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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 Univ 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 i n THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ' June 1980 .^Richard Bruce Cameron, 1980 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f _ T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Abstract The purpose of t h i s paper i s to present 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 manifolds. We begin with the Poincar£ conjecture and i t s p a r t i a l s o l u t i o n using the h-cobordism theorem. We review next the work of Kervaire and Milnor, concerned with the diffeomorphism classes of homotopy spheres. The surgery problem developed from t h e i r work, and we present i t s s o l u t i o n i n the simply-connected case, by Browder. This s o l u t i o n amounts to the surgery invariant theorem, the fundamental surgery theorem and associated r e s u l t s . We end our discussion with the plumbing theorem, and several important c l a s s i f i c a t i o n theorems of Browder, Novikov and Wall. i i i Table of Contents T i t l e page. i Abstract i i Chapter I. The Poincare Conjecture. §1. The Poincare Conjecture and the h-cobordism theorem. 1 §2. Exotic D i f f e r e n t i a l Structures on the 7-Sphere. 4 §3. Groups of Homotopy Spheres. 11 Chapter I I . The Fundamental Surgery Theorem. §4. The Surgery Problem. 15 §5. The Surgery Invariant. 17 §6. Surgery below the Middle Dimension. 33 §7. I n i t i a l Results i n the Middle Dimension. 45 §8. The Proof of the Fundamental Theorem for m odd. 53 §9. The Proof of the Fundamental Theorem for m even. 61 Chapter I I I . Plumbing and the C l a s s i f i c a t i o n of Manifolds. §10. Intersection and Plumbing. 71 §11. The Homotopy Types of Smooth Manifolds and C l a s s i f i c a t i o n 76 Bibliography 80 1 Chapter I. The Poincare Conjecture. § 1 . The Poincare Conjecture and the h-cobordism Theorem. The o r i g i n a l form of the Poincare conjecture was the following: 1 . 1 If M i s a closed 3-manifold such that ( M ) f i H A ( S 3 ) , ' " t h e n ' M = S 3 . This was shown to be f a l s e , through the following counter-example: The binary icosahedral group I i s defined by the generators A, B, ft and C, and r e l a t i o n s A3=B2=C5=ABC between them. I i s perfect, and i s ft ft a subgroup of S 3 . Define a closed 3-manifold M=S 3/I . Then TTI ( M)=I , ft and Hi ( M ^ T T J ( M ) = 1 = 1 . ' By Poincarg d u a l i t y , H 2 ( M ) = 1 . Thus, H J J . ( M)=H A(S 3) , but M i s not homeomorphic to S 3, because TTJ_ ( M)=I , whereas T T I ( S 3 ) = 1 . The f a i l u r e of the o r i g i n a l conjecture led to an amended formulation 1 . 2 If M i s a closed, simply-connected 3-manifold, then M - S 3 . Note that, by the Hurewicz isomorphism theorem, the Poincare d u a l i t y theorem, and the un i v e r s a l c o e f f i c i e n t theorem, the hypothesis that M i s simply-connected implies that i n fac t -n ( M ) - T T ^ (S 3) , and hence that M - S 3 . Although there have been p a r t i a l r e s u l t s concerning t h i s conjecture, i t has not yet been completely s e t t l e d . The Poincare conjecture can be extended to dimensions other than 3 : 1 . 3 If M i s a closed n-manifold which i s homotopically equivalent to S N , i t i s homeomorphic to S N . This statement has been proved for n ^ 3 , 4 . In f a c t , 1 . 3 can be stated i n an apparently weaker form which i s , by the Hurewicz isomorphism theorem, a c t u a l l y equivalent to 1 . 3 : 1 . 4 If M i s a closed, simply-connected n-manifold with the 2 i n t e g r a l homology of S n, then M i s homeomorphic to S n. We w i l l prove the generalized Poincare conjecture i n dimensions greater than 4 by means of the h-cobordism theorem. A smooth manifold t r i a d i s defined to be a t r i p l e (W;V,V')S where W i s a compact, smooth manifold, and the boundary of W i s the d i s j o i n t union of two open and closed submanifolds V and V'. 1.5 Theorem (h-cobordism theorem) : Suppose the t r i a d (W;V,V) has the properties: (1) W,V, and V' are simply-connected, (2) HA(W,V)=0, (3) dim W=n>6. Then W i s diffeomorphic to V x [ 0 , l ] . The following proposition i s c e n t r a l to the proof of the generalized conjecture: 1.6 Proposition: Suppose W i s a compact simply-connected smooth n-manifold, n>6, with a simply-connected boundary V. Then the following four assertions are equivalent: (1) W i s diffeomorphic to 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 point. Proof: I t i s clear that (l)->-(2)->-(3)->-(4) , so that we need only prove (4)-Kl). I f D 0 i s a smooth n-disc imbedded i n intW, then (W\intD 0,3D 0,V) s a t i s f i e s the conditions of the h-cobordism theorem. In p a r t i c u l a r , by excision (W\intD 0 , 3D0)=H^. (W,D0)=0. Since the cobordism (W;c(>,V) i s the composition of (Dg ; (J>, 9r/0) with a product cobordism (W\intD 0;9Dg,V), W i s homeomorphic to Dg. A theorem of Milnor shows that the composition preserves d i f f e r e n t i a b l e structures, so that W i s i n fac t diffeomorphic to Dg. QED 3 We are now ready to prove the generalized conjecture. Proof of 1.4: Case 1: n>5. If D Q £ M i s a smooth n-disc, then M\intDg s a t i s f i e s the hypothesis of 1.6\ In p a r t i c u l a r , H ( M \ i n t D 0 ) = H N _ : L ( M \ i n t D 0 ,3D0) by Poincare d u a l i t y i - H N ( M , D Q ) by excision ' 0 i f i>0 Hi i f i=0 by the exact cohomology sequence. Consequently, M = ( M \ i n t D g ) U D Q i s diffeomorphic to a union of two copies D ^ , D ^ of the n-disc with the boundaries i d e n t i f i e d under a diffeomorphism I K S D ^ - ^ D ^ . Such a manifold i s c a l l e d a twisted sphere. The proof i s completed by showing that any twisted sphere M = D^u, D ^ J i s homeomorphic 1 h 2 to S n. Let S i ^ D ^ S 1 1 be an embedding onto the southern hemisphere of S n £ & n + 1 . (I.e. the set {x| ||x|=l, x n + 1 - ° } - ) E a c t l P°int of fi^ may be written tv, 0<t<l, veSD^. Define g : M+S n by g(u)=g 1(u) for ueD^, g(tv)= sinC^g^^Ch 1 ( v ) ) + c o s ( ^ | ) e n + 1 , where e n + 1 = ( ° »• • • ,0',l)'eff n + 1, f o r a l l points tveD^'. Then g i s a well-defined i n f e c t i v e continuous map onto S n, and i s hence a homeomorphism. This completes the proof for case 1. Case 2: n=5. We use here: 1.7 Theorem: Suppose i s a closed, simply-connected smooth manifold with the homology of S n. 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 manifold. Thus, 1.7 and 1.6 imply "tha't"MS""bb'unds a manifold homeomorphic to D 6 , so that M 5 i s homeomorphic to S 5. Remark: The generalized conjecture holds i n dimensions 1 and 2 as w e l l . The proof i s t r i v i a l , because of the well-known c l a s s i f i c a t i o n of 1- and 2-manifolds. 4 By using 1.7 and 1.6 one can show that i n f a c t a simply-connected homology n-sphere i s diffeomorphic to S n, for n=5,6. However, Milnor has proved that t h i s i s not true for n=7. The next section w i l l be devoted to an examination of t h i s r e s u l t . § 2. Exotic D i f f e r e n t i a l ' Structures on the 7-Sphere. The inva r i a n t X(M7) For every closed oriented smooth 7-manifold s a t i s f y i n g the hypothesis 2.1 H3(M)=Htt(M)=0 we w i l l define a residue class X(M) modulo 7. According to Thorn every closed smooth 7-manifold M i s the boundary of a smooth 8-manifold,B. The inv a r i a n t X(M) w i l l be defined as a function of the index T and the Pontrjagin class p.]_ of B 8 . If yeHy(M 7) i s the distinguished generator, then an o r i e n t a t i o n veH 8(B 8,M 7) i s determined by the r e l a t i o n s h i p 8v=u. Define a quadratic form over the group H^(B 8,M 7)/torsion by the formula a-><v,a2>. Let T ( B 8 ) be the index of t h i s form (the number of p o s i t i v e terms minus the number of negative terms when the form i s diagonalized over R). Let p eH^CB 8) be the f i r s t Pontrjagin class of the tangent bundle of B 8. (For the d e f i n i t i o n of Pontrjagin classes, see [Milnor 1974].) The hypothesis 2.1 (together with the long cohomology sequence of the p a i r (B 8,M 7)) implies that the i n c l u s i o n homomorphism i i H ^ ( B 8 j M 7 ) - ^ (B 8) i s an isomorphism. Therefore, we can define a 'Pontrjagin number' q( B 8 ) = < v , ( x _ 1 P l ) 2 > . 2.2 Theorem; The residue class of 2q(B 8)—r(B 8) modulo 7 does not depend on the choice of the manifold B 8. Define A(M 7) as t h i s residue c l a s s . As an immediate consequence, we have: 2 . 3 C o r o l l a r y : I f A(M7)*0 then M i s not the boundary of an 8-manifold with fourth B e t t i number zero. Proof of Theorem ' 2 . 2 : Let Bf , B | be manifolds both having boundary M7. (We may assume they are d i s j o i n t . ) Then C 8=B^U M7B 8 i s a closed 8-manifo which possesses a d i f f e r e n t i a b l e structure compatible with that of Bf and B 8 . Choose that o r i e n t a t i o n v for 6 8.which i s consistent with the o r i e n t a t i o n vj_ of B 8 (and therefore consistent with - V 2 ) . Let q(C 8) denote the Pontrjagin number <v,p 2(C 8)>. According to [Thorn 1954] we have T(C 8)=<v,^-(7p 2(C 8)-p 2(C 8)>, and therefore 45T(C 8)+q(C 8)=7<v,p 2(C 8)>=0 (mod 7) This implies (1) 2 q ( C 8 ) - T ( C 8 ) E 0 (mod 7) 2.4 Lemma: Under the above conditions we have ( 2 ) T ( C 8 ) = T ( B 8 ) - T ( B 8 ) , and ( 3 ) q(C 8)=q(B 8)-q(B 8). Formulae ( 1 ) , ( 2 ) , and ( 3 ) c l e a r l y imply that 2 q ( B 8 ) - T ( B 8 ) E 2 q ( B 8 ) - T ( B 8 ) , (mod 7) which i s j u s t the statement of the theorem. Proof of Lemma 2.4: Consider the diagram: H n(B 1,M)eH n(B 2,M) «—|L_ Hn(C,M) ± l ® ± 2 j H n ( B 1 ) ® H n ( B 2 ) < H n(C) Note that for n=4 these homomorphisms are a l l isomorphisms. 6 If a=jh 1 ( c q S K^XeH 1* (C) , then (4) <v,a2>=<v,jh 1(a 2ea 2)>=<v 1®(-v 2),a 2®a|>=<v 1,a 2>-<v 2,a 2> Thus the quadratic form of C i s the 'direct sum' of the quadratic form:j of B\ and the negative of the quadratic form of B 2. This c l e a r l y implies formula (2). Define a j = i i 1 ( B ^ ) and a 2 = i 2 1 p 1 ( B 2 ) . Then the r e l a t i o n k(Pl (C))=Pi (B^ )©Pi (B 2) implies that j h 1(a^$a 2)=Pi(C). The computation (4) now shows that <v,p 2(C)>=<v 1,a 2>-<v 2,a 2>, which i s j u s t formula (3). This completes the proof of the lemma and of the theorem. The following property of the invariant A i s c l e a r : 2.5 Lemma: If the o r i e n t a t i o n of M i s reversed, 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 7)*0 then M7 possesses no orientation-reversing diffeomorphism onto i t s e l f . A p a r t i a l c h a r a c t e r i s a t i o n of the n-sphere Consider the following hypothesis concerning a closed mani f o l d M11: 2.7 There e x i s t s a d i f f e r e n t i a b l e function f:M>R having only two c r i t i c a l points X Q . Furthermore, these c r i t i c a l points are non-degenerate. (That i s , i f uj,...,u are l o c a l coordinates i n a neighbourhood of x 0 (or X } ) then the matrix (32f/9u^8u^.) i s nonsingular at X Q (or x^)) . 2.8 Theorem: I f M11 sa t i s f i e s hypothesis 2.7 then there ex i s t s a homeomorphism of M onto S n which i s a diffeomorphism except possibly 7 at a si n g l e point. Proof: This r e s u l t i s e n t i r e l y due to [Reeb 1952]. The proof w i l l be based on the orthogonal t r a j e c t o r i e s of the manifolds f=constant. Normalise the function so that f( X Q)=0,f(x^)=1. According to [Morse 1925, Lemma 4] there e x i s t l o c a l coordinates v i ,. . . ,v i n a neighbourhood V/"of xn so that f(x )=v i+...+v for xeV. n n (Morse assumes that f i s of class C 3, and constructs coordinates of class C , but the same proof works i n the C case.) The expression ds 2=dv? + ..,+dv2 defines a Riemannian metric i n the neighbourhood V. i n Choose a d i f f e r e n t i a b l e Riemannian metric f o r M* which coincides with t h i s one i n some neighbourhood V' of xg. (This i s possible by [Steenrod 1951, 6.7 and 12.2].) Now the gradient of f can be considered as a contravariant vector f i e l d . Following Morse we consider the d i f f e r e n t i a l equation •^r= grad f/|grad f | 2 . In the neighbourhood V' this equation has solutions (vi ( t ) , . . . , v (t)) = (ai St,...,a St) for 0<t<e , n n where a=(a^,..., a^eH 1 1 i s any n-tuple with Za 2=l. These can be extended uniquely to solutions x (t) for O ^ t ^ l . Note that these 3. solutions s a t i s f y the i d e n t i t y f ( x (t))=t. Map the i n t e r i o r of the unit sphere of i ? n into M*1 by the map ( a - i / t , . . . ,a St)—>x (t) . n a It i s e a s i l y v e r i f i e d that t h i s defines a diffeomorphism of the open n - c e l l onto M\{xi). The assertion of the theorem now follows. Given any diffeomorphism g:S n ^ —>-Sn \ an n-manifold can be obtained as follows. 2.9 Construction: Let M^g) be the manifold obtained from two copies of i? n by matching the subsets i?n\{0} under the diffeomorphism (Such a manifold i s c l e a r l y homeomorphic to S n. If g i s the i d e n t i t y map, then M^g) i s diffeomorphic to S .) 