0,+1))  T h i s completes t h e proof o f Lemma 2.13.  11 §3.  Groups of Homotopy Spheres.  The f o l l o w i n g r e s u l t s about homotopy n-spheres a r e proved i n [Kervaire, Milnor  1963]:  (1) The h-cobordism c l a s s e s of homotopy n-spheres form an a b e l i a n group 9^ under the connected sum  operation.  (2) The h-cobordism c l a s s e s o f homotopy n-spheres which bound p a r a l l e l i s a b l e m a n i f o l d s form a subgroup bP , of 9 . n+1 n w i l l be proved below.) n  (3) The q u o t i e n t the c o k e r n e l J  group ® / b P - ^ n  (This  i s o m o r p h i c t o a subgroup o f  n+  o f the Hopf-Whitehead  homomorphism J  (where  V :TT (SO, )—>TT (S ) ) , and i s f i n i t e , n n k n+k  (4) The group b P zero  ^ i - finite, s  n +  f o r n even, and f i n i t e  f o r n*3.  cyclic  (In p a r t i c u l a r , i t i s  f o r n odd, n*3.)  (5) Thus, the group 0 of (h-cobordism c l a s s e s o f ) homotopy n n-spheres i s f i n i t e , We r e c a l l  f o r n*3.  from above t h a t every homotopy n-sphere, n*3,4, i s  homeomorphic t o S . n  [Smale 1962] has shown t h a t two homotopy n—spheres,  n*3,4, a r e h-cobordant i f and o n l y i f they a r e d i f f e o m o r p h i c . (for  n^3,4 a t l e a s t ) the group 0 can be d e s c r i b e d n  Thus  as the s e t of  d i f f e o m o r p h i s m c l a s s e s of d i f f e r e n t i a b l e s t r u c t u r e s on S , and the n  last  r e s u l t above can be i n t e r p r e t e d as s t a t i n g t h a t t h e r e a r e o n l y  finitely  many e s s e n t i a l l y d i f f e r e n t such s t r u c t u r e s , f o r each n, n*3,4.  We w i l l now prove a s s e r t i o n s  (2) and (3) above.  L e t M be an s - p a r a l l e l i s a b l e c l o s e d n - m a n i f o l d . t r i v i a l , where line  ( I . e . T^Oe i s 1  i s t h e tangent bundle o f M, and e i s the t r i v i a l n+k bundle.) Choose an embedding i:M—>S , w i t h k>n+l. Such an 1  12 embedding e x i s t s and i s unique up to d i f f e r e n t i a b l e  isotopy.  3.1  n+k Lemma ( K e r v a i r e , M i l n o r ) : An n-:dimensional submanifold o f S ,  n<k,  i s s - p a r a l l e l i s a b l e i f and o n l y i f i t s normal bundle i s t r i v i a l . Thus v,, i s t r i v i a l . M  L e t d> be a t r i v i a l i s a t i o n o f v . M n+k k w  Then the  —>-S . The  Pontrjagin-Thom c o n s t r u c t i o n y i e l d s a map  p(M,cj>):S  homotopy c l a s s of p(M,cf)) i s a w e l l - d e f i n e d  element o f t h e s t a b l e  k homotopy group n ^ i r ^ ( S ) . A l l o w i n g the t r i v i a l i s a t i o n +  to vary,  we o b t a i n a s e t p (M)={p (M,<j>) }cn . n 3.2  Lemma: p(M)<fI  c o n t a i n s t h e zero of II i f and o n l y i f M bounds n  n  a p a r a l l e l i s a b l e manifold. P r o o f : <=.  I f M=8W and W i s a p a r a l l e l i s a b l e m a n i f o l d , then, because n+k  of  d i m e n s i o n a l c o n s i d e r a t i o n s , the embedding i:M—>S  to  an embedding of W i n t o D " ^" " , and W w i l l have t r i v i a l 11  1  1  Choose a t r i v i a l i s a t i o n ip o f v  can be extended  1  and l e t <J>=IJJ|M.  normal b u n d l e .  The P o n t r jagin-Thom map  w  n+k k n+k+1 p(M,<f>):S —>-S extends over D , and hence i s n u l l - h o m o t o p i c . /w , \ ^ , -r, „n+k+l_„n+k „ ,, ,„n+k , _,k =>-. I f p(M,cJ>) = 0, we have a map F:D =S x[0,l]/S xl—^-S r  which s a t i s f i e s F| S ^x0=p(M,<j>) , and F | s n+  n +  ^xl=e , A  the c o n s t a n t map. n+k  F  can be made r e g u l a r a t * (the base p o i n t ) , r e l a t i v e to S  we s h a l l assume, w i t h o u t l o s s o f g e n e r a l i t y , t h a t i t i s . F  1  xO, so  Then  (*)£D k "'" is a submanifold W, and < f > can be extended!'to a t r i v i a l i s a t i o n n+  +  ^ on W. 3.3  By Lemma 3.1 above and the f o l l o w i n g lemma, W i s p a r a l l e l i s a b l e .  Lemma: A connected m a n i f o l d w i t h non-vacuous boundary i s  s - p a r a l l e l i s a b l e i f and o n l y i f i t i s p a r a l l e l i s a b l e . T h i s completes 3.4  the p r o o f of Lemma 3.2.  Lemma: I f Mg i s h-cobordant  P r o o f : I f M +(-M )=9W, 0  M — q  n+k x q M  [Kervaire, Milnor]  1  f o r q=0,l. ^ '  t o MT_ , then p(Mg)=p(M^). n+k  choose an embedding of W i n S Then a t r i v i a l i s a t i o n d> q  x [ 0 , l ] such t h a t  of v., extends to a M q  t r i v i a l i s a t i o n ib on W , which r e s t r i c t s to a t r i v i a l i s a t i o n <b, l-q Y  Clearly 3.5  g i v e s r i s e to a homotopy between  (W,IJJ)  Lemma: I f M and M' a r e s - p a r a l l e l i s a b l e  P r o o f : C o n s t r u c t a m a n i f o l d W w i t h boundary  p(Mg,<|>o) and  on M. 1-q  p(M^,(j)l).  then p (M)+p (M' ) <=p (M#M ) c n . 1  (-M)u(-M')u(M#M') as f o l l o w s  b e g i n n i n g w i t h Mx[0,l]uM'x[0,1], j o i n the boundary components M*l and M'xl by a smooth connected sum..  T h i s sum. can be extended smoothly over  neighbourhoods of the j o i n e d p o r t i o n s , i n Mx[0,l] and M ' x [ 0 , l ] . details  of t h i s  construction are given i n [Kervaire, Milnor  The m a n i f o l d W has the homotopy type of the o n e - p o i n t u n i o n  (The  1963].) MvM'.  n+k Embed.W  in S  x [ 0 , l ] such t h a t  w e l l - s e p a r a t e d s u b m a n i f o l d s of S  n +  (-M)  and  (-M')  a r e mapped  into  ^ x 0 , and such t h a t the image of  M#M'  n+k lies of  in S  (-M)  defined  Given t r i v i a l i s a t i o n s $ and <j>' of the normal bundles  xl.  and  (-M'), i t i s not. h a r d to see t h a t t h e r e e x i s t s  throughout W .  extension.  L e t X/J denote the r e s t r i c t i o n t o M#M'  Then c l e a r l y p (M,<j))+p (M' ,<j>' )  an e x t e n s i o n of  this  i s homotopic t o p(M#M',i(j).  T h i s completes the p r o o f . 3.6 Lemma: The s e t p ( S ) c n  i s a subgroup of the s t a b l e homotopy group n F o r any homotopy sphere Z the s e t p(Z) i s a c o s e t of t h i s subgroup n  11^.  p(S ). n  II  Thus the correspondence E—>-p(Z) d e f i n e s a homomorphism p' from  to the q u o t i e n t group I I / p ( S ) . n n  n  P r o o f : Combining the p r e v i o u s lemma w i t h the i d e n t i t i e s (1)  SN#S sS  (1)  p(S )+p(S )<=p(SN) ,  (2)  p(S )+p(Z)cp(z),  n  (2) S //ZSZ  n  n  n  n  n  (3) Z # ( - Z ) - S , we  obtain  n  which shows t h a t p ( S ) i s a subgroup of I I n  ,  which shows t h a t p(Z) i s a u n i o n of c o s e t s of  t h i s subgroup, and (3)  p(E)+p(-Z)cp(sn),  which shows t h a t p(Z) must be a s i n g l e  coset  .14 T h i s completes the p r o o f o f Lemma 3.6. By Lemma 3.2 the k e r n e l of p':0 —HI / p ( S ) c o n s i s t s e x a c t l y o f a l l n n h-cobordism c l a s s e s of homotopy n-spheres which bound p a r a l l e l i s a b l e n  manifolds. It follows  Thus, these elements form a group which we denote by bP , c0 . n+1 n t h a t b P ^ ^ i s i s o m o r p h i c t o a subgroup o f n ^ / p t S ) . 11  +  is finite and ,{3) .  Since  [Serre 1951], t h i s completes the p r o o f of a s s e r t i o n s (2)  (The r e l a t i o n s h i p w i t h the Hopf-Whitehead homomorphism,  mentioned i n a s s e r t i o n ( 3 ) , i s e s t a b l i s h e d i n [ K e r v a i r e 1959,  p.349].)  15 Chapter I I . §4.  The  The Fundamental  Theorem of Surgery.  Surgery Problem.  The t e c h n i q u e of s u r g e r y , which K e r v a i r e and M i l n o r used to o b t a i n t h e i r r e s u l t s on homotopy s p h e r e s , d i s c u s s e d , above, was element i n Browder's  a l s o a key  s o l u t i o n of the s u r g e r y problem,(which was  based  on  work by K e r v a i r e / M i l n o r , and N o v i k o v ) . Very i n f o r m a l l y , t h i s problem can be s t a t e d as G i v e n a map to f  follows:  f:M—>X between m a n i f o l d s , when can f and M be m o d i f i e d  and M'  such t h a t f ' :M'—>-X i s a homotopy e q u i v a l e n c e ?  To s t a t e a more p r e c i s e v e r s i o n of t h i s problem, we a few  shall first  need  definitions. A Poincare pair  (X,Y) of dimension m i s a p a i r of CW  t h a t t h e r e i s an element  [X]n :H^(X)—HH  m  c a l l e d P o i n c a r e d u a l i t y , and Let  [X] i s c a l l e d  This property i s  the o r i e n t a t i o n c l a s s of (X,Y).  (X,Y) be a P o i n c a r e p a i r of dimension m  (Y may  be  (M,9M) a smooth compact o r i e n t e d m-manifold w i t h boundary, f:(M,3M)—*-(X,Y) a map. smooth compact and F|M=f.  such  [X]eH (X,Y) of i n f i n i t e o r d e r f o r which m  (X,Y) i s an isomorphism f o r a l l q.  ;  complexes  A cobordism of f i s a p a i r  (W,F)  empty), and  where W i s a  (m+l)-manif o l d , 9w=MuU" uM' , 9U=9Mu9M', F: (W,U)—»-(X,Y) , m  m  I f U=9MxI and F ( x , t ) = f ( x ) f o r xe9M, t e l , then (W,F)  will  be  c a l l e d a cobordism of f r e l Y. L e t us assume t h a t k » m k w i t h normal bundle v m4-k—1 in S .  and t h a t  (M,9M) i s embedded i n ( D ^ , S ^ m +  "*")  i , so t h a t v|9M  i s e q u a l to the normal bundle of  9M  k Let E  be a k-plane bundle over X.  A normal map k  f:(M,9M)—>-(X,Y) of degree 1 t o g e t h e r w i t h a bundle map f.  m +  A normal cobordism  is a k  b:v —>-E  (W,F,B) of ( f , b ) i s a cobordism (W,F)  map  covering  of f ,  16 k k k together w i t h an e x t e n s i o n -B-:CJ —»-E o f b, where m i s the normal of  i nD  m + k  x l , where the embedding i s such t h a t  (M,3M)c(D  m+k  bundle xO,S  m+k  ~ xO), 1  /wi / ^ k T „m+k-l , , m+k-1 (M',9M')c(D x l , s x l ) and UcS xi. m +  TT  T  A normal cobordism r e l Y i s a cobordism r e l Y such t h a t i t i s a normal cobordism and B(v,t)=b(v) f o r vev|3M, tel. The p r e c i s e v e r s i o n o f the s u r g e r y problem i s : k .Problem:Given  k  a normal map ( f , b ) , f:(M,3M)—KX,Y) > b:v —»-E , when i s  ( f , b ) n o r m a l l y cobordant t o a homotopy e q u i v a l e n c e o f p a i r s ? A r e l a t e d q u e s t i o n i s the R e s t r i c t e d Problem: Given a normal map ( f , b ) , f : (M, 3M)—> (X,Y) , b : v — , when i s ( f , b ) n o r m a l l y cobordant r e l Y t o ( f ' , b ' ) , where f':M'—>-X i s a homotopy e q u i v a l e n c e ? The s o l u t i o n t o t h e r e s t r i c t e d problem i s g i v e n by the f o l l o w i n g two  theorems:  4.1  The I n v a r i a n t Theorem: L e t ( f , b ) be a normal map, as above, such  t h a t f| cM induces an isomorphism i n homology. a ( f , b ) d e f i n e d , a=0 i f m i s odd, ae  i f m=0  Then t h e r e i s an i n v a r i a n t (mod 4) and oe  i f m=2 (mod 4)  such that a(f,b)=0 i f ( f , b ) i s n o r m a l l y cobordant t o a map i n d u c i n g a homology . f f i ^ r  isomorphism.  •-The Fundamental/Surgery  and suppose  Theorem: L e t ( f , b ) be a normal map, as above,  (1) f|3M induces an isomorphism i n homology, (2) X i s  s i m p l y - c o n n e c t e d , and (3) m>5.  I f m i s odd then (f ,b) i s n o r m a l l y cobordant  r e l Y to a homotopy e q u i v a l e n c e f':M'—>X.  I f m i s even, then ( f , b ) i s  n o r m a l l y cobordant. r e l Y t o ( f ' , b ' ) such t h a t f ' :M'—>-X i s a homotopy e q u i v a l e n c e if and o n l y i f a ( f , b ) = 0 . Our d i s c u s s i o n o f s u r g e r y f o l l o w s v e r y c l o s e l y the treatment of [Browder  1972], and c o n s i s t s o f the d e f i n i t i o n of the i n v a r i a n t a ,  the statement and p r o o f  o f c e r t a i n p r o p e r t i e s i t has, the proof  I n v a r i a n t and Fundamental consequences  theorems, and .the. statement o f c e r t a i n  o f the Fundamental, theorem, p a r t i c u l a r l y  plumbing and the Plumbing  Theorem.  The Surgery  the technique o f  F i n a l l y we w i l l use the l a t t e r to  d e r i v e some c l a s s i f i c a t i o n r e s u l t s f o r  §5.  o f the  manifolds.  Invariant.  B e f o r e d e f i n i n g a we s h a l l r e c a l l some p e r t i n e n t f a c t s about and b i l i n e a r forms over Z and Z 2 .  quadratic  A symmetric b i l i n e a r form (•»•) on a Z-module V s a t i s f i e s : (1)  (x,y)=(y,x)  x,x',yeV. A=(a  If{bA  ) represents  and (2)  (Xx+X'x',y)=X(x,y)+X (x',y) f o r X,A'eZ, 1  i s a b a s i s f o r V and a_^ = (b^ ,b^) , then the m a t r i x (•,•) i n the sense t h a t  on the r i g h t a r e r e p r e s e n t a t i o n s  (x,y)=xAy  t  (where x and y  of the elements i n the b a s i s  {b^}).  I f we pass to a new b a s i s by an i n v e r t i b l e m a t r i x M, so t h a t b'=Mb, l&hen i n terms o f the new b a s i s  (•»•) i s . r e p r e s e n t e d  The b i l i n e a r form (•,•) d e f i n e s a q u a d r a t i c q(x) = ( x , x ) . from q.  by MAM .  form q:V—>Z by  We have (x,y)=i-(q(x4y)-q(x)-q(y)) so t h a t  Each o f q and (•,•) i s s a i d to be a s s o c i a t e d  The form (•,•) a l s o d e f i n e s n a t u r a l l y a b i l i n e a r form 5.1  P r o p o s i t i o n : I f (•»•) i s a symmetric  dimensional vector  associated  (•,•) i s d e r i v a b l e t o the o t h e r . (? , •_) :VxQ—>-Q.  b i l i n e a r form on a f i n i t e  space V over Q i n t o Q, then t h e r e i s a b a s i s f o r  V such t h a t the m a t r i x r e p r e s e n t i n g Define  t  the s i g n a t u r e quadratic  (*,•) i n t h a t b a s i s i s d i a g o n a l .  of a b i l i n e a r form (and hence of the  form) to be the number of p o s i t i v e  e n t r i e s minus the number of n e g a t i v e diagonal matrix representing  diagonal  the form.  diagonal  e n t r i e s , using a  The s i g n a t u r e  i s , i n fact,  i n v a r i a n t under a change o f b a s i s , and we s h a l l t h i n k o f i t as an i n v a r i a n t of q u a d r a t i c forms over Z, t a k i n g v a l u e s A quadratic  i n Z.  (or b i l i n e a r ) form over Z i s c a l l e d n o n s i n g u l a r i f  the determinant o f the m a t r i x A r e p r e s e n t i n g i t i s it  i s c a l l e d nonsingular  5.2  Over a f i e l d  i f the determinant i s nonzero.  P r o p o s i t i o n : L e t q be a n o n s i n g u l a r  dimensional  1.  v e c t o r space V over R.  q u a d r a t i c form on a f i n i t e  Then sgn(q)=0 i f and o n l y i f  t h e r e i s a subspace UcV such t h a t : (1) dim U=^dim V D  and (2) (x,y)=0 f o r x,yeU.  0  Some r e s u l t s we w i l l use f o l l o w . 5.3  P r o p o s i t i o n : L e t q be a n o n s i n g u l a r  suppose q i s i n d e f i n i t e  q u a d r a t i c form V—>-Z and  ( i . e . n e i t h e r p o s i t i v e nor n e g a t i v e  definite).  Then t h e r e i s xeV, x*0 such t h a t q(x)=0. 5.4  P r o p o s i t i o n : L e t q be a n o n s i n g u l a r  q u a d r a t i c form V  suppose 2|q(x,x) f o r a l l xeV (q i s c a l l e d even). A q u a d r a t i c form q on a Z - v e c t o r 2  space  and  Then 8|sgn(q).  V i s a f u n c t i o n q:V—>-Z  2  such t h a t q(0)=0 and q(x+y)-q(x)-q(y)=(x,y) i s b i l i n e a r .  Two  quadratic  forms q,q' on V a r e e q u i v a l e n t i f t h e r e i s an automorphism a:V—»V such t h a t q=q'°a. and If  Under., t h i s d e f i n i t i o n , i t i s c l e a r t h a t  (x,x)=q(2x)-2q(x)=0 so that (•,•) i s n o n s i n g u l a r ,  (*,•) i s a s y m p l e c t i c b i l i n e a r  classify Z -vector 2  form.  i t f o l l o w s t h a t V i s o f even dimension, and  t h a t we may f i n d a b a s i s {a.,b.} f o r V such t h a t . i i (a_^ ,a. ) = (b^ ,b_. )=0 .  (x,y) = (y,x)  (a.,b.)=6,., l j ±3  Such a b a s i s i s c a l l e d s y m p l e c t i c .  spaces w i t h n o n s i n g u l a r  We s h a l l now  q u a d r a t i c forms, and  thereby  d e f i n e the A r f i n v a r i a n t o f such forms. L e t U be the 2-dimensional Z - v e c t o r 2  that  (a,a)=(b,b)=0, ( a , b ) = l .  space, w i t h b a s i s a,b, such  There a r e two q u a d r a t i c forms on U  compatible w i t h (•,•): qo qo(b)=0.  n  q i > d e f i n e d by  d  Note t h a t f o r b o t h q ( a + b ) = l .  ( a ) = q i ( b ) = l , qo(a)=  (The n o t a t i o n s U,qo and qi  i  w i l l remain f i x e d 5.5  a  throughout § 5.)  Lemma: Any n o n s i n g u l a r q u a d r a t i c form on a 2-dimensional Z - v e c t o r 2  space i s e q u i v a l e n t t o qg or q^. S i n c e such a space has o n l y 4 elements, the isomorphism i s easy to c o n s t r u c t . I f q and q' a r e q u a d r a t i c forms on spaces V and V', then q®q' is 5.6  the q u a d r a t i c form on VOV' g i v e n by  (q®q')(v,v')=q(v)+q'(v ). 1  Lemma: On U9U, qo^qo i s i s o m o r p h i c t o q ^ q j . The p r o o f c o n s i s t s of a s i m p l e rearrangement of b a s e s . Now we can b e g i n c l a s s i f y i n g  5.7  forms.  P r o p o s i t i o n : A n o n s i n g u l a r q u a d r a t i c form q on a Z - v e c t o r space 2  (which must have even dimension 2m) i s e q u i v a l e n t e i t h e r t o „, m-1 . qi$($ q ) 0  or t o  m 9 q . 0  P r o o f : L e t {a_^,b_^}, i=l,...,m be a s y m p l e c t i c b a s i s of V, and l e t be the subspace spanned by ^»b^>  and l e t  a  ^=q|v^.  Then by the n a t u r e  m of the b a s i s , q ^ ^ j  and by Lemma 5.5 ^  By Lemma 5.6 qi$qi-qo®qo>  s  o  q I  s  i s e q u i v a l e n t to qo or q^ .  e q u i v a l e n t t o e i t h e r © q o or m  TU~~1  qi®(®  q .) •  QED  0  To complete the c l a s s i f i c a t i o n , we must show t h a t <f>o $ qo I =  not e q u i v a l e n t t o <}>i=q;L®(e  m  5.8  "*"qo)-  m  s  T h i s i s c l e a r from the  P r o p o s i t i o n : The q u a d r a t i c form cf^ on V sends a m a j o r i t y o f  elements of V t o l e Z , w h i l e §Q sends a m a j o r i t y o f elements to 0 e Z . 