2.10 C o r o l l a r y : A manifold M n can be obtained by the construction 2.9 i f and only i f i t s a t i s f i e s the hypothesis 2.7. Proof: If M n(g) i s obtained by the construction 2.9, then the function . F ( x > = . l " l , v - 1 ... . n x ; (1+ u z ) (1+ v z ) w i l l s a t i s f y the hypothesis 2.7. The converse can be established by a s l i g h t modification of the proof of theorem 2.8. Examples of 7-manifolds Consider 3-sphere bundles over the 4-sphere, with the r o t a t i o n group S0(4) as s t r u c t u r a l group. The equivalence classes of such bundles are i n one-to-one correspondence (by [Steenrod, 1951, §18]) with the elements of the group ^3(S0(4))=Z®Z. A s p e c i f i c isomorphism between the groups i s obtained as follows. For each (h,j)eZ©Z, l e t f ^ : S — y S0(4) be defined by (u) • v=u^« v u ^ , for veR . (Quaternion m u l t i p l i c a t i o n i s understood on the r i g h t of the equation.) Let i be the standard generator for ^ ( S 1 * ) . Let E, . be the sphere bundle corresponding to [f^.. ] £ ^ 3 (SO (4)) . 2.11 Lemma: The Pontrjagin class p^(E ) equals ±2(h-j)i. (The proof w i l l be given l a t e r . One can show that the c h a r a c t e r i s t i c class ca h j) (see [Steenrod 1951]) i s equal to ( h + J ) x ) For each odd integer k l e t M7^ be the t o t a l space of the bundle E^j > where h and j are determined by the equations h+j=l, h-j=k. This manifold has a natural d i f f e r e n t i a b l e structure and o r i e n t a t i o n , which w i l l be described l a t e r . 9 2.12 Lemma: The invar i a n t X (M7^) i s the residue class modulo 7 of k 2 - l . 2.13 Lemma: The manifold s a t i s f i e s the hypothesis 2.7. Combining these we have: 2.14 Theorem: For k 2 ^ l mod 7 the manifold M7' i s homeomorphic, but not diffeomorphic, to S 7. (For k=±l the manifold M 7 i s diffeomorphic to S 7, but i t i s not known whether t h i s i s true for any other k with k 2 = l mod 7.) Cle a r l y any d i f f e r e n t i a b l e structure on S 7 can be extended throughout i? 8\{0}. However: 2.15 C o r o l l a r y : There ex i s t s a d i f f e r e n t i a b l e structure on which g cannot be extended throughout 'R . This follows immediately from the preceding assertions, together with c o r o l l a r y 2.3. Proof of Lemma 2.11: I t i s clear that the Pontrjagin class P l ^ ^ j ) i s a l i n e a r function of h and j . Furthermore i t i s known to be independent of the or i e n t a t i o n of the f i b r e . But i f the o r i e n t a t i o n of S 3 i s reversed, then E, . i s replaced by £ . , . This shows that hj * ' - j , - h Pl( 5 ^ j ) i s given by an expression of the form c ( h - j ) i . Here c i s a constant which w i l l be evaluated l a t e r . Proof of Lemma 2.12: Associated with each 3-sphere bundle —vS1* there i s a 4 - c e l l bundle p, :B8—>S h. The t o t a l space B 8 of th i s k k k bundle i s a d i f f e r e n t i a b l e manifold with boundary M . The cohomology group H^CB8) i s generated by the element ct=p, ( i ) . Choose orientations y,v for M 7 and B 8 so that <v,(i _ 1a) 2>=+l. Then the index T(B 8) w i l l be 1. g The tangent bundle of B^ i s the Whitney sum of (1) the bundle of vectors tangent to the f i b r e , and (2) the bundle of vectors normal to the f i b r e . The f i r s t bundle (1) i s induced (under p^) from the bundle 10 E^ _. > a n { i therefore has Pontrjagin class p^=p^(c (h-j ) i )=cka. The second i s induced from the tangent bundle of S 4, and therefore has f i r s t Pontrjagin class zero. Now by the Whitney product theorem: p x(B 8)=cka+0. 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 quaternion p r o j e c t i v e plane QP with an 8 - c e l l removed. But the Pontrjagin class p^(QP 2) i s known to be twice a generator of H^CQP2). Therefore the constant c must be ±2, which completes the proof of 2.11. Now q(B 8)=<v,(i _ 1(±2ka)) 2>=4k 2, and 2q-x=8k 2-l=k 2-l (mod 7). K. This completes the proof of Lemma 2.12. Proof of Lemma 2.13: As coordinate neighbourhoods i n the base space take the complement of the north pole, and the complement of the south pole. These can be i d e n t i f i e d with the Euclidean space B.^ under stereographic p r o j e c t i o n . Then a point which corresponds to ueff.^ under one pr o j e c t i o n w i l l correspond to u' =— p | ^ - p r under the other. The t o t a l space M7 can now be obtained as follows (cf. [Steenrod 1951 §18]). Take two copies of RkxS3 and i d e n t i f y the subsets (ff' +\{0})xS 3 by h J u u vu r^ F v'~nrr , (using quaternion the d i f f eomorphism (u,v)—>-(u',v') = m u l t i p l i c a t i o n ) . This makes the d i f f e r e n t i a b l e structure of M7 precise. Replace the coordinates (u',v') by (u",v"), where u"=u'(v') Consider the function f :M7—« defined by f ( x ) = — R ^ U — , where R(v) denotes the r e a l part of the quaternion v. I t i s e a s i l y v e r i f i e d that f has only two c r i t i c a l points (namely (u,v)=(0,+1)) and that these are non-degenerate. This completes the proof of Lemma 2.13. 11 §3. Groups of Homotopy Spheres. The following r e s u l t s about homotopy n-spheres are proved i n [Kervaire, Milnor 1963]: (1) The h-cobordism classes of homotopy n-spheres form an abelian group 9^ under the connected sum operation. (2) The h-cobordism classes of homotopy n-spheres which bound p a r a l l e l i s a b l e manifolds form a subgroup bP , n of 9 . (This n+1 n w i l l be proved below.) (3) The quotient group ® n/bP n +-^ isomorphic to a subgroup of the cokernel of the Hopf-Whitehead homomorphism J (where V J :TT (SO, )—>TT (S ) ), and i s f i n i t e , n n k n+k (4) The group b P n + ^ i - s f i n i t e , for n*3. (In p a r t i c u l a r , i t i s zero for n even, and f i n i t e c y c l i c for n odd, n*3.) (5) Thus, the group 0 of (h-cobordism classes of) homotopy n n-spheres i s f i n i t e , for n*3. We r e c a l l from above that every homotopy n-sphere, n*3,4, i s homeomorphic to S n. [Smale 1962] has shown that two homotopy n—spheres, n*3,4, are h-cobordant i f and only i f they are diffeomorphic. Thus (for n^3,4 at least) the group 0 can be described as the set of n diffeomorphism classes of d i f f e r e n t i a b l e structures on S n, and the l a s t r e s u l t above can be interpreted as s t a t i n g that there are only f i n i t e l y many e s s e n t i a l l y d i f f e r e n t such structures, for each n, n*3,4. We w i l l now prove assertions (2) and (3) above. Let M be an s - p a r a l l e l i s a b l e closed n-manifold. (I.e. T^Oe 1 i s t r i v i a l , where i s the tangent bundle of M, and e 1 i s the t r i v i a l n+k l i n e bundle.) Choose an embedding i:M—>S , with k>n+l. Such an 12 embedding ex 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. n+k 3.1 Lemma (Kervaire, M i l n o r ) : An n-:dimensional submanifold of S , n<k, i s s - p a r a l l e l i s a b l e i f and only 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 . Let d> be a t r i v i a l i s a t i o n of v w . Then the M M n+k k Pontrjagin-Thom construction y i e l d s a map p(M,cj>):S —>-S . The homotopy class of p(M,cf)) i s a well-defined element of the stable k homotopy group n ^ i r + ^ ( S ). Allowing the t r i v i a l i s a t i o n to vary, we obtain a set p (M)={p (M,<j>) }cn . n 3.2 Lemma: p(M)<fI contains the zero of II i f and only i f M bounds n n a p a r a l l e l i s a b l e manifold. Proof: <=. If M=8W and W i s a p a r a l l e l i s a b l e manifold, then, because n+k of dimensional considerations, the embedding i:M—>S can be extended to an embedding of W into D11"1^"1"1, and W w i l l have t r i v i a l normal bundle. Choose a t r i v i a l i s a t i o n ip of v and l e t <J>=IJJ|M. The Pontr jagin-Thom map w n+k k n+k+1 p(M,<f>):S —>-S extends over D , and hence i s null-homotopic. /w , \ ^ , -r, „n+k+l_„n+k r„ ,, ,„n+k , _,k =>-. If p(M,cJ>) = 0, we have a map F:D =S x[0,l]/S xl—^-S which s a t i s f i e s F| Sn+^x0=p(M,<j>) , and F | s n + ^ x l = e A , the constant map. n+k F can be made regular at * (the base p o i n t ) , r e l a t i v e to S xO, so we s h a l l assume, without loss of generality, that i t i s . Then F 1 (*)£Dn+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 ^ on W. By Lemma 3.1 above and the following lemma, W i s p a r a l l e l i s a b l e . 3.3 Lemma: A connected manifold with non-vacuous boundary i s s - p a r a l l e l i s a b l e i f and only i f i t i s p a r a l l e l i s a b l e . [Kervaire, Milnor] This completes the proof of Lemma 3.2. 3.4 Lemma: If Mg i s h-cobordant to MT_ , then p(Mg)=p(M^). n+k Proof: If M0+(-M1)=9W, choose an embedding of W i n S x[0,l] such that n+k M — x q for q=0,l. Then a t r i v i a l i s a t i o n d> of v., extends to a q M ^ ' q 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, on M. Y l - q 1-q Cle a r l y ( W , I J J ) gives r i s e to a homotopy between p(Mg,<|>o) and p(M^ , ( j)l). 3.5 Lemma: If M and M' are s - p a r a l l e l i s a b l e then p (M)+p (M' ) <=p (M#M1) c n . Proof: Construct a manifold W with boundary (-M)u(-M')u(M#M') as follows beginning with Mx[0,l]uM'x[0,1], j o i n the boundary components M*l and M'xl by a smooth connected sum.. This sum. can be extended smoothly over neighbourhoods of the joined portions, i n Mx[0,l] and M'x[0,l]. (The d e t a i l s of t h i s construction are given i n [Kervaire, Milnor 1963].) The manifold W has the homotopy type of the one-point union MvM'. n+k Embed.W i n S x[0,l] such that (-M) and (-M') are mapped into well-separated submanifolds of S n + ^ x 0 , and such that the image of M#M' n+k l i e s i n S x l . Given t r i v i a l i s a t i o n s $ and <j>' of the normal bundles of (-M) and (-M'), i t i s not. hard to see that there exists an extension defined throughout W . Let X/J denote the r e s t r i c t i o n to M#M' of t h i s extension. Then c l e a r l y p (M,<j))+p (M' ,<j>' ) i s homotopic to p(M#M ' , i(j) . This completes the proof. 3.6 Lemma: The set p(S n)cn i s a subgroup of the stable homotopy group n 11^. For any homotopy sphere Z the set p(Z) i s a coset of t h i s subgroup p ( S n ) . Thus the correspondence E—>-p(Z) defines a homomorphism p' from I I to the quotient group I I / p ( S n ) . n n Proof: Combining the previous lemma with the i d e n t i t i e s (1) S N # S n s S n (2) Sn//ZSZ (3) Z#(-Z)-S n, we obtain (1) p(S n)+p ( S n)<=p ( S N ) , which shows that p(S n) i s a subgroup of I I , (2) p(S n)+p(Z)cp ( z), which shows that p(Z) i s a union of cosets of t h i s subgroup, and (3) p(E)+p(-Z)cp ( s n ) , which shows that p(Z) must be a s i n g l e coset .14 This completes the proof of Lemma 3.6. By Lemma 3.2 the kernel of p':0 —HI /p(S n) consists exactly of a l l n n h-cobordism classes of homotopy n-spheres which bound p a r a l l e l i s a b l e manifolds. Thus, these elements form a group which we denote by bP , c0 . n+1 n It follows that bP^ +^ i s isomorphic to a subgroup of n^/ptS 1 1). Since i s f i n i t e [Serre 1951], t h i s completes the proof of assertions (2) and ,{3) . (The r e l a t i o n s h i p with the Hopf-Whitehead homomorphism, mentioned i n assertion (3), i s established i n [Kervaire 1959, p.349].) 15 Chapter I I . The Fundamental Theorem of Surgery. §4. The Surgery Problem. The technique of surgery, which Kervaire and Milnor used to obtain th e i r r e s u l t s on homotopy spheres, discussed, above, was also a key element i n Browder's s o l u t i o n of the surgery problem,(which was based on work by Kervaire/Milnor, and Novikov). Very informally, t h i s problem can be stated as follows: Given a map f:M—>X between manifolds, when can f and M be modified to f and M' such that f' :M'—>-X i s a homotopy equivalence? To state a more precise version of t h i s problem, we s h a l l f i r s t need a few d e f i n i t i o n s . A Poincare p a i r (X,Y) of dimension m i s a p a i r of CW complexes such that there i s an element [X]eH (X,Y) of i n f i n i t e order for which m [X]n ; :H^(X)—H H m (X,Y) i s an isomorphism for a l l q. This property i s c a l l e d Poincare d u a l i t y , and [X] i s c a l l e d the o r i e n t a t i o n class of (X,Y). Let (X,Y) be a Poincare p a i r of dimension m (Y may be empty), (M,9M) a smooth compact oriented m-manifold with boundary, and f:(M,3M)—*-(X,Y) a map. A cobordism of f i s a pair (W,F) where W i s a smooth compact (m+l)-manif old, 9w=MuU"muM'm, 9U=9Mu9M', F: (W,U)—»-(X,Y) , and F|M=f. If U=9MxI and F(x,t)=f(x) for xe9M, t e l , then (W,F) w i l l be c a l l e d a cobordism of f r e l Y. Let us assume that k » m and that (M,9M) i s embedded i n (D m +^, S m + ^ "*") k i with normal bundle v , so that v|9M i s equal to the normal bundle of 9M m4-k—1 k i n S . Let E be a k-plane bundle over X. A normal map i s a map k k f:(M,9M)—>-(X,Y) of degree 1 together with a bundle map b:v —>-E covering f. A normal cobordism (W,F,B) of (f,b) i s a cobordism (W,F) of f, 16 k k k together with an extension -B-:CJ —»-E of b, where m i s the normal bundle of i n D m + k x l , where the embedding i s such that (M,3M)c(D m + kxO,S m + k~ 1xO), / w i /^ m +k T „m+k-l , , T T m+k-1 T (M',9M')c(D x l , s xl) and