2  2  The p r o o f i s by i n d u c t i o n on the dimension o f V. U s i n g t h i s n o t a t i o n , we d e f i n e the Arf. i n v a r i a n t of a n o n s i n g u l a r q u a d r a t i c form q on V as f o l l o w s :  20 r  0 i f q=<t>o  Arf(q)= { 1 i f q=<h Thus we have: 5.9  Theorem:(Arf) Two n o n s i n g u l a r  Z -vector 2  q u a d r a t i c forms on a f i n i t e  dimensional  space a r e e q u i v a l e n t i f and o n l y i f they have the same A r f  invariant. In analogy w i t h a p r e v i o u s  result  concerning  q u a d r a t i c forms  over  Z, we have the 5.10  P r o p o s i t i o n : L e t q be a n o n s i n g u l a r  space V.  q u a d r a t i c form on t h e Z - v e c t o r 2  Then A r f ( q ) = 0 i f and o n l y i f t h e r e i s a subspace UcV such t h a t  (1) rank„ U=^rank„ V, and (2) q(x)=0 f o r a l l xeU. z z z< 2  Given radical If  2  a bilinear  form (•,*) on a v e c t o r space V, d e f i n e R, the  o f V, to be {xev| (x,y)=0 f o r a l l yeV}. q:V—>-Z  2  i s a q u a d r a t i c form w i t h  (•,•) as a s s o c i a t e d  form, we have d e f i n e d A r f ( q ) o n l y i f R=0.  I f q|R50, i t i s e a s i l y  t h a t q d e f i n e s q' on V/R, and the r a d i c a l of V/R i s z e r o . we may d e f i n e A r f ( q ) to be A r f ( q ' ) . d e f i n e the A r f i n v a r i a n t ,  and i n f a c t  bilinear  In this  seen case  I f q|R^0, i t doesn't make sense t o the e q u i v a l e n c e  o f the form i s  determined by rankV and rankR. Thus we have: 5.11 of  Theorem: L e t q:V—>-Z be a q u a d r a t i c form over 2  the a s s o c i a t e d b i l i n e a r  form.  d e f i n e d i f and o n l y i f q|R=0. up  Z , R the r a d i c a l 2  Then the A r f i n v a r i a n t  Arf(q) i s  I n g e n e r a l , i f q|RsO, then q i s determined  t o isomorphism by rank  V, rank„ R, and A r f ( q ) , Z ^2 then q i s determined by rank„ V and rank R.  while  i f qlRiO,  2  Zi  2  ^2  Note: Browder uses the n o t a t i o n c(q) f o r the A r f i n v a r i a n t . We w i l l now d e f i n e an i n v a r i a n t  I which d e t e c t s maps i n the  21 cobordism c l a s s of a homology isomorphism. A map f:(X,Y)—>-(A,B) between P o i n c a r e p a i r s of the same dimension i s said  t o be, of degree 1 i f f ^ [ X ] = [A], where f ^ :H^ (X,Y)—^-H^ (A,B)  i s the map i n homology induced by f . We denote.the map induced H (X)'i—••H^ (A) by f , and s i m i l a r n o t a t i o n i n cohomology. #  A  5.12  Theorem: Maps o f degree 1 s p l i t , i . e . w i t h n o t a t i o n as above,  there are a a  A  :H (A,B-)-H^ (X,Y) ,  :H^ (A)-*H*(X) ,  A  ft *  ft  ft  ft  :H (X,Y)—>H (A,B),  ft  g :H (X)—*H ( A ) ,  such t h a t f * c ^ = l , f * B * = l , a f =1, 3 f =1. The s p l i t t i n g s a r e d e f i n e d s t r a i g h t f o r w a r d l y u s i n g the P o i n c a r e d u a l i t y isomorphisms, and t h e i r  inverses.  I t f o l l o w s from t h i s theorem t h a t t h e r e a r e d i r e c t sum s p l i t t i n g s  £^®lm  H^(X,Y)=ker  H^(X)=ker  c ^ ,  H (X,Y)=im f Qker a ,  f S>im A  H (X)=im f eker  ft  Thus we e s t a b l i s h the f o l l o w i n g n o t a t i o n : K ( X , Y ) = ( k e r f ^ ) cH (X,Y),  K ( X ) = ( k e r f ) cH (X),  K (X,Y)=(ker a * ) c H ( X , Y ) ,  K (X)=(ker 0 * ) c H ( X ) 7  q  q  q  q  q  #  q  q  q  (and s i m i l a r l y f o r (co)homology w i t h c o e f f i c i e n t s ) . K  q  and  have the f o l l o w i n g property.  In the exact homology and cohomology  sequences of the p a i r (X,Y),  a l l t h e maps p r e s e r v e the d i r e c t sum s p l i t t i n g , so induce a diagram, commutative  up to s i g n , w i t h exact  rows:  . . .-i!K - (Y)-^K (X Y-)- —»K (X)— q  1  q  J  q  V  9[X]n 9  [X]n-  [X]n-  •K (Y)—*K (X) *->K (X,Y) m-q m-q m-q  -K (Y)-^—. . . q  9[X]n«  jl  m-q-1  (Y)  22 The p r o o f o f t h i s p r o p e r t y c o n s i s t s o f the p r o o f t h a t the d i r e c t sum s p l i t t i n g s a r e p r e s e r v e d by the Poincare' d u a l i t y map ([X]n«) and the homology maps. From t h i s sequence, and u s i n g the d e f i n i t i o n of the K develop  q  groups, we  the f o l l o w i n g diagram, w i t h exact rows and columns: 0  0  (Y)< H (Y-y q  0  0  R |X)^-R (X,Y)^  K |(Y)<  H ( X ) ^ - H (X, Y)«  H  H  H  q  q  q _ 1  (Y)-  f  (f|Y)* H (B)« i 0 q  H (A)^ + 0  H (A,B)-« + 0  q  H  q  q  (B)-< + 0  1  . ..  Suppose m=dim(X,Y)=4k and c o n s i d e r the p a i r i n g K  (X,Y;S)®K *(X,Y;5)-H3  g i v e n by (x,y)=(xuy)[X].  T h i s i s symmetric because the dimension i s even. 2k D e f i n e 1 ( f ) to be the s i g n a t u r e of (•,•) on K  (X,Y;<3).  Note t h a t  (•,•) i s the r a t i o n a l form o f the i n t e g r a l form d e f i n e d on K ( X , Y ) / t o r s i o n by the same formula. I f (f|Y) :H (B;<2)—>H (Y;Q) i s * 2k 2k an isomorphism, then so i s j :K (X,Y;Q)—>K ( X ; Q ) , and so (xuy)[X]=((j*x)uy)[X]. But we have the f o l l o w i n g p r o p e r t y of the K 5.13  groups:  P r o p o s i t i o n : Under the p a i r i n g H ( X ; F ) ® H ~ ( X , Y ; F ) q  (x,y)=(xuy)[X], where F i s a r i n g ,  q  q  q  K (X;F)®K  m - q  (X,Y;F)  m  q  F g i v e n by  ( X , Y ; F ) i s o r t h o g o n a l to  f * ( H ( A ; F ) ) , K ( X ; F ) i s o r t h o g o n a l to f * ( H q  q  m _ q  ( A , B ; F ) ) , and on  the p a i r i n g i s n o n s i n g u l a r i f F i s a f i e l d .  I f F=Z, i t i s n o n s i n g u l a r on K ( X ) / t o r s i o n ® K ~ ( X , Y ) / t o r s i o n . q  m  q  The p r o o f i s s t r a i g h t f o r w a r d v e r i f i c a t i o n , depending on c e r t a i n elementary p r o p e r t i e s o f the cup and cap p r o d u c t s .  Taking q=2k and ~F=Q, we  s e e ' t h a t the p a i r i n g (•,•) .  i s nonsingular.  ft  ft  defined  above  ft  S i m i l a r l y i f ( f | Y ) :H (B)—>H (Y) i s an isomorphism,  then the i n t e g r a l form i s n o n s i n g u l a r .  In p a r t i c u l a r t h i s i s the case  i f Y=B=<J>. 5.14  Theorem; L e t f:(X,Y)—*(A,B) be a map  p a i r s of dimension m=4k+l.  of degree 1 between P o i n c a r e  Then I(f|Y)=0.  P r o o f : The p r o o f i s an a p p l i c a t i o n of P r o p o s i t i o n  5.2.  5.15  P r o p o s i t i o n : Under the hypotheses of the theorem we * 2k 1 2k * 2k 2k r a n k f i m i ) =±rank„K (Y;Q), where i :K (X;§)—>K (Y;Q) H 2 (ci  have i s induced  from the i n c l u s i o n i:Y—>-X. P r o o f : We have a diagram, commutative •..-.  HC  2V  ?k I ! ?k+l ^K (Y;S)^K (X,Y;«)  i -(X-;#)—  Z K  Z K  [Y]n«|.  [X]n8  *'^ 2k+1 K  ( X  ' ' Y  0  up to s i g n :  s  K  2k  ( Y 5  *  •. . .  X  [X]n« .  ^O-^—^K^CX;Q)  In t h i s diagram the rows a r e exact and the v e r t i c a l maps a r e isomorphisms.  ft 2k Hence (im i ) =(ker i j . ) *  0 1  zk  .  I t i s e a s i l y shown t h a t the  ft U n i v e r s a l C o e f f i c i e n t Formulae h o l d f o r K and K , and thus, s i n c e Q 2k 2k is a field, K (Y;e)SHom(K (Y;Q),Q), K (X;Q)=Hom(K (X;Q),Q), ft ft 2k and i ^ o m C i ^ , 1 ) . Hence r a n k ^ ( i m i ) =rank^(im i ^ ^ ^ ' ^ A  2k  2k  a n (  2k  (Y;Q).  rank^(im i ) + r a n k ^ ( k e r i ^ ) ^ r a n k ^ K ^ Y ; £ ) = r a n k ^ K A  2 k  rank_(im i*) =-^rank J C ( Y ; S ) .  Hence,  2k  QED  2 k  ft 2k 5.16  Lemma: With the hypotheses of 5.15,  (im i )  2k <=K  (Y;Q)  a n n i h i l a t e s i t s e l f under the p a i r i n g (*,*)• Proof: (i*x,i*y)=((i*x)u(i*y))[Y]=(i*(xuy))[Y]=(xuy)(i*[Y])=0 ijY]=i^[X]=0 in H  4 k  (X). * 2k  Proof of Theorem 5.14: By 5.15, 1 rank =-—rank K  since  (im i )  2k cK  (Y;Q) i s a subspace of  2k  (Y;Q),  and by 5.16  i t a n n i h i l a t e s i t s e l f under the  24 pairing..  Hence by P r o p o s i t i o n 5.2, sgn(',«) 0  on K  =  2k  (Y;Q) , s o - t h a t  I(f|Y)=0.  QED  The sum o f P o i n c a r 6 p a i r s i s d e f i n e d as f o l l o w s : If  (X^,X uY_^) i = l , 2 a r e P o i n c a r e p a i r s o f dimension m, such t h a t 0  XinX2=Xo, Y^nXo=Yg, and (Xg,Yo) i s a P o i n c a r e p a i r of dimension m-1, then i t f o l l o w s  [Browder 1972, p.13]  p a i r of dimension m, c a l l e d If ±  (XiUX ,YiuY ) i s a Poincare 2  2  the sum of (X^,XgUY_^) a l o n g  (X,Y) and (A,B) a r e the sums, r e s p e c t i v e l y , of  (A ,B uA ), i  that  0  and f:(X,Y)—^(A,B) w i t h f ( X ) c A , ±  i  ( X Q , Y Q ) .  (X^JY^UXQ)  and  then t h e f o l l o w i n g  are equivalent: (1) f has degree 1 (2) f = f | ( X , Y ) has degree 1 0  0  (3) f = f | ±  0  ( X ^ Y ^ X Q )  ( a l l with appropriate  have degree 1  orientations).  We say t h a t f i s the sum of fi and f 5.17  Theorem: Suppose  2  .  f : (X,Y)—>(A,B), a degree 1 map, i s the sum o f  two maps f ^ : ( X ^ X g u Y . ^ ) — K A ^ J A Q . U B ^ ) , i = l , 2 , and suppose t h a t the map on t h e i n t e r s e c t i o n f Q : H ( A , B Q ; Q ) - + R ( X , Y ; Q ) 0  Then  0  0  i s an isomorphism.  I(f)=I(f )+I(f ). 1  If  2  (X,Y) i s a P o i n c a r e p a i r o f dimension m=4k we may c o n s i d e r the 2k  symmetric p a i r i n g  H  2k (X,Y;Q)®H  (X,Y;Q)—*Q  g i v e n by ( x , y ) = ( x u y ) [ X ] , 2k  and we d e f i n e I(X,Y) t o be the s i g n a t u r e of (•,•) on H 5.18  (X,Y;<3).  Theorem: I ( f ) = I ( X , Y ) - I ( A , B ) . Thus we have the important theorem  5.19  Theorem: L e t f : ( X , Y ) — K A , B ) be a map of degree 1 between P o i n c a r e  p a i r s o f dimension m=4k.  Suppose  ( f | Y ) :H (B;<2)—>-H (Y;Q) i s an isomorphism  and t h a t f i s cobordant r e l Y to f':(X',Y)—>(A,B)  such t h a t  25 :H (A;Q)—*-H (X' ;Q)  f  i s an isomorphism.  Then l ( f ) = 0 .  P r o o f : L e t U be the.cobordism r e l Y between X and X', so t h a t 9U=XuX', XriX'=Y, (U,3U)is a P o i n c a r e p a i r of dimension m+1, and F i s the map F as a map  (U,Y)—>(A,B) such t h a t F | x = f , F | x ' = f .  of degree 1 G: (U,XuX')—>-(AxI ,AxOuBxIuAxl) .  I(G|XuX')=0, and by Theorem 5.17 since f  compatibly  We may  I(G|XuX')=I(f)-I(f').  Now  l(f')=0  (X,Y) s a t i s f i e s  2  P o i n c a r e d u a l i t y f o r homology w i t h c o e f f i c i e n t s i n Z ) . 2  m  1  i  2  by J l ( x ) = ( S q x ) [X] , where S q 1  2  By P o i n c a r e \ d u a l i t y , H ( X ; Z ) ® H 1  t  Steenrod  h  [X]eH (X,Y;Zo) i s the o r i e n t a t i o n m  m_1  2  Define a l i n e a r  i s the i  1  i  square (see [Steenrod 1962]) and  5.14,  QED  (X,Y) be a Z - P o i n c a r e p a i r of dimension m ( i . e .  £ :H ~ (X,Y;Z )—*Z  consider  By Theorem  i s an isomorphism, and hence l ( f ) = 0 .  Let  map  oriented,  (X,Y;Z )—>Z 2  g i v e n by (x,y) = (xuy)[X] i s  2  a n o n s i n g u l a r p a i r i n g , so t h a t H ( X ; Z ) i s i s o m o r p h i c , u s i n g t h i s  pairing  X  2  to H o m ( H for a l l  m-1  ( X , Y ; Z ) ,Z ) , and hence £ ( x ) = ( x , v ) f o r a unique v ^ H  xeH  2  m _ 1  2  1  1  (X;Z ) , 2  (X,Y;Z ). 2  D e f i n e the Wu 5.20  ±  class.  c l a s s of X to be V=l+v +v +. . . , v^eH"*"(X;Z ) as above. 2  P r o p o s i t i o n : L e t (X,Y) and (A,B) be Z - P o i n c a r e p a i r s of dimension 2  f : (X,Y)—>(A,B) a map on homology w i t h Z  2  m,  of degree 1 (mod 2) ( i . e . f*[X]=[A] f o r f ^ d e f i n e d  coefficients).  Then v^(X)=v_^+f ( v ^ ( A ) ) ,  where  v eK (X). ±  i  * The p r o o f c o n s i s t s of a c a l c u l a t i o n to show t h a t v ^ ( X ) - f 5.21  P r o p o s i t i o n : With n o t a t i o n as i n 5.20, suppose m=2q.  pairing  (•,•)  on K^(X,Y;Z ) 2  i s symplectic  (i.e.  i (v^(A))eK (X)  Then the  (x,x)=0 f o r a l l  x)  if  and o n l y i f f * v (A)=v ( X ) . q q Proof: (x,x)=x [X]=(Sq x)[X]=(xuv (X))[X]=(x,v (X)) for xeH (X,Y;Z ), q * q and s i n c e K (X,Y;Z ) and (im f ) a r e o r t h o g o n a l by P r o p o s i t i o n 5.13, 2  q  q  q  q  2  2  (x,f*v (A))=0 f o r xeK (X,Y;Z ). q  ±  2  Hence f o r x e K ( X , Y ; Z ) , (x,x)=(x,v ) q  2  by P r o p o s i t i o n 5.20.  5.22  Then  (x,x)=0 i f and o n l y i f v =v ( X ) - f v (A)=0. q q q  C o r o l l a r y : L e t (X,Y) and (A,B) be o r i e n t e d P o i n c a r g d u a l i t y  of d i m e n s i o n m=4£ , and l e t f : (X,Y)—>(A,B) = v ^ ( X ) , then the p a i r i n g  be of degree 1.  pairs  If f  (A)  (x,y)=(xuy)[X] ( f o r x,yeK ( X , Y ) / t o r s i o n ) i s  2  even ( i . e . 2|(x,x) f o r a l l x ) . T h i s f o l l o w s from the f a c t t h a t  (x,x) reduced mod 2 i s zero by 5.21  and thus (x,x) must be even. 5.23  C o r o l l a r y : L e t (X,Y) and (A,B) be o r i e n t e d P o i n c a r e p a i r s of  dimension m=4£ , f : (X,Y)—>(A,B)  o f degree 1 such t h a t  (f | Y) :H^ (Y)—*-H^ (B) A  * i s an isomorphism.  I f f ( v ^ ^ ( A ) ) = v ^ ^ ( X ) , then 1 ( f ) i s d i v i s i b l e by 8.  This follows d i r e c t l y  from 5.22 and P r o p o s i t i o n 5.4.  L e t us now i n v e s t i g a t e t h e Wu c l a s s , w i t h t h e aim of showing it  that  i s p r e s e r v e d by normal maps. k  (X,Y) be a p a i r of s p a c e s , and E a f i b r e bundle over X w i t h k-1 f i b r e F such t h a t H^(F;Z )=H^(S ; Z ) . Then we may d e f i n e the Thorn space T(£)=XucE(£) u s i n g t h e p r o j e c t i o n o f E as a t t a c h i n g map. There Let  2  2  i s a Thorn c l a s s UeH (T(£);Z ) such t h a t 2  •UU:H (X; Z ) - * H q  2  .  q + k  (T(E);Z )  •uU:H (X,Y;Z )->H q  2  q+k  2  (T(£),T(?|Y);Z ) 2  •nU:H (T(£),T(?|Y);Z )->H (X,Y;Z ) s s—k 2  •nU:H (T(£>;Z )->H  (X;Z )  2  S  are isomorphisms. mod 2. 5.24  2  S ~~K.  L e t h:iT (A,B)—>H (A,B;Z ) r  r  2  be the Hurewicz homomorphism  We have the f o l l o w i n g important theorem o f S p i v a k :  Theorem: L e t (X,Y) be an n - d i m e n s i o n a l P o i n c a r e p a i r , w i t h X  s i m p l y - c o n n e c t e d and Y a f i n i t e is  2  complex up t o homotopy type.  Then  there  a s p h e r i c a l f i b r e space E, w i t h X as base space, i t s f i b r e a homotopy  ( k - l ) - s p h e r e , and an element aeit  , (T(E) ,T(E | Y)) such t h a t h(a)nU=[X].  27 The f i b r e bundle E i s c a l l e d  the Spivak normal f i b r e space of X,  and can a l s o be d e f i n e d f o r homology w i t h 5.25 E Z  k.  2  coefficients.  P r o p o s i t i o n : L e t (X,Y) be a Z - P o i n c a r e p a i r of dimension  m,  2  a Z  2  Spivak normal  f i b r e space over X ( i . e . the f i b r e of E i s a  homology ( k - 1 ) - s p h e r e ) , aerr „ (T(E) ,T(E IY)) such t h a t h(a)nU=[X] m+k.  in H  m  (X,Y;Z ). 2  We  Then V ( X ) u U = S q ( U ) . -1  r e c a l l the f a c t  t h a t the Thorn c l a s s UeH  c h a r a c t e r i s e d by the f a c t  (T(E);Z ) i s 2  * k t h a t j (U) generates H ( E F ; Z ) = Z , 2  2  where  j :EF—KC(E) i s the i n c l u s i o n of the Thorn complex over a p o i n t i n t o the whole Thorn complex. 5.26  P r o p o s i t i o n : L e t b :E™>E' be a map  of f i b r e spaces c o v e r i n g f:X—*-X',  where E and E' have f i b r e F, H ^ ( F ; Z ) = H ( S 2  k - 1  A  ;Z ). 2  Then b induces a  map  of Thorn complexes T(b) :T(E)—»-T(E ) , and T(b) U'=U,  are  the Thorn c l a s s e s of E and  r  where U and  U'  E'.  Proof,: L e t E,E' be the t o t a l spaces of E,E' r e s p . , so t h a t the f o l l o w i n g TT  diagram commutes :  F  >E  >-X  F  »-E-  1  Hence, f,b induce T(b) :Xu-' cE—>-X' u ,cE', and the diagram EF IT  commutes.  Hence j T(b) U'=j'  J  >T(E)  TT  U',  so that j T(b) U' * generates H ( E F ; Z ) , and thus T(b) U'=U. QED  I ., + EF-^—KT (E ' ) w  k  2  5.27  C o r o l l a r y : L e t (X,Y) and  (A,B) be Z  2  P o i n c a r e p a i r s of dimension  E' a f i b r e space over A w i t h f i b r e F a ( k - 1 ) - d i m e n s i o n a l Z sphere. E=f  L e t f:(X,Y)—>-(A,B) be of degree 1 i n Z  (£').  h(ct)nU=[X].  Suppose t h e r e i s an element a e T T  m + k  2  2  m,  homology  homology, and l e t  ( T ( E ) ,T(E | Y)) such t h a t  Then f * ( V ( A ) ) = V ( X ) , i n p a r t i c u l a r f * v (A)=v^(X) f o r a l l q.  P r o o f : By 5.26,  i f b:E—>E' i s the n a t u r a l map,  T(b) U'=U.  S e t t i n g V(X)=V,  V ( A ) = V , we have, u s i n g 5.25,  T(b)  (V'uU')=f  =T(b)*(Sq~ U')=Sq~ T(b)*U'=Sq~ U=VuU. i  5.28  1  1  Theorem: L e t (X,Y) and  V'uT(b) U'=f  Hence f*V'=V.  (A,B) be o r i e n t e d P o i n c a r e p a i r s of  dimension m=4£ , f : (X,Y)—*-(A,B) of degree 1 such t h a t  (f | Y) ^ i s an  isomorphism, and E' a f i b r e space over A w i t h f i b r e F a Z (k-l)-sphere.  Set E=f E  and suppose  1  (V')uU  t h e r e i s aeir  k  2  homology  ( T ( E ) ,T(E | Y) )  such t h a t h(a)nU e q u a l s the o r i e n t a t i o n c l a s s of (X,Y) reduced mod Then 1 ( f ) i s d i v i s i b l e by P r o o f : By 5.27, Let  f*v  8.  (A)=v  ( X ) , so by 5.23  ( f , b ) be a normal map,  1 ( f ) i s d i v i s i b l e by  (A,B) an o r i e n t e d P o i n c a r e p a i r of  dimension m, m=4£ , and b:v—>r\ a l i n e a r bundle map ifl~l~lc normal bundle of (M,9M)c(D  c o v e r i n g f , v the  mH~lc• 1 —  ,S  ) , n a k-plane bundle over A.  C o r o l l a r y : I f ( f , b ) i s a normal map  then 1 ( f ) i s d i v i s i b l e by  8.  f:(M,9M)—>(A,B) of degree 1, M a smooth  o r i e n t e d m-manifold w i t h boundary,  5.29  2.  w i t h (f|9M)^ an  isomorphism,  8.  P r o o f : The p a i r , ( f , b ) s a t i s f i e s  the c o n d i t i o n s of 5.28,  where E  l =  n is  a l i n e a r bundle over (A,B). Thus, we may Let  make the f o l l o w i n g  ( f , b ) be a normal map  isomorphism, m=4£  (X,Y) and  f:(M,3M)—>(A,B), e t c . w i t h (f | 9M)^  the dimension of M.  I n v a r i a n t Theorem f o r m=4£ Let  definition:  D e f i n e a (f ,b)=%[ ( f ) . o f o l l o w s from Theorem 5.19.  an  Then the  (A,B) be o r i e n t e d P o i n c a r e p a i r s of dimension m=2q,  and l e t f : (X,Y)—*-(A,B) be a map  of degree 1.  L e t E be the Spivak  normal f i b r e space o f (X,Y), and n t h a t of (A,B), and l e t aen  ,. (T(E) ,T(E I Y)) , Beir (T(n) ,T(n I B) ) be the elements d e f i n e d m+k ' m+k  such  t h a t h(a)nU =[X], h(B)nU =[A], where U ,U a r e the Thorn c l a s s e s of E,n E n E n and h i s the Hurewicz homomorphism.  L e t b:E"~*ri be a map  of f i b r e  spaces  29 covering  f.  We s h a l l c a l l  the p a i r ( f , b ) a normal map of P o i n c a r e p a i r s .  Note t h a t t h i s d e f i n i t i o n i s analogous t o t h a t o f a normal map above.  