UcS x i . A normal cobordism r e l Y i s a cobordism r e l Y such that i t i s a normal cobordism and B(v,t)=b(v) for vev|3M, tel. The precise version of the surgery problem i s : k k .Problem:Given a normal map ( f , b ) , f:(M,3M)—KX,Y) > b:v —»-E , when i s (f,b) normally cobordant to a homotopy equivalence of pairs? A r e l a t e d question i s the Restricted Problem: Given a normal map ( f , b ) , f : (M, 3M)—> (X,Y) , b : v — , when i s (f,b) normally cobordant r e l Y to ( f ' , b ' ) , where f':M'—>-X i s a homotopy equivalence? The s o l u t i o n to the r e s t r i c t e d problem i s given by the following two theorems: 4.1 The Invariant Theorem: Let (f,b) be a normal map, as above, such that f| cM induces an isomorphism i n homology. Then there i s an invar i a n t a(f,b) defined, a=0 i f m i s odd, ae i f m=0 (mod 4) and oe i f m=2 (mod 4) such that a(f,b)=0 i f (f,b) i s normally cobordant to a map inducing a homology isomorphism. . f f i ^ r •-The Fundamental/Surgery Theorem: Let (f,b) be a normal map, as above, and suppose (1) f|3M induces an isomorphism i n homology, (2) X i s simply-connected, and (3) m>5. If m i s odd then (f ,b) i s normally cobordant r e l Y to a homotopy equivalence f':M'—>X. I f m i s even, then (f,b) i s normally cobordant. r e l Y to (f',b') such that f' :M'—>-X i s a homotopy equivalence if and only i f a(f,b)=0. Our discussion of surgery follows very c l o s e l y the treatment of [Browder 1972], and consists of the d e f i n i t i o n of the inv a r i a n t a, the statement and proof of c e r t a i n properties i t has, the proof of the Invariant and Fundamental theorems, and .the. statement of c e r t a i n consequences of the Fundamental, theorem, p a r t i c u l a r l y the technique of plumbing and the Plumbing Theorem. F i n a l l y we w i l l use the l a t t e r to derive some c l a s s i f i c a t i o n r e s u l t s for manifolds. §5. The Surgery Invariant. Before d e f i n i n g a we s h a l l r e c a l l some pertinent facts about quadratic and b i l i n e a r forms over Z and Z 2 . 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) and (2) (Xx+X'x',y)=X(x,y)+X 1(x',y) for X,A'eZ, x,x',yeV. If{bA i s a basis for V and a_^ = (b^ ,b^) , then the matrix A=(a ) represents (•,•) i n the sense that (x,y)=xAy t (where x and y on the r i g h t are representations of the elements i n the basis {b^}). I f we pass to a new basis by an i n v e r t i b l e matrix M, so that b'=Mb, l&hen i n terms of the new basis (•»•) is. represented by MAMt. The b i l i n e a r form (•,•) defines a quadratic form q:V—>Z by q(x) = (x,x). We have (x,y)=i-(q(x4y)-q(x)-q(y)) so that (•,•) i s derivable from q. Each of q and (•,•) i s said to be associated to the other. The form (•,•) also defines n a t u r a l l y a b i l i n e a r form (? , •_) :VxQ—>-Q. 5.1 Proposition: I f (•»•) i s a symmetric b i l i n e a r form on a f i n i t e dimensional vector space V over Q into Q, then there i s a basis for V such that the matrix representing (*,•) i n that basis i s diagonal. Define the signature of a b i l i n e a r form (and hence of the associated quadratic form) to be the number of p o s i t i v e diagonal entries minus the number of negative diagonal e n t r i e s , using a diagonal matrix representing the form. The signature i s , i n f a c t , invariant under a change of ba s i s , and we s h a l l think of i t as an invariant of quadratic forms over Z, taking values i n Z. A quadratic (or b i l i n e a r ) form over Z i s c a l l e d nonsingular i f the determinant of the matrix A representing i t i s 1. Over a f i e l d i t i s c a l l e d nonsingular i f the determinant i s nonzero. 5.2 Proposition: Let q be a nonsingular quadratic form on a f i n i t e dimensional vector space V over R. Then sgn(q)=0 i f and only i f there i s a subspace UcV such that: (1) dimDU=^dim0V and (2) (x,y)=0 f or x,yeU. Some r e s u l t s we w i l l use follow. 5.3 Proposition: Let q be a nonsingular quadratic form V—>-Z and suppose q i s i n d e f i n i t e ( i . e . neither p o s i t i v e nor negative d e f i n i t e ) . Then there i s xeV, x*0 such that q(x)=0. 5.4 Proposition: Let q be a nonsingular quadratic form V and suppose 2|q(x,x) for a l l xeV (q i s c a l l e d even). Then 8|sgn(q). A quadratic form q on a Z 2-vector space V i s a function q:V—>-Z2 such that 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 are equivalent i f there i s an automorphism a:V—»V such that q=q'°a. Under., this d e f i n i t i o n , i t i s clear that (x,y) = (y,x) and (x,x)=q(2x)-2q(x)=0 so that (*,•) i s a symplectic b i l i n e a r form. I f (•,•) i s nonsingular, i t follows that V i s of even dimension, and that we may f i n d a basis {a.,b.} for V such that (a.,b.)=6,., . i i l j ±3 (a_^ ,a. ) = (b^ ,b_. )=0 . Such a basis i s c a l l e d symplectic. We s h a l l now c l a s s i f y Z 2-vector spaces with nonsingular quadratic forms, and thereby define the Arf invar i a n t of such forms. Let U be the 2-dimensional Z 2-vector space, with basis a,b, such that (a,a)=(b,b)=0, (a,b)=l. There are two quadratic forms on U compatible with (•,•): qo a n d q i > defined by (a)=qi(b)=l, qo(a)= qo(b)=0. Note that f or both q i(a+b)=l. (The notations U,qo and qi w i l l remain f i x e d throughout § 5.) 5.5 Lemma: Any nonsingular quadratic form on a 2-dimensional Z 2-vector space i s equivalent to qg or q^. Since such a space has only 4 elements, the isomorphism i s easy to construct. If q and q' are quadratic forms on spaces V and V', then q®q' i s the quadratic form on VOV' given by (q®q')(v,v')=q(v)+q'(v 1). 5.6 Lemma: On U9U, qo^qo i s isomorphic to q ^ q j . The proof consists of a simple rearrangement of bases. Now we can begin c l a s s i f y i n g forms. 5.7 Proposition : A nonsingular quadratic form q on a Z 2-vector space (which must have even dimension 2m) i s equivalent either to „, m-1 . m qi$($ q 0) or to 9 q 0 . Proof: Let {a_^ ,b_^ }, i=l,...,m be a symplectic basis of V, and l e t be the subspace spanned by a^»b^> and l e t ^ =q|v^. Then by the nature m of the basis, q ^ ^ j and by Lemma 5.5 ^  i s equivalent to qo or q^ . By Lemma 5.6 qi$qi-qo®qo> s o q I s equivalent to either © mqo or TU~~1 qi®(® q0.) • QED To complete the c l a s s i f i c a t i o n , we must show that <f>o=$mqo I s not equivalent to <}>i=q;L®(em "*"qo)- This i s clear from the 5.8 Proposition : The quadratic form cf^  on V sends a majority of elements of V to l e Z 2 , while §Q sends a majority of elements to 0eZ 2. The proof i s by induction on the dimension of V. Using t h i s notation, we define the Arf. inva r i a n t of a nonsingular quadratic form q on V as follows: 20 Arf(q)= r 0 i f q=<t>o { 1 i f q=<h Thus we have: 5.9 Theorem:(Arf) Two nonsingular quadratic forms on a f i n i t e dimensional Z 2-vector space are equivalent i f and only i f they have the same Arf inv a r i a n t . In analogy with a previous r e s u l t concerning quadratic forms over Z, we have the 5.10 Proposition: Let q be a nonsingular quadratic form on the Z 2-vector space V. Then Arf(q)=0 i f and only i f there i s a subspace UcV such that (1) rank„ U=^rank„ V, and (2) q(x)=0 for a l l xeU. z 2 z z<2 Given a b i l i n e a r form (•,*) on a vector space V, define R, the r a d i c a l of V, to be {xev| (x,y)=0 f o r a l l yeV}. If q:V—>-Z2 i s a quadratic form with (•,•) as associated b i l i n e a r form, we have defined Arf(q) only i f R=0. If q|R50, i t i s e a s i l y seen that q defines q' on V/R, and the r a d i c a l of V/R i s zero. In t h i s case we may define Arf(q) to be A r f ( q ' ) . If q|R^0, i t doesn't make sense to define the Arf in v a r i a n t , and i n fac t the equivalence of the form i s determined by rankV and rankR. Thus we have: 5.11 Theorem: Let q:V—>-Z2 be a quadratic form over Z 2 , R the r a d i c a l of the associated b i l i n e a r form. Then the Arf invar i a n t Arf(q) i s defined i f and only i f q|R=0. In general, i f q|RsO, then q i s determined up to isomorphism by rank V, rank„ R, and A r f ( q ) , while i f qlRiO, Z 2 ^2 then q i s determined by rank„ V and rank R. Zi 2 ^2 Note: Browder uses the notation c(q) for the Arf in v a r i a n t . We w i l l now define an invariant I which detects maps i n the 21 cobordism class of a homology isomorphism. A map f:(X,Y)—>-(A,B) between Poincare pai r s of the same dimension i s said to 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 A , and s i m i l a r notation i n cohomology. 5.12 Theorem: Maps of degree 1 s p l i t , i . e . with notation as above, there are a A :HA(A,B-)-H^ (X,Y) , :H^  (A)-*H*(X) , ft * ft ft ft ft a :H (X,Y)—>H (A,B), g :H (X)—*H (A), such that f*c^=l, f*B*=l, a f =1, 3 f =1. The s p l i t t i n g s are defined straightforwardly using the Poincare d u a l i t y isomorphisms, and t h e i r inverses. It follows from t h i s theorem that there are d i r e c t sum s p l i t t i n g s H^(X,Y)=ker £^®lm c ^ , H^(X)=ker fAS>im H (X,Y)=im f Qker a , H (X)=im f eker ft Thus we e s t a b l i s h the following notation: K q(X,Y)=(ker f^) cH (X,Y), K q(X)=(ker f # ) cH (X), K q(X,Y)=(ker a*) qcH q(X,Y), K q(X)=(ker 0 * ) q c H q ( X ) 7 (and s i m i l a r l y for (co)homology with c o e f f i c i e n t s ) . K q and have the following property. In the exact homology and cohomology sequences of the pair (X,Y), a l l the maps preserve the d i r e c t sum s p l i t t i n g , so induce a diagram, commutative up to sign, with exact rows: . . . - i ! K q - 1 ( Y ) - ^ K q ( X V Y - ) - J — » K q ( X ) — - K q ( Y ) - ^ — . . . 9[X]n 9 [X]n-•K (Y)—*K m-q m-q [X]n-(X)jl*->K (X,Y) m-q 9[X]n« m-q-1 (Y) 22 The proof of th i s property consists of the proof that the d i r e c t sum s p l i t t i n g s are preserved by the Poincare' d u a l i t y map ([X]n«) and the homology maps. From t h i s sequence, and using the d e f i n i t i o n of the K q groups, we develop the following diagram, with exact rows and columns: 0 0 0 0 (Y)< R H|X)^-R H(X,Y)^ K |(Y)< H q (Y-y H q ( X ) ^ - H q (X, Y)« H q _ 1 (Y)-( f | Y ) * f H q(B)« H q(A)^ Hq(A,B)-« H q 1(B)-< . .. i + + + 0 0 0 0 Suppose m=dim(X,Y)=4k and consider the p a i r i n g K (X,Y;S)®K *(X,Y;5)-H3 given by (x,y)=(xuy)[X]. This i s symmetric because the dimension i s even. 2k Define 1(f) to be the signature of (•,•) on K (X,Y;<3). Note that (•,•) i s the r a t i o n a l form of the i n t e g r a l form defined 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 following property of the K q groups: 5.13 Proposition: Under the p a i r i n g H q(X;F)®H m~ q(X,Y;F) F given by (x,y)=(xuy)[X], where F i s a r i n g , q(X,Y;F) i s orthogonal to f*(H q ( A ; F ) ) , K q(X;F) i s orthogonal to f * ( H m _ q ( A , B ; F ) ) , and on K q(X;F)®K m - q(X,Y;F) the p a i r i n g i s nonsingular i f F i s a f i e l d . I f F=Z, i t i s nonsingular on K q(X)/torsion®K m~ q(X,Y)/torsion. The proof i s straightforward v e r i f i c a t i o n , depending on c e r t a i n elementary properties of the cup and cap products. Taking q=2k and ~F=Q, we see'that the p a i r i n g (•,•) defined above . ft ft ft i s nonsingular. 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 nonsingular. 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; Let f:(X,Y)—*(A,B) be a map of degree 1 between Poincare pai r s of dimension m=4k+l. Then I(f|Y)=0. Proof: The proof i s an a p p l i c a t i o n of Proposition 5.2. 5.15 Proposition: Under the hypotheses of the theorem we have * 2k 1 2k * 2k 2k r a n k f i m i ) =±rank„K (Y;Q), where i :K (X;§)—>K (Y;Q) i s induced H 2 (ci from the i n c l u s i o n i:Y—>-X. Proof: We have a diagram, commutative up to sign: 2V i ?k Is! ?k+l •..-. HC -(X-;#)— ^ K Z K ( Y ; S ) ^ K Z K X(X,Y;«) • . . . [X]n- [Y]n«|. [X]n« * 8 * ' ^ K 2 k + 1 ( X ' Y ' 0 K2k ( Y 5 ^ O-^—^K^CX;Q) . In t h i s diagram the rows are exact and the v e r t i c a l maps are ft 2k isomorphisms. Hence (im i ) =(ker i j . ) 0 1 . I t i s e a s i l y shown that the * z k ft Universal C o e f f i c i e n t Formulae hold for K and K A, and thus, since Q 2k 2k i s a f i e l d , K (Y;e)SHom(K 2 k(Y;Q),Q), K (X;Q)=Hom(K 2 k(X;Q),Q), ft ft 2k and i ^omCi^,1) . Hence rank^(im i ) =rank^(im i ^ ^ ^ ' a n (^ 2k rank^(im i A ) 2 k + r a n k ^ ( k e r i ^ ) ^ r a n k ^ K ^ Y ; £ ) = r a n k ^ K (Y;Q). Hence, rank_(im i*) 2 k=-^rank J C 2 k ( Y ; S ) . QED ft 2k 2k 5.16 Lemma: With the hypotheses of 5.15, (im i ) <=K (Y;Q) annihilates 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 since i j Y ] = i ^ [ X ] = 0 i n H 4 k ( X ) . * 2k 2k Proof of Theorem 5.14: By 5.15, (im i ) cK (Y;Q) i s a subspace of 1 2k rank =-—rank K (Y;Q), and by 5.16 i t annihilates i t s e l f under the 24 2k pairing.. Hence by Proposition 5.2, sgn(',«) =0 on K (Y;Q) , so-that I(f|Y)=0. QED The sum of Poincar6 pai r s i s defined as follows: If (X^,X0uY_^) i=l,2 are Poincare pai r s of dimension m, such that XinX2=Xo, Y^nXo=Yg, and (Xg,Yo) i s a Poincare p a i r of dimension m-1, then i t follows [Browder 1972, p.13] that (XiUX 2,YiuY 2) i s a Poincare p a i r of dimension m, c a l l e d the sum of (X^,XgUY_^) along ( X Q , Y Q ) . If (X,Y) and (A,B) are the sums, re s p e c t i v e l y , of ( X ^ J Y ^ U X Q ) and (A i,B ±uA 0), and f:(X,Y)—^(A,B) with f ( X ± ) c A i , then the following are equivalent: (1) f has degree 1 (2) f 0 = f | ( X 0 , Y 0 ) has degree 1 (3) f ±=f | ( X ^ Y ^ X Q ) have degree 1 ( a l l with appropriate o r i e n t a t i o n s ) . We say that f i s the sum of fi and f 2 . 5.17 Theorem: Suppose f : (X,Y)—>(A,B), a degree 1 map, i s the sum of two maps f ^ : (X^XguY.^)—KA^ JAQ.UB^ ) , i=l,2, and suppose that the map on the i n t e r s e c t i o n fQ:H (A 0,BQ ; Q ) - + R (X 0,Y 0 ;Q) i s an isomorphism. Then I ( f ) = I ( f 1 ) + I ( f 2 ) . If (X,Y) i s a Poincare p a i r of dimension m=4k we may consider the 2k 2k symmetric p a i r i n g H (X,Y;Q)®H (X,Y;Q)—*Q given by (x,y)=(xuy)[X], 2k and we define I(X,Y) to be the signature of (•,•) on H (X,Y;<3). 5.18 Theorem: I(f)=I(X,Y)-I(A,B). Thus we have the important theorem 5.19 Theorem: Let f:(X,Y)—KA,B) be a map of degree 1 between Poincare p a i r s of dimension m=4k. Suppose (f|Y) :H (B;<2)—>-H (Y;Q) i s an isomorphism and that f i s cobordant r e l Y to f':(X',Y)—>(A,B) such that 25 f :H (A;Q)—*-H (X' ;Q) i s an isomorphism. Then l( f ) = 0 . Proof: Let U be the.cobordism r e l Y between X and X', so that 9U=XuX', XriX'=Y, (U,3U)is a Poincare p a i r of dimension m+1, compatibly oriented, and F i s the map (U,Y)—>(A,B) such that F | x=f, F | x ' = f . We may consider F as a map of degree 1 G: (U,XuX')—>-(AxI ,AxOuBxIuAxl) . By Theorem 5.14, I(G|XuX')=0, and by Theorem 5.17 I(G|XuX')=I(f)-I(f'). Now l(f')=0 since f i s an isomorphism, and hence l(f)=0. QED Let (X,Y) be a Z 2-Poincare p a i r of dimension m ( i . e . (X,Y) s a t i s f i e s Poincare d u a l i t y f o r homology with c o e f f i c i e n t s i n Z 2 ) . Define a l i n e a r map £ i:H m~ 1(X,Y;Z 2)—*Z 2 by Jl i(x) = (Sq 1x) [X] , where Sq 1 i s the i t h Steenrod square (see [Steenrod 1962]) and [X]eH (X,Y;Zo) i s the o r i e n t a t i o n c l a s s . m By Poincare\ d u a l i t y , H 1(X;Z 2)®H m _ 1(X,Y;Z 2)—>Z 2 given by (x,y) = (xuy)[X] i s a nonsingular p a i r i n g , so that H X(X;Z 2) i s isomorphic, using t h i s p a i r i n g to Hom(H m - 1(X,Y;Z 2) ,Z 2) , and hence £ ±(x)=(x,v 1) for a unique v ^ H 1 (X;Z 2) , for a l l xeH m _ 1(X,Y;Z 2). Define the Wu class of X to be V=l+v +v +. . . , v^eH"*"(X;Z2) as above. 5.20 Proposition: Let (X,Y) and (A,B) be Z 2-Poincare pa i r s of dimension m, f : (X,Y)—>(A,B) a map of degree 1 (mod 2) ( i . e . f*[X]=[A] for f ^ defined on homology with Z 2 c o e f f i c i e n t s ) . Then v^(X)=v_^+f (v^(A)), where v i e K ± ( X ) . * i The proof consists of a c a l c u l a t i o n to show that v^(X)-f (v^(A))eK (X) 5.21 Proposition: With notation as i n 5.20, suppose m=2q. Then the p a i r i n g (•,•) on K^(X,Y;Z 2) i s symplectic ( i . e . (x,x)=0 for a l l x) i f and only i f f*v (A)=v (X). q q Proof: (x,x)=x 2[X]=(Sq qx)[X]=(xuv q(X))[X]=(x,v q(X)) for xeH q(X,Y;Z 2), q * q and since K (X,Y;Z 2) and (im f ) are orthogonal by Proposition 5.13, (x,f*v ±(A))=0 f or xeK q(X,Y;Z 2). Hence f o r xeK q(X,Y;Z 2), (x,x)=(x,v ) by Proposition 5.20. Then (x,x)=0 i f and only i f v =v (X)-f v (A)=0. q q q 5.22 C o r o l l a r y : Let (X,Y) and (A,B) be oriented Poincarg d u a l i t y p a i r s of dimension m=4£ , and l e t f: (X,Y)—>(A,B) be of degree 1. If f (A) =v 2^(X), then the p a i r i n g (x,y)=(xuy)[X] (for x,yeK (X,Y)/torsion) i s even ( i . e . 2|(x,x) for a l l x ) . This follows from the fac t that (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 : Let (X,Y) and (A,B) be oriented Poincare pa i r s of dimension m=4£ , f : (X,Y)—>(A,B) of degree 1 such that (f | Y) A :H^  (Y)—*-H^ (B) * i s an isomorphism. If 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 Proposition 5.4. Let us now investigate the Wu c l a s s , with the aim of showing that i t i s preserved by normal maps. k Let (X,Y) be a pa i r of spaces, and E a f i b r e bundle over X with k-1 f i b r e F such that H^(F;Z 2)=H^(S ; Z 2 ) . Then we may define the Thorn space T(£)=XucE(£) using the pr o j e c t i o n of E as attaching map. There i s a Thorn class UeH (T(£);Z 2) such that •UU:Hq(X; Z 2 ) - * H q + k ( T ( E ) ; Z 2 ) . •uU:H q(X,Y;Z 2)->H q + k(T(£),T(?|Y);Z 2) •nU:H (T(£),T(?|Y);Z 2)->H (X,Y;Z 2) s s—k •nU:H (T(£>;Z 2)->H (X;Z 2) S S ~~K. are isomorphisms. Let h : i T r(A,B)—>H r(A,B;Z 2) be the Hurewicz homomorphism mod 2. We have the following important theorem of Spivak: 5.24 Theorem: Let (X,Y) be an n-dimensional Poincare p a i r , with X simply-connected and Y a f i n i t e complex up to homotopy type. Then there i s a s p h e r i c a l f i b r e space E, with X as base space, i t s f i b r e a homotopy (k-l)-sphere, and an element aeit , (T(E) ,T(E | Y)) such that 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 also be defined for homology with c o e f f i c i e n t s . 5.25 Proposition : Let (X,Y) be a Z 2-Poincare p a i r of dimension m, k. E 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 Z 2 homology (k-1)-sphere), aerr „ (T(E) ,T(E IY)) such that h(a)nU=[X] m+k. i n H (X,Y;Z 2). Then V(X)uU=Sq - 1(U). m We r e c a l l the fa c t that the Thorn class UeH (T(E);Z 2) i s * k characterised by the fact that j (U) generates H (EF;Z 2)=Z 2, where j :EF—KC(E) i s the i n c l u s i o n of the Thorn complex over a point into the whole Thorn complex. 5.26 Proposition: Let b :E™>E' be a map of f i b r e spaces covering f:X—*-X', where E and E' have f i b r e F, H ^ ( F ; Z 2 ) = H A ( S k - 1 ; Z 2 ) . Then b induces a map of Thorn complexes T(b) :T(E)—»-T(Er) , and T(b) U'=U, where U and U' are the Thorn classes of E and E'. Proof,: Let E,E' be the t o t a l spaces of E,E' resp. , so that the following TT diagram commutes : F >E >-X F »-E-1 Hence, f,b induce T(b) :Xu-' cE—>-X' u ,cE', and the diagram EF J >T(E) IT TT commutes. Hence j T(b) U'=j' U', so that j T(b) U' I ., + w k * EF-^—KT (E ' ) generates H (EF;Z 2), and thus T(b) U'=U. QED 5.27 C o r o l l a r y : Let (X,Y) and (A,B) be Z 2 Poincare pa i r s of dimension m, E' a f i b r e space over A with f i b r e F a (k-1)-dimensional Z 2 homology sphere. Let f:(X,Y)—>-(A,B) be of degree 1 i n Z 2 homology, and l e t E=f (£'). Suppose there i s an element a e T T m + k ( T ( E ) ,T(E | Y)) such that h(ct)nU=[X]. Then f*(V(A))=V(X), i n p a r t i c u l a r f*v (A)=v^(X) for a l l q. Proof: By 5.26, i f b:E—>E' i s the natural map, T(b) U'=U. Setting V(X)=V, V(A)=V, we have, using 5.25, T(b) (V'uU')=f V'uT(b) U'=f (V')uU =T(b)*(Sq~ iU')=Sq~ 1T(b)*U'=Sq~ 1U=VuU. Hence f*V'=V. 5.28 Theorem: Let (X,Y) and (A,B) be oriented Poincare pa i r s of dimension m=4£ , f : (X,Y)—*-(A,B) of degree 1 such that (f | Y) ^  i s an isomorphism, and E' a f i b r e space over A with f i b r e F a Z 2 homology (k-l)-sphere. Set E=f E 1 and suppose there i s aeir k ( T ( E ) ,T(E | Y) ) such that h(a)nU equals the o r i e n t a t i o n class of (X,Y) reduced mod 2. Then 1(f) i s d i v i s i b l e by 8. Proof: By 5.27, f*v (A)=v (X), so by 5.23 1(f) i s d i v i s i b l e by 8. Let (f,b) be a normal map, f:(M,9M)—>(A,B) of degree 1, M a smooth oriented m-manifold with boundary, (A,B) an oriented Poincare p a i r of dimension m, m=4£ , and b:v—>r\ a l i n e a r bundle map covering f, v the ifl~l~lc mH~lc—• 1 normal bundle of (M,9M)c(D ,S ) , n a k-plane bundle over A. 5.29 C o r o l l a r y : I f (f,b) i s a normal map with (f|9M)^ an isomorphism, then 1(f) i s d i v i s i b l e by 8. Proof: The pair, (f,b) s a t i s f i e s the conditions of 5.28, where E l = n i s a l i n e a r bundle over (A,B). Thus, we may make the following d e f i n i t i o n : Let (f,b) be a normal map f:(M,3M)—>(A,B), etc. with (f | 9M)^ an isomorphism, m=4£ the dimension of M. Define a (f ,b)=%[ (f) . Then the o Invariant Theorem for m=4£ follows from Theorem 5.19. Let (X,Y) and (A,B) be oriented Poincare pa i r s of dimension m=2q, and l e t f : (X,Y)—*-(A,B) be a map of degree 1. Let E be the Spivak normal f i b r e space of (X,Y), and n that of (A,B), and l e t aen ,. (T(E) ,T(E I Y)) , Beir (T(n) ,T(n I B) ) be the elements defined such m+k ' m+k that h(a)nU =[X], h(B)nU =[A], where U ,U are the Thorn classes of E,n E n E n and h i s the Hurewicz homomorphism. Let 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 Poincare p a i r s . Note that t h i s d e f i n i t i o n i s analogous to that of a normal map given above. We also define normal cobordism.and normal cobordism r e l B of Poincare p a i r s by the same analogy. Browder [1972, III.4] defines, using Spanier and Whitehead's S-theory, a quadratic form \|i :K q(X,Y;Z 2)-^-Z 2 with associated b i l i n e a r form (•,•)> where (x,y) = (xuy)[X] for x,y K q(X,Y;Z 2). I f (f|Y)*:H*(B;Z 2)—>H*(Y;Z 2) i s an isomorphism, i t follows from Proposition 5.13 that (•,•) i s nonsingular on K q(X,Y;Z 2) (=K q(X;Z 2)). Then the Arf in v a r i a n t of ij; i s defined. Let (f,b) be a normal map of Poincare complexes, f:(X,Y)—>-(A,B) , and suppose that (f | Y) :H (B;Z2)—>-H (Y;Z 2) i s an isomorphism. Then define the Kervaire i n v a r i a n t c ( f ,b)=Arf (IJJ) . Now we w i l l develop some properties of the Kervaire i n v a r i a n t . Let (f,b) be a normal map, f:(X,Y)—y(A,B), etc. and suppose i n addition that Y and B are sums of Poincarg pairs along the boundaries, and that f sends summands. into summands. In p a r t i c u l a r , suppose that Y=Y1.uY2, Y 0=Y 1nY 2, B=B 1uB 2, B 0=B 1nB 2, f(Y )cB , and that (B ±,Bg) and (Y_^,Yg) are Poincare p a i r s compatibly oriented with (X,Y) and (A,B) . If E,n are the Spivak normal f i b r e spaces of (X,Y) and (A,B), then E|Y_^ , n | B^ are the corresponding Spivak normal f i b r e spaces, so that i f f =f|Y , b i = b | | Y ^ ) , then ( f ^ * ^ ) are a l l normal maps, i=0,l,2. A A A Note that i f f 2 :H ( B 2 ; Z 2 ) — ( Y 2 ; Z 2 ) i s an isomorphism then i t A A A follows that fo:H (Bg;Z2)—>H (Y 0;Z 2) i s also an isomorphism. 5.30 Theorem: Let (f,b) be a normal map as above, so that fj'Y i s the sum i A A A of f j and f 2 on Y\ and Y 2, etc. Suppose f 2 :H (B2;Z2)—>-H (Y 2;Z 2) i s an isomorphism. Then c(f|,bi)=0. 30 This theorem has the following c o r o l l a r i e s : 5.31 C o r o l l a r y : I f (f,b) i s a normal map and i s normally cobordant r e l Y to ( f ' , b ' ) } f l*:H*(A,B;Z2)—*-H*(X* ,Y;Z2) an isomorphism, then c(f,b)=0. 5.32 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 derived from the theorem by using the normal cobordism as a normal map, the second by taking Y2=cf>. The proof of Theorem 5.30 r e l i e s on the d e f i n i t i o n of , and i s given i n [Browder 1972, I I I . 4 ] . Let ( f , b ) , f:(X,Y)—>-(A,B) be a normal map of Poincare .pairs, and suppose (X,Y) and (A,B). are sums of Poincare p a i r s , i . e . X=X^uX2, A=AXUA2, X0=X!nX2, A 0=A 1nA 2, Y =X±nY, B ^ A ^ B , f ( X ± ) c A , and (X±,XQUY±) , (A^,AQUB^) are Poincare pairs oriented compatibly with (X,Y) and (A,B). Set f 1=f[x i:(X 1,X 0uY i)->(A : L,AoUB i), f 0=f | X 0 : (X 0 ,Y 0)^(A 0 ,B0) , and b_^  the appropriate r e s t r i c t i o n of b. Now suppose that (f | Y) :H (B;Z 2)—>H (Y;Z 2) and f 0 :H (A 0;Z 2)—*H (X 0;Z 2) are isomorphisms. I t follows e a s i l y from arguments with the Mayer-Vietoris sequence that (f_JX 0uY^) are isomorphisms, so c ( f , b ) , c ( f i , b x ) , and c ( f 2 , b 2 ) are a l l defined. 5.33 Theorem: c(f,b)=c(f, ,b, ) + c ( f t , b j ) . Proof: We s h a l l present a p a r t i a l proof here; the balance i s to be found i n [Browder 1972] . Let and 4*2 be the quadratic forms defined on K q(X,Y), K^CX j J X Q U Y I ) and K q(X 2,X 0uY 2) r e s p e c t i v e l y . An argument with the Mayer-Vietoris sequence (which i s r e a l l y the exact sequence of the t r i p l e of p a i r s ( X Q , Y 0 ) C ( X , Y ) C ( X , Y U X O ) , where the l a s t p a i r i s replaced by the exc i s i v e pair (Xj,XguYj)u(X 2, XQU Y 2 )) gives an isomorphism p! ®p 2 : K q (XT. , X 0 uYi ) <3>Kq ( X 2 , X 0 uY 2 )->Kq ( X,Y) , where p^ i s defined by the diagram K ^ X j ,X0uYi)-« —K q(X,X 2uY) 4-K q(X,Y) where the isomorphism comes from an excision, and the v e r t i c a l arrow i s induced by i n c l u s i o n ( s i m i l a r l y for P 2 ) . I t remains to show (p^x)=ip Ax) , xeK q(X^,X 0uY ) . Then ip i s isomorphic to the d i r e c t sum l j ^ e i j ^ , so that Arf (ip)=Ar£(jp 1 )+Arf (TJJ2) . The remainder of the proof i s given on pp. 72-73 of Browder. Now suppose (A,B) i s a Poincare 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 covering f, v i s the normal bundle of (M,3M) i n ( D m + k , S m + k 1). - j _ > e < (f,b) i s a normal map i n the o r i g i n a l sense. Then by Theorem 1.4.19 of Browder, there i s a f i b r e homotopy equivalence (unique up to homotopy) b' : E—HY such that T(b')^(T(b)^(a) ) = B, where aeir ,, (T(v),T(v|9M)) and gerr „ (T(n) ,T(n I B)) are the elements such that m+k m+k h(a)nU =[M] and h (B)nU =[A]. Then b'b:v—*T\ , and (f,b'b) i s a normal v ri map of Poincare p a i r s , and we define a(f,b)=c(f,b'b)eZ2 i f m=4k+2 and (f|9M) on Z2 cohomology i s an isomorphism. 