given  We a l s o d e f i n e normal cobordism.and normal cobordism r e l B o f  P o i n c a r e p a i r s by the same analogy. Browder  [1972, I I I . 4 ] d e f i n e s , u s i n g S p a n i e r and Whitehead's  a quadratic  form \|i :K (X,Y;Z )-^-Z  w i t h a s s o c i a t e d b i l i n e a r form (•,•)>  q  2  2  where (x,y) = (xuy)[X] f o r x,y K ( X , Y ; Z ) .  If  q  2  i s an isomorphism, i t f o l l o w s  (f|Y)*:H*(B;Z )—>H*(Y;Z ) 2  from P r o p o s i t i o n 5.13 t h a t  n o n s i n g u l a r on K ( X , Y ; Z ) ( = K ( X ; Z ) ) . q  S-theory,  (•,•) i s  Then the A r f i n v a r i a n t of ij; i s  q  2  2  2  defined. Let  ( f , b ) be a normal map of P o i n c a r e complexes, f:(X,Y)—>-(A,B) ,  and suppose t h a t define  (f | Y)  :H (B;Z )—>-H ( Y ; Z ) i s an isomorphism. 2  Then  2  i n v a r i a n t c ( f ,b)=Arf (IJJ) .  the Kervaire  Now we w i l l develop some p r o p e r t i e s o f the K e r v a i r e i n v a r i a n t . Let  ( f , b ) be a normal map, f : ( X , Y ) — y ( A , B ) , e t c . and suppose i n  a d d i t i o n t h a t Y and B a r e sums o f P o i n c a r g p a i r s a l o n g the b o u n d a r i e s , and t h a t f sends summands. i n t o summands. Y=Y .uY , Y = Y n Y , 1  2  0  1  2  B=B uB , B = B n B , 1  2  0  1  2  In p a r t i c u l a r , suppose  f ( Y )cB , and t h a t  that  (B ,Bg) and ±  (Y_^,Yg) a r e P o i n c a r e p a i r s compatibly o r i e n t e d w i t h (X,Y) and (A,B) . I f E,n a r e the Spivak normal f i b r e spaces of (X,Y) and (A,B), then E|Y_^, n | B^ a r e the c o r r e s p o n d i n g Spivak normal f i b r e s p a c e s , so t h a t if  f =f|Y , b = b | | Y ^ ) , then ( f ^ * ^ ) a r e a l l normal maps,  i=0,l,2.  i  A  A  A  Note t h a t i f f :H ( B ; Z ) — ( Y ; Z ) i s an isomorphism then i t 2  A follows 5.30  A  2  2  2  2  A  t h a t f o : H (Bg;Z )—>H ( Y ; Z ) i s a l s o an isomorphism. 2  0  2  Theorem: L e t ( f , b ) be a normal map as above, so t h a t fj'Y i s the sum i  A of f j and f  2  isomorphism.  on Y\ Then  and Y , e t c . 2  c(f|,bi)=0.  Suppose  A  A  f :H (B ;Z )—>-H ( Y ; Z ) 2  2  2  2  2  i s an  30 T h i s theorem has the f o l l o w i n g 5.31 to  C o r o l l a r y : I f ( f , b ) i s a normal map  (f',b')  5.32  corollaries:  f *:H*(A,B;Z )—*-H*(X* l  }  2  and i s n o r m a l l y cobordant r e l Y  ,Y;Z ) an isomorphism, then c ( f , b ) = 0 . 2  C o r o l l a r y : I f ( f , b ) i s a normal map,  f : (X,Y)—>(A,B), then  c(f|Y,b|(E|Y))=0. The f i r s t  c o r o l l a r y i s d e r i v e d from the theorem by u s i n g the normal  cobordism as a normal map, The p r o o f o f Theorem g i v e n i n [Browder 1972, Let  the second by t a k i n g Y =cf>. 2  5.30  relies  on the d e f i n i t i o n o f  III.4].  ( f , b ) , f:(X,Y)—>-(A,B) be a normal map  suppose (X,Y) and  , and i s  of P o i n c a r e . p a i r s , and  (A,B). a r e sums of P o i n c a r e p a i r s , i . e . X=X^uX , 2  A=A UA , X =X!nX , A = A n A , Y =X nY, B ^ A ^ B , f ( X ) c A  , and  (A^,AQUB^) are Poincare p a i r s oriented  (X,Y) and  X  2  0  2  0  1  2  ±  ±  compatibly w i t h  Set f = f [ x : ( X , X u Y ) - > ( A , A o U B ) , 1  i  1  0  i  : L  i  (X ,XQUY ) ±  ,  ±  (A,B).  f = f | X : (X , Y ) ^ ( A ,B ) , and 0  0  0  0  0  0  b_^ the a p p r o p r i a t e r e s t r i c t i o n of b. Now  suppose that  a r e isomorphisms. sequence t h a t  5.33  (Y;Z ) and f :H ( A ; Z ) — * H 2  0  0  2  (X ;Z ) 0  x  and  defined. t  s h a l l p r e s e n t a p a r t i a l p r o o f h e r e ; the b a l a n c e i s to be found  [Browder 1972] . Let  and 4*2 be the q u a d r a t i c forms d e f i n e d on K ( X , Y ) , q  K ^ C X j J X Q U Y I ) and K ( X , X u Y ) r e s p e c t i v e l y . q  2  M a y e r - V i e t o r i s sequence of p a i r s  0  2  (which i s r e a l l y  (XQ,Y )C(X,Y)C(X,YUXO),  excisive pair  0  An argument w i t h the  the exact sequence of the t r i p l e  where the l a s t p a i r i s r e p l a c e d by the  ( X j , X g u Y j ) u ( X , X Q U Y 2 ) ) g i v e s an isomorphism 2  2  from arguments w i t h the M a y e r - V i e t o r i s  Theorem: c ( f , b ) = c ( f , ,b, ) + c ( f , b j ) .  P r o o f : We in  2  a r e isomorphisms, so c ( f , b ) , c ( f i , b ) ,  0  2  :H (B;Z )—>H  I t follows e a s i l y  (f_JX uY^)  c ( f , b ) are a l l 2  (f | Y)  p! ®p : K (XT. , X 0 uYi ) <3>K ( X 2 , X 0 uY )->K q  q  2  2  q  (X,Y) ,  where p^ i s d e f i n e d by the diagram K ^ X j ,X uYi)-«  —K (X,X uY) q  0  2  4-  K (X,Y) q  where the isomorphism comes from an e x c i s i o n , and the v e r t i c a l i s induced by i n c l u s i o n I t remains to show  (similarly  arrow  for P 2 ) .  (p^x)=ip Ax) , x e K ( X ^ , X u Y ) . Then ip i s q  0  i s o m o r p h i c to the d i r e c t sum l j ^ e i j ^ , so t h a t A r f (ip)=Ar£(jp 1 )+Arf (TJJ ) . 2  The remainder of the p r o o f i s g i v e n on pp. 72-73 of Browder. Now  suppose  (A,B) i s a P o i n c a r e complex  of dimension m,  and E  i s a l i n e a r bundle over A, f : (M,3M)—>(A,B) i s of degree l,and  b:v—  i s a l i n e a r bundle map  c o v e r i n g f , v i s the normal bundle of (M,3M)  in  ( f , b ) i s a normal map  (D  m +  k,S  m + k  1).  -j_  > e <  i n the o r i g i n a l  sense.  Then by Theorem 1.4.19 o f Browder, t h e r e i s a f i b r e homotopy e q u i v a l e n c e (unique up to homotopy) b ' : E — H Y such t h a t T ( b ' ) ^ ( T ( b ) ^ ( a ) ) = B , where aeir  ,, (T(v),T(v|9M)) and gerr „ (T(n) ,T(n I B)) a r e the elements such t h a t m+k m+k  h(a)nU =[M] v map  and h ( B ) n U =[A]. ri  of P o i n c a r e p a i r s , and we on Z2  (f|9M)  Then b'b:v—*T\ , and  (f,b'b) i s a normal  d e f i n e a(f,b)=c(f,b'b)eZ  cohomology i s an  2  i f m=4k+2 and  isomorphism.  5.34  P r o p o s i t i o n : The v a l u e of  &eiT  (T(n) ,T(n I B)) , and thus depends o n l y on the normal map  m+k  The p r o o f of 5.34  ( f , b ) i s independent of the c h o i c e of  i s p r o v i d e d i n [Browder  With t h i s d e f i n i t i o n of a ( f , b ) f o r m=2 C o r o l l a r y 5.31  1972].  (mod  4 ) , we  see t h a t  p r o v i d e s the p r o o f of the I n v a r i a n t Theorem f o r m^2  and thus completes the p r o o f of that We  (f,b).  (mod  4)  theorem.  have a l s o proved the f o l l o w i n g two p r o p e r t i e s of the i n v a r i a n t  a:  32 5.35  Proposition:  ( A d d i t i o n P r o p e r t y ) Suppose ( f ,b) i s a normal map  which i s the sum o f two normal maps ( f , b i ) and x  f|.9M, f|9M_^ i = l , 2 , and f | Mg induce isomorphisms  and such  ^ 2 ^ 2 ) 5  i n homology.  that  Then  a(f,b)=a(fi,bi)4c(f ,b ). 2  2  T h i s p r o p e r t y i s proved f o r m=4& by Theorem 5.17, and f o r m=4£+2 by Theorem 5.33. 5.36  I t i s v a c u o u s l y t r u e f o r m=2q+l.  Proposition:  (Cobordism P r o p e r t y ) L e t (f ,b) be a normal map,  f : (M,""9M)->(X,Y) , b :v->E , and s e t f'=f|9M:9M-*Y, b b | (v | 9M) :v | 9M—>E | Y. , =  I f m=2"k+l then  (f',b')=0.  T h i s p r o p e r t y f o l l o w s from Theorem 5.14 f o r the case m=4&+l and from C o r o l l a r y 5.32 f o r the case Let  m=4£+3.  us c a l l t h e q u a n t i t y I(X,Y) d e f i n e d above the index o f X.  Then by the H i r z e b r u c h Index Theorem [ H i r z e b r u c h 1966], we have Index M=L (p!(E ) , . . . , p (E ) ) [ X ] , and Theorem 5.18 g i v e s us d i r e c t l y 1  1  K.  K.  the f o l l o w i n g 5.37  Proposition:  (Index P r o p e r t y ) I f Y=<f>, m=4k, ( f , b ) a normal map,  then 8a(f,b)=index M-index X, and index X equals t h e s i g n a t u r e of the 2k q u a d r a t i c form on H  (X;#) g i v e n by <xux,[X]>, where [X] i s the o r i e n t a t i o n  .  c l a s s i n E^(X;Q)  F i n a l l y we s t a t e w i t h o u t p r o o f the 5.38  P r o p o s i t i o n : (Product Formulae) L e t ( f  f . :(M. ,9M.)—KX. >9X.) . X  X  X  X  Suppose  a(fnxf  2  l 9  .bixbo)  bi),  ,  2  2  a(f1 ,bi)=ai, and  I  a(f ,b2)=a2 are a l l defined ( i . e . f j x f | 9 ( M x M ) , 2  2  isomorphisms  ( f , b ) be normal maps  1  2  f^^M  a  r  e a  1  1  homology  with appropriate c o e f f i c i e n t s ) .  Then (1) a(f xf ,b xb )=I(X )a +I(X )a +8a1a2 :  2  1  2  x  2  2  1  when M xM ]  2  i s of  dimension 4k, where I(X^) i s the index of X^, (2) a ( f } X f , b i x b 2 ) = x ( X 1 ) a 2 + x ( X 2 ) a 1 when M^xM 2  2  i s of dimension 4k+2  33 where x ( X ^ )  i s the E u l e r c h a r a c t e r i s t i c o f  .  Note t h a t I(X)=0 by d e f i n i t i o n i f dim X^O  § 6.  (mod 4)  Surgery below t h e M i d d l e Dimension. j  We w i l l now d e s c r i b e the technique will  o f s u r g e r y , the use o f which  enable us t o s o l v e the s u r g e r y problem. Suppose t h a t <Ji :S* xD "'"—•M™, p+q+l=m, i s a d i f f e r e n t i a b l e >  i n t o t h e i n t e r i o r o f M i f 9M*<j>. Define M'=M0U  ({)  D  P+1  L e t Mo=M\int(im <j>) .  x s , w i t h (j)(x,y) i d e n t i f i e d q  Then M' i s a m a n i f o l d , 9M'=3M, and M' i s s a i d u s i n g (j), on M.  I t i s sometimes denoted  i d e n t i f i e d with  P + 1  xD  0  to ( x , y ) e S x S = 9 ( D P  q  P + 1  xS ). q  t o be the r e s u l t o f s u r g e r y  by x(M,4>) (e.g. by M i l n o r ) .  <P  q + 1  ) such  that ( x , y ) e S x D P  (cj) (x,y) ,l)eMxI.  the t r a c e o f the s u r g e r y . m a n i f o l d w i t h boundary.  Then 9M = 9Mucj> ( S P * s q ) .  nH~l W. between M and M' as f o l l o w s :  We may d e f i n e a cobordism W =Mx[0,l]u(D  embedding,  q+  Clearly  q + 1  c9(D  P + 1  xD  q + 1  )is  gW^Mu (9MxI) uM' , and W^ i s c a l l e d  As we have d e f i n e d i t , W, i s n o t a smooth  However, i t has a c a n o n i c a l smooth s t r u c t u r e  ( i . e . i t i s PL-homeomorphic t o a smooth m a n i f o l d ) which i s d e s c r i b e d in  [ M i l n o r 1961].  ( M i l n o r c a l l s W^ io(M,<(>).)  I f w"1"*"1 i s a m a n i f o l d w i t h 9W=Mu(9MxI)uM' and W' has 9W' = M'u(9M'xI)uM", then we may d e f i n e t h e sum o f the two cobordisms by t a k i n g W=WuW' and i d e n t i f y i n g M'c9W w i t h M'c9W'.  Then i t i s c l e a r t h a t  9W=Mu(9MxI)uM". 6.1  Theorem: L e t W be a cobordism w i t h 9W=Mu(9MxI)uM'.  Then t h e r e i s a  sequence o f s u r g e r i e s based on embeddings <J>^, i = l , . . . , k , each s u r g e r y b e i n g on the m a n i f o l d which r e s u l t s t h a t W i s the sum o f W, ,...,W, . *1 <*> k  from the p r e v i o u s s u r g e r y , and such  34 The p r o o f i s an immediate  consequence  o f the Morse Lemma, and a  l u c i d p r o o f may be found i n [ M i l n o r . 1 9 6 1 ] . 6.2  P r o p o s i t i o n : I f M' i s the r e s u l t o f s u r g e r y on M based on an  embedding <j> :S^xD "'"—*-M, then M i s the r e s u l t of s u r g e r y on M' based q+  on an embedding TJJ : S xD^ "'"—>M' such t h a t t h e t r a c e s o f the two s u r g e r i e s q  l+  are the same. 6.3  P r o p o s i t i o n : L e t cf>:S *D ^—*-M be a smooth embedding i n t h e i n t e r i o r P  q+  m  of M, p+q+l=m, and l e t W, be the t r a c e o f the s u r g e r y based on (f>.  Then  9  W, has Mu-D "'' as a d e f o r m a t i o n r e t r a c t , where <j>=<|> I S xO • 9 <P P+  P  P r o o f : W, = (Mx.I) u (D ^xD ' ~'*') , image <j>cMxI, so we may deform Mxl t o Mxl P  9  P + 1  ,  9  l e a v i n g Mxlu (D  q  (D "'"xD "'") f i x e d . P+  9  xO)u(S xD P  q + 1  Then D "'"xD ' ' may be deformed  q+  ),  P+  q+  onto  1  l e a v i n g t h i s l a t t e r subspace f i x e d .  T h i s then y i e l d s  the d e f o r m a t i o n r e t r a c t i o n of W, t o Mu-D^"'".  <J) 6.4  Proposition:  m-manifold,  9  (a) L e t f:(M,3M)—v(A,B) be a map, M an o r i e n t e d smooth  (A,B) a p a i r o f s p a c e s , and l e t cj>:S xo —>-int M P  q+1  be a smooth  embedding, p+q+l=m.  Then f extends t o F: (W^ ,9MxI)—>-(AxI,BxI) to get a — p cobordism o f f i f and o n l y i f f°<j> i s homotopic to the c o n s t a n t map S —>A. (b) Suppose i n a d d i t i o n t h a t n k  i s a l i n e a r k-plane bundle over A,  k  b :v —Hn  i s a l i n e a r bundle map c o v e r i n g f , v the normal bundle of  (M,3M)c(D  m + k  ,S  , k>>m.  m + k  a) i s the normal bundle of to a ) | D  P+1  xn,  covering F|D  P + 1  Then b extends to b:o)—>n c o v e r i n g F, where cD  m + k  x1  i f and only i f b| (v|(f>(S )) extends P  5  xO.  P r o o f : S i n c e Mu-D "'" i s a d e f o r m a t i o n r e t r a c t of W, , i t f o l l o w s t h a t f <l> 9 extends t o W i f and o n l y i f f extends t o Mu-D ''". But t h e l a t t e r i s P+  P+  9  9  t r u e i f and o n l y i f f°$ i s n u l l - h o m o t o p i c , which proves ( a ) . For  ( b ) , i t f o l l o w s from the bundle c o v e r i n g homotopy p r o p e r t y ,  and the f a c t t h a t MU-D ^ i s a d e f o r m a t i o n r e t r a c t of W,, t h a t b extends P+  9  9  to co i f and o n l y i f b extends t o O J | D P ' ' " X O .  QED  +  If  ( f , b ) i s a normal map, (J> :S xD "'"—KLnt M™, p+q+l=m, P  q+  f:(M,9M)—KA,.B) , and i f the t r a c e o f <j) can be made a normal cobordism by  e x t e n d i n g f and b over W^ , we w i l l  say t h a t t h e s u r g e r y based on tf>  i s a normal s u r g e r y on ( f , b ) . From Theorem 6.1, i t f o l l o w s e a s i l y is  t h a t any normal cobordism r e l B  the composite o f normal s u r g e r i e s . Let  <)>: S x D P  q + 1  —>int  be an embedding, w i t h p+q+l=m.  t r a c e , and M' t h e r e s u l t o f the c o r r e s p o n d i n g s u r g e r y . i n v e s t i g a t e the e f f e c t we w i l l  i s the  We w i l l  o f s u r g e r y on the homotopy o f M; i n p a r t i c u l a r ,  examine the r e l a t i o n between the homotopy  groups o f M and M',  below t h e 'middle dimension'. 6.5  Theorem: I f P ^ p <  1t  n  e  n  Tr^M^-Tr^CM)  f o r i<p, and  Tr (M')^rr (M)/{^ TT (S )}, P  p  p  #  p  where {G} denotes the Z[iri(M)]  submodule o f TT (M) generated by G. P P+l P r o o f : By 6.3, W^ i s o f t h e same homotopy type as Mu-D . Hence TT. (W. )=TT  (M) f o r i<p, and TT ( W J ^ T T (M)/{<J>„TT ( S ) } . P  By 6.2 and 6.3,  i c p i p <p p ?fp we have a l s o t h a t W,=W, =M'u-D \ where y: S^D*** —>M' g i v e s the s u r g e r y q+  <> j  1  ip  which r e v e r s e s the e f f e c t o f s u r g e r y base on t}>. Hence TT ^ (W^ )-n\ (M') f o r i < q , TT q  (Wj=Tr q>  (M' ) / U ,,TT ( S ) } . ffq q  q  f o r i<p and the r e s u l t  S i n c e p< ^r , q>p , so TT . (M')-TT . (W ) z l l <p 5  1  follows.  QED k  Let  k  ( f , b ) be such t h a t f : (M, 9M)—>(A,B), b :v —>-n , k>>m, n a l i n e a r  bundle over A, v the normal bundle o f ( M , 9 M ) c ( D <{>:S —KLnt M be a smooth embedding. P  where M=Mu-D "'". P+  m + k  ,S  m + k  "*") , and l e t  Suppose t h a t f extends t o F:M—>A,  We c o n s i d e r the problem o f ' t h i c k e n i n g M t o a normal  cobordism', i . e . o f e x t e n d i n g cj> t o a smooth embedding $ : S xD "'"—>-int M™, P  q+  p+q+l=m such t h a t <j>=<j> | S xQ, and so t h a t F: (W^ ,.3MxI)—>(AxI ,BxI) P  can be  covered by a bundle map  b :w—>T) e x t e n d i n g b, where u i s the normal  bundle of W"  and F i s the e x t e n s i o n of F, unique up to  in D  <P  homotopy. kill  xT  (When t h i s i s p o s s i b l e , normal s u r g e r y based oriuj) w i l l  the c l a s s of 4 i n i (M).) P  normal k-frames 6.6 if  m + k  L e t V. ,., be the space of o r t h o k,q+l  i n R^ ^ ^-. +<  +  Theorem: There i s an o b s t r u c t i o n Oe-rr (V, ,..) such t h a t p k,q+l  and. o n l y i f <}> extends to <(> such t h a t F:W^—>-A can be covered by  b:o)—>r] e x t e n d i n g b as  above.  P r o o f : S i n c e k i s v e r y l a r g e , we may to  0=0  Mu-D "'"cD xi P+  m+k  t  perpendicularly.  with D  extend the embedding M c D  smoothly embedded and meeting  P + 1  The normal bundle y of D "*"cD P+  m+k  D  m + k  m + k  x0  xI is trivial.  F d e f i n e s a homotopy of f°<|> to a p o i n t , which i s covered by a bundle homotopy b on v|c|>(S ), ending w i t h a map  of v|c(>(S ) i n t o  P  a s i n g l e - f i b r e of n, i . e . a t r i v i a l i s a t i o n w e l l - d e f i n e d up to homotopy.  P  of v|<f>(S ), which i s P  T h i s t r i v i a l i s a t i o n of v | $ ( S ) , P  which i s a subbundle of Y | < K S ) , which i s a l s o t r i v i a l , P  d e f i n e s a map  a of S  P  i n t o the k-f rames of I?  which g i v e s an element aeir (V, , , ) . Now p k,q+l  q+k+  "'", a:S —^-V^ q+]_> P  i f d> extends to A and b  extends to b as above, then the normal bundle to of W, to  restricted  D ^", OO|D "'* i s a subbundle of y e x t e n d i n g v | ^ ( S ) , and b P+  P+  P  Hence 0=0 i n  d e f i n e s an e x t e n s i o n of a to a':D "'"—•V, k,q+l TT (V. , . ) . p k,q+l P+  C o n v e r s e l y , i f 0=0, J  a' d e f i n e s a t r i v i a l v|c)>(S ). P  then a extends to a':D"'"—>-V, , , and k,q+l P +  n  subbundle co' of dimension k i n y , e x t e n d i n g  The subbundle as" o r t h o g o n a l to oo  a b u n d l e over D the  therefore  1  P + 1  )  t o t a l space of y  in y is trivial  and the t o t a l space of to" i s D ( a l l up to homeomorphism) .  P + 1  xi?  q + 1  (being cD  P + 1  S i n c e a)"|cj)(S ) P  xi?  q + k + 1 5  equals t h e normal bundle o f <f>(S ) i n M, t h i s embedding  defines  P  <j):S x "'"—yu, and a ' d e f i n e s P  the e x t e n s i o n  q+  of b t o b:w—>-n , where  w| D "*"=a)' by c o n s t r u c t i o n .  QED  P+  We s h a l l now study V, ,, i n order k,q+l  to analyse the o b s t r u c t i o n  (0 w i l l o f t e n be r e f e r r e d to as 'the o b s t r u c t i o n  0.  to t h i c k e n i n g (M,F)  to a normal cobordism'.) R e c a l l t h a t t h e group S0(k+q+l) a c t s orthonormal k-frames i n # k l + l +c  given  frame f i x e d .  topologised  Hence V  n  d S0(q+1) i s the subgroup l e a v i n g a = S0(k+q+l)/S0(q+1) , and V  to make t h i s a homeomorphism.  