5.34 Proposition: The value of (f,b) i s independent of the choice of &eiT (T(n) ,T(n I B)) , and thus depends only on the normal map ( f , b ) . m+k The proof of 5.34 i s provided i n [Browder 1972]. With t h i s d e f i n i t i o n of a(f,b) for m=2 (mod 4), we see that Co r o l l a r y 5.31 provides the proof of the Invariant Theorem for m^ 2 (mod 4) and thus completes the proof of that theorem. We have also proved the following two properties of the invariant a: 32 5.35 Proposition: (Addition Property) Suppose (f ,b) i s a normal map which i s the sum of two normal maps ( f x , b i ) and ^ 2 ^ 2 ) 5 and such that f|.9M, f|9M_^  i=l,2, and f | Mg induce isomorphisms i n homology. Then a ( f , b ) = a ( f i , b i ) 4 c ( f 2 , b 2 ) . This property i s proved for m=4& by Theorem 5.17, and for m=4£+2 by Theorem 5.33. I t i s vacuously true for m=2q+l. 5.36 Proposition: (Cobordism Property) Let (f ,b) be a normal map, f : (M,""9M)->(X,Y) , b :v->E , and set f'=f|9M:9M-*Y, b , = b | (v | 9M) :v | 9M—>E | Y. If m=2"k+l then (f',b')=0. This property follows from Theorem 5.14 for the case m=4&+l and from Corollary 5.32 for the case m=4£+3. Let us c a l l the quantity I(X,Y) defined above the index of X. Then by the Hirzebruch Index Theorem [Hirzebruch 1966], we have Index M=L (p!(E 1),...,p (E 1 ) ) [ X ] , and Theorem 5.18 gives us d i r e c t l y K. K. the following 5.37 Proposition: (Index Property) I f Y=<f>, m=4k, (f,b) a normal map, then 8a(f,b)=index M-index X, and index X equals the signature of the 2k quadratic form on H (X;#) given by <xux,[X]>, where [X] i s the ori e n t a t i o n class i n E^(X;Q) . F i n a l l y we state without proof the 5.38 Proposition: (Product Formulae) Let ( f l 9 b i ) , ( f 2 , b 2 ) be normal maps f. :(M. ,9M.)—KX. >9X.) . Suppose a ( f n x f 2 . b i x b o ) , a(f1 ,bi)=ai, and X X X X I a(f 2,b 2 ) = a 2 are a l l defined ( i . e . f j x f 2 | 9 ( M 1 x M 2 ) , f ^ ^ M a r e a 1 1 homology isomorphisms with appropriate c o e f f i c i e n t s ) . Then (1) a(f:xf2,b1xb2)=I(Xx)a2+I(X2)a1+8a1a2 when M]xM2 i s of dimension 4k, where I(X^) i s the index of X^, (2) a ( f } X f 2 , b i x b 2 ) = x ( X 1 ) a 2 + x ( X 2 ) a 1 when M^xM2 i s of dimension 4k+2 33 where x(X^) i s the Euler c h a r a c t e r i s t i c of . Note that I(X)=0 by d e f i n i t i o n i f dim X^O (mod 4) § 6. Surgery below the Middle Dimension. j We w i l l now describe the technique of surgery, the use of which w i l l enable us to solve the surgery problem. Suppose that <Ji :S*>xDq+"'"—•M™, p+q+l=m, i s a d i f f e r e n t i a b l e embedding, into the i n t e r i o r of M i f 9M*<j>. L e t Mo=M\int(im <j>) . Then 9M0 = 9Mucj> ( S P * s q ) . Define M'=M 0 U ( { )D P + 1xs q, with (j)(x,y) i d e n t i f i e d to (x,y)eS PxS q=9(D P + 1xS q). Then M' i s a manifold, 9M'=3M, and M' i s said to be the r e s u l t of surgery using (j), on M. I t i s sometimes denoted by x(M,4>) (e.g. by Milnor) . nH~l We may define a cobordism W. between M and M' as follows: <P W =Mx[0,l]u(D P + 1xD q + 1) such that ( x , y ) e S P x D q + 1 c 9 ( D P + 1 x D q + 1 ) i s i d e n t i f i e d with (cj) (x,y) ,l)eMxI. C l e a r l y gW^Mu (9MxI) uM' , and W^  i s c a l l e d the trace of the surgery. As we have defined i t , W, i s not a smooth manifold with boundary. However, i t has a canonical smooth structure ( i . e . i t i s PL-homeomorphic to a smooth manifold) which i s described i n [Milnor 1961]. (Milnor c a l l s W^  io(M,<(>).) If w"1"*"1 i s a manifold with 9W=Mu(9MxI)uM' and W' has 9W' = M'u(9M'xI)uM", then we may define the sum of the two cobordisms by taking W=WuW' and i d e n t i f y i n g M'c9W with M'c9W'. Then i t i s cl e a r that 9W=Mu(9MxI)uM". 6.1 Theorem: Let W be a cobordism with 9W=Mu(9MxI)uM'. Then there i s a sequence of surgeries based on embeddings <J>^ , i=l,...,k, each surgery being on the manifold which r e s u l t s from the previous surgery, and such that W i s the sum of W, ,...,W, . *1 <*>k 34 The proof i s an immediate consequence of the Morse Lemma, and a l u c i d proof may be found i n [Milnor.1961]. 6.2 Proposition : If M' i s the r e s u l t of surgery on M based on an embedding <j> :S^ xDq+"'"—*-M, then M i s the r e s u l t of surgery on M' based on an embedding TJJ : SqxD^l+"'"—>M' such that the traces of the two surgeries are the same. 6.3 Proposition: Let cf>:SP*Dq+^—*-Mm be a smooth embedding i n the i n t e r i o r of M, p+q+l=m, and l e t W, be the trace of the surgery based on (f>. Then 9 W, has Mu-DP+"'' as a deformation r e t r a c t , where <j>=<|> I S PxO • 9 <P Proof: W, = (Mx.I) u (DP^xDq',~'*') , image <j>cMxI, so we may deform Mxl to Mxl 9 9 leaving Mxlu (DP+"'"xDq+"'") f i x e d . Then DP+"'"xDq+'1' may be deformed onto 9 ( D P + 1 x O ) u ( S P x D q + 1 ) , leaving t h i s l a t t e r subspace f i x e d . This then y i e l d s the deformation r e t r a c t i o n of W, to Mu-D^ "'". <J) 9 6.4 Proposition: (a) Let f:(M,3M)—v(A,B) be a map, M an oriented smooth m-manifold, (A,B) a pair of spaces, and l e t cj>:SPxoq+1—>-int M be a smooth embedding, p+q+l=m. Then f extends to F: (W^  ,9MxI)—>-(AxI,BxI) to get a — p cobordism of f i f and only i f f°<j> i s homotopic to the constant map S —>A. (b) Suppose i n addition that n i s a l i n e a r k-plane bundle over A, k k b :v —Hn i s a l i n e a r bundle map covering f, v the normal bundle of (M,3M)c(D m + k,S m + k , k>>m. Then b extends to b:o)—>n covering F, where a) i s the normal bundle of c D m + k x 1 5 i f and only i f b| (v|(f>(SP)) extends to a)|D P + 1xn, covering F|D P + 1xO. Proof: Since Mu-DP+"'" i s a deformation r e t r a c t of W, , i t follows that f <l> 9 extends to W i f and only i f f extends to Mu-DP+''". But the l a t t e r i s 9 9 true i f and only i f f°$ i s null-homotopic, which proves (a). For (b), i t follows from the bundle covering homotopy property, and the fa c t that MU-DP+^ i s a deformation r e t r a c t of W,, that b extends 9 9 to co i f and only i f b extends to OJ|D P + ' ' " XO. QED If (f,b) i s a normal map, (J> :SPxDq+"'"—KLnt M™, p+q+l=m, f:(M,9M)—KA,.B) , and i f the trace of <j) can be made a normal cobordism by extending f and b over W^  , we w i l l say that the surgery based on tf> i s a normal surgery on ( f , b ) . From Theorem 6.1, i t follows e a s i l y that any normal cobordism r e l B i s the composite of normal surgeries. Let <)>: S P x D q + 1 — > i n t be an embedding, with p+q+l=m. i s the trace, and M' the r e s u l t of the corresponding surgery. We w i l l i n vestigate the e f f e c t of surgery on the homotopy of M; i n p a r t i c u l a r , we w i l l examine the r e l a t i o n between the homotopy groups of M and M', below the 'middle dimension'. 6.5 Theorem: I f P < ^ p 1 t n e n T r ^ M ^ - T r ^ C M ) for i<p, and T r p ( M ' ) ^ r r p ( M ) / { ^ # T T p ( S P ) } , where {G} denotes the Z[iri(M)] submodule of TT (M) generated by G. P P+l Proof: By 6.3, W^  i s of the same homotopy type as Mu-D . Hence TT. (W. )=TT (M) for i<p, and TT (W J^TT (M)/{<J>„TT (S P)}. By 6.2 and 6.3, i c p i p <p p ?fp we have also that W,=W, =M'u-Dq+\ where y: S^D***1—>M' gives the surgery <j> ip which reverses the e f f e c t of surgery base on t}>. Hence TT ^  (W^  )-n\ (M') for i<q, TT (Wj = T r (M' )/U ,,TT (S q)}. Since p<5^r1, q>p , so TT . (M')-TT . (W ) q q> q ffq z l l <p for i<p and the r e s u l t follows. QED k k Let (f,b) be such that 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 of (M,9M)c(D m + k,S m + k "*") , and l e t <{>:SP—KLnt M be a smooth embedding. Suppose that f extends to F:M—>A, where M=Mu-DP+"'". We consider the problem of 'thickening M to a normal cobordism', i . e . of extending cj> to a smooth embedding $ : SPxDq+"'"—>-int M™, p+q+l=m such that <j>=<j> | S PxQ, and so that F: (W^  ,.3MxI)—>(AxI ,BxI) can be covered by a bundle map b :w—>T) extending b, where u i s the normal bundle of W" i n D m + k x T and F i s the extension of F, unique up to <P homotopy. (When t h i s i s possib l e , normal surgery based oriuj) w i l l k i l l the class of 4 i n i (M).) Let V. ,., be the space of ortho-P k,q+l normal k-frames i n R^+<^+^-. 6.6 Theorem: There i s an obstruction Oe-rr (V, ,..) such that 0=0 p k,q+l i f and. only i f <}> extends to <(> such that F:W^ —>-A can be covered by b:o)—>r] extending b as above. Proof: Since k i s very large, we may extend the embedding McD m + k to Mu-D P +"'"cD m + kxi t with D P + 1 smoothly embedded and meeting D m + k x 0 perpendicularly. The normal bundle y of D P +"*"cD m + kx I i s t r i v i a l . F defines a homotopy of f°<|> to a point, which i s covered by a bundle homotopy b on v|c|>(SP), ending with a map of v|c(>(SP) into 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 of v|<f>(SP), which i s well-defined up to homotopy. This 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 P ) , which i s also t r i v i a l , therefore defines a map a of S P into the k-f rames of I? q + k +"'", a:SP—^-V^ q+]_> which gives an element aeir (V, , , ) . Now i f d> extends to A and b p k,q+l extends to b as above, then the normal bundle to of W, r e s t r i c t e d to DP+^", OO|DP+"'* i s a subbundle of y extending v | ^ ( S P ) , and b defines an extension of a to a':DP+"'"—•V, Hence 0=0 i n k,q+l TT (V. , . ) . p k,q+l Conversely, i f 0=0, then a extends to a':DP+"'"—>-V, , n , and J k,q+l a' defines a t r i v i a l subbundle co' of dimension k i n y, extending v|c)>(SP). The subbundle as" orthogonal to oo1 i n y i s t r i v i a l (being a bundle over D P + 1 ) and the t o t a l space of to" i s D P + 1 x i ? q + 1 c D P + 1 x i ? q + k + 1 5 the t o t a l space of y ( a l l up to homeomorphism) . Since a)"|cj)(SP) equals the normal bundle of <f>(SP) i n M, t h i s embedding defines <j):SPx q+"'"—yu, and a' defines the extension of b to b:w—>-n , where w| DP+"*"=a)' by construction. QED We s h a l l now study V, ,, i n order to analyse the obstruction 0. k,q+l (0 w i l l often be ref e r r e d to as 'the obstruction to thickening (M,F) to a normal cobordism'.) Recall that the group S0(k+q+l) acts t r a n s i t i v e l y on the set of orthonormal k-frames i n #k + cl+l a n d S0(q+1) i s the subgroup leaving a given frame f i x e d . Hence V = S0(k+q+l)/S0(q+1) , and V . i s topologised to make th i s a homeomorphism. Further, we r e c a l l that SO (n)-^-SO (n+l)-P-^-Sn i s a f i b r e bundle map, where p i s the map which evaluates an orthogonal transformation on the unit vector V Q=(1,0, . .. , 0 ) e S n c i ? n + 1 , i . e . P ( T ) = T ( V Q ) . (For t h i s material, reference may be made to [Husemoller 1966].) 6.7 Lemma: i :ir. (S0(n))—HT . (SO (n+1) ) i s an isomorphism for i < n - l , • * l i and a s u r j e c t i o n for i < n - l . Proof: iT^(S n)=0 for i<n, so the r e s u l t follows from the exact homotopy sequence of the f i b r a t i o n SO (n+1)—^S 1 1: . . .-nr ( S n - ) - ^ i F . ( S 0 ( n ) ) ^ # ^ T r . ( S 0 ( n + l ) ) - ^ / ^ T T . ( S n ) - ^ . .. QED l + l i i l 6.8 Lemma: The map p: SO (n+1)—>-Sn i s the proje c t i o n of the p r i n c i p a l S0(n) bundle associated with the oriented tangent bundle of S n. Proof: Let f = ( f i , . . . , f ) be a tangent frame to S n at v n = ( l , 0 , ...,0) . n Define a map e:SO(n+1)—>F, the bundle of frames of S n, by e(T) i s the frame (T(f^ ),...,T(f )) at T ( v 0 ) e S n . 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 follows. 6.9 Lemma: The composite ir ( S n ) — ^ - > i r .. (S0(n))-P-#^-TT ., (S n 1 ) i s the n n-1 n-1 boundary i n the exact sequence of the tangent S n ^ bundle to S n, and 38 i s 0 i f n i s odd, and m u l t i p l i c a t i o n by 2 i f n i s even. Proof: The tangent S11 bundle i s obtained from the bundle of frames by taking the quotient by SO(n-l) SO(n), the structure group of the bundle. Hence we 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 — i ,n It follows that i n the exact sequence for the r i g h t hand bundle, 3=p ,,3 :TT . (S n)—*-rr, , ( S n "S . Now by the Euler-Poincare Theorem the r# l i - l tangent sphere bundle has a cross-section (there i s a nonsingular tangent vector f i e l d ) i f and only i f the Euler c h a r a c t e r i s t i c x ( M ) i s zero. More p r e c i s e l y , the only obstruction to a cross-section to the tangent sphere bundle of a manifold i s x ( M ) g > where geHm(M;Z) i s the class dual to the o r i e n t a t i o n class of M. Now i f M=Sn, the obstruction to a cross-section can also be i d e n t i f i e d with — n n—1 the c h a r a c t e r i s t i c map (see [Steenrod 1951, 23.4]) 8 :TT (S )->ir (S ) n n-1 Hence 3=0 i f n i s odd, m u l t i p l i c a t i o n by 2 i f n i s even. 6.10 Theorem: p„ :TT (SO(n+1))—>TT (S n) i s s u r j e c t i v e i f and only i f if n n n=l,3, or 7. Proof: If p^ i s s u r j e c t i v e , then there i s a map a:Sn—>-S0(n+l) such that poa^l, and hence the p r i n c i p a l bundle of x n has a section and i s therefore t r i v i a l , i . e . S n i s p a r a l l e l i s a b l e . But i t i s known that S n i s p a r a l l e l i s a b l e i f and only i f n=l,3 or 7. 