SO (n)-^-SO (n+l)- -^-S P  evaluates  a  n  n  .  F u r t h e r , we r e c a l l  is that  i s a f i b r e bundle map, where p i s the map which  an o r t h o g o n a l t r a n s f o r m a t i o n  V Q = ( 1 , 0 , . .. , 0 ) e S c i ?  t r a n s i t i v e l y on t h e s e t o f  n + 1  ,  on the u n i t  i.e. P(T)=T(VQ).  vector  (For t h i s m a t e r i a l ,  reference  may be made t o [Husemoller 1966].) 6.7  Lemma: i :ir. ( S 0 ( n ) ) — H T . (SO (n+1) ) i s an isomorphism f o r i < n - l , •  and  *  l  i  a surjection for i<n-l.  P r o o f : i T ^ ( S ) = 0 f o r i<n, n  so the r e s u l t  follows  from the e x a c t homotopy  sequence of the f i b r a t i o n SO ( n + 1 ) — ^ S : . . .-nr ( S - ) - ^ i F . ( S 0 ( n ) ) ^ # ^ T r . ( S 0 ( n + l ) ) - ^ / ^ T T . ( S ) - ^ . .. l+l i i l 11  6.8 S0(n)  QED  n  n  Lemma: The map p: SO (n+1)—>-S i s the p r o j e c t i o n of the p r i n c i p a l n  b u n d l e a s s o c i a t e d w i t h the o r i e n t e d  tangent bundle o f S . n  P r o o f : L e t f = ( f i , . . . , f ) be a tangent frame t o S n  a t v n = ( l , 0 , ...,0) .  n  D e f i n e a map e:SO(n+1)—>F, the bundle o f frames of S , by e(T)  i s the  n  frame ( T ( f ^ ) , . . . , T ( f  )) a t T ( v ) e S . n  0  Then e i s s u r j e c t i v e , and i n j e c t i v e .  Hence e i s a homeomorphism, and the lemma f o l l o w s . 6.9  Lemma: The composite ir ( S ) — ^ - > i r .. (S0(n))- -#^-TT ., ( S n n-1 n-1 n  P  boundary i n the e x a c t sequence of the tangent S  n  n  1  ) i s the  ^ bundle t o S , and n  38 i s 0 i f n i s odd, Proof: by  The  and m u l t i p l i c a t i o n by  tangent S  bundle i s o b t a i n e d  11  t a k i n g the q u o t i e n t  bundle.  Hence we  2 i f n i s even.  by  from the bundle of  frames  SO(n-l) SO(n), the s t r u c t u r e group of  the  have the commutative diagram;  SO(n)—2—>SO(n)/SO(n-l)=S ~ n  1  i SO (n+l)-^->SO (n+1) /SO  (n-1)  P ,n  — i It follows  t h a t i n the exact sequence f o r the r i g h t hand b u n d l e ,  3=p ,,3 :TT . (S )—*-rr, , ( S # l i - l n  r  n  "S .  tangent sphere bundle has tangent v e c t o r is  zero.  Now  by  the E u l e r - P o i n c a r e  a cross-section  f i e l d ) i f and  Theorem  (there i s a  nonsingular  o n l y i f the E u l e r c h a r a c t e r i s t i c x  More p r e c i s e l y , the o n l y o b s t r u c t i o n  to a  where  geH (M;Z) i s the c l a s s d u a l to the o r i e n t a t i o n c l a s s of M.  Now  m  M=S , the o b s t r u c t i o n  to a c r o s s - s e c t i o n can  n  the c h a r a c t e r i s t i c map  (see  Hence 3=0  m u l t i p l i c a t i o n by  6.10  i f n i s odd,  Theorem: p„ :TT (SO(n+1))—>TT ( S ) if n n n  n=l,3, or Proof:  I f p^ i s s u r j e c t i v e , then t h e r e  therefore n  6.11 Z  2  a l s o be  identified  if with  — n n—1 23.4]) 8 :TT (S )->ir (S ) n n-1 2 i f n i s even.  1951,  i s s u r j e c t i v e i f and  only i f  7.  t h a t p o a ^ l , and  S  [Steenrod  (M)  cross-section  is x(M)g>  to the tangent sphere bundle of a m a n i f o l d  the  a:S —>-S0(n+l) such n  hence the p r i n c i p a l bundle of x n has  trivial,  i.e. S  n  i s p a r a l l e l i s a b l e i f and C o r o l l a r y : ker i f n i s odd  i s a map  and  i : T  it  is parallelisable. o n l y i f n=l,3 or  But  a s e c t i o n and  i t i s known t h a t  7.  . (S0(n))—*ir .(SO (n+1)) i s Z i f n-i n-1  n * l , 3 , 7 , and  0 i f n=l,3,7.  is  n  i s even,  39 P r o o f : ker ±„=3ir  if  (S )=7r  (S )/p„rr  n  n  n  n  if  n  (S0(n+1)) .  P„TT (S0(n+l))22Tr ( S ) , and by 6.10 If n. n  the i n c l u s i o n i s s t r i c t ,  n  hence ^ ( S ) / p i r ( S 0 ( n + l ) ) = Z  i f n i s odd, n * l , 3 , 7 .  n  n  #  n  I f n i s odd, by  2  6.9 i f n*l,3,7,  I f n=l,3, or 7,  p^ i s s u r j e c t i v e , so k e r ±^=0 I f n i s even, by 6.9  p °9 i s a monomorphism, so 9:TT ( S ) — > K n  J1  if  n  i s a monomorphism, so ker  n-1 QED  Z. IT ( V , )=Z m k,m ^  6.12  Theorem: TT . ( V , )=0 f o r i<m, l k,m  k>2.  F u r t h e r j„ :TT.(V, )—Hr. ( V , ,.. ) i s an isomorphism f o r i<m, # l k,m l k+l,m  2  ..(SO(n))  i f m i s odd, Z i f m i s even,  J  k^2,  and j,, :tf- ( V . )=TT (S™)—*-TT ( V , ) i s s u r j e c t i v e , and an isomorphism i f ff m 1, m m mk,m m i s even, where j i s i n c l u s i o n . Proof: F i r s t ,  so t h a t V  take k=2,  f i b r a t i o n over S  m+1  2  m  =S0(m+2)/S0(m) and we have a n a t u r a l  =S0(m+2)/S0(m+l) w i t h f i b r e S =S0(m+l)/S0(m). m  we have a commutative S0(m+1)  Also  diagram of f i b r e b u n d l e s : P  .  >S  .  m  j >-V, 2 ,m  S0(m+2) P m + 1  —i->s  m+1  By" the n a t u r a l i t y of the homotopy e x a c t sequences we /P  m+1  1  N  _  have:  / m+lx c  m+1  t ' 9  TT C S O C n r i - l ) ) - ^ (S™) m m 2  By 6.9  p^°9=0 i f m i s even, P^°3  i s m u l t i p l i c a t i o n by 2 i f m i s odd,  Hence 8'=p^°3, and from the e x a c t homotopy sequence of the f i b r e b u n d l e , l+l we  (S  deduce  m + 1  ) - ^ r r . (Sm)-W.-(V_ )-Hr, ( S l i I ,m i  that j # J  i s surjective J  f o r i<m,  TT ( V „ ) = Z i f m i s even, TT ( V . )=Z m 2,m m 2,m  2  m + 1  )=0  f o r i<m,  and T T . ( V „ )=0 f o r i<m, l 2,m  i f m i s odd.  C o n s i d e r next the n a t u r a l i n c l u s i o n V , —>-V. ,.. g i v e n by k,m k+l,m  i n c l u d i n g SO (m+k)—>-SO(m+k+l.) i n such a way t h a t the subgroup preserved.  We have t h e commutative SO(m)-  SO(m) i s  diagram:  -*S0 (m)  S 0(m+k)-i*S 0(m+k+1)  k,m  "* k+l,m V  and a c o r r e s p o n d i n g diagram i n c o r p o r a t i n g the e x a c t sequences, .-. .—tif.. (SO (m>)——»TT. (S0(m+k>) - • . > 77,-(V.) *TT . . (SO (m) )k,m' i - l  TT, (S6(m)-)  H7, (SO (m+k+1)-)  >-7r . (V •) •try -, (S0(m))-*. i k+l,m' i - l v  By~Lemma 6.7, i„ i s an isomorphism f o r i<m+k-l, and, s i n c e k>2, it  follows that  QED  i s an isomorphism f o r i<m.  The f o l l o w i n g theorem d e s c r i b e s what can be accomplished  toward  s o l u t i o n of the s u r g e r y problem, by the use of s u r g e r y below the middle dimension. 6.13  Theorem: L e t (M,9M) be a smooth compact m-manifold w i t h boundary, V  m>4, v finite  Tn~f"T<"  t h e normal bundle f o r (M,9M)c(D complex,  B£A, n  Tn-r-V 1 ,S  ), k » m .  L e t A be a  a k-plane bundle over A, l e t f:(M,9M)—>-(A,B) ,  and l e t b:v—>-n. be a l i n e a r bundle map c o v e r i n g f . Then t h e r e . i s a cobordism W of M, w i t h 9W=Mu(9MxI)uM', 9M'=3Mxl, an e x t e n s i o n F o f f , F : ( W , 9 M x I ) — > ( A , B )  w i t h F|9Mxt=f | 9M f o r each t e l ,  and an e x t e n s i o n b o f b , b:u>—HI, where to i s the normal bundle of W i n D  m + k  xi,  such t h a t f'=F|M':M'—>A i s [y]-connected ( [ a ] i s the g r e a t e s t  integer not l a r g e r  than a ) .  P r o o f : The p r o o f i s by i n d u c t i o n : we s h a l l assume t h a t f :M—*-A i s n-connected, n + l < [ y ] , and show how t o c o n s t r u c t W , F , e t c . as above, w i t h f':M'—>-A (n+1)-connected  (n+1 i s any n o n n e g a t i v e i n t e g e r ) .  I f n+l=0, we need o n l y show how t o make the map induced on TTQ surjective. of  S i n c e A i s a f i n i t e complex, A has o n l y a f i n i t e number  components, A=A]U A u . . .uA^ .  L e t a_^eA_^, and take M'=MuS u. . .MS™, m  2  where S ^ ' i s the m-sphere.  L e t W=MxIuD? ''"u.. .uD "^ and l e t F:W—>-A r 1+  m+  1  1  be d e f i n e d by F|Mxt=f f o r each t e l , F(D™ "'")=a^.  S i n c e the normal  +  bundle o f D "'' i s t r i v i a l , l  and the e x t e n s i o n c o n d i t i o n on the bundle  m+  ni-hl map  i s easy t o f u l f i l l on the D^  to b over W.  , i t f o l l o w s d i r e c t l y t h a t b extends  C l e a r l y the map induced by f'=F|M' i s onto TTQ (A) , which  proves the i n i t i a l step o f our i n d u c t i o n . Now  assume n + l = l , f :M—>-A i s 0-connected.  Let  and M  be two  2  components o f M such t h a t f(M^) and f ( M ) a r e i n the same component 2  of  A.  Take two p o i n t s x ^ e i n t M_^, 1=1,2, and d e f i n e cj>:S^—*M by  <Kl)=xi , <K-l)=x . 2  it  S i n c e f(<KS )) l i e s i n a s i n g l e component o f A, 0  f o l l o w s t h a t f :M—>A extends t o f iMu-D —>-A. 1  Then, s i n c e m>4, i t  f o l l o w s from Theorems 6.6 and 6.12 t h a t $ extends t o <J>:S°xD —>M m  d e f i n i n g a normal cobordism o f f t o f components o f M.  1  and r e d u c i n g the number o f  U s i n g t h i s argument r e p e a t e d l y , we a r r i v e a t a  1 - t o - l correspondence o f components. Now  we c o n s i d e r the fundamental groups.  Let  {a^,...,a ;rj,...,v^} g  and {xi , .. . , x ^ ; y i ,.. . ,y^} be p r e s e n t a t i o n s o f TT]_ (A) and TTJ (M) , resp . Let s c o p i e s o f S^ be embedded d i s j o i n t l y i n an m - c e l l D™ i n t M, *': S°—>M, and assume the base p o i n t o f M.is i n D and f ( D )=*, the s base p o i n t o f A.  L e t M=Mu^ , (uD ) . 1  f r e e group on s g e n e r a t o r s g i , . . . , g  Then TT^ (M)=TT^ (M)*F, where F i s a » where each g_^ i s the homotopy  c l a s s o f a loop i n D u ( u D ) c o n s i s t i n g o f a p a t h i n D , one o f the m  1  D-^'s, and another path i n D™.  m  Hence TT| (M)={X^ ,. .. ,x^,gi ,. .. ,g ;y\ , ..  D e f i n e f:M—>A e x t e n d i n g f by l e t t i n g the image o f the i  g  D  t r a v e r s e a loop r e p r e s e n t i n g the g e n e r a t o r a_^. is  Then f^:ni(M)—HTT_ ( A )  s u r j e c t i v e , and f u r t h e r m o r e we may r e p r e s e n t f^-'on the f r e e groups  {xj ,.. . ,x ,gi ,. . . ,g } and' {aj_ , . . . ,a } by a f u n c t i o n a , w i t h a (x. )=x! , x! a word x  i n t h e a., and a(g.)=a.. j i x  Then as above, we may extend <j>'  to <)): (uS°)xD —>-M t o d e f i n e a normal cobordism o f f , and w i t h s m  and F:W,—>A homotopic t o f:M—>A.  W,=M, cp  (Here W, i s the t r a c e o f the s i m u l -  9  9  taneous s u r g e r i e s . )  By P r o p o s i t i o n 6.2, TT j (M' )-TTT_ (W ) , where M'  is  d e f i n e d by 3w"^=Mu (3MxI)DM', and hence f ^ :TTT_ (M' )—>TT^ ( A ) i s s u r j e c t i v e , TTI(M') has the same p r e s e n t a t i o n as TT^ (M) , and f | i s a l s o by a on the f r e e groups.  represented  I n p a r t i c u l a r , f ' i s 1-connected.  L e t us c o n s i d e r t h e exact sequence o f the map f:M—>A  i n homotopy,  vrr , v ( f ) HT (M) nr ( A ) Mr ( f ) — n+1 n n n R e c a l l t h a t t h e elements of the groups TT ...(f) a r e d e f i n e d by commutative n+1 where k i s the i n c l u s i o n o f the boundary, dxagrams : S >-M f  (*)  n+1 3 D ->A and a l l maps and homotopies p r e s e r v e base p o i n t s .  Thus f3 d e f i n e s a map  n+1 f :Mu- D »-A e x t e n d i n g f . a 6.14  Xemma: L e t f :M—>A be n-cdnnected, n>0, and l e t ( $ , , $ ) £ T r ^ ( f ) be n +  the element r e p r e s e n t e d by the above diagram (*) .  I f f iMu^D^-^A i s  d e f i n e d by f3 as above, then TT . (f )=rr. (f )=0 f o r i<n, and TT (f)=Tr ^ (f)/K, x x n+1 n+1 where K i s a normal subgroup c o n t a i n i n g the TT (M) module g e n e r a t e d by J  x  the element  (R,a) i n TT  (f) .  n+1 P r o o f o f Lemma 6.14: C o n s i d e r the commutative diagram: H^->TT ( A ) ->TT v(M')f, £+1  ai  ••  ) — \ (MU D a  N+1  )-^/->TT  £  (A)-  1  43 Here i:M—>Mu^D "^ i s i n c l u s i o n , and j ^ i s induced by ( l , i ) on the n+  diagram  (*) ( i . e . j ^ [ 3 ' » a ' ] = [ 3 ' , i ° a ' ] ) .  for  £<n, and s u r j e c t i v e  for  I n (by the F i v e  f o r &=n, so i t f o l l o w s  any p a i r  Jt  i s an isomorphism  e a s i l y that  TT^ (f )=TT^ (f )=0  Lemma).  C l e a r l y any map o f S so t h a t  Clearly, ±  n  i n t o Mu^D "'" i s homotopic n+  „n a S  (3',a')  _n+l KMu D  t o a map i n t o M,  , ,. xs homotopxc t o a p a i r or  a f  D the  form  n+1 g*  (3",i°a"): „n  a  l ^n+1 ->M——>Mu D w  w  a  n+1 B" D" • —yA—^->A Hence 3^  :7l  i^^—^n+l  Clearly from  ^  n+  S  s  u  J  r  e  c  ^  t  v  e  #  ( 3 , a ) i s i n the k e r n e l of j ^ and hence e v e r y t h i n g  ( 3 , a ) by the a c t i o n  of TT^ (M) i s a l s o  obtained  i n k e r j ^ , which proves t h e  lemma.  QED  We have a l r e a d y shown t h a t we may assume, w i t h o u t l o s s of g e n e r a l i t y , that  f:M—>A i s 1-connected, and t h a t  the fundamental groups have  p r e s e n t a t i o n s TTI (M)={x! , . . . j X ^ g j ,. . . ,g ;yj ,. .. ,y^} , y x  l 5  ...,x  k  o n l y , TTJ (A) = {a! ,.. . , a ; r g  p r e s e n t e d by the f u n c t i o n a word i n 6.15  1  ,.. . ,r > , w i t h f ^ t  : i r  ±  words i n  i  OO—(A)  (on the f r e e groups) a(x.)=x!(a^,...,a )  , . . . ,a , j = l , . . . , k ,  and a(g^)=a_^, 1=1,  Lemma: k e r f ^ i s the s m a l l e s t normal subgroup  ,s. c o n t a i n i n g the  -1 words x. ( x ! ( g ! , . . . , g ) ) , j = l , . . . , k and r ( g , . . . , g ) , 1=1,...,t. 3 3 i s x  s  P r o o f o f Lemma 6.15: Adding t h e r e l a t i o n s x. ( x ! ( g ) ) makes J 3 i n t o a s e t of g e n e r a t o r s . with a defining  gi,...,g  Adding the r ^ ( g ) makes the group i n t o  the isomorphism.  The map a a n n i h i l a t e s  Xj^(Xj(g))  s  (A),  44 and r ^ ( g ) , so t h a t these elements generate k e r f ^ as a normal For  each element x_."*"(x^ (g)) x = X J ^ ( X J (g)) ,  such t h a t  and r_^(g) choose an element  subgroup. x^.,r^eTr (f) 2  r_^=r^(g) , and choose embeddings S —>M t o 1  r e p r e s e n t the x. and r , ( a l s o denoted by x. and r . ) such t h a t J i J i  their  images.are a l l d i s j o i n t , which i s p o s s i b l e by g e n e r a l p o s i t i o n , m 4. It  L e t M=MU (^^D?) ,with the 2 - d i s c s a t t a c h e d by these  since  embeddings.  f o l l o w s from Lemma 6.14 t h a t f^:Tri(M)—MT]_ (A) i s an isomorphism.  U s i n g a g a i n Theorems 6.6 and 6.12, i t f o l l o w s t h a t t h e r e i s a normal cobordism W, and a map F:W—>-A such t h a t McW i s a d e f o r m a t i o n r e t r a c t and F|M=f, so t h a t F^:TTI(W)—>-TT (A) i s an isomorphism. x  By P r o p o s i t i o n s  6.2 and 6.3, i t f o l l o w s t h a t i f M' i s the r e s u l t o f s u r g e r y , then f ^ : T r ( M ; ' ) — ( A ) i s an isomorphism, and hence TT (A)—MT ( f ) x  2  and thus TT ( f ) 2  2  i s surjective,  i s abelian.  We now p r o c e e d t o the i n d u c t i o n s t e p .  Suppose f :M—>A i s n-connected,  n>0, and i f n = l suppose TT^ (M)—S-TTI (A) i s an isomorphism, so t h a t is  rr (f) 2  abelian.  6.16  Lemma:  17  1  S  a  finitely  generated module over :TTT_ (M) .  T h i s lemma i s proved u s i n g u n i v e r s a l c o v e r i n g spaces  [Browder  1972].  Now we may r e p r e s e n t each o f t h i s f i n i t e number o f g e n e r a t o r s i n TT , _ ( f ) by a diagram n+1  S^  -*-M  1  .  oi i .  a  I f n+l<[™-], then n<^- and i t f o l l o w s 2  f T.n+1 D — B  from Whitney's (3.,a.) i i  embedding theorem  >k i ('general p o s i t i o n ' ) t h a t we may choose  so t h a t t h e a . have d i s j o i n t l  images.  S e t t i n g M=Mu ( u D ^ " * " ) , i i  D^ + ^" a t t a c h e d by a^, f :M—>A d e f i n e d by the 6^, we may a p p l y Theorems 6.6 and 6.12 t o t h i c k e n M t o a normal cobordism W o f M, and u s i n g 6.14, Tr^(f)=0 f o r £<n+l.  I f M' i s the r e s u l t of t h e s u r g e r i e s  45 ( i . e . 9W=Mu (9MxI)uM') , from P r o p o s i t i o n s 6.2 and 6.3 i t f o l l o w s IT ( f ' ) = T r  (f)=0 f o r i < n + l .  that  T h i s completes the p r o o f o f Theorem 6.13. QED  Note t h a t we have always used the low d i m e n s i o n a l i t y o f the groups i n v o l v e d t o ensure t h a t 0 was zero  (by Theorem 6.12) and t o f i n d  r e p r e s e n t a t i v e s o f elements o f ^ ^ ( f ) which were embeddings. d e r i v e r e s u l t s i n h i g h e r dimensions, we s h a l l have t o f i n d  To  other  means o f d e a l i n g w i t h these o b s t a c l e s .  § 7.  I n i t i a l R e s u l t s i n the M i d d l e Dimension.  Let  (A,B) be an o r i e n t e d P o i n c a r e p a i r o f dimension m, l e t M be  an o r i e n t e d smooth compact  m-manifold w i t h boundary  f:(M,9M)—>-(A,B) be a map o f degree 1. bundle over A, k>>m, and l e t v ) . Suppose b:v—m  Let n  9M, and l e t  be a l i n e a r k-plane  be the normal bundle o f (M,9M) i n  (D  ,S  i s a l i n e a r bundle map c o v e r i n g f .  Then  (f ,b) i s what we have c a l l e d a normal map.  ( R e c a l l t h a t we  d e f i n e d a normal cobordism o f ( f , b ) r e l B t o be an (m+1)-manifold W w i t h 9W=Mu(9MxI)uM , together w i t h an e x t e n s i o n o f f , T  F:(W,3Mxl)—>(A,B) f o r which F|9Mxt=f|3M f o r each t e l , and an e x t e n s i o n b o f b t o the normal bundle OJ of W i n D ^ ^ x l . ) Suppose f u r t h e r t h a t A i s a s i m p l y - c o n n e c t e d CW complex, m>5, and t h a t 7.1  (f | 9M)^:H^(9M)—*-H (B) i s an isomorphism. A  Theorem: There i s a normal cobordism r e l B o f ( f , b ) t o ( f ' , b ' )  such t h a t f.' :M'—>A i s f ^ + l - c o n n e c t e d i f and o n l y i f a ( f , b ) = 0 . In p a r t i c u l a r , t h i s i s t r u e i f m i s odd. The p r o o f o f t h i s theorem w i l l occupy the b a l a n c e o f the p r e s e n t chapter.  F i r s t note the u l t i m a t e  corollary.  7.2  Corollary:  (Fundamental Theorem of Surgery)  The map  f ' above  i s a homotopy e q u i v a l e n c e . Hence, ( f , b ) i s n o r m a l l y cobordant r e l B to a homotopy e q u i v a l e n c e i f and only i f a ( f ,b)=0.  In p a r t i c u l a r ,  t h e r e i s such a normal cobordism i f m i s odd. Proof of C o r o l l a r y 7.2: By the n a t u r a l i t y of the e x a c t homology sequence of p a i r s , we . .*H.  have  (9M')~—>E. (M')-—7*-H_, (M' ,9M'-)-—-*-H. .. (3M'-)  ii  i  i  i i  ( f ' | 9M'). ..  H (B-)  Since  (f|9M)^:H (9M)—(B>  -Hl^A)—  i - l |  ( f |3M')*  ^H (A,B)-  V  ±  *V-1(B)—  ;  f  ^H _ (A)-»-. . ±  1  i s an isomorphism, and 9M'=9M,  A  f'|9M'=f|9M, we see t h a t  i - l |  41. , (M')-  ( f | 9 M ' ) ^ i s an isomorphism i n . e a c h dimension  By 7.1, f':M —*A i s [ y ] + l - c o n n e c t e d , so t h a t f ^ H ^ M ' ) — ^ H ( A ) i s an r  ±  isomorphism f o r i<y.  Thus by the F i v e Lemma,  i s an isomorphism f o r i ^ y -  Since f * i  from P o i n c a r g d u a l i t y t h a t f j>m^=S-.  Now f  i s a map i  :H (M* ,3M- )—*-H (A,B) 1  ±  ±  of degree 1, i t f o l l o w s  :H (A)—>-H (M') i s an isomorphism f o r  f '* :H (A)—>H^ (M') i s g i v e n by J  J  =Hom(f.^ , Z ) + E x t ( f ^ _ , Z ) , a c c o r d i n g to the U n i v e r s a l  J  1  C o e f f i c i e n t Theorem, where f ^ :H (M')—^-Hy(A) , e t c . in Since f ' i s an isomorphism f o r i ^ — , i t f o l l o w s t h a t f isomorphism f o r j<y, and hence f  *k "l  i s an  :H^ (A)—*-H^ (M') i s an isomorphism  A for a l l j .  Thus, H (f')=0, and the U n i v e r s a l C o e f f i c i e n t Theorem  i m p l i e s t h a t H^(f')=0.  But M  1  and A a r e s i m p l y - c o n n e c t e d , so t h a t  by the R e l a t i v e Hurewicz Theorem and the.Theorem have the r e s u l t :  of Whitehead  we  f':M'—*-A i s a homotopy e q u i v a l e n c e . T h i s e s t a b l i s h e  the c o r o l l a r y . We  s h a l l develop c e r t a i n p r e l i m i n a r y r e s u l t s b e f o r e p r o c e e d i n g  w i t h the p r o o f of Theorem  7.1.  By Theorem 6.13, we may assume t h a t f:M—>k i s [ ^ - c o n n e c t e d , i . e . IT (f)=0 f o r i<[^-]. it  Set A=[^].  Since A and M a r e s i m p l y - c o n n e c t e d ,  f o l l o w s from t h e R e l a t i v e Hurewicz Theorem t h a t ^ . n ^ f )  T h i s g i v e s a commutative £+1  (f )  s i  ^ ^(f ) . +  diagram: >-TT (M)-^/->TT (A)->0 N  £  h ...->H  £+1  (f)  ^H ( M ) - ^ - ^ H (A)-vO £  £  where h i s the Hurewicz homomorphism, and f ^ i s the map induced by f i n homotopy.  R e c a l l t h a t f ^ is-.s.urjective, and s p l i t s by Theorem 5.12.  It follows that  (ker f . ) =h(ker f,,)„. * Xit Jo  Whitney's embedding theorem s t a t e s :  'Let c:V  a continuous  map o f smooth m a n i f o l d s , m>2n, m-n>2, M s i m p l y - c o n n e c t e d , V connected. Then c i s homotopic t o a smooth embedding.' [Milnor  (A p r o o f can be found i n  1965].)  S i n c e &^y> i t f o l l o w s from Whitney's embedding theorem t h a t any  -  - £  element x£ir _ _^(f) may be r e p r e s e n t e d by (£>,<)>), where $:S —>-int M i s £+1 — — £+1 - a smooth embedding, and 3 :D —*-A, 3°i=f°(f>. Set M=Mu-D , f:M—>A £  the  (  extension of f defined using 3 . We should l i k e  to thicken  (M,f) t o a normal cobordism; i . e . t o  perform normal s u r g e r y u s i n g <j>, and t o examine ^ £ + 1 ^ ' ^ ' where f*  is  the  map on the r e s u l t o f the s u r g e r y , w i t h the hope o f h a v i n g k i l l e d  the  homotopy c l a s s o f c(>. However, t h e r e a r e two d i f f i c u l t i e s we must  face:  First,  i f m=2£, then a c c o r d i n g t o Theorems 6.6 and 6.12,  t h e r e i s an o b s t r u c t i o n 0 t o t h i c k e n i n g  (M,f) t o a normal cobordism,  which l i e s i n a n o n t r i v i a l group TT^ (V^ ) . £  Second, a l t h o u g h we may compute ^ ^ ^ C f ) u s i n g Lemma 6.14, it  i s no l o n g e r c l e a r how t h i s group i s r e l a t e d  to ^ . ^ ( f ' ) '  i f ^=[^"1  We s h a l l f i r s t  direct  our a t t e n t i o n toward the second d i f f i c u l t y .  U n l e s s s t a t e d o t h e r w i s e , we s h a l l assume h e n c e f o r t h t h a t a normal map s t i s f y i n g  the hypotheses of Theorem  (f,b) i s  7.1, and f :M—*-A i s  q-connected, where q=[^], i . e . m=2q o r 2q+l. 7.3  Lemma: f i s (q+1)-connected i f and o n l y i f f^:H  (A) i s  (M)—>H  an isomorphism, i . e . i f and o n l y i f K (M)=0. q  P r o o f : By t h e R e l a t i v e Hurewicz Theorem, Theorem H  q + 1  ^~ q+l^^'  17  H  a n c  *  5.12, f..:H ,, (M)—»-H ,., (A) i s s u r j e c t i v e , so t h a t * q+1 q+1 (f)*<ker  f „ )  Q  - K  Q  "  ( M ) V  QED  Thus we need n o t examine homotopy, b u t w i l l study the e f f e c t of  s u r g e r y on homology.  our  The f o l l o w i n g lemma w i l l a l l o w us t o s i m p l i f y  arguments by c o n s i d e r i n g only the case o f c l o s e d Let  (f_^,b_^), ±=1,2, be two d±sjo±nt cop±es o f the normal map  ( f , b ) , so t h a t f : ( M ,31^)—>(A  ,B ) , ±=1,2, ±s j u s t f renamed.  by the Sum•.-•Theorem f o r Po±ncare pa±rs A =A^uA2  w i t h B} ±dent±f±ed  3  double o f A).,  M3=MT_UM2,  to B  Since  3  3  sequences ±mply t h a t  q+1 3 (f  ) S K  ±s a P o i n c a r e complex  (flSM)^  b =biub2 3  ±s an ±somorph±sm,  the Mayer-V±etor±s  H_^(f )=0 f o r ±<q+l, and 3  q 3 )*\Q*l)<3>K ( M 2 ) ; . (M  q  q  0  q  and, by i n c l u s i o n , on M  such t h a t  3  (with r e s p e c t to ( f 1 , b ) and ( f , b 3 ) ) . x  3  a prime denotes the r e s u l t o f s u r g e r y , we have M =MiuM 3  K (M )=K (M ')©K (M ) . q  m  (the c o n s t a n t map), and such t h a t 9 d e f i n e s a normal s u r g e r y  f l ? - ^  If  closed  d e f i n e a normal map  Now suppose ()):S xD — > ± n t Mi_ ±s a smooth embedding  on  ( c a l l e d the  2  2  (f 3 ,b ) : M — 4 - A 3 .  2  3  Then  [Browder 1972, 1.3.2],  u n i t e d a l o n g 3Mi=9M , ±s a smooth  o r i e n t e d man±fold, and f 3 = f T _ u f ,  H  manifolds.  q  1  q  has n o t a f f e c t e d  2  2  and  T h i s f o l l o w s from the f a c t t h a t the s u r g e r y  the f a c t o r M  2  i n the d e c o m p o s i t i o n o f M . 3  49 Thus we have: 7.4  P r o p o s i t i o n : The e f f e c t o f normal s u r g e r y on K^(M) i s t h e same  as t h e e f f e c t o f the induced s u r g e r y on K^(M3), and hence t o compute i t s e f f e c t , we may assume 9M=B=cj>. This construction w i l l simplify  the a l g e b r a i n our d i s c u s s i o n .  L e t <J>:S xD — * - i n t M be a smooth embedding which d e f i n e s a normal q  m  q  s u r g e r y on M (with r e s p e c t t o ( f , b ) ) . M ^ M Q U D ^ X S ^ -  9(D ^xS q +  m  q  1  "S .  ,  SO t h a t  H S ^ S ^  1  Set Mg=M\int im <f>, and l e t )  i s i d e n t i f i e d with  S ^ S  Then M' i s the r e s u l t o f the s u r g e r y on M.  1  " " ^  S i n c e c|>  d e f i n e s a normal s u r g e r y , H (M')-H (A)<$K (M'), and we wish t o determine  9 q q how K (M) changes t o K (M ) (which i s the same as t h e change o f H (M) q q q to H (M')). q T  We f o r m u l a t e some u s e f u l r e s u l t s c o n c e r n i n g the r e l a t i o n between P o i n c a r e d u a l i t y i n m a n i f o l d s and s u b m a n i f o l d s . 7.5  P r o p o s i t i o n : L e t U and W be compact m-manifolds w i t h boundary,  f:U—>int W, g:(W,9W)—>-(W,W\int U) embeddings, w i t h o r i e n t a t i o n s  compatible.  Then the f o l l o w i n g diagram commutes: H (W,9W)< q  g ;  - H (W,W\int U ) — > - H ( U , 9 U ) q  (g*[W])n-  [W]nH  q  •(W)— m-q  1 — — ( W ) - ( m-q  [U]n-*——H ••• (U) m-q  so t h a t f o r x £ H ( U / 9 U ) , f^([U]nx)=[W]ng*(x), where g":W/9W->U/9U. q  P r o o f : I f f:(U,3U)—>(W,W\int o r i e n t e d U and W c o m p a t i b l y .  U ) , then f^[U]=g^[W], s i n c e we have Then the commutativity  follows  the n a t u r a l i t y o f the cap p r o d u c t . 7.6  from QED  C o r o l l a r y : Set E e q u a l t o the normal tube o f f:N —>W" , N c l o s e d n  m  and o r i e n t e d , and l e t g:W/9W—>E/9E=T(v), where v i s the normal bundle of N ^ w ™ . L e t U € H  m _ n  ( T ( v ) ) be the Thorn c l a s s .  Then  [W]ng*U=f^[N].  Proof: Since  [E]nU=[N] by 7.5,  f ([E]nU)=f^[N]=[W]n(g U).  QED  A  The i n t e r s e c t i o n p a i r i n g i n homology, • :H (M)®H (M,9M)—>Z q m-q is are  d e f i n e d by x*y=(x',y')=(x'uy')[M], where x'eH d u a l to x,y, i . e . [M]nx'=x, [M]ny'=y.  product  •:H  (M)®H  q  m-q  m_q  (M,3M) , y'eH (M) q  T h i s induces an  intersection  (M)—-*Z by x*y=x«j.(y), where j:M—>(M,3M) i s i n c l u s i o n , *  The p r o p e r t i e s of the b i l i n e a r  form (•,•)  on cohomology  induce  analogous p r o p e r t i e s f o r the i n t e r s e c t i o n p a i r i n g , such as (a)  With c o e f f i c i e n t s i n a f i e l d  (M;F)®H (M,3M;F)—>F i s q m-q (This" a l s o h o l d s over Z, modulo t o r s i o n . )  a nonsingular p a i r i n g . (b) I f xeH  (M), yeH q  7.7  F, H  (M) , x-y= ( - l )  P r o p o s i t i o n : L e t xeH (M), yeH [M]nx'=x, [M]ny'=y.  Then  m  _  q  )  y x.  m_q  ( M , 3 M ) , y'eH (M)  u s i n g elementary p r o p e r t i e s  the cup and cap p r o d u c t s . •'Now l e t  (}>:S xD q  m _ q  ,  0  q+1  0  Following  M be a smooth embedding.  —>int  E ) , M- =M u(D  M=M\cf> ( i n t  xS  I,[1  "*") > the r e s u l t  q  [ K e r v a i r e , M i l n o r 1963] we w i l l ( M , M Q ) and  sequences of the p a i r s  m  q  c o n s i d e r the e x a c t  (M ,Mg). f  0  as the normal tube of S c M , q  l e t UeH  m _ q  groups.  q  q  y x = U ( x ) f o r any x e H _ ^ ( E , 3 E ) m  H _ ^ ( E , 3 E ) — > Z by p r o p e r t y m  by 7.7.  (a) above.  Thinking  ( E , 3 E ) = Z be the Thorn c l a s s ,  I f y = [ E ] n U , then ] i = i ^ [ S ] , i : S — > E ,  a g e n e r a t o r ( c f . 7.6).  7.8  q  of s u r g e r y based on $.  isomorphisms on the r e l a t i v e homology and cohomology E  Set E = S x D ~ ,  the e x c i s i o n <j):(E,3E)—>-(M,M ) which i n d u c e s  As u s u a l , we have  of  q  x*y=x'(y).  P r o o f : x-y=(x'uy')[M]=x'([M]ny')=x'(y), of  (  (M,3M), x ' e H  q  be such t h a t  q  m-q  and  T h i s i n d u c e s an isomorphism L e t j:M—*-(M,Mg) be the i n c l u s i o n .  P r o p o s i t i o n : u* ( j ( y ) ) = ((})^(u))«y.  Proof: y  A  ( J ( y > ) = U ( j ( y ) ) = (j*U)  identifying  A  A  (y) = (<|>*(y))-y, u s i n g 7.7  : H ^ ( M ) — ( M , M g ) w i t h the c o l l a p s i n g  map  and 7.6,  and  51 J*:H*(M)-H^(M/M )=H^(E/3E) 0  7.9  C o r o l l a r y : The. f o l l o w i n g sequence 0  >-H (M ) m-q  (M ) ( M ) - - > Z — i •>H m-q-1  Htl m-q  0  u  i s exact: HI  X  0  where x=<t>.(y), yeH ( S x n * q q  m  q  m-q-  (M)  •O,  1  ) i s the image of [ S ] , the o r i e n t a t i o n q  c l a s s of S P r o o f : The sequence i s t h a t of ( M , M Q ) , r e p l a c i n g H _ (M,Mg) by m  q  H (E,3E). . .,, >H • (M,Mn) m-q m-q • u  u s i n g the diagram  and u s i n g 7.8 t o i d e n t i f y x*  y  Thus t h e r e i s an exact sequence 0  _,, q+1  .. ( M ' ) - ^ - v Z - ^ H ~(M )—*+H (M') q+1 q q  where y = i ^ ( y ' ) , V *=K [ S ~ m  ] generates H _ _ ( D  q _ 1  m  q  q + 1  1  x  i s the n a t u r a l embedding, and k '  ^ i D ^ x S ^ ^ ^ M ' is  ^0  0  •  u  m S  " " ) , q  1  :s " " ->-D m  q  1  q+1  xS ~ " m  q  inclusion. Let X £ H  (S xD q  r + 1  and s i m i l a r l y  r + 1  ,S xs )=Z q  f o r A'.  be the g e n e r a t o r such t h a t U(X)=1,  r  (We s h a l l a l l o w A and y A , A' and y'»A' to be  confused.) 7.10  Lemma: i^d'(A')=9^(y)=x and i ; d ( A ) = ^ ( y ' ) = y . A  P r o o f : L e t m=q+r+l. .. .—>H  (S xD q  r + 1  We have a commutative r + 1  diagram:  ,S xS )-^l-^HjS xS )-i *-,H .(S xD r q  r  q  r  1  q  r + 1  T  90. ->H -(M,M )r+1  Clearly, if' A £ H 3iA=l®[S ]£H (S xS ). r  q  r  (S x S q  r  r  r  (S xD q  r + 1  r + 1  ,S xS ) q  q + 1  r  0  such t h a t U(A)=1, then  r  We a l s o have the commutative  *->H. ( D  ->H (M)-  ->H (M )-  0  diagram  x S ) r  and ±  2  ->H;(M')  (1®[S ])=y' .  )-  1  i j d (X) = ± ^ 3 * * ( X )  Hence  ( X ) =ip*i * (1® [ S ])  (y ' ) =y.  r  0  2  A s i m i l a r argument proves the other a s s e r t i o n . 7.11  Theorem: L e t <j>: S x D q  m=q+r+l, q<r+l.  QED  -*M be an embedding, M a c l o s e d  r  m-manifold,  Suppose <j> [S]=<j> (y)=x generates an i n f i n i t e  cyclic  q  A  d i r e c t summand o f H (M).. Then rank H (M')<rank H (M) , and t o r s i o n H (M') q q q q t o r s i o n H (M), i . e . the f r e e p a r t of H (M) i s reduced and the t o r s i o n q q p a r t i s not i n c r e a s e d . 7.12  F u r t h e r H^(M')=H_^(M) f o r i < q .  C o r o l l a r y : L e t ( f , b ) be a normal map, f : (M,3M)—>(A,B) , (f|3M)^  an isomorphism, and l e t <j> :S xD "'"—>-int M be an embedding which q  defines  r+  a normal cobordism o f ( f , b ) , q<r+l.  Suppose <(>.,. (y)=x generates an i n f i n i t  c y c l i c d i r e c t summand o f K (M). Then rank K (M')<rank K (M), and q q q t o r s i o n K (M')=torsion R (M), w h i l e K.(M')=K.(M) f o r i < q . q q i i The c o r o l l a r y f o l l o w s d i r e c t l y With a f i e l d 7.13  from 7.11 and P r o p o s i t i o n 7.4.  of c o e f f i c i e n t s we have analogous  results:  Theorem: L e t <J> ,M be as i n 7.11, and suppose <j) (y )=x*0 i n H^(M;F). A  Then .rank_H (M';F)<rank H (M;F), and H.(M';F)=H.(M;F) f o r i < q . F q r q i . i• 7.14 C o r o l l a r y : With the hypotheses of 7.12, suppose o n l y t h a t <|>.(y)=x*0 i n K (M;F) . * q  Then r a n k j t  r q  (M' ;F)<rank K (M;F) and b q  K (M';F)=K (M;F) f o r i < q . ±  ±  The p r o o f of 7.14 i s s i m i l a r  t o t h a t o f 7.12.  P r o o f of, Theorem 7.11.: C o n s i d e r the e x a c t sequence o f C o r o l l a r y 7.9: 0  ^H  r + 1  (MQ  ) — ^ - ^ H  R  +  1  (M)———yZ  - ^-H^ (MQ )  S i n c e x generates an i n f i n i t e c y c l i c d i r e c t property  (a) o f the i n t e r s e c t i o n p a i r i n g  ^  (M)  K)  .  summand, i t f o l l o w s  from  t h a t t h e r e i s an element  y e H ^ ( M ) such t h a t x*y=l ( s i n c e 3M=<}>). r+  Hence x* i s s u r j e c t i v e and we get i :H (M )=H (M) A  r  0  r  0  (M ) - ^ ^ H 0  r  +  1  (M)—•Z—>0  (1)  C o n s i d e r the exact sequence of C o r o l l a r y 7.9 diagram from Lemma 0  f o r (M*,M  ) and the  7.10:  >-H ,.,(]%)—>H ,-(M'")—^—>Z——>H (M )—^*-*H (M') q+1 q+1 ' ^ q| q  •()  0  (2)  u  ** H  (M) q  where i^d'(X')=x. it  S i n c e x generates an i n f i n i t e c y c l i c d i r e c t  follows t h a t i ^ d '  splits,  H (M )=Z®H (M') q  0  i;:H  q  so t h a t d' s p l i t s , q + 1  (M )=H 0  q + 1  summand,  and  (M')  (3)  From (3) i t f o l l o w s t h a t rank H (M')=rank H ( M ) - l , and s i n c e q q q=r or r+1, from (1) i t f o l l o w s t h a t rank H (M)>rank H ( M ) , so t h a t q q rank H (M')<rank H (M) (the d i f f e r e n c e b e i n g 1 i f q=r, 2 i f q=r+l). q q From (1) i t f o l l o w s t h a t t o r s i o n H ( M Q ) i s i s o m o r p h i c t o t o r s i o n H (M), q q 0  0  and from (3) i t f o l l o w s t h a t t o r s i o n H ( M o ) = t o r s i o n H (M'). q  q  Hence  t o r s i o n H (M')=torsion H (M). q q The p r o o f of 7.13  QED  i s almost i d e n t i c a l , u s i n g  w i t h c o e f f i c i e n t s i n F, and u s i n g p r o p e r t y c o e f f i c i e n t s i n F.  ( 1 ) , ( 2 ) , and  (3)  (a) o f i n t e r s e c t i o n w i t h  The d e t a i l s a r e o m i t t e d .  To proceed f u r t h e r i n the p r o o f of the Fundamental  Theorem, we  must c o n s i d e r d i f f e r e n t dimensions s e p a r a t e l y ; i n p a r t i c u l a r , we must d i s t i n g u i s h 3 c a s e s : m odd, m=0 §8.  The P r o o f of the Fundamental  From C o r o l l a r y "7.12 we may 8.1  (mod  4 ) , and m=2  Theorem f o r m odd.  deduce  the f o l l o w i n g  Theorem: L e t ( f , b ) be a normal map,  connected,  (mod 4 ) .  theorem.  f : (M, 3M)—>-(A,B) , A s i m p l y -  ( f j S M ) ^ an isomorphism, m=2q+l>5.  There i s a normal  cobordism r e l B of ( f , b ) to ( f ' , b ' ) , such t h a t f':M'—>-A i s q-connected,  and K (M')=torsion K (M). q q P r o o f : By Theorem 7.11, we may f i r s t  f i n d a normal cobordism r e l B t o  (fl»bi), such t h a t f]_:Mi—>-A i s q-connected.  We note t h a t the  surgeries  used i n 7.11 a r e on embedded spheres o f dimension l e s s than q, so t h a t 6.2 and 6.3 t h a t K ( M i ) = K (M)©F, where F q q  it  f o l l o w s from P r o p o s i t i o n s  is  the f r e e a b e l i a n group produced by k i l l i n g  torsion classes i n K  Thus we may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t  (M)  f i s q-connected.  L e t x e K (M) be a g e n e r a t o r of an i n f i n i t e c y c l i c d i r e c t summand. q S i n c e f i s q-connected, i t f o l l o w s from the R e l a t i v e Hurewicz Theorem t h a t TT ,.,(f)=H , . ( f ) , and H ... (f ) = K (M) by Theorem 5.12. S i n c e q<^, q+1 q+± q+i q Z it  follows  X'STT  from the Whitney  1 ... f|:" such t h a t a i s a smooth embedding.  ...(f) by ( 3 ,a) ,  q Then 3 d e f i n e s + 1  Embedding Theorem t h a t we may r e p r e s e n t  i q A a map f :M—>-A where M=Mu D +  1  q  +  \ and by Theorem  6.12,  s i n c e q<m-q, the o b s t r u c t i o n t o t h i c k e n i n g M t o a normal cobordism is  zero.  Corollary  I f x'err 7.11,  q+1  ( f ) i s such t h a t a r e p r e s e n t s  x e K (M) , then by q  K (M ) has rank one. l e s s than K (M) , and the same ~^ q q  t o r s i o n subgroup.  1  I t e r a t i n g t h i s procedure u n t i l  the rank i s zero  proves the theorem.  QED  We d e r i v e an important diagram by u n i t i n g the two e x a c t of C o r o l l a r y 7.9. 8.2 Lemma: We have a diagram:  sequences  H ^(M') q+  Ty d 0  H I  . - ( M )  X  ' >Z—^-VH  q+1  ( M  qi  H  1  0  ) — ^ - > H  ( M )  q"  HD  55 where i ^ d ' (A ' )=x=<j>^(y) , i ; d ( A ) = y = ^ ( y ' ) , y i s a g e n e r a t o r  of  H (S xD q  q + 1  q  ) ,  y' of H ( D x S ) , etc. q Hence, H (M')/ (i'dZ)=H ( M ) / ( i . d ' Z ) . q « q « q + 1  q  Proof: This follows d i r e c t l y fact  from C o r o l l a r y 7.9 and Lemma 7.10, and the  t h a t H (M) / ( i . d Z)=H (M ) / (d * ZS>dZ)=H ( M ' ) / ( i ! d Z ) . q * q q  QED  r  0  I f x=i^d'(A') i s a t o r s i o n element of o r d e r  s, then x* i s the zero  map, so t h a t p a r t of the diagram of 8.2 becomes the s h o r t exact 0  Since i  >Z  A  d  >H (M ) q 0  —>^-*H (M) q  K)  i s a homomorphism, sd'(A')eker  (1)  ia i =  d, so we have:  m  s d ' ( A ' ) = d ( n ) = d ( ( - t ) A ) = - t d ( A ) , and sd'(A')+td(A)=0 i n H (M ), q n  8.3  Lemma; Suppose x i s a t o r s i o n element o f f i n i t e o r d e r s i n H^(M).  