6.11 C o r o l l a r y : ker i : T . (S0(n))—*ir .(SO (n+1)) i s Z i f n i s even, it n - i n-1 Z 2 i f n i s odd and n*l,3,7, and 0 i f n=l,3,7. 39 Proof: ker ±„=3ir (Sn)=7r (Sn)/p„rr (S0(n+1)) . I f n i s odd, by 6.9 if n n if n P„TT (S0(n+l ) ) 2 2 T r (S n) , and by 6.10 the i n c l u s i o n i s s t r i c t , i f n*l,3,7, If n. n hence ^ n ( S n ) / p # i r n ( S 0 ( n + l ) ) = Z 2 i f n i s odd, n*l,3,7. If n=l,3, or 7, p^ i s s u r j e c t i v e , so ker ±^=0 -If n i s even, by 6.9 p J 1°9 i s a monomorphism, so 9:TT (S n)—>K ..(SO(n)) if n n-1 i s a monomorphism, so ker Z. QED 6.12 Theorem: TT . ( V , )=0 for i<m, IT ( V , )=Z 2 i f m i s odd, Z i f m i s even, l k,m m k,m ^ k>2. Further j„ :TT.(V, )—Hr. ( V , ,.. ) i s an isomorphism for i<m, k^2, J# l k,m l k+l,m 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 , take k=2, so that V 2 m=S0(m+2)/S0(m) and we have a natural f i b r a t i o n over S m + 1=S0(m+2)/S0(m+l) with f i b r e Sm=S0(m+l)/S0(m). Also we have a commutative diagram of f i b r e bundles: S0(m+1) P. >Sm . j S0(m+2) >-V, P m + 1 — i - > s 2 ,m m+1 By" the n a t u r a l i t y of the homotopy exact sequences we have: /Pm+1N 1 _ /cm+lx m+1 t 9 ' TT CSOCnri-l))- 2^ (S™) m m 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 exact homotopy sequence of the f i b r e bundle, ( S m + 1 ) - ^ r r . (Sm)-W.-(V_ )-Hr, (S m + 1)=0 for i<m, l + l l i I ,m i we deduce that j i s s u r j e c t i v e for i<m, and TT.(V „ )=0 for i<m, J# J l 2,m TT ( V „ ) = Z i f m i s even, TT ( V . )=Z 2 i f m i s odd. m 2,m m 2,m Consider next the natural i n c l u s i o n V , — > - V . ,.. given by k,m k+l,m including SO (m+k)—>-SO(m+k+l.) i n such a way that the subgroup SO(m) i s preserved. We have the commutative diagram: SO(m)- -*S0 (m) S 0(m+k)-i*S 0(m+k+1) k,m "*Vk+l,m and a corresponding diagram incorporating the exact sequences, .-. .—tif.. (SO (m>)——»TT. (S0(m+k>) - • . > 77,-(V.- ) *TT . . (SO (m) )-TT, (S6(m)-) H7, (SO (m+k+1)-) >-7r . (V k,m' i - l •) •try -, (S0(m))-*. i v k+l,m' i - l By~Lemma 6.7, i„ i s an isomorphism for i<m+k-l, and, since k>2, i t follows that i s an isomorphism for i<m. QED The following theorem describes what can be accomplished toward so l u t i o n of the surgery problem, by the use of surgery below the middle dimension. 6.13 Theorem: Let (M,9M) be a smooth compact m-manifold with boundary, V Tn~f"T<" Tn-r-V 1 m>4, v the normal bundle for (M,9M)c(D ,S ), k » m . Let A be a f i n i t e complex, B£A, n 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 covering f. Then there.is a cobordism W of M, with 9W=Mu(9MxI)uM', 9M'=3Mxl, an extension F of f, F : ( W ,9MxI)—>(A,B) with F |9Mxt=f | 9M for each t e l , and an extension b of b, b:u>—HI, where to i s the normal bundle of W i n D m + k x i , such that f'=F|M':M'—>A i s [y]-connected ([a] i s the greatest integer not larger than a). Proof: The proof i s by induction: we s h a l l assume that f :M—*-A i s n-connected, n+l<[y], and show how to construct W , F , etc. as above, with f':M'—>-A (n+1)-connected (n+1 i s any nonnegative i n t e g e r ) . If n+l=0, we need only show how to make the map induced on TTQ s u r j e c t i v e . Since A i s a f i n i t e complex, A has only a f i n i t e number of components, A=A]U A 2u . . .uA^ . Let a_^ eA_^ , and take M'=MuSmu. . .MS™, where S^'is the m-sphere. Let W=MxIuD?1+''"u.. .uDm+"^ and l e t F:W—>-A 1 1 r be defined by F|Mxt=f for each t e l , F(D™+"'")=a^. Since the normal bundle of Dm+"'' i s t r i v i a l , and the extension condition on the bundle l ni-hl map i s easy to f u l f i l l on the D^ , i t follows d i r e c t l y that b extends to b over W. Cl 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 of our induction. Now assume n+l=l, f :M—>-A i s 0-connected. Let and M2 be two components of M such that f(M^) and f(M 2) are i n the same component of A. Take two points x^eint M_^ , 1=1,2, and define cj>:S^ —*M by <Kl)=xi , <K-l)=x 2. Since f(<KS 0)) l i e s i n a si n g l e component of A, i t follows that f :M—>A extends to f iMu-D1—>-A. Then, since m>4, i t follows from Theorems 6.6 and 6.12 that $ extends to <J>:S°xDm—>M defining a normal cobordism of f to f 1 and reducing the number of components of M. Using t h i s argument repeatedly, we a r r i v e at a 1-to-l correspondence of components. Now we consider the fundamental groups. Let { a ^ , . . . , a g ; r j , . . . , v ^ } and {xi , .. . ,x^;yi ,.. . ,y^} be presentations of TT]_ (A) and TTJ (M) , resp . Let s copies of S^  be embedded d i s j o i n t l y i n an m-cell D™ i n t M, *': S°—>M, and assume the base point of M.is i n D and f(D )=*, the s base point of A. Let M=Mu^  , (uD 1) . Then TT^  (M)=TT^ (M)*F, where F i s a free group on s generators gi, . . . , g » where each g_^  i s the homotopy class of a loop i n D mu(uD 1) co n s i s t i n g of a path i n D m, one of the D-^'s, and another path i n D™. Hence TT| (M)={X^ ,. .. ,x^,gi ,. .. ,gg;y\ , .. Define f:M—>A extending f by l e t t i n g the image of the i D traverse a loop representing the generator a_^ . Then f^:ni(M)—HTT_ (A) i s s u r j e c t i v e , and furthermore we may represent f^-'on the free groups {xj ,.. . ,x ,gi ,. . . ,g } and' {aj_ , . . . ,a } by a function a, with a (x. )=x! , x! a word i n the a., and a(g.)=a.. Then as above, we may extend <j>' x j i x to <)): (uS°)xDm—>-M to define a normal cobordism of f, and with W,=M, s cp and F:W,—>A homotopic to f:M—>A. (Here W, i s the trace of the simul-9 9 taneous surgeries.) By Proposition 6.2, TT j (M' )-TTT_ (W ) , where M' i s defined 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 presentation as TT^  (M) , and f | i s also represented by a on the free groups. In p a r t i c u l a r , f ' i s 1-connected. Let us consider the exact sequence of the map f:M—>A i n homotopy, vrr ,v(f) HT (M) nr ( A ) Mr ( f ) — n+1 n n n Rec a l l that the elements of the groups TT ...(f) are defined by commutative n+1 dxagrams : S >-M where k i s the i n c l u s i o n of the boundary, D n+1 3 f (*) ->A and a l l maps and homotopies preserve base points. Thus f3 defines a map n+1 f :Mu- D a »-A extending f . 6.14 Xemma: Let f :M—>A be n-cdnnected, n>0, and l e t ($,,$) £ T r n +^(f) be the element represented by the above diagram (*) . I f f iMu^D^-^A i s defined by f3 as above, then TT . (f )=rr. (f )=0 for i<n, and TT ( f ) = T r J ^ 1 ( f ) / K , x x n+1 n+1 where K i s a normal subgroup containing the TTx (M) module generated by the element (R,a) i n TT (f) . n+1 Proof of Lemma 6.14: Consider the commutative diagram: f, £+1 ->TT v(M')- H^ ->TT ( A ) -a i • • ) — \ (MU aD N + 1)-^/->TT £ ( A ) -43 Here i:M—>Mu^Dn+"^ i s i n c l u s i o n , and j ^ i s induced by ( l , i ) on the diagram (*) ( i . e . j^[3'»a']=[3',i°a']). C l e a r l y , ± J t i s an isomorphism for £<n, and s u r j e c t i v e for &=n, so i t follows e a s i l y that TT^  (f )=TT^ (f )=0 for I n (by the Five Lemma). Cle a r l y any map of S n into Mu^Dn+"'" i s homotopic to a map into M, so that any pair (3',a') „n a _n+l , ,. S KMu D xs homotopxc to a pair or a f D n+1 g* the form ( 3 " , i ° a " ) : „n a n+1 B" w l w ^n+1 ->M——>Mu D a D" • —yA—^->A Hence 3^:7ln+i^^—^n+l ^ S s u r J e c t ^ v e # C l e a r l y (3,a) i s i n the kernel of j ^ and hence everything obtained from (3,a) by the action of TT^  (M) i s also i n ker j ^ , which proves the lemma. QED We have already shown that we may assume, without loss of generality, that f:M—>A i s 1-connected, and that the fundamental groups have presentations TTI (M)={x! , . . . j X ^ g j ,. . . ,g ;yj ,. .. ,y^} , y± words i n x l 5...,x k only, TTJ (A) = {a! ,.. . , a g ; r 1 ,.. . ,r t> , with f ^ : i r i OO—(A) presented by the function (on the free groups) a(x.)=x!(a^,...,a ) a word i n , . . . ,a , j=l,...,k, and a(g^)=a_^, 1=1, ,s. 6.15 Lemma: ker f ^ i s the smallest normal subgroup containing the -1 words x. (x!(g!,...,g ) ) , j=l,...,k and r (g x,...,g ), 1=1,...,t. 3 3 s i s Proof of Lemma 6.15: Adding the r e l a t i o n s x. (x!(g)) makes gi , . . . , g J 3 s into a set of generators. Adding the r^(g) makes the group into (A), with a defining the isomorphism. The map a annihilates X j ^ ( X j ( g ) ) 44 and r ^ ( g ) , so that these elements generate ker f ^ as a normal subgroup. For each element x_."*"(x^  (g)) and r_^(g) choose an element x ^ . , r ^ e T r 2 ( f ) such that x = X J ^ ( X J (g)) , r_^=r^(g) , and choose embeddings S1—>M to represent the x. and r , (also denoted by x. and r.) such that t h e i r J i J i images.are a l l d i s j o i n t , which i s possible by general p o s i t i o n , since m 4. Let M=MU (^^D?) ,with the 2-discs attached by these embeddings. I t follows from Lemma 6.14 that f^:Tri(M)—MT]_ (A) i s an isomorphism. Using again Theorems 6.6 and 6.12, i t follows that there i s a normal cobordism W, and a map F:W—>-A such that McW i s a deformation r e t r a c t and F|M=f, so that F^:TTI(W)—>-TT x (A) i s an isomorphism. By Propositions 6.2 and 6.3, i t follows that i f M' i s the r e s u l t of surgery, then f ^ : T r x ( M ; ' )— ( A ) i s an isomorphism, and hence TT2 (A)—MT2 (f) i s s u r j e c t i v e , and thus TT2 (f) i s abelian. We now proceed to the induction step. 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 that r r 2 ( f ) i s abelian. 6.16 Lemma: 1 7 1 S a f i n i t e l y generated module over :TTT_ (M) . This lemma i s proved using u n i v e r s a l covering spaces [Browder 1972]. Now we may represent each of th i s f i n i t e number of generators i n TT ,_(f) by a diagram S1^ -*-M If n+l<[™-], then n<^ - and i t follows n+1 . o i . a i 2 f T.n+1 D — >k -B i from Whitney's embedding theorem ('general position') that we may choose (3.,a . ) so that the a . have d i s j o i n t images. Setting M=Mu ( u D ^ " * " ) , i i l i i D^+^" attached by a^, f :M—>A defined by the 6^, we may apply Theorems 6.6 and 6.12 to thicken M to a normal cobordism W of M, and using 6.14, T r^(f)=0 for £<n+l. If M' i s the r e s u l t of t h e surgeries 45 ( i . e . 9W=Mu (9MxI)uM') , from Propositions 6.2 and 6.3 i t follows that IT (f ' ) = T r (f)=0 for i<n+l. This completes the proof of Theorem 6.13. QED Note that we have always used the low dimensionality of the groups involved to ensure that 0 was zero (by Theorem 6.12) and to f i n d representatives of elements of ^ ^ ( f ) which were embeddings. To derive r e s u l t s i n higher dimensions, we s h a l l have to f i n d other means of dealing with these obstacles. § 7. I n i t i a l Results i n the Middle Dimension. Let (A,B) be an oriented Poincare p a i r of dimension m, l e t M be an oriented smooth compact m-manifold with boundary 9M, and l e t f:(M,9M)—>-(A,B) be a map of degree 1. Let n be a l i n e a r k-plane bundle over A, k>>m, and l e t v be the normal bundle of (M,9M) i n (D ,S ). Suppose b:v—m i s a l i n e a r bundle map covering f . Then (f ,b) i s what we have c a l l e d a normal map. (Recall that we defined a normal cobordism of (f,b) r e l B to be an (m+1)-manifold W with 9W=Mu(9MxI)uMT, together with an extension of f , F:(W,3Mxl)—>(A,B) for which F|9Mxt=f|3M for each t e l , and an extension b of b to the normal bundle OJ of W i n D ^ ^ x l . ) Suppose further that A i s a simply-connected CW complex, m>5, and that (f | 9M)^:H^(9M)—*-HA(B) i s an isomorphism. 7.1 Theorem: There i s a normal cobordism r e l B of (f,b) to (f',b') such that f.' :M'—>A i s f^+l-connected i f and only i f a(f,b)=0. In p a r t i c u l a r , t h i s i s true i f m i s odd. The proof of th i s theorem w i l l occupy the balance of the present chapter. F i r s t note the ultimate c o r o l l a r y . 7.2 C o r o l l a r y : (Fundamental Theorem of Surgery) The map f' above i s a homotopy equivalence. Hence, (f,b) i s normally cobordant r e l B to a homotopy equivalence i f and only i f a ( f ,b)=0. In p a r t i c u l a r , there i s such a normal cobordism i f m i s odd. Proof of Corollary 7.2: By the n a t u r a l i t y of the exact homology sequence of p a i r s , we have . . * H . (9M')~—>E. (M')-—7*-H_, (M' ,9M'-)-—-*-H. .. (3M'-) 41. , (M')-i i i i i i i - l | i - l | (f' | 9M'). ( f |3M')* f; . . H (B-) - H l ^ A ) — ^H±(A,B)- V*V-1(B)— ^H±_1(A)-»-. . Since ( f|9M)^:H A(9M)—(B> i s an isomorphism, and 9M'=9M, f'|9M'=f|9M, we see that ( f | 9 M ' ) ^ i s an isomorphism in.each dimension By 7.1, f':Mr—*A i s [y]+l-connected, so that f^H^M')—^H ±(A) i s an isomorphism for i<y. Thus by the Five Lemma, :H±(M* ,3M-1)—*-H±(A,B) i s an isomorphism for i ^ y - Since f i s a map of degree 1, i t follows * i i from Poincarg d u a l i t y that f :H (A)—>-H (M') i s an isomorphism for j>m^=S-. Now f '*J :HJ (A)—>H^ (M') i s given by f J=Hom(f.^ ,Z)+Ext(f^_ 1,Z) , according to the Universal C o e f f i c i e n t Theorem, where f ^ :H (M')—^-Hy(A) , etc. in *k "l Since f' i s an isomorphism for i ^ — , i t follows that f i s an isomorphism for j<y, and hence f :H^  (A)—*-H^ (M') i s an isomorphism A for a l l j . Thus, H (f')=0, and the Universal C o e f f i c i e n t Theorem implies that H^(f')=0. But M1 and A are simply-connected, so that by the Relative Hurewicz Theorem and the.Theorem of Whitehead we have the r e s u l t : f':M'—*-A i s a homotopy equivalence. This establishe the c o r o l l a r y . We s h a l l develop c e r t a i n preliminary r e s u l t s before proceeding with the proof of Theorem 7.1. By Theorem 6.13, we may assume that f:M—>k i s [^-connected, i . e . IT (f)=0 for i<[^-]. Set A=[^]. Since A and M are simply-connected, i t follows from the Relative Hurewicz Theorem that ^ . n ^ f ) s i ^ + ^ ( f ) . This gives a commutative diagram: £+1 h (f ) >-TTN (M)-^/->TT£ (A)->0 ...->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 that f ^ is-.s.urjective, and s p l i t s by Theorem 5.12. It follows that (ker f.) =h(ker f,,)„. * X- it Jo Whitney's embedding theorem sta t e s : 'Let c:V a continuous map of smooth manifolds, m>2n, m-n>2, M simply-connected, V connected. Then c i s homotopic to a smooth embedding.' (A proof can be found i n [Milnor 1965].) Since &^ y> i t follows from Whitney's embedding theorem that any - - £ element x£ir£_(_^(f) may be represented 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) to a normal cobordism; i . e . to perform normal surgery using <j>, and to examine ^ £ + 1 ^ ' ^ ' where f* i s the map on the r e s u l t of the surgery, with the hope of having k i l l e d the homotopy class of c(>. However, there are two d i f f i c u l t i e s we must face: F i r s t , i f m=2£, then according to Theorems 6.6 and 6.12, there i s an obstruction 0 to thickening (M,f) to 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, although we may compute ^ ^ ^ C f ) using Lemma 6.14, i t i s no longer clear how t h i s group i s related to ^ . ^ ( f ' ) ' i f ^=[^"1 We s h a l l f i r s t d i r e c t our attention toward the second d i f f i c u l t y . Unless stated otherwise, we s h a l l assume henceforth that (f,b) i s a normal map s t i s f y i n g the hypotheses of Theorem 7.1, and f :M—*-A i s q-connected, where q=[^], i . e . m=2q or 2q+l. 7.3 Lemma: f i s (q+1)-connected i f and only i f f^:H ( M ) — > H (A) i s an isomorphism, i . e . i f and only i f K q(M)=0. Proof: By the Relative Hurewicz Theorem, 1 7 ^ ~ H q + l ^ ^ ' a n c * Theorem 5.12, f..:H ,, (M)—»-H ,., (A) i s s u r j e c t i v e , so that * q+1 q+1 H q + 1 ( f ) * < k e r f „ ) Q - K Q ( M ) V " QED Thus we need not examine homotopy, but w i l l study the e f f e c t of surgery on homology. The following lemma w i l l allow us to simp l i f y our arguments by considering only the case of closed manifolds. Let (f_^,b_^), ±=1,2, be two d±sjo±nt cop±es of the normal map ( f , b ) , so that f : ( M ,31^)—>(A ,B ), ±=1,2, ±s j u s t f renamed. Then by the Sum•.-•Theorem for Po±ncare pa±rs [Browder 1972, 1.3.2], A 3=A^uA2 with B} ±dent±f±ed to B 2 ±s a Poincare complex ( c a l l e d the double of A)., M 3 = M T _ U M 2 , united along 3Mi=9M2, ±s a smooth closed oriented man±fold, and f 3 = f T _ u f 2 , b 3=biub2 define a normal map (f 3 ,b 3) : M 3 — 4 - A 3 . Since ( f l S M ) ^ ±s an ±somorph±sm, the Mayer-V±etor±s sequences ±mply that H_^(f3)=0 for ±<q+l, and Hq+1 ( f 3 ) S K q ( M3 )*\Q*l)<3>Kq ( M 2 ) ; . Now suppose ()):SqxDm q—>±nt Mi_ ±s a smooth embedding such that f l 0 ? - ^ (the constant map), and such that 9 defines a normal surgery on and, by i n c l u s i o n , on M3 (with respect to (f 1,b x) and ( f 3 , b 3 ) ) . I f a prime denotes the r e s u l t of surgery, we have M3=MiuM2 and K q(M 3)=K q(M 1')©K q(M 2) . This follows from the fac t that the surgery has not affected the factor M2 i n the decomposition of M3. 49 Thus we have: 7.4 Proposition: The e f f e c t of normal surgery on K^(M) i s the same as the e f f e c t of the induced surgery on K^(M3), and hence to compute i t s e f f e c t , we may assume 9M=B=cj>. This construction w i l l s i m p l i f y the algebra i n our discussion. Let <J>:SqxDm q — * - i n t M be a smooth embedding which defines a normal surgery on M (with respect to ( f , b ) ) . Set Mg=M\int im <f>, and l e t M ^ M Q U D ^ X S ^ - 1 , SO that H S ^ S ^ 1 ) i s i d e n t i f i e d with S ^ S 1 " " ^ 9(D q +^xS m q "S . Then M' i s the r e s u l t of the surgery on M. Since c|> defines a normal surgery, H (M')-H (A)<$K (M'), and we wish to determine 9 q q how K (M) changes to K (MT) (which i s the same as the change of H (M) q q q to H (M')). q We formulate some us e f u l r e s u l t s concerning the r e l a t i o n between Poincare d u a l i t y i n manifolds and submanifolds. 7.5 Proposition: Let U and W be compact m-manifolds with boundary, f:U—>int W, g:(W,9W)—>-(W,W\int U) embeddings, with orientations compatible. Then the following diagram commutes: Hq(W,9W)< g ; - Hq(W,W\int U)—>-H q(U,9U) [W]n-1 (g*[W])n- [U]n-H • ( W ) — — — ( W ) - ( -*——H ••• (U) m-q m-q m-q so that f o r x£H q(U/9U), f^([U]nx)=[W]ng*(x), where g":W/9W->U/9U. Proof: If f:(U,3U)—>(W,W\int U), then f^[U]=g^[W], since we have oriented U and W compatibly. Then the commutativity follows from the n a t u r a l i t y of the cap product. QED 7.6 Co r o l l a r y : Set E equal to the normal tube of f:Nn—>W"m, N closed and oriented, and l e t g:W/9W—>E/9E=T(v), where v i s the normal bundle of N ^ w ™ . Let 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 A( [E ]nU)=f^[N]=[W]n(g U). Q E D 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 i s defined by x*y=(x',y')=(x'uy')[M], where x'eH m _ q(M,3M) , y'eH q(M) are dual to x,y, i . e . [M]nx'=x, [M]ny'=y. This induces an i n t e r s e c t i o n product •:H (M)®H (M)—-*Z by x*y=x«j.(y), where j:M—>(M,3M) i s i n c l u s i o n , q m-q * The properties of the b i l i n e a r form (•,•) on cohomology induce analogous properties for 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 F, H (M;F)®H (M,3M;F)—>F i s q m-q a nonsingular p a i r i n g . (This" also holds over Z, modulo torsion.) (b) If xeH (M), yeH (M) , x-y= ( - l ) q ( m _ q ) y x. q m-q 7.7 Proposition: Let xeH q(M), yeH (M,3M), x'eH m _ q(M,3M), y'eH q(M) be such that [M]nx'=x, [M]ny'=y. Then x*y=x'(y). Proof : x-y=(x'uy')[M]=x'([M]ny')=x'(y), using elementary properties of the cup and cap products. •'Now l e t ( } > : S q x D m _ q — > i n t M be a smooth embedding. Set E = S q x D m ~ q , M0=M\cf> (in t E ) , M- ,=M 0u (D q + 1 xS I , [ 1 q "*") > the r e s u l t of surgery based on $. Following [Kervaire, Milnor 1963] we w i l l consider the exact sequences of the pai r s (M , M Q ) and (Mf,Mg). As usual, we have the e x c i s i o n <j):(E,3E)—>-(M,M0) which induces isomorphisms on the r e l a t i v e homology and cohomology groups. Thinking of E as the normal tube of S q c M , l e t UeH m _ q ( E ,3E)=Z be the Thorn c l a s s , a generator ( c f . 7.6). If y= [ E]nU, then ] i = i ^ [ S q ] , i : S q — > E , and yx=U(x) for any xeH m _^(E,3E) by 7.7. This induces an isomorphism H m _^(E,3E)—>Z by property (a) above. Let j:M—*-(M,Mg) be the i n c l u s i o n . 7.8 Proposition: u* ( j A ( y ) ) = ((})^(u))«y. Proof: y (J A(y>)=U(j A(y)) = (j*U) (y) = (<|>*(y))-y, using 7.7 and 7.6, and i d e n t i f y i n g :H^(M)—(M,Mg) with the c o l l a p s i n g map J*:H*(M)-H^(M/M0)=H^(E/3E) 7.9 C o r o l l a r y : The. following sequence i s exact: 0 >-H (M0) Htl ( M ) - X - > Z — i m-q u m-q •>H (M0) HI (M) •O, m-q-1 m-q-1 where x=<t>.(y), yeH ( S q x n m q ) i s the image of [ S q ] , the o r i e n t a t i o n * q class of S 51 Proof: The sequence i s that of ( M , M Q ) , replacing Hm_q(M,Mg) by m-q using the diagram y H (E,3E). . .,, >H • (M,Mn) • m-q u and using 7.8 to i d e n t i f y x* Thus there i s an exact sequence 0 _,, .. (M')-^-vZ-^H ~(M0 )—*+H (M') 0^ q+1 u • q+1 q q where y = i ^ ( y ' ) , V *=K [ S m ~ q _ 1 ] generates H m _ q _ 1 ( D q + 1 x S m " q " 1 ) , ^ i D ^ x S ^ ^ ^ M ' i s the natural embedding, and k ' : s m" q" 1->-D q + 1xS m~ q" 1 i s i n c l u s i o n . Let X £ H r + 1 ( S q x D r + 1 , S q x s r ) = Z be the generator such that U(X)=1, and s i m i l a r l y for A'. (We s h a l l allow A and y A , A' and y'»A' to be confused.) 7.10 Lemma: i^d'(A')=9^(y)=x and i;d(A)=^ A(y')=y. Proof: Let m=q+r+l. We have a commutative diagram: . . . — > H r + 1 ( S q x D r + 1 , S q x S r ) - ^ l - ^ H j S q x S r ) - i 1 * - , H T . ( S q x D r + 1 ) -r 90. ->Hr+1-(M,M0)- ->Hr(M0)- ->Hr (M)-C l e a r l y , i f ' A £ H r + 1 ( S q x D r + 1 , S q x S r ) such that U(A)=1, then 3iA=l®[S r]£H r(S qxS r). We also have the commutative diagram (S qx S r *->H. ( D q + 1 x S r ) ->H;(M') and ±2 (1®[S ])=y' . Hence i j d (X) = ± ^ 3 * * ( X ) 0 ( X ) =ip*i 2 * (1® [S r ]) (y ' ) =y. A s i m i l a r argument proves the other as s e r t i o n . QED 7.11 Theorem: Let <j>: S qxD r -*M be an embedding, M a closed m-manifold, m=q+r+l, q<r+l. Suppose <j>A [Sq]=<j> (y)=x generates an i n f i n i t e c y c l i c d i r e c t summand of H (M).. Then rank H (M')<rank H (M) , and to r s i o n H (M') q q q q torsio n H (M), i . e . the free part of H (M) i s reduced and the to r s i o n q q part i s not increased. Further H^(M')=H_^(M) for i<q. 7.12 C o r o l l a r y : Let (f,b) be a normal map, f : (M,3M)—>(A,B) , (f|3M)^ an isomorphism, and l e t <j> :SqxDr+"'"—>-int M be an embedding which defines a normal cobordism of ( 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 of K (M). Then rank K (M')<rank K (M), and q q q torsio n K (M')=torsion R (M), while K.(M')=K.(M) for i<q. q q i i The c o r o l l a r y follows d i r e c t l y from 7.11 and Proposition 7.4. With a f i e l d of c o e f f i c i e n t s we have analogous r e s u l t s : 7.13 Theorem: Let <J> ,M be as i n 7.11, and suppose <j) A (y )=x*0 i n H^(M;F). Then .rank_H (M';F)<rank H (M;F), and H.(M';F)=H.(M;F) for i<q. F q r q i . i • 7.14 Co r o l l a r y : With the hypotheses of 7.12, suppose only that <|>.(y)=x*0 i n K (M;F) . Then rankjt (M' ;F)<rank K (M;F) and * q r q b q K ±(M';F)=K±(M;F) for i<q. The proof of 7.14 i s s i m i l a r to that of 7.12. Proof of, Theorem 7.11.: Consider the exact sequence of Corollary 7.9: 0 ^H r + 1 (MQ ) — ^ - ^ H R + 1 (M)———yZ - -^H^  (MQ ) ^ (M) K ) . Since x generates an i n f i n i t e c y c l i c d i r e c t summand, i t follows from property (a) of the i n t e r s e c t i o n p a i r i n g that there i s an element yeH r +^(M) such that x*y=l (since 3M=<}>). Hence x* i s s u r j e c t i v e and we get i A:H r(M 0)=H r(M) 0 (M0 ) - ^ ^ H r + 1 (M)—•Z—>0 (1) Consider the exact sequence of Corollary 7.9 for (M*,M ) and the diagram from Lemma 7.10: 0 >-H ,.,(]%)—>H ,-(M'")—^—>Z——>H (M 0)—^*-*H (M') •() (2) q+1 u q+1 ' ^ q| q ** H (M) q where i^d'(X')=x. Since x generates an i n f i n i t e c y c l i c d i r e c t summand, i t follows t h a t i ^ d ' s p l i t s , so that d' s p l i t s , and H q(M 0)=Z®H q(M') i;:H q + 1(M 0)=H q + 1(M') (3) From (3) i t follows that rank H (M')=rank H ( M 0 ) - l , and since q q q=r or r+1, from (1) i t follows that rank H (M)>rank H (M 0), so that q q rank H (M')<rank H (M) (the diffe r e n c e being 1 i f q=r, 2 i f q=r+l). q q From (1) i t follows that t o r s i o n H ( M Q ) i s isomorphic to t o r s i o n H (M), q q and from (3) i t follows that t o r s i o n H q(Mo)=torsion H q(M'). Hence torsi o n H (M')=torsion H (M). QED q q The proof of 7.13 i s almost i d e n t i c a l , using (1), (2), and (3) with c o e f f i c i e n t s i n F, and using property (a) of i n t e r s e c t i o n with c o e f f i c i e n t s i n F. The d e t a i l s are omitted. To proceed further i n the proof of the Fundamental Theorem, we must consider d i f f e r e n t dimensions separately; 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 cases: m odd, m=0 (mod 4), and m=2 (mod 4). §8. The Proof of the Fundamental Theorem for m odd. From Corollary "7.12 we may deduce the following theorem. 8.1 Theorem: Let (f,b) be a normal map, f : (M, 3M)—>-(A,B) , A simply-connected, (fjSM)^ an isomorphism, m=2q+l>5. There i s a normal cobordism r e l B of (f,b) to ( f ' , b ' ) , such that f':M'—>-A i s q-connected, and K (M')=torsion K (M). q q Proof: By Theorem 7.11, we may f i r s t f i n d a normal cobordism r e l B to (fl»bi), such that f]_:Mi—>-A i s q-connected. We note that the surgeries used i n 7.11 are on embedded spheres of dimension less than q, so that i t follows from Propositions 6.2 and 6.3 that K ( M i ) = K (M)©F, where F q q i s the free abelian group produced by k i l l i n g torsion classes i n K (M) Thus we may assume without loss of generality that f i s q-connected. Let x e K (M) be a generator of an i n f i n i t e c y c l i c d i r e c t summand. q Since f i s q-connected, i t follows from the Relative Hurewicz Theorem that TT ,.,(f)=H , . ( f ) , and H ... (f ) = K (M) by Theorem 5.12. Since q<^, q+1 q+± q+i q Z i t follows from the Whitney Embedding Theorem that we may represent X'STT ...(f) by (3 ,a) , 1 ... f|:" such that a i s a smooth embedding. q + 1 i q + 1 A Then 3 defines a map f :M—>-A where M=Mu D q + \ and by Theorem 6.12, since q<m-q, the obstruction to thickening M to a normal cobordism i s zero. I f x ' e r r (f) i s such that a represents x e K (M) , then by q+1 q Corollary 7.11, K (M1) has rank one. less than K (M) , and the same ~^ q q tors i o n subgroup. 