Proof:  S i n c e d(A)  is infective), ( s i n c e s*0) . using  (2). If  i s of i n f i n i t e  order  t i f t*0.  o r d e r by (I) (which i m p l i e s t h a t d  (2) shows t h a t d'(A') i s a l s o of i n f i n i t e C l e a r l y t y = t i ^ d ( A ) = i ^ (-sd' (A ' ) )=0,  order  i f t*0  s i n c e i^.°d'=0, and  Hence (order y ) | t .  t'y=0, then t ' i ^ d ( A ) = i ^ ( t ' d ( A ) ) = 0 ,  so t'd(A)eker  i*=im d',  t'd(A)=-s'd'(A') f o r some seZ, o r s'd'(A')+t'd(A)=0 i n H ( M ) . q  Applying  i^,  we g e t s ' i d'(A')=s'x=0, so  S'=JG«S.  (2) from s'd'(A')+t'd(A)=0 we g e t ( t ' - J U ) d ( A ) = 0 . o r d e r , so t'-£t=0, o r t'=£t.  i s infective Consider  0  Subtracting I But d(A)  times  i s of i n f i n i t e  Hence t | t ' , and t=order y .  Suppose t=0 so t h a t sd'(A')=0. i^  (2)  f o r some t e Z .  Then y i s of i n f i n i t e -'order i f t=0, and of ( f i n i t e )  and  sequence:  Then k e r  iRetorsion  on>dZ, and hence y=i^d(A) i s of i n f i n i t e  H^(M ), so 0  o r d e r i n H^(M ). T  the commutative diagram on the next page, i n which d and d'  a r e from the exact  sequences of C o r o l l a r y 7.9.  H  -^-H  (S xS )«q  q  q  ^(D q+1  q + 1  x  q S  ,S xS )=Z q  q  (3) Z=H ^ ( S x D q+1 q  R e c a l l that X e H X*eH  q + 1  (D  q + 1  q + 1  (S xD q  q + 1  ,S x q  q S  ->H (M )  )-^-  q  , s x S ) i s such t h a t 3 X = l ® [ S ] , and  q + 1  q  q  q  x S , S x S ) I s such t h a t q  q  3'X'=[S ]®l.  q  q  Suppose M i s c l o s e d , so t h a t 3M =S xS , and <j) : S x S — > - M q  q  q  0  i n c l u s i o n o f the boundary.  i s the  q  0  Then we have the exact sequence diagram  of P o i n c a r e d u a l i t y : '•—>-H- (MQ)  .. .  ^  q  q  q  q + 1  S  q  0  (Mp  ,S xS )—>. . . q  q  q  (M ,S xS )q  q+1  q  [S xs ]n-  [M ]n->H  ,H (S x )-^-vH  ( S  q  0  q  q  x S  q  ) — > H  (M )0  q  , qx .c T n ( i m <}> * )=ker <t> . Thus, [ S Sq ] H  (4)  H  0  0;%  ( 3 ) , d a')=(|) ^8 (X')=<(. ([S ]®l),  By  ,  ,  and  q  0  0A  d(X)=cj) 0 A a(X)=<j) ^(l®[S ]), q  0  so t h a t 8.4  (2) can be r e w r i t t e n as <j>o*(s ([S ]®l)+t(1® [ S ] ) )=0. q  Lemma: L e t q be even.  q  Then <J> *( ( [S ]®l)+t (1® [ S ] ) )=0 S  q  q  0  implies  e i t h e r s=0 or t=0. P r o o f : L e t U e H ( S ) be such t h a t U [ S ] = l . q  and  q  [S xS ]n(l®U)=[S ]®l, i n H ( S x S ) . q  Then  q  q  q  q  [S xS ]n(U®1)=1®[S ] q  q  q  Hence  q  q  [S xS ]n(s(l®U)+t(U®l))=s([S ]®l)+t(l®[S ]), q  q  q  q  and by (4) i t f o l l o w s t h a t s (l®U)+t (U®l)=(}>o (z) f o r some z e H ( M ) . H  0  But 9 * : H ( M o ) — > H ( S x s ) 2 q  2 q  q  q  i s z e r o , as <j>  0  (connected) boundary of M .  if  (s(l®U)+t(U®l))  2  s  the i n c l u s i o n o f the  Hence (s (l®U)+t (U®1) )=<J>* ( z )=0. 2  0  But  i  =2st(U®U) i f q=dim U i s even.  2  Hence i t i s zero  and o n l y i f s=0 or 't=0.  Proof o f Theorem  QED  7.1 f o r m=2q+l, q even: By Theorem 8.1, we may assume  f:M—>A i s q-connected and K (M) i s a t o r s i o n group. q  L e t xeK (M) be the q  g e n e r a t o r o f a c y c l i c . summand o f order s.  L e t <j>: S xD ^—>-M be an q  q+  embedding w i t h c|) (u)=x, and d e f i n i n g a normal cobordism A  Assume M i s c l o s e d , u s i n g P r o p o s i t i o n 7.4. of Lemma 8.2. Z £ H (M). s q  By Leimia 7 .10 , i ^ d ' (A ') =x,  Consider  of ( f , b ) .  the diagram  a generator  o f a summand  By (2) and Lemma 8.4, sd'(A')=0, so d'(A ' ) generates a  c y c l i c d i r e c t summand Z <=H (M ) . s- q From (1) i t f o l l o w s t h a t t o r s i o n subgroup of t o r s i o n  ( M ) , and s i n c e  H  q  w i t h a t l e a s t one c y c l i c for K (M'). q  H  (M )/d'Z,  (M')=H  q  t h a t t o r s i o n H ^ ( M ' ) i s isomorphic  i s isomorphic  H^.(MQ)  n  to a  i t follows  q  t o a subgroup o f t o r s i o n H ^ ( M )  summand Z^ missing,,,so  the same i s t r u e  ( I t f o l l o w s a l s o t h a t rank H (M*)=rank H (M)+l.) q q  By Theorem 8.1 we may f i n d a normal cobordism  of ( f ' , b ' ) to  ( f " , b " ) w i t h K (M").=tors i o n K • ( M ) . q q 1  Iterating K^(M)  these c o n s t r u c t i o n s a f i n i t e number o f times  i s finitely  generated)  w i l l . p r o d u c e an ( f i , b j )  (since  normally  cobordant  t o (f ,b) w i t h K ^ ( M i ) = 0 , and f ^ (q+1)-connected.  completes  the proof f o r m=l (mod 4 ) .  This QED  Proof o f Theorem 7.1 f o r m=2q+l, q odd: L e t <j>:S xD ^"—*-M be an embedding q  which d e f i n e s a normal cobordism,  q+  i . e . so t h a t ( f , b ) extend  t r a c e of the s u r g e r y based on d>, W, .  over the  L e t oo :S —>S0(q+l) , w i t h SO (q+1) q  <P  a c t i n g on D "*" from  the r i g h t , and d e f i n e a new embedding <J>^: S xD "'"—*M  q+  q  by < } > (x,t)=<J> (x, tm(x) ) .  Then < f >  d e f i n e s a s u r g e r y w i t h the r e s u l t  M'=Mgli^iD ' "'"xs , where M Q cOmes from s u r g e r y u s i n g C  +  q  d i f f eomorphism S x s — * s x s q  8.5  q  q  q  q+  and oo' i s the  g i v e n by OJ ' (x,y)= (x,yoo (x) ) .  Lemma: The t r a c e o f the surgery, based on < j > a l s o d e f i n e s a normal 0)  cobordism  i f and o n l y i f the homotopy c l a s s  [oo] goes t o zero i n  Tr (S0(q+k+l)) , i . e . i [oo]=0 where i : SO (q+1)—*S0(q+k+l) i s i n c l u s i o n . q  #  58 P r o o f : The map <f>. : S xD xi? —>Mx.ff q  q+1  k  g i v e n by <j>. (x, t ,r) = (9 (x, tto (x) ) ,r)  k  io)  =(9  (x,t),r)  io)  d e f i n e s a new f r a m i n g of the normal bundle t o S  in D  q  m + k  ,  CO of v | s $ v ' ,  i.e.  where v i s t h e normal bundle of M c D  q  bundle of S cM.  m + k  , v ' the normal  Then <j> d e f i n e s a normal cobordism i f and o n l y i f the to  q  framing extends t o a f r a m i n g o f the normal b u n d l e of D ^* i n D q+  m + k  xl,  so t h a t t h e f i r s t p a r t o f the frame d e f i n e s an embedding o f D ^xD "'" q+  in  D  m + k  q+  x l e x t e n d i n g 9 :S xD "'"cMcD , and the second p a r t o f the frame •to ~ q  q+  m+k  extends the t r i v i a l i s a t i o n o f v | 9 (S xD "*") d e f i n e d by b:v—>ri, t o a q  trivialisation MxIuD  q + 1  x-  q + 1 D  q+  o f the normal bundle of D  q +  ^*D  q +  \  and hence  .  Now S = 3 D q  q + 1  , D  q + 1  cD  m + k  x I such t h a t the normal bundle o f S i n q  D  m + k  x 0 i s the r e s t r i c t i o n to S  D  m + k  xi.  o f y> t h e normal bundle of D  q  Now y has a f r a m i n g d e f i n e d on S  q  by the map $ : S x D q  9 ( x , t ,r)== (9 (x, t) ,r) s i n c e 9 d e f i n e d a normal cobordism. of  these two framings i s a map o f S Hence the frame <f>  iw  to  into  so t h a t k e r i ^ , i-^iir  the boundary  k  S0(q+k+l) which i s o b v i o u s l y ito.  QED (S0(q+r+l)) i s an isomorphism f o r r > l ,  (S0(q+1))—*-TT (S0(q+k+l)) , i s the same f o r a l l k > l . q  SO (q+1)—^-s-SO (q+2) that  xi? —s-E(v) ,  q+  k = l , the e x a c t homotopy sequence  g i v e s the r e s u l t  q+1  ^ in  extends over D ''" i f and o n l y i f ito i s homotopic  By Lemma 6.7, IT (S0(q+r))—VJT  is  q  q +  The d i f f e r e n c e  zero i n S0(q+k+l).  For  that of  (ker / / )  ^S  1  = 3 q  0  o f the f i b r e  space  q + 1  7 r q + 1  o f the e x a c t sequence.  (  s q + 1  ) » where 3 : T r 0  q + 1  (Sq+1)-^Tr (SO(q+l>) q  Hence from Lemma 8.5, i f  9:S xD ^—>-M d e f i n e s a normal cobordism, then we may change 9 by q  q+  to:S —>S0(q+l) q  i f [to]e30TT  q + 1  (S  q + 1  ) , and 9 ^ w i l l s t i l l  d e f i n e a normal  cobordism. Now we w i l l compare the e f f e c t  o f the s u r g e r i e s based on 9 and <j> .  59 Let gJ=[S ]®l, g 2 = l ® [ S ] e H ( S x ) . 4  H  H  M  H  S  8.6  Lemma:. L e t g be a generator of ^q+i^^"*"^) , and l e t [co]=m8o(g),  • =<I> .  +6"^Cg£)=* #<8i) •  Then <f>o ^(gi)=<)>o (8i)+2m<j) (gl) ,  !  ,  UJ  #  0#  0  P r o o f : R e c a l l t h a t Lemma 6.19 says t h a t the composition '  * 1  ., ( S  q + 1  (SG(q+l))—^/-^Tr  -)—^-»ir  q+i  q  q  Now, 4>Q i s r e p r e s e n t e d by t h e  •s xS -—-°—>M -,  q  q  q  q  i s m u l t i p l i c a t i o n by 2, i f . q i s . odd. composition" S x S - ^  (S )  where oo' i s g i v e n by (x,y)—Kx,yco(x)) .  q  0  I f y i s taken t o be the base p o i n t y Q e S , then by d e f i n i t i o n ygco (x)=poo (x) , q  where p : SO (q+1)—>S  i s the bundle p r o j e c t i o n .  q  Hence on S x y , <t>'(x,yo) q  0  =<f>0 (x,pco(x)) , so <po P0 (l pw)A on S x y , where A : S — > S x S =C  x  q  q  q  i s g i v e n by  q  0  x—*(x,x). I f g £ T r ( S ) i s the g e n e r a t o r , i  (x)= ( x , y ) , i  q  q  :  0  2  (x) = ( y ,x) , g..= (i...)-^, 0  then A^g=gi+g , and h(g^.)=g^, where h i s the Hurewicz homomorphism. 2  Thus, < } > 6 <8l ) 0 (lxpw) A (g) =<() o C l p o J ) Cgi +"g2 ) =4> 0 # (Si + S 2 ) 2m  x  #  #  #  #  #  #  =<('0#(gl) f 0#(g2) • +2m<  ,  Since co(y ) i s the i d e n t i t y o f S0(q+1), we have <j>(51 yo S =cf> | y o S , x  q  0  x  q  0  so <t>^^(g )=<j>oft(§2) •  The r e s u l t i n homology  2  f o l l o w s by a p p l y i n g h.  QED  R e t u r n i n g t o the diagram of Lemma 8.2, where d(X)=(()o (l®[S ]) q  A  =h<j>o^(g2)> and ' d''(A )=h(|>o^(g]_) , I f 1  w  c o s t r u c t the analogous diagram  e  u s i n g $ i n s t e a d o f $, we f i n d d (A)=h<(> ,, ( g )=d (A) , and d'(A')=h(j) n ( g i ) CO CO CO If CO CO If =d'(A')+2md(A), o r d(A)=d ( A ) , d*(A')=d'(A')-2md ( A ) . Hence (2) becomes CO CO CO s(d'(A')-2md (A))+td (A)=0, or sd'(A')+(t-2ms)d (A)=0. (5) CO CO CO CO CO 8.7 P r o p o s i t i o n : L e t p be a prime and l e t xeK^(M) be an element o f 0  f i n i t e order such t h a t mod  p.  4>^(y) ) =x  2  (x) *0 i n K (M;Z ) , where (•) denotes r e d u c t i o n P q P P  L e t <|):S xD ^—>-int M be an embedding which r e p r e s e n t s x, i . e . q  a  n  q+  d which d e f i n e s a normal s u r g e r y o f ( f , b ) .  Then one may  choose co: S —*S0 (q+1) so t h a t < J > : S xD ' "-->int M a l s o d e f i n e s a normal q  q  q+  1  surgery  of ( f , b ) , o r d e r ( t o r s i o n K ( M * ) ) < o r d e r ( t o r s i o n  K (M)), and  q to q rank„ K (M';Z )<rank„ K (M;Z ) . (The order o f a t o r s i o n group T i s L q to p L q p P P the s m a l l e s t p o s i t i v e i n t e g e r n such t h a t nx=0 i n T f o r a l l ' xeT.) Proof:  By Lemma 8.2,  (M)/(x)=H (M')/(y), where (x) i n d i c a t e s the q  subgroup generated by x.  I f t h e order  o f x i s s, then (2) g i v e s  sd'(A')+td(A)=0, and Lemma 8.3 s t a t e s t h a t the order and  o f y i s t i f t*0,  i s i n f i n i t e i f t=0. By Lemma 8.5 we may change < f > so that (2)  becomes ( 5 ) :  sd'(A')+(t-2ms)d  to  (A)=0,  to  w i t h o r d e r y =t-2ms i f t-2ms 0, and y  to  •'to  y  so t h a t H (M)/(x)=H (M')/(y ) q o f i n f i n i t e order  q to to i f t-2ms=0.  Choose m so t h a t -s<(t-2ms.)<s, which guarantees t h a t o r d e r y^<order x or y  i s of i n f i n i t e order.  Hence, o r d e r ( t o r s i o n H (M'))  i s not  to q to l a r g e r than o r d e r ( t o r s i o n H (M)), and so o r d e r ( t o r s i o n K (M'))  i s less  q q to than o r e q u a l t o o r d e r ( t o r s i o n K (M)). But i f (x) *0, then by q P C o r o l l a r y 7.14, rank„ K (M';Z )<rank„ K (M;Z ) . QED a q to p 6 q p P P We a r e now a b l e Let  to complete the proof  o f Theorem 7.1 f o r m=3 (mod 4)  (f,b) be a normal map, and by Theorem 8.1 we may assume f i s  q-connected, and K^(M) i s a t o r s i o n group. d i v i d i n g order  L e t p be the l a r g e s t prime  K (M), and l e t xeK (M) be an element such t h a t q q  (x) *0 p  i n K (M;'Zp)*r' By Whitney's embedding theorem we may f i n d an embedded q  i n t M "'" r e p r e s e n t i n g 2q+  x, and by Theorems 6.6 and 6.12, we may  extend t h i s embedding t o an embedding <j> : S x D ^ — * i n t M such t h a t q  $ d e f i n e s a normal surgery  q +  on ( f , b ) .  By P r o p o s i t i o n 8.7, $ may be chosen so t h a t o r d e r ( t o r s i o n K (M')) q  <order(torsion  K (M)), and rank„ K (M':Z )<rank„ K (M;Z ) . q Z q p Z q p p  p  P r o c e e d i n g i n t h i s f a s h i o n step by s t e p , we w i l l f i n d a f t e r a f i n i t e number o f such s u r g e r i e s , a normal cobordism of ( f , b ) t o ( f ^ . b ^ ) such t h a t fi i s q-connected, o r d e r ( t o r s i o n K ^ ( M ) ) < o r d e r ( t o r s i o n K ( M ) ) , x  q  61 and  rank„ K ( M i ; Z )=0. P  *  q  Since  the U n i v e r s a l C o e f f i c i e n t Theorem h o l d s  P  for  t h e K^, K  groups, K ( M i ; Z ^ ) = K ^ ( M ) ® Z , because K ( M i ) = 0 f o r i < q ,  and  i t f o l l o w s t h a t K^(Mi) i s a t o r s i o n group o f order prime t o p, and  q  order K (Mi)<order K (M). q q  x  p  ±  Since K (M) has p - t o r s i o n , i t f o l l o w s t h a t , q  i n f a c t , o r d e r K ( M i ) < o r d e r K^(M). q  Hence we have reduced  the order o f  the k e r n e l , and so a f i n i t e number o f i t e r a t i o n s w i l l make the order of the k e r n a l z e r o , thus p r o d u c i n g  a normal cobordism o f ( f , b ) w i t h  some ( f , b ) , where f i s q-connected and K (M)=0.  Hence f i s a c t u a l l y  q  (q+1)-connected, which proves Theorem 7.1 f o r m=3 (mod 4 ) . completes the p r o o f §9.  The P r o o f  Set m=2q. that  o f Theorem 7.1 f o r m odd.  o f the Fundamental Theorem f o r m even.  L e t ( f , b ) . be a normal map w i t h  i<q.  Since  the  and K  duality  groups s a t i s f y  Theorem, i t f o l l o w s t h a t K^M) =0 f o r i>q, xeK^(M) be r e p r e s e n t e d and  f:(M,9M)—KA,B) such  (f | SM)^ :H^(9M)—>-H^(B) i s an isomorphism, and f i s q-connected.  Then K (M)=0 f o r i<q, and by P o i n c a r e for  This also  d e f i n e M=MU D ot  q+1  K  m_1  (M,9M)=K  m-1  (M)=0  the U n i v e r s a l C o e f f i c i e n t  and K^(M) i s f r e e . L e t  by an embedding a:S —>-int M, s o t h a t ' ( g , a ) e T r ^ ( f ) , q  q +  , f:M—*A e x t e n d i n g  f , d e f i n e d u s i n g 3 :D "^—>A. q+  By Theorem 6.6, t h e r e i s an o b s t r u c t i o n Oeu (V, ) (which Is Z i f q q K,q i s even, Z thickened  2  i f q I s odd) such t h a t 0=0 i f and o n l y i f f:M—>A can be  t o a normal cobordism.  [M]nx'=xeK (M). q surgery  L e t x'eK (M,9M) be d e f i n e d by q  R e c a l l t h a t , as p a r t o f our d e f i n i t i o n above o f the  i n v a r i a n t a ( f , b ) , we d e f i n e d a b i l i n e a r p a i r i n g (*,•) on  K (M,3M), and made use o f a q u a d r a t i c form ip :K (M, 9M; Z ) — > Z . q  q  2  2  Theorem: The o b s t r u c t i o n 0 to t h i c k e n i n g f:M—»A to a normal  ^•1  cobordism i s g i v e n by 0=(x',x')  where (*): denotes r e d u c t i o n mod 2  Theorem 4.1  f  2  i f q i s odd,  2.  B e f o r e p r o v i n g Theorem 9.1, we p r o o f of Theorem  0=ty((x ) )  i f q i s even,  s h a l l use i t to complete  the  7.1. s t a t e s t h a t i f ( f , b ) i s n o r m a l l y cobordant r e l B  to a homotopy e q u i v a l e n c e , then a ( f , b ) = 0 .  Thus, our i n t e n t i s to •  assume t h a t a ( f , b ) = 0 , and then to c o n s t r u c t a normal cobordism of ( f , b ) to a homotopy e q u i v a l e n c e . F i r s t , suppose  q i s even.  Then  (f ,b)=-^-I(f) , so t h a t i f o  (f,b)=0, i t f o l l o w s t h a t 1 ( f ) , the s i g n a t u r e of (•,•) is  zero.  By Theorem 6.13  i<q, and i s f r e e f o r i=q. such t h a t by  (x',x')=0,  <f>:S xD —>int M, q  q  we may  assume t h a t  t h e r e i s an x'eK (M,9M) q  [M]nx'=xeK (M) can be r e p r e s e n t e d q  ( i . e . <JK(y)=x, y the generator of H ( S * D ) ) , * q q  such t h a t the s u r g e r y based on § d e f i n e s a normal But we may  q  cobordism of  (f,b).  choose x' to be i n d i v i s i b l e . ( f o r o t h e r w i s e , x'=kx", where  x" i s i n d i v i s i b l e , and  (x',x )=0=(kx",'kx")=k (x",x") , so ,  2  so the g e n e r a t o r of a d i r e c t summand of K (M,9M). q  7.12,  q  (M^K"*"(M,9M)=0 f o r  By P r o p o s i t i o n 5.3,  so by 9.1,  on K (M,9M),  rank K (M')<rank K (M), and f q q  is still  (x",x")=0),  Hence, by  Corollary  q-connected, where  f':(M',9M')—KA,B) r e s u l t s from normal s u r g e r y based on <(> ( i n f a c t , the rank decreases by 2: see Lemma 8.2).  S i n c e ( f , b ) and  (f',b')  a r e n o r m a l l y cobordant, I ( f ' ) = I ( f ) = 0 (see Theorem 5.14), and we r e p e a t the p r o c e d u r e .  In f a c t , i f we  i s reduced t o z e r o , the r e s u l t i n g map  may  i t e r a t e the p r o c e s s u n t i l i s (q+1)-connected,  as d e s i r e d .  63 Now take q odd.  Then a ( f , b ) = c ( f , b ) i s the A r f i n v a r i a n t o f  on K ( M , 9 M ; Z ) .  I f a ( f , b ) = 0 , then t h e r e i s c e r t a i n l y some yeK (M,9M;Z )  f o r which iKy)=0  (see f o r example P r o p s i t i o n 5.8 o r 5.10).  q  q  2  2  q-connected, then K (M,9M;Z )=K (M,9M)®Z , and y = ( x ' ) q  q  2  i n d i v i s i b l e x'eK (M,9M). q  2  2  If f is  f o r some  By 9.1, x=[M]nx' i s r e p r e s e n t e d by  <j>:S xD —>-int M such t h a t 9 d e f i n e s a normal cobordism, and by C o r o l l a r y q  q  7.12,  rank K (M')<rank K^(M), w i t h f ' s t i l l q  a ( f ' ,b' )=a('f ,b)=0, s i n c e  q-connected.  But  ( f ' , b ' ) i s n o r m a l l y cobordant to ( f , b ) ,  so we may proceed as above to.produce a (q+1)-connected map. completes  This  t h e p r o o f o f Theorem 7.1, and hence o f the Fundamental  Theorem. The b a l a n c e o f t h i s s e c t i o n w i l l be taken up by the p r o o f o f Theorem 9.1. Let  ( f , b ) be a normal map, f:(M',9M)—»-(A,B) , M i s of dimension  m=2q, and f i s q-connected. by an embedding a:S —KLnt q  Choose an x e K ( M ) , and l e t i t be r e p r e s e n t e d q  M.  Let ?  image o f a i n M, and s e t M='Mu D ' ^. q  +  q  denote the normal bundle o f the Then f may be extended  t o f:M—>A.  L e t Oeu (V, ) be the o b s t r u c t i o n to t h i c k e n i n g M and f to a normal q k>q cobordism ( c f . Theorem 6.6), and l e t 9 :TT (V, )—*-TT (S0(q)) be the q >q q-i c o n n e c t i n g homomorphism i n the e x a c t sequence o f the f i b r e bundle k  p:S0(k+q)—>V, =S0(k+q)/S0(q), k., q  with f i b r e S0(q).  We d e f i n e the c h a r a c t e r i s t i c map of a k-plane bundle over a sphere k  n  as f o l l o w s : l e t E =(E,S ,TT) be a k - d i m e n s i o n a l o r i e n t a b l e v e c t o r bundle. I f S ={ (x_^)ei? "'"| x§+xf+ .. .+x = l} , then we may d e f i n e two subsets n  n+  2  and  D_ such t h a t D ( r e s p . D_) i s t h e hemisphere c e n t r e d on the N ( r e s p . S) p o l e of S , i . e . D ^ = { ( x . ) e S i x SO}, and s i m i l a r l y f o r D . Clearly + i n — +  n  n  S =D^uD , and i t i s easy to show t h a t D^nD^-S n  n  n  n  (the 'equator' o f S ) . n  Since  the r e s t r i c t i o n s of ? t o  choose t r i v i a l i s a t i o n s T  and  a r e both t r i v i a l , we may  and T_ such t h a t x :E|D^—>D^xfl  +  (similarly  k  +  k for  T_) .  defined  Since T  and T  +  a r e f i b r e i s o m o r p h i c , the map s^ii? —hff  f o r each xeS* "*"=D^nD^ by T °T_|_^"(x,y) = (x,s  (y)) i s i n f a c t an  1  orientation-preserving l i n e a r transformation Thus, we have d e f i n e d This i s c a l l e d  k  a map c ( E ) : S • n  of R , i . e . s^eSCKk).  —>-SO(k) g i v e n by c ( £ ) ( x ) = s . x  the c h a r a c t e r i s t i c map o f E , and a l t h o u g h i t i s n o t  unique, i t i s w e l l - d e f i n e d  up t o homotopy.  (Thus i t can be h e l d  that  the c h a r a c t e r i s t i c 'map' i s n o t r e a l l y a map, b u t only an element of TT (SO(k)).) n—l With Z, and 0 d e f i n e d 9.2  as above, we have  P r o p o s i t i o n : 30 i s the c h a r a c t e r i s t i c map of z,, an element of  TT (S0(q)). q-1 Proof:  Choose a base p o i n t J g S 0 ( q + k ) , a (q+k)-frame i n R  q+k  . Let  e  p:SO(q+k)—>V  -SO(q+k)/SO(q) k., q  be the p r o j e c t i o n , g i v e n by s e l e c t i n g  the f i r s t k elements of a (k+q-)—frame. such t h a t , i f h:S —*-S0(q+k)/S0(q)=V  L e t xgeS  be a base p o i n t  q  represents  q  0, then h ( x ) = p ( J o ) 0  K. ,q  Divide S  q  i n t o two c e l l s ,  S =D uD , so t h a t q  q  x eD nD =S  q  q  q  q_1  0  =3D =3D . q  q  Without l o s s of g e n e r a l i t y , we may assume t h a t h(D )=p (</Q ) , s i n c e q  D  q  i s contractible.  p°h=h on D . q  L e t h:D ^-*S0(q+k) be such t h a t h ( x ) = J and q  0  Then p h ( S  q  elements o f h ( y ) f o r y e S  1  q  )=h(S  q  1  )=p(J )»  s  o  t  h  a  t  t  h  e  0  0  first k  ^ make up t h e base frame of V,  . Let  k,q i:S0(q)—>S0(q+k) be the r e p r e s e n t a t i o n  of S0(q) a c t i n g on the subspace  q+k of R o r t h o g o n a l t o the space spanned by p(J"o). Then t h e r e i s a map Y :-S —>S0(q) such t h a t h(y)=J" (i°Y (y) ) • By the d e f i n i t i o n o f 3, q-1  0  Y represents  30eiT _ (S0(q)) q  1  (see [Steenrod  1951]).  65 Now  ? i s the o r h t o g o n a l bundle to the t r i v i a l bundle spanned by S i n c e h (D )=p (C7Q ) > the l a s t q v e c t o r s  h(x), for xeS . q  trivialisation  of t, over D , and s i n c e p°h=h, the l a s t q v e c t o r s of  h(x), f o r xeD , give a t r i v i a l i s a t i o n q  yeS  follows 1951,  of C over D . q  S i n c e y(y), f o r  sends the l a s t p a r t o f JQ i n t o the l a s t p a r t o f h ( y ) , i t  \  q  i n JQ g i v e a  q  t h a t y i s c ( £ ) , the c h a r a c t e r i s t i c map  of t, (see [Steenrod  (18.1)]).  QED  From our d i s c u s s i o n above of the homotopy p r o p e r t i e s of SO(n), we 9.3 for  derive  the f o l l o w i n g  P r o p o s i t i o n : The boundary  3 : 7 r  q(^  q)—^  ^(SO(q)) i s a monomorphism  q * l , 3 , or 7.  P r o o f : By comparing v a r i o u s f o l l o w i n g commutative SO(q)  —  r e l a t e d f i b r e b u n d l e s , we produce the  diagram: >-SO(q)  —*SO(q)/SO(q-l)=S  q - 1  13  SO (q+1).  ^ >S0 (q+k)-  k+1,9"  P2  'Pi  s-=v  1  P3  q  k,q  i,q  k,q  where the p^ a r e the p r o j e c t i o n s of f i b r e b u n d l e s , and i _ . a r e i n c l u s i o n s of f i b r e s .  Let 9  be the c o n n e c t i n g homomorphism i n the homotopy exact  sequence of the bundle w i t h p r o j e c t i o n p^. . p ! , 3 i :TT (S )—*-TT ,(S V q q-1 q  q  By Lemma 6.9,  "*") i s m u l t i p l i c a t i o n by two, and i s thus i n j e c t i v e .  But by the commutativity of the diagram, P^°9]. =83 °j^ • monomorphism, and s i n c e by Theorem 6.12 follows  i f q i s even,  Hence j ^ i s a  TT (V, )=Z i f q i s even, i t q k,q  t h a t 83=8 i s a monomorphism i f q i s even.  I f q * l , 3 , or 7, and q i s odd, then by C o r o l l a r y 6.11 where i ^ : T T _ ( S O ( q ) ) — ( S O q  1  (q+1)) .  Hence 9  T  ker i^=Z , 2  i s onto Z 2 C T f _ ( S O ( q ) ) , q  1  66 and s i n c e j ^ : i T ( S ) — ^ ^ 0 ^ ^) i s s u r j e c t i v e by Theorem 6.12,  3i 33°j^5  q  =  q  it  f o l l o w s t h a t 3 (TT (V ))^Z . q k, q 3  S i n c e TT (V )=Z q k, q  Z  f o r q odd  2  (by  we have 83=8 a monomorphism f o r q * l , 3 , or 7.  6.12),  QED  Thus f o r q * l , 3 , or 7, the o b s t r u c t i o n 0 to d o i n g normal s u r g e r y on a particular S  q  embedded i n M^  q  can be i d e n t i f i e d w i t h the c h a r a c t e r i s t i c  of C , the normal bundle of the chosen S  map  i n M, Oeker  q  ..(SO(q)), q-1  i,,cTr  it and i s t h e r e f o r e z e r o i f t, i s t r i v i a l .  Now  ker i ^  i s generated by  3j(i),  where i € T T ( S ) i s the c l a s s of the i d e n t i t y , so t h a t 31 ( t ) i s the q  q  c h a r a c t e r i s t i c map  f o r the tangent bundle T of S .  I t follows  q  that  0 = X ( 3 ( i ) ) f o r some XeZ. 1  I f q i s even, the E u l e r c l a s s x ( ) = 2 g e H ( S ) , where g i s the g e n e r a t o r T  f o r which or may  g[S ]=l. q  be deduced  q  q  T h i s f o l l o w s from the g e n e r a l formula X ( f o r M=S , q even, u s i n g the f a c t q  to the normal bundle of the d i a g o n a l M i n M*M. Thorn c l a s s , i t f o l l o w s from C o r o l l a r y 7.6  that  T  ) X( )gJ =  M  M  i s equivalent  For i f U e H ( E , E ) i s the q  0  that  [S xS ]nri*U=[S ]®l+l®[S ] , q  q  q  q  the homology c l a s s of the d i a g o n a l , where ri:S xs —>-E/Eg i s the n a t u r a l q  c o l l a p s i n g map. if  q i s even.  Hence n U=g®l+l®g, Since n  q  and n (U )=(ri U) =(g®l+l®g) =2g®g, 2Q 9 i s an isomorphism on H , i t f o l l o w s t h a t U =2gU, 2  so x ( T ) 2 g , s i n c e by d e f i n i t i o n x ( £ ) U ^ = ( U ^ ) =  2  2  2  f o r a bundle  The E u l e r c l a s s i s r e p r e s e n t e d by the u n i v e r s a l E u l e r XeH (BSO(q)), q  bundles  E. class  where BSO(q) i s the c l a s s i f y i n g space f o r o r i e n t e d  (see [Husemoller 1966]  or [Steenrod 1951]).  c:X—>BS0(q) i s the c l a s s i f y i n g map  q-plane  That i s , i f  of a q-plane bundle E over X, c ( Y ) = E »  where y i s the u n i v e r s a l q-plane bundle over BSO(q), then x ( C ) = q * I f c:S —>-BS0(q) r e p r e s e n t s T q, then c ( x ) 2 g as above, but i f  c  (x) •  =  c'  *-BS0(q) r e p r e s e n t s X (T q) i n the homotopy group TT  Xc and c' a r e homotopic,  i . e . [Xc] = [c'] i n T T ( B S 0 ( q ) ) . q  (S0(q)),then Hence c' =Xc- ,  67 so we have: 9.4  Lemma: I f q i s even and 3 0=A3;L ( i ) , then x ( ? ) 2 A g , where £ i s the =  2  normal bundle of a ( S ) i n M , q  (M) , 0 the  r e p r e s e n t i n g an element i n  2q  o b s t r u c t i o n t o d o i n g a normal s u r g e r y on t h i s S , q  9.5  Lemma: y ( O [ S ] = ( x ' , x ' ) , where [M]nx'=x, a:S —>M q  q  representing xeK (M), q  Proof: x(?)U=U class. [E]eH  i s an embedding  Zq  t, the normal bundle o f o t ( S ) , as above. q  by d e f i n i t i o n of x » where U e H ( E ( ? ) / E ( O )  2  q  0  Clearly  i s the Thorn  ( ( ? ) ) [ S ] = ( ( C ) U ) [ E ] = U [ E ] = ( n * U ) [ M ] , where q  2  x  2  x  ( E ( ? ) / E ( ? ) ) i s the o r i e n t a t i o n c l a s s , so [E]=n*[M], where  2  0  TI :M/3M—>E/EQ i s the n a t u r a l c o l l a p s i n g map. By C o r o l l a r y  7.6, [M]nn U=x, so t h a t n U=x'.  Hence  X(?)[S ]=(n*U) [M]=(x') [M]=(x',x'). q  2  By 9.4 and 9.5 f o r q even, 3  2  QED  2  (x*,x')=2A  where 3 0 = A 3 ( i ) . 2  1  i s a monomorphism f o r q even, so we may i d e n t i f y 0 w i t h  By 9.3  (x',x'),  which proves Theorem 9.1 f o r q even. F i n a l l y . , we t u r n our a t t e n t i o n to the case of q odd. -  L e t a_^:S —>-M. , i = l , 2 , be 'embe'da'ings r e p r e s e n t i n g q  x^eK (M),  Zq  q  where, as u s u a l , K^(M) i s d e f i n e d u s i n g a normal map ( f , b ) , f:(M,3M)—>-(A,B), (f | 3M) :H (3M)—>-H^(B) an isomorphism. A  Suppose  A  images, and l e t 0\ on ai(S ) q  disjoint  the  have  disjoint  and c9 be the o b s t r u c t i o n s t o d o i n g normal s u r g e r y 2  and a ( S ) r e s p e c t i v e l y . q  2  Join  ( S ) t o a ( S ) by an a r c , q  q  2  (except, o f c o u r s e , a t i t s e n d p o i n t s ) from b o t h images.  By  t h i c k e n i n g t h i s to a tube T - D x [ l , 2 ] we may take q  ((*! ( S ) \ ( D x l ) ) U3 TU (<x 2 ( S ) \ (D x2) ) , q  q  q  q  0  where 3 T=3D x [1,2] , D xi=Tna q  0  q  (S ). q  T h i s subset o f M i s homeomorphic  to S , and so g i v e s us an embedding a:S —>-M r e p r e s e n t i n g x^+x , which q  q  2  can be made d i f f e r e n t i a b l e by 'rounding the c o r n e r s ' .  9.6  Lemma: c9=c9i+t9o i n ir (V, ) , where 0 i s the o b s t r u c t i o n to d o i n g q k,q  •  1  &  a  s u r g e r y on a(S  ).  P r o o f : S i n c e TcM, I f we  have M c D  q +  m + k  " ^ meets D  may  m + k  q + 1  ={D  xl  and by composing embeddings  t  q  +  to o b t a i n Tx [0 ,e ]<=MxI.  1  cD  m  +  k  xi  such t h a t a  x 0 transversally i n a^(S ).  m + k  (S ) i n D  of a  q  q  +  i s g i v e n by a  1  (S )=9D q  Then we  q  we  may  q + 1  ,  assume  (S )x[0,ej. q  \(D xlx[0,e])}u{(8D x[l,2]x[0,e])u(D x[l,2]x )}  q + 1  q  q  q  e  u{D This i s a  m + k  Choose D  xi.  t h a t a neighbourhood Set D  [0 ,e]  m u l t i p l y T by  , then M x l < = D  Tx[0,e]cD  produce and D  we  q + 1  \(D x2x[0,e])}. q  D  ( q + l ) - r c e l l meeting  may  smooth t h i s D " ' " ,  The  smoothed D " ' "  x 0 t r a n s v e r s a l l y i n a ( S ) , and  we  q  t o g e t h e r w i t h a ( S ) , by  'rounding c o r n e r s ' .  q  q +  q +  m + k  i s the union of t h r e e c e l l s , D ^"=AiuBuA , which q+  2  correspond to the t h r e e e x p r e s s i o n s i n b r a c e s , i n the e x p r e s s i o n for D  q +  ^ " above, a f t e r c l o s u r e and smoothing.  C =D.\int A i s a ( q + 1 ) - c e l l , 3C.n9D =F , F i l i l i i i J  J  1  B n A ^ g ^ n A ^ g B and  3B\((3C nA )u(9C nA )) = S 1  1  2  Assume A . < = D ' ' " . a q-cell in q - 1  2  c h o i c e of the f r a m i n g of the normal bundle y of D  the framings over D  q  +  q  +  D  \  ^ and D  q  l  +  q +  "'",  3D., l  xI.  S i n c e the d e f i n i t i o n of the o b s t r u c t i o n 0 doesn't  t h a t the framings over D  Then  q+  q  +  depend on  \ we  may  assume  and D " ' " have been chosen  so  +  2  ^ c o i n c i d e over A.. l  the  F u r t h e r we  that  may  assume t h a t the framings o f v, the normal bundle o f M i n D ™ ^ , over a ( S ) , a i ( S ) , and a ( S ) , induced by b, have been chosen q  q  q  2  so  over F^ .they a r e a l l the same, coming from a framing of v|T that T i s a c e l l ) , to extend  and  the framings of y, Y l J  a n  d  Y2  m a  y  that (note  be assumed  t h a t of v over Tna ( S ) , Tna_^(S ) (as i s a p p r o p r i a t e ) . q  q  Thus the t h r e e maps 3 , 3 . , i = l , 2, 3 :a ( S ) — * V . , 3 . :a, (S )—•V, l k,q l i k,q d e f i n i n g 0 and 0^, may be taken to be the base k-frame over T n a ( S ) , q  q  q  Tn  ( S ) , and 3 | ( a q  a i  (S )na(S ))=3 |(a q  (S )na(S )).  q  q  ±  f o r the homotopy c l a s s e s ,  [ 3 ] = [ 3 i ]+[621  I t follows that  q  ±  n  ( q  77  v v  )» k,q  o  0=0i+0  r  QED  Lemma: I f 0=0, then I|J ( ( x ) ) = 0 , w i t h n o t a t i o n as above.  9.7  T  2  P r o o f : S i n c e 0=0, we can perform normal s u r g e r y based so t h a t the t r a c e i s a normal cobordism if  .  2  W "'", 2 q +  on a:S —>-M , q  2q  W=Mu (3M*I)uM' , and  i:3W—*W and k:M—>3W a r e i n c l u s i o n s , i.k.x=0.  I t f o l l o w s from  * elementary  r e s u l t s about  and K  * (see p. 21 above),  t h a t x"=i z,  zeK (W), where x"ek (3W) i s d e f i n e d by [3W]nx"=k.,.x, and K (W) comes q  q  q  from the map F:W—>AxI extending f on M. I f K (3W;Z ) i s d e f i n e d f o r the map 3F:3W—*Ax0uBxIuAxl, and q  2  T|JQ i s t h e q u a d r a t i c form K (3W;Z )—>-Z  used  q  2  2  i n t h e d e f i n i t i o n of the  K e r v a i r e i n v a r i a n t , i t f o l l o w s from a lemma i n [Browder 1972, III.4.13]  i>oX{i  that and  z) )=iK(x") )=0. 2  Now 3F i s c l e a r l y  2  ( f ' , b ' ) on M' (the r e s u l t of s u r g e r y ) .  the sum of ( f , b ) on M  By an i n t e r m e d i a t e r e s u l t  i n t h e p r o o f o f Theorem 5.12, i|> (n* (x' ) )=iji ( (x* ) ) , x'eK (M,3M), so q  0  it  2  remains t o show t h a t n ( x ' ) = ( x " ) 2  2  (where n:3W—*M/3M).  2  Consider k^x=k^ ( [M] nx' )=k^(ri^ [3W] nx' ) = [3W] nri x', u s i n g i d e n t i t i e s of the cap product it  ( c f . C o r o l l a r y 7.6),  so t h a t s i n c e [SWjnx'^k^x,  f o l l o w s t h a t x"=n x', and hence i K ( x ' ) ) = 0 .  QED  2  Now we prove  that 0=iK(x') )..  I f 0=0, then i K ( x ' ) ) = 0 by 9.7,  2  2  so i t remains t o show t h a t i f 0=1 then 4 ( ( x ' ) ) = l . )  2  By t a k i n g the connected  sum w i t h the map S x S — * S , or a l t e r n a t e l y q  doing a normal s u r g e r y on. a S module on two g e n e r a t o r s a  q  "*"cD  /  q  cM  2  q  ,  q  2 q  we may add to K^(M) the f r e e  and a , c o r r e s p o n d i n g to [S ]®l and l ® [ S ] q  x  q  2  i n H ( S x S ) , and add to K (M,3M) t h e elements g , g such t h a t q  q  q  q  :  [M#(S xs )]ng =a , with q  q  i  ±  2  ( g , g ) = l , ( g , g ) = 0 , 1=1,2, o r t h o g o n a l t o the 1  2  i  ±  o r i g i n a l K (M,3M), and such t h a t T/J ( g )=ij> ( g ) = 0 . q  x  2  Hence if) ( g i + g ) = ^ (gi)+4<(82)+(gl >82)=12  I f g r S ^ M i (S^xS^) r e p r e s e n t s from 9.7 zero,  t h a t the o b s t r u c t i o n c9  t o surgery  then i(>(gT+g2) would be zero..  representing 0"=1+1=O..  the d i a g o n a l  class ai+a , i t follows 2  on $ i s 1, s i n c e i f i t were  Then on the sum embedding a+3  x + ( a i + a ) , the o b s t r u c t i o n 0"=0+0' by Lemma 9.6, so t h a t 2  Hence if) ( ( x ' ) + ( g l + g 2 ) )=0 by 9.7. 2  * ( (x') 2 - K g l +g ):) 2  we see t h a t i p ( ( x ' ) ) = l . 2  T h i V completes the p r o o f the Fundamental Theorem.  But s i n c e  (• (x' ) 2 ) n ( gi +82 )  ( ( x ' ) , ( g i + g ) )=0, 2  2  ( (x' ) 2 ) +1=0 > QED  of Theorems 9.1 and 7.1, and thus of  71 Chapter I I I . §10.  Plumbing and  I n t e r s e c t i o n and  L e t Ni and  N  Manifolds.  Plumbing.  be  2  the C l a s s i f i c a t i o n of  smooth submanifolds of dimension p  of a smooth m-manifold M,  such t h a t p+q=m.  (resp.  A p o i n t xeNjnN  q)  will  2  be  c a l l e d d i s c r e t e i f t h e r e i s an open neighbourhood V of x i n M such t h a t VnNi nN2={'x} .  Note t h a t i f every p o i n t i n N^ nN  then N'f"nN "is a d i s c r e t e subset 2  I f xeNinN and  2  of  M.  i s d i s c r e t e and V i s as above ( i . e . V i s open i n M  VnNinN ={x}) then (V\Ni)u(V\N )=V\{x}. 2  Thus we  2  H (V,V\N )®H (V,V\N )—>H q  P  1  P+q  2  Suppose t h a t M,  ( V , V \ { . x } ) g i v e n by  Ni and  N  2  are o r i e n t e d , and  [ N ] e H (N ,N \{z}) be  with  L e t E_^, i = l , 2 , be  N  ±  p  the o r i e n t a t i o n s .  i n M,  so H  E°=E_ \N_ . L  2  2  x  i  i  I f the E. are o r i e n t e d , and  r . denotes i  ( V , V \ N ^ ) c ( E , E ° ) , then by  the Thorn Isomorphism Theorem  i  q  • nUj  ) is a  q  a r e isomorphisms ( s i m i l a r l y f o r N ) .  xeN.. l c o n d i t i o n s we  may  of a d i s c r e t e p o i n t xeN^nN above.  We  by  homologically  Note t h a t g e o m e t r i c a l l y transverse.  there  generator,  s h a l l a l s o assume nr.U =[N.] i i I x  *  sgn(x)=(riUiUr U )[M]^, 2  s h a l l c a l l x a (homologically)  sgn(x)=±l.  x has  2  the  d e f i n e the s i g n or o r i e n t a t i o n  *  if  [M] x  Under the p r e c e d i n g  if  We  2  t h a t the o r i e n t a t i o n s are c o m p a t i b l e , i . e . so t h a t for  i s an e x c i s i o n ,  1  J  'UTJi,  compatible  a t u b u l a r neighbourhood of  i s an element Ui eH (E]_ ,E? ) such t h a t r * U i e H ( V , V and  m  the generators  2  product.  l e t [M] eH (M,M\{x}),  Then the i n c l u s i o n (E ,E°)c(M,M\N >  L  (M,M\N )=H (E.,E°). i l l  inclusion  z  have a p a i r i n g  the r e l a t i v e cup  [ N j ] e H ( N ! , N i \ { y } ) and y  is discrete,  2  2  using  the p a i r i n g  t r a n s v e r s e p o i n t of i n t e r s e c t i o n  t r a n s v e r s e p o i n t s are a l s o  (A p o i n t xeNjnN  2  i s geometrically  transverse  an open neighbourhood V i n M such t h a t t h e r e i s a d i f f e o m o r p h i s m  (V.VnNi ,VnN )—>(i? ,i? xO,Oxi? ) .) m  P  q  2  I f N} i s compact and N}nN n3M i s empty, i t has been shown t h a t 2  g i v e n an e>0 t h e r e i s a d i f f e o m o r p h i s m h:M—HM, which i s the i d e n t i t y on 9M, and i s e - i s o t o p i c t o 1^, such t h a t h ( N i ) n N of  2  consists solely  (geometrically) transverse points. On p.50 above we d e f i n e d a p a i r i n g • : H^ (M) ®H^ (M)—>Z by x*y=(x'uy') [M.].,  by  where x'eH (M,3M), y'eH (M) are q  P  defined  [M]nx'=x, [M]ny'=j y, and j i s i n c l u s i o n . A  L e t N ,- N P  2  be compact o r i e n t e d submanifolds o f  a compact '  o r i e n t e d m a n i f o l d w i t h boundary, m=p+q, and suppose Ni_ i s c l o s e d i n M, 3MnNj=cf>, and . 8MnN = 8N . 2  (homologically)  transver s a l l y .  We s t a t e w i t h o u t p r o o f 10.1  Assume f u r t h e r t h a t Nj and N  2  intersect  2  L e t i :Nj—»-M denote . the i n c l u s i o n s .  the f o l l o w i n g theorem from  [Browder 1972],  Theorem: ( i i ^ t N i ] ) • ( i ^ [ N ] ) = Z s g n ( x ) , where the sum i s taken 2  over a l l p o i n t s xeN^  2  nN . 2  Thus, the i n t e r s e c t i o n o f the o r i e n t a t i o n c l a s s e s counts the number of i n t e r s e c t i o n p o i n t s , w i t h If N  q  sign.  i s a c l o s e d submanifold  l y i n g i n the i n t e r i o r o f M , 2q  normal bundle ? , then we may c o n s i d e r how N i n t e r s e c t s i t s e l f . q  possible itself  (see above) t o change N by an e - i s o t o p y  transversally.  the sum running  with It i s  so t h a t i t i n t e r s e c t s  Then Theorem 10.1 g i v e s us:  i^[N]•i^[N]=Isgn(x),  over the p o i n t s o f s e l f - i n t e r s e c t i o n .  However, we c a n  a l s o i n t e r p r e t t h i s r e s u l t u s i n g the normal bundle £: 10.2  P r o p o s i t i o n : i ^ [ N ] • i ^ [ N ] = x ( ? ) [ N ] , where x ( ? ) 'is the E u l e r c l a s s o f We a r e now prepared  t o d e s c r i b e the c o n s t r u c t i o n known as plumbing  d i s c bundles. L e t t,. be a q-plane bundle over a smooth q-manifold  , and l e t E  be the t o t a l space o f the c l o s e d d i s c bundle a s s o c i a t e d  t o ?.. 1  Suppose t h a t  and  are oriented  compatibly f o r 1=1,2.  Choose x.eN. and B.cN. a q - c e l l w i t h x . e i n t B_, . Since B. 1 1 1 1 i i l is  c o n t r a c t i b l e , t,. B. i s t r i v i a l , i i  over B. i s d i f f e o m o r p h i c l  and t h a t p a r t of E, l y i n g i  t o B,xD., where D. i s a q - d i s c , i l l  t h a t the f i b r e s a r e mapped t o x*D^. h_,h :B ->D , +  1  such  We may choose d i f f e o m o r p h i s m s  k_,k :D!->B ,  2  +  2  where a s u b s c r i p t e d + i n d i c a t e s o r i e n t a t i o n - p r e s e r v i n g , and a indicates orientation-reversing. We plumb E]_ w i t h E the d i s j o i n t u n i o n E i u E  a t x^ and x  2  2  by i d e n t i f y i n g the s u b s e t s of  g i v e n by B^xD^ and B x D  2  2  I (x,.y) = (k y ,h x) or the map I_(x,y)=(k_y ,h_x) . +  +  the plumbing i s w i t h s i g n +1 i f .1 i s used.  The r e s u l t i n g m a n i f o l d  using  2  the map  We s h a l l say t h a t  i s used, and w i t h s i g n -1 i f I _  i s denoted by ET_DE , and i t can 2  be smoothed i n a c a n o n i c a l way. S i n c e b o t h of I reverse  and I  i t i f q i s odd, Ei_DE  N , and C  2  2  If 1  l  p r e s e r v e o r i e n t a t i o n i f q i s even, and 2  can be o r i e n t e d  compatibly w i t h  Nj.^i,  even, and w i t h Ni,r, ,-N , and c; i f q i s odd.  s  n  2  2  N o t e . t h a t N^cE^cE DE , where the i n c l u s i o n s a r e o b v i o u s , and that ,  1  N}nN ={x^}={x } 2  and  2  2  ( i n E ^ D E ) , which i s a t r a n s v e r s a l i n t e r s e c t i o n , 2  t h a t the s i g n o f x i s the same as the s i g n of the plumbing.  (Of c o u r s e , a l l o f t h i s d i s c u s s i o n can be a p p l i e d t o the case o f plumbing one m a n i f o l d in  with i t s e l f ,  i f we choose two d i s t i n c t  i t and take E}=E .) 2  I f we choose s e v e r a l p a i r s / o f p o i n t s Ei  points  and E  We w i l l  2  together  still  repeatedly,  in  and N , we may plumb 2  choosing the s i g n o f each plumbing.  denote the r e s u l t by E i D E , and we see from 10.1 t h a t 2  i-l * [Ni ] * i-z-k t^2 ] i s determined plumbings. n2 X  by the way we choose the s i g n of the  Thus, i f we choose a number n\2,  and plumb- E i w i t h E a t 2  p o i n t s , always w i t h s i g n +1, then we have i ^ [NT_ ] • ±2^[N  2  ]=nj2 •  We may go on t o plumb w i t h o t h e r d i s c b u n d l e s , by making s u r e the p o i n t s i n NT_UN  that  we choose t o plumb a t a r e w e l l away from t h e  2  f i n i t e number o f p o i n t s i n Ni_nN , and by c h o o s i n g the s i g n s of the 2  plumbings,  we may cause  i . [N. ] • i . [N. ] = n , j * k , t o take on any v a l u e M  *  we l i k e .  (Note  3  *  k  t h a t we must have n  = ( - l ) n . , .) Jk q  kj  are determined  jk  by the E u l e r c l a s s x ( £ ^ ) >  The s e l f - i n t e r s e c t i o n s  a c c o r d i n g t o P r o p o s i t i o n 10.2.  Thus, we a r r i v e a t t h e remarkable 10.3  Theorem: L e t M be a symmetric n><n m a t r i x w i t h i n t e g e r  entries, 4k  and w i t h even d i a g o n a l e n t r i e s .  Then f o r k > l t h e r e i s a m a n i f o l d W  w i t h boundary such t h a t W i s ( 2 k - l ) - c o n n e c t e d , 8W i s  (2k-2)-connected,  H ( W ) i s f r e e a b e l i a n , the m a t r i x of t h e i n t e r s e c t i o n 2k  * * 2 k  8  ^ 2 k — ^  bilinear  form 4k  f : (W, 3W)—KD  ^  S  S^-  Ven  pairing  by M ( o r e q u i v a l e n t l y , M i s t h e m a t r i x of t h e  (•,•) on H.(W,8W)), and t h e r e i s a normal map ( f , b ) , w i t h 4k-1 ,S  ) f o r which M i s the i n t e r s e c t i o n m a t r i x on ^^(W) .  The p r o o f i s p r o v i d e d i n d e t a i l i n [Browder 1972]. We have from 10.4 and  the same s o u r c e the  Lemma: I n the c o n s t r u c t i o n of 10.3, 3W i s a homotopy sphere i f o n l y i f the determinant  0  of M i s ±1.  2 1 C o n s i d e r the f o l l o w i n g 8x8 m a t r i x due t o H i r z e b r u c h : 1 2 1 1 2 1 1 2 1 M = 1 2 1 0 1 1 2 1 0 0 1 2 0 1 0 0 2 0  0  T h i s m a t r i x i s , as r e q u i r e d , symmetric and even on the d i a g o n a l . Simple  computation  shows t h a t |MQ|=1 and t h a t the s i g n a t u r e o f Mg i s 8.  We may q u i c k l y prove 10.5_  Theorem: L e t k > l .  f:(W,8W) =^(D :  4k  the f o l l o w i n g theorem o f M i l n o r . There i s a m a n i f o l d W and a normal map  (f,b),  4k-l j ,S ) such t h a t (f|3W) i s a homotopy e q u i v a l e n c e , and  a(f,b)=l. P r o o f : L e t W be the 4k-manifold w i t h boundary c o n s t r u c t e d i n Theorem 10.3 u s i n g the m a t r i x Mg.  S i n c e f M Q [ = 1 , we have by 10.4 t h a t 3W i s a 2k  homotopy sphere.  By 10.3, the b i l i n e a r  m a t r i x Mg, and sgn Mg=8. it  follows, that  form  (*,*) on K  (W,3W) has  Thus, i f ( f , b ) i s the normal map of 10.3,  a ( f ,b)=~rl (f )=isgn o  M =l.  QED  0  o  A somewhat.different  c o n s t r u c t i o n i n dimensions  congruent t o  2 mod 4 g i v e s us the f o l l o w i n g theorem o f K e r v a i r e . 10.6  Theorem: F o r q odd t h e r e i s a m a n i f o l d U and a normal map (g,c)  such t h a t g : ( U , 3 U ) — > ( D , S 2 q  2 q _ 1  ) with a(g,c)=l.  T a k i n g Theorems 10.5 and 10.6 t o g e t h e r w i t h P r o p o s i t i o n 5.35 (the A d d i t i o n P r o p e r t y o f a), we d e r i v e immediately 10.7  Theorem: I f m=2k>4, then t h e r e i s an m-manifold M w i t h boundary,  and a normal map ( g , c ) , g:(M,3M)—>-(D ,S t r i v i a l bundle a(g,c)  ; \1.  the Plumbing Theorem:  ) , c:v —>-e  (where e  i s the  over D ™ ) , w i t h g|9M a homotopy e q u i v a l e n c e and w i t h  t a k i n g on any d e s i r e d v a l u e .  The HomotoT-v Types of Sicocth x l & n i f o H n and Cl ass!i' cation.. :  It: has  si-iows by [Browdar 1-962]. a-i'': [Ncvikov 1964] •• t h * ,  - ': v.in. n ^ r . ^ ^ - c o n d j tions-.-k^s  ;  % space  totbe-J-off. t*:-  h-yjsotr,?^  t. f r  76 § 11.  The Homotopy Types o f Smooth M a n i f o l d s and C l a s s i f i c a t i o n .  I t has been shown by [Browder 1962]  and [Novikov 1964]  that  certain  n e c e s s a r y c o n d i t i o n s f o r a space t o be o f the homotopy type o f a smooth m a n i f o l d are sometimes a l s o  sufficient.  In the theorem we w i l l use the f o l l o w i n g n o t a t i o n : h:Tr.—>-H. i s the i  i  Hurewicz homomorphism, E i s an o r i e n t e d k-plane bundle over a space X, UeH  (T(£)) i s i t s Thorn c l a s s , p. are i t s P o n t r j a g i n c l a s s e s , and L,  i  are  k  the H i r z e b r u c h p o l y n o m i a l s . 11.1  Theorem: L e t X be a simply-connected P o i n c a r d complex o f dimension  m>5, E an o r i e n t e d k-plane bundle over X, k>m+l, aeir  h(a)nU=[X].  ,. (T(E ) ) such m+k.  that  I f (1) m i s odd, or (2) m=4k and index X = ( L ^ ( p i , P 2 , . . . , p ^ ) ) [ X ] ,  then t h e r e i s a homotopy e q u i v a l e n c e f :M—*-X, f o r some smooth m-manifold * m+k M, such t h a t v=f (E) i s the normal bundle o f an embedding McS , and f can be found i n the normal  cobordism c l a s s r e p r e s e n t e d by a .  O u t l i n e of Proof: A r e p r e s e n t a t i v e f : S manifold.M  f:M—*X, b:v—>E.  (4.2), (f,b)  t o a homotopy e q u i v a l e n c e i f and o n l y  But i f m=4k, then by the Index P r o p e r t y ( P r o p o s i t i o n 5.35),  (p ,...,p :  K.  ) ) [ X ] - i n d e x X, which i s zerotwhen  (2) h o l d s . QED  K.  Remark: I f m=6,14,30, o r 62 (none o f which are covered by 11.1), w i t h the above hypotheses f  via f  t o a homotopy e q u i v a l e n c e i f m i s odd, and i f m=2q  ( f , b ) i s n o r m a l l y cobordant  i f a(f,b)=0.  m + k  The map f induces a normal map ( f , b ) w i t h  Then by the Fundamental Theorem o f Surgery  i s n o r m a l l y cobordant  a(f,b)=(L  — > T ( E ) o f a i s chosen, and the  i s d e f i n e d by p u l l i n g X back t o a submanifold o f S  ( a f t e r some m o d i f i c a t i o n s ) .  then  m + k  then  t h e r e i s a homotopy e q u i v a l e n c e f:M—>X with"  (E)=v, b u t f may not be n o r m a l l y cobordant  t o a map r e p r e s e n t i n g a.  ,77 We have d e f i n e d ab ove for  the purpose  i  sum of P o i n c a r e  the A d d i t i o n Property.  of  k-plane bundles ( f , b ) such  the connected  complexes  Given P o i n c a r e p a i r s  over X_^, smooth m a n i f o l d s  (X_^,Y^),  (M^,3M_^), and normal maps  t h a t f : (M,SM^)—+ ( X , Y ) , we have the Poincar£ p a i r  i  ±  ±  ±  (Xi#X2JYIUYJ),  the smooth m a n i f o l d M^#M2 w i t h boundary 3 M T U 3 M , and  the normal map  (f j #f .bj #b ) such  2  2  and bj#b2 j~Ki#?2  » where  :v  2  x  2  i s the normal bundle  3MT_IJ 3M  2  )—»-(X! #X , Y T U Y 2  (components o f ) the boundary.  2  of MjZ/M^ i n D™"^.  and Y ^ a r e a l l nonempty, we may d e f i n e the connected  If along  t h a t f i #f : (M #M ,  2  sum  See [Browder 1972] f o r d e t a i l s .  We produce analogous c o n s t r u c t s : M^iLM , X]UX2, and maps f i J l f , b|JLb2 • 2  Note t h a t 3  2  3M} #8M , and t h a t .(XiliX  (MTOII^ ) =  2  2  JY\#Y2)  form a P o i n c a r g  pair.  Then (fjlLf2 ,b] JLb2) i s a normal map. r  11.2  P r o p o s i t i o n : L e t ( f , b ) , ( g , c ) be normal maps w i t h f : (M,3M)—>-(X,Y) ,  g: ( N , 3 N ) — K D , S m  m - 1  ) • . Then ( f i l g , b i l c ) i s n o r m a l l y  cobordant  to  (f,b).  T h i s p r o p o s i t i o n t o g e t h e r w i t h p r e v i o u s r e s u l t s l e a d s t o the 11.3  Theorem: L e t ( X , Y ) be a m-dimensional Poincar£ p a i r w i t h X and Y nonempty, mfc5, and l e t ( f , b ) be a normal map  simply-connecetd  w i t h f : (M,3M)-*(X,Y) and ( f | 3 M ) ^ an isomorphism. normal map such In  ( g , c ) , g:(U,3U)—»-(D ,S  t h a t (fjLg,hu.c) i s n o r m a l l y  particular,  ( f ,b) i s n o r m a l l y  Then t h e r e i s a  ) w i t h g|3U a homotopy e q u i v a l e n c e ,  cobordant cobordant  r e l Y to a homotopy e q u i v a l e n c e . to a homotopy e q u i v a l e n c e .  P r o o f : By the Plumbing Theorem (10.7) t h e r e i s a (g,c) as above w i t h a ( g , c ) = - c ( f ,b) .  By the A d d i t i o n P r o p e r t y , P r o p o s i t i o n 5.35, a ( f j i g , b j L c )  =a(f,b)+a(g,c)=0, s o - b y normally  (f,b)  r e l Y t o ( f ' , b ' ) , where f':M'—>X i s a homotopy  cobordant  equivalence.  (Note  i s normally  the Fundamental Theorem (4.2) (fjLg,b!Lc) i s  that ( X A D M , Y # S  cobordant  m _ 1  )=(X,Y)).  t o (f',b').  Then 11.2 shows t h a t QED  )  Recall 3McTJ,  that  a cobordism W between M and M' ( i . e . 9W=MuUuM',  3M'cTj) i s an h - c o b o r d i s m i f t h e i n c l u s i o n s McW, M'cW,  9McTJ,  9 M ' C T J a r e a l l homotopy e q u i v a l e n c e s .  and  With t h i s d e f i n i t i o n we can s t a t e of Novikov, 11.4  the c l a s s i f i c a t i o n theorem  and i t s c o r o l l a r y .  Theorem; L e t X be a simply-connected  dimension  P o i n c a r e complex o f  m>4, and (f-£> ^) f o r i = 0 , l , be normal maps w i t h f_^:M_j—->-X, D  where  i s a smooth m-manifold.  equivalences.  Suppose t h a t  I f f g i s n o r m a l l y cobordant  f g and f ^ a r e homotopy  t o f i , then t h e r e i s a  normal map (g,c) w i t h g:(U,9U)—>(T> ^~,S ) , where g| 9U i s a homotopy m+  e q u i v a l e n c e , such t h a t  t o (f\ g| 3 U , b i  (fg,bg) i s h-cobordant  In p a r t i c u l a r , Mg i s h-cobordant if  m  c|9U).  t o Mj_ i f m i s even, and t o M j # ( 9 U )  m i s odd.  11.5  Corollary:  L e t M and M' be c l o s e d  m a n i f o l d s o f dimension  not less  that  smooth  5.  simply-connected  A homotopy e q u i v a l e n c e  f :M—*-M' i s homotopic t o a d i f f eomorphism f':M#E—>-M' f o r some homotopy sphere E = 9 U , U p a r a l l e l i s a b l e (thus M i s homeomorphic to M#Z) i f and o n l y i f t h e r e i s a bundle map b:v—*v ' c o v e r i n g f such t h a t a  £  %  +  k  (  T  (  v  T ( b ) ^ ( a ) = a ' , where a,a' a r e the n a t u r a l )  )  >  a  '-m k  ( T ( V , ) )  +  collapsing  maps  -  F i n a l l y we have a theorem o f W a l l and i t s c o r o l l a r y . 11.6  Theorem; L e t (X,Y) be a P o i n c a r e p a i r o f dimension  both.X and Y s i m p l y - c o n n e c t e d , Y nonempty. (T(E) ,T(E IY))  over X, and choose aeir  m>6, w i t h  L e t E be a k-plane  such t h a t  h(a)nU=[X].  bundle Then  m+k the normal map r e p r e s e n t e d by a i s n o r m a l l y cobordant equivalence  ( f , b ) , f:(M,9M)—>(X,Y), which i s unique  In p a r t i c u l a r ,  t o a homotopy  up t o h-cobordism.  (X,Y) has the homotopy type 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 ,  79 unique up to h-cobordism i n the g i v e n normal cobordism  class.  We w i l l prove the e x i s t e n c e p a r t o f t h i s theorem. uniqueness  (as w e l l as the other p r o o f s omitted  to be found  The p r o o f o f  from t h i s s e c t i o n ) i s  i n [Browder 1972, I I . 3 ] .  P r o o f : L e t ( f ' , b ' ) w i t h f':(M ,9M')—>(X,Y) be a normal map r e p r e s e n t i n g r  a.  By the Cobordism P r o p e r t y , 5.36, a(f'|9M',b'|9M')=0,  Fundamental Theorem (4.2) (f'|9M',b'[9M') i s n o r m a l l y homotopy e q u i v a l e n c e . of  T h i s normal cobordism  ( f ' , b ' ) to some ( f " , b " ) such  so t h a t by the  cobordant  extends to a normal  t h a t f"|9M" i s a homotopy  By Theorem 11.3, ( f " , b " ) i s n o r m a l l y  to a  cobordant  cobordism  equivalence.  to a homotopy  equivalence,  (f,b). 11.7  C o r o l l a r y : L e t M and M' be compact smooth  manifolds nonempty.  of dimension  m>6,  simply-connected  w i t h 9M and 9M' simply-connected  Then a homotopy e q u i v a l e n c e f : (M, 9M)—• (M' , 9M ) 1  and i s isotopic  to a d i f f eomorphism f ' :M—>-M' i f and o n l y i f t h e r e i s a bundle map b:v—>-v' c o v e r i n g f such t h a t T ( b ) ^ ( a ) = a ' , where v ,v ' a r e the normal bundles,  and aeir  t1  m+k  c o l l a p s i n g maps.  (T(v) ,T(v I 9M) ) , a'eiT  m+k  (T (v ' ) ,T(v ' | 9M' ) ) a r e the  Bibliography Browder,W.: Homotopy of  the Aarhus  type of d i f f e r e n t i a b l e m a n i f o l d s .  Proceedings  Symposium, 1962, 42-46.  Surgery on simply-connected m a n i f o l d s . B e r l i n - H e i d e l b e r g - N e w York: S p r i n g e r 1972. H i r z e b r u c h , F . : New Ed.  t o p o l o g i c a l methods i n a l g e b r a i c geometry.  3rd  B e r l i n - H e i d e l b e r g - N e w York: S p r i n g e r 1966.  Husemoller,D.:  F i b r e bundles.  New York: McGraw H i l l 1966.  K e r v a i r e , M . : An i n t e r p r e t a t i o n of G. Whitehead's g e n e r a l i s a t i o n o f the  Hopf i n v a r i a n t .  Ann. Math. 69 (1959), 345-364.  — M i l n o r , J . : Groups of homotopy spheres I .  Ann. Math. 77  (1963),  504-537. M i l n o r , J . : On m a n i f o l d s homeomorphic to the 7-sphere.  Ann. Math.  64 (1956), 399-405. A procedure f o r k i l l i n g manifolds.  the homotopy groups of d i f f e r e n t i a b l e  Symposia i n Pure Math., Amer. Math. Soc. 3 (1961),  39-55. L e c t u r e s on the h-cobordism and J . Sondow.  theorem, notes by L. Siebenmann  P r i n c e t o n : U n i v e r s i t y P r e s s 1965.  Characteristic classes.  P r i n c e t o n : U n i v e r s i t y P r e s s 1974.  Morse,M. : R e l a t i o n s between the numbers of c r i t i c a l p o i n t s of a r e a l f u n c t i o n of n independent v a r i a b l e s .  T r a n s . Amer. Math. Soc.  27 (1925), 345-396. Novikov,S.P.:  Homotopy e q u i v a l e n t smooth m a n i f o l d s I .  AMS  Tranlations  48 (1965), 271-396. Reeb,G.: Sur c e r t a i n p r o p r i i t e s  t o p o l o g i q u e s des variet£s  A c t u a l , s c i . i n d u s t r . 1183, P a r i s , 1952, 91-154.  feuillet£es,  S e r r e , J . - P . : Homologie s i n g u l i e r e des espaces f i b r e s .  Applications,  Ann. Math. 54 (1951), 425-505. 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