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 derive an important diagram by uni t i n g the two exact sequences of C o r o l l a r y 7.9. 8.2 Lemma: We have a diagram: H q +^(M') T y 0 H I . - ( M ) X ' >Z—^- V H ( M 0 ) — ^ - > H ( M ) H D q+1 q i " H d 1 q 55 where i ^ d ' (A ' )=x=<j>^ (y) , i;d(A)=y=^(y') , y i s a generator of H q ( S q x D q + 1 ) , y' of H ( D q + 1 x S q ) , etc. q Hence, H (M')/ (i'dZ)=H (M)/(i.d'Z). q « q « Proof: This follows d i r e c t l y from Cor o l l a r y 7.9 and Lemma 7.10, and the fact that H (M) / ( i . d rZ)=H (M0 ) / (d * ZS>dZ)=H (M')/(i!dZ). QED q * q q If x=i^d'(A') i s a torsi o n element of order s, then x* i s the zero map, so that part of the diagram of 8.2 becomes the short exact sequence: 0 >Z d >H (M0) —>^ -*H (M) K) (1) q q Since i A i s a homomorphism, sd'(A')eker i a = i m d, so we have: sd'(A')=d(n)=d((-t)A)=-td(A), and sd'(A')+td(A)=0 (2) i n H (M n), for some teZ. q 8.3 Lemma; Suppose x i s a to r s i o n element of f i n i t e order s i n H^(M). 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 ) order t i f t*0. Proof: Since d(A) i s of i n f i n i t e order by (I) (which implies that d i s i n f e c t i v e ) , (2) shows that d'(A') i s also of i n f i n i t e order i f t*0 (since s*0) . C l e a r l y ty=ti^d (A)=i^ (-sd' (A ' ) )=0, since i^.°d'=0, and using (2). Hence (order y ) | t . If t'y=0, then t'i^d(A)=i^(t'd(A))=0, so t'd(A)eker i*=im d', and t'd(A)=-s'd'(A') for some seZ, or s'd'(A')+t'd(A)=0 i n H q(M 0). Applying i ^ , we get s ' i d'(A')=s'x=0, so S ' = J G « S . Subtracting I times (2) from s'd'(A')+t'd(A)=0 we get (t ' - J U ) d ( A ) = 0 . But d(A) i s of i n f i n i t e order, so t'-£t=0, or t'=£t. Hence t | t ' , and t=order y. Suppose t=0 so that sd'(A')=0. Then ker iRetorsion H^(M 0), so i ^ i s i n f e c t i v e on>dZ, and hence y=i^d(A) i s of i n f i n i t e order i n H^(M T). Consider the commutative diagram on the next page, i n which d and d' are from the exact sequences of Corollary 7.9. H (S qxS q)«-q -^-H ^ ( D q + 1 x S q , S q x S q ) = Z q+1 Z=H ^ ( S q x D q + 1 , S q x S q ) - ^ -q+1 (3) ->H (M ) q R e c a l l that X e H q + 1 ( S q x D q + 1 , s q x S q ) i s such that 3X=l®[S q], and X * e H q + 1 ( D q + 1 x S q , S q x S q ) Is such that 3'X'=[S q]®l. Suppose M i s closed, so that 3M 0=S qxS q, and <j)0 :S qxS q—> - M i s the in c l u s i o n of the boundary. Then we have the exact sequence diagram of Poincare d u a l i t y : . . . '•—>-H-q(MQ) ^ ,H q ( S q x S q ) - ^ - v H q + 1 ( M p ,S qxS q)—>. . . [M 0]n-->H q + 1(M 0,S qxS q)-,q.cqT * [S qxs q]n-( S q x S q ) — > H (M0)-q q (4) Thus, [S HxS H]n(im <}>0)=ker <t>0;%. By (3), d ,a')=(|) 0^8 ,(X')=<(. 0 A([S q]®l), and d(X)=cj) 0 A a(X)=<j) 0^(l®[S q]), so that (2) can be rewritten as <j>o*(s ([S q]®l)+t(1® [S q]) )=0. 8.4 Lemma: Let q be even. Then <J>0*(S( [S q]®l)+t (1® [S q]) )=0 implies either s=0 or t=0. Proof: Let UeH q(S q) be such that U[S q]=l. Then [S qxS q]n(U®1)=1®[S q] and [S qxS q]n(l®U)=[S q]®l, i n H q ( S q x S q ) . Hence [S qxS q]n(s(l®U)+t(U®l))=s([S q]®l)+t(l®[S q]), and by (4) i t follows that s (l®U)+t (U®l)=(}>o (z) for some z e H H ( M 0 ) . But 9 * :H 2 q(Mo)—>H 2 q(S qxs q) i s zero, as <j>0 i s the i n c l u s i o n of the (connected) boundary of M 0. Hence (s (l®U)+t (U®1) )2=<J>* ( z 2 )=0. But (s(l®U)+t(U®l)) 2 =2st(U®U) i f q=dim U i s even. Hence i t i s zero i f and only i f s=0 or 't=0. QED Proof of Theorem 7.1 for 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 torsi o n group. Let xeK (M) be the q q generator of a c y c l i c . summand of order s. Let <j>: SqxDq+^—>-M be an embedding with c|)A(u)=x, and defi n i n g a normal cobordism of ( f , b ) . Assume M i s closed, using Proposition 7.4. Consider the diagram of Lemma 8.2. By Leimia 7 .10 , i ^ d ' (A ') =x, a generator of a summand Z £ H (M). By (2) and Lemma 8.4, sd'(A')=0, so d'(A ' ) generates a s q c y c l i c d i r e c t summand Z <=H (M ) . s - q From (1) i t follows that t o r s i o n H ^ . ( M Q ) i s isomorphic to a subgroup of torsi o n H (M), and since H ( M ' ) = H (M n)/d'Z, i t follows q q q that to r s i o n H^(M') i s isomorphic to a subgroup of torsi o n H^(M) with at l e a s t one c y c l i c summand Z^ missing,,,so the same i s true for K (M'). (It follows also that rank H (M*)=rank H (M)+l.) q q q By Theorem 8.1 we may f i n d a normal cobordism of (f',b') to (f",b") with K (M").=tors ion K • (M 1) . q q I t e r a t i n g these constructions a f i n i t e number of times (since K^(M) i s f i n i t e l y generated) will.produce an ( f i , b j ) normally cobordant to (f ,b) with K^(Mi)=0, and f^ (q+1)-connected. This completes the proof for m=l (mod 4). QED Proof of Theorem 7.1 for m=2q+l, q odd: Let <j>:SqxDq+^"—*-M be an embedding which defines a normal cobordism, i . e . so that (f,b) extend over the trace of the surgery based on d>, W, . Let oo :Sq—>S0(q+l) , with SO (q+1) <P acting on Dq+"*" from the r i g h t , and define a new embedding <J>^: SqxDq+"'"—*M by <}> (x,t)=<J> (x, tm(x) ) . Then <f> defines a surgery with the r e s u l t M'=Mgli^iDC'+"'"xsq, where M Q cOmes from surgery using and oo' i s the d i f f eomorphism S q x s q — * s q x s q given by OJ ' (x,y)= (x,yoo (x) ) . 8.5 Lemma: The trace of the surgery, based on <j> also defines a normal 0) cobordism i f and only i f the homotopy class [oo] goes to zero i n T r q(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 . 58 Proof: The map <f>. : S qxD q + 1xi? k—>Mx.ff k given by <j>. (x, t ,r) = (9 (x, tto (x) ) ,r) io) io) = ( 9 (x,t),r) defines a new framing of the normal bundle to S q i n D m + k , CO i . e . of v | s q $ v ' , where v i s the normal bundle of McD m + k, v' the normal bundle of S qcM. Then <j> defines a normal cobordism i f and only i f the to framing extends to a framing of the normal bundle of Dq+^* i n D m + k x l , so that the f i r s t part of the frame defines an embedding of Dq+^xDq+"'" i n D m + k x l extending 9 :SqxDq+"'"cMcDm+k, and the second part of the frame • t o ~ extends the t r i v i a l i s a t i o n of v | 9 (SqxDq+"*") defined by b:v—>ri, to a t r i v i a l i s a t i o n of the normal bundle of D q +^*D q +\ and hence that of MxIuD q + 1x- D q + 1. Now S q=3D q + 1, D q + 1 c D m + k x I such that the normal bundle of S q i n D m + k x 0 i s the r e s t r i c t i o n to S q of y> the normal bundle of D q +^ i n D m + k x i . Now y has a framing defined on S q by the map $ :S qxD q + 1xi? k—s-E(v) , 9 (x,t ,r)== (9 (x, t) ,r) since 9 defined a normal cobordism. The differe n c e of these two framings i s a map of S q into S0(q+k+l) which i s obviously ito. Hence the frame <f>iw extends over Dq+''" i f and only i f ito i s homotopic to zero i n S0(q+k+l). QED By Lemma 6.7, IT (S0(q+r))—VJT (S0(q+r+l)) i s an isomorphism for r > l , so that ker i ^ , i-^iir (S0(q+1))—*-TTq(S0(q+k+l)) , i s the same for a l l k>l. For k=l, the exact homotopy sequence of the f i b r e space SO (q+1)—^-s-SO (q+2) ^ S q + 1 gives the r e s u l t that (ker 1 / / ) q = 3 0 7 r q + 1 ( s q + 1 ) » where 3 0 : T r q + 1(S q + 1 ) - ^ T r q(SO(q+l>) i s the boundary of the exact sequence. Hence from Lemma 8.5, i f 9:SqxDq+^—>-M defines a normal cobordism, then we may change 9 by to:Sq—>S0(q+l) i f [ t o ] e 3 0 T T q + 1 ( S q + 1 ) , and 9 ^ w i l l s t i l l define a normal cobordism. Now we w i l l compare the e f f e c t of the surgeries based on 9 and <j> . 59 Let gJ=[S 4]®l, g 2 = l ® [ S H ] e H H ( S M x S H ) . 8.6 Lemma:. Let g be a generator of ^ q+i^^"*"^) , and l e t [co]=m8o(g), • !=<I>UJ. Then <f>o,^(gi)=<)>o#(8i)+2m<j)0#(gl) , +6"^Cg£)=*0#<8i) • Proof: R e c a l l that Lemma 6.19 says that the composition ' * ., (S q + 1-)—^-»ir ( S G ( q + l ) ) — ^ / - ^ T r (S q) 1q+i 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. Now, 4>Q i s represented by the composition" S q x S q - ^ •s qxS q-—-°—>M 0-, where oo' i s given by (x,y)—Kx,yco(x)) . If y i s taken to be the base point yQeS q, then by d e f i n i t i o n ygco (x)=poo (x) , where p : SO (q+1)—>Sq i s the bundle p r o j e c t i o n . Hence on S qxy 0 , <t>'(x,yo) =<f>0 (x,pco(x)) , so <po =CP0 (l xpw)A on S qxy 0 , where A:S q—>S qxS q i s given by x—*(x,x). If g £ T r q ( S q ) i s the generator, i : (x)= (x,y 0) , i 2 (x) = (y 0 ,x) , g..= (i...)-^, then A^g=gi+g 2 , and h(g^.)=g^, where h i s the Hurewicz homomorphism. Thus, <}> 6 # <8l ) 0 # (lxpw) #A # (g) =<() o # C l x p o J ) # Cgi +"g2 ) =4> 0 # (Si + 2 mS2 ) =<('0#(gl)+2m<f,0#(g2) • Since co(y0) i s the i d e n t i t y of S0(q+1), we have <j>(51 yoxSq=cf>0 | y o x S q , so <t>^^(g2)=<j>oft(§2) • The r e s u l t i n homology follows by applying h. QED Returning to the diagram of Lemma 8.2, where d(X)=(()oA(l®[Sq]) =h<j>o^(g2)> and ' d''(A 1 )=h(|>o^(g]_) , I f w e costruct the analogous diagram using $ instead of $, we f i n d d (A)=h<(> 0 ,, (g 2 )=d (A) , and d'(A')=h(j) n ( g i ) CO CO CO If CO CO If =d'(A')+2md(A), or 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 Proposition: Let p be a prime and l e t xeK^(M) be an element of f i n i t e order such that (x) *0 i n K (M;Z ), where (•) denotes reduction P q P P mod p. Let <|):SqxDq+^—>-int M be an embedding which represents x, i . e . 4>^(y) = x) a n d which defines a normal surgery of ( f , b ) . Then one may choose co: Sq—*S0 (q+1) so that <J> : SqxDq+'1"-->int M also defines a normal surgery of ( f , b ) , order(torsion K (M*))<order(torsion K (M)), and q to q rank„ K (M';Z )<rank„ K (M;Z ). (The order of a to r s i o n group T i s L q to p L q p P P the smallest p o s i t i v e integer n such that nx=0 i n T for a l l ' xeT.) Proof: By Lemma 8.2, (M)/(x)=H q(M')/(y), where (x) indicates the subgroup generated by x. If the order of x i s s, then (2) gives sd'(A')+td(A)=0, and Lemma 8.3 states that the order of y i s t i f t*0, and 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 (A)=0, so that H (M)/(x)=H (M')/(y ) to to q q to to with order y =t-2ms i f t-2ms 0, and y of i n f i n i t e order i f t-2ms=0. yto •'to Choose m so that -s<(t-2ms.)<s, which guarantees that order y^<order x or y i s of i n f i n i t e order. Hence, order(torsion H (M')) i s not to q to larger than order(torsion H (M)), and so order(torsion K (M')) i s less q q to than or equal to order(torsion K (M)). But i f (x) *0, then by q P Corollary 7.14, rank„ K (M';Z )<rank„ K (M;Z ). QED a q to p 6 q p P P We are now able to complete the proof of Theorem 7.1 for m=3 (mod 4) Let (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 torsion group. Let p be the largest prime d i v i d i n g order K (M), and l e t xeK (M) be an element such that (x) *0 q q p i n Kq(M;'Zp)*r' By Whitney's embedding theorem we may f i n d an embedded i n t M2q+"'" representing x, and by Theorems 6.6 and 6.12, we may extend t h i s embedding to an embedding <j> :S qxD q +^—*int M such that $ defines a normal surgery on ( f , b ) . By Proposition 8.7, $ may be chosen so that order(torsion K q(M')) <order(torsion K (M)), and rank„ K (M':Z )<rank„ K (M;Z ). q Z p q p Z p q p Proceeding i n th i s fashion step by step, we w i l l f i n d a f t e r a f i n i t e number of such surgeries, a normal cobordism of (f,b) to (f^.b^) such that fi i s q-connected, order(torsion K^(M x))<order(torsion K q(M)), 61 and rank„ K ( M i ; Z )=0. Since the Universal C o e f f i c i e n t Theorem holds P q * P for the K^, K groups, K q(Mi;Z^)=K^(M x)®Z p, because K ±(Mi)=0 for i<q, and i t follows that K^(Mi) i s a torsion group of order prime to p, and order K (Mi)<order K (M). Since K (M) has p-torsion, i t follows that, q q q in f a c t , order K q(Mi)<order K^(M). Hence we have reduced the order of the kernel, and so a f i n i t e number of i t e r a t i o n s w i l l make the order of the kernal zero, thus producing a normal cobordism of (f,b) with some ( f , b ) , where f i s q-connected and K q(M)=0. Hence f i s a c t u a l l y (q+1)-connected, which proves Theorem 7.1 for m=3 (mod 4). This also completes the proof of Theorem 7.1 for m odd. §9. The Proof of the Fundamental Theorem for m even. Set m=2q. Let (f,b). be a normal map with f:(M,9M)—KA,B) such that (f | SM)^ :H^(9M)—>-H^(B) i s an isomorphism, and f i s q-connected. Then K (M)=0 for i<q, and by Poincare d u a l i t y K m _ 1(M,9M)=K m - 1(M)=0 for i<q. Since the and K groups s a t i s f y the Universal C o e f f i c i e n t Theorem, i t follows that K^M) =0 for i>q, and K^(M) i s fr e e . Let xeK^(M) be represented by an embedding a:Sq—>-int M, so that' (g , a ) e T r q + ^ ( f ) , and define M=MUotDq+1, f:M—*A extending f, defined using 3 :Dq+"^—>A. By Theorem 6.6, there i s an obstruction Oeu (V, ) (which Is Z i f q q K,q i s even, Z 2 i f q Is odd) such that 0=0 i f and only i f f:M—>A can be thickened to a normal cobordism. Let x'eKq(M,9M) be defined by [M]nx'=xeK (M). R e c a l l that, as part of our d e f i n i t i o n above of the q surgery i n v a r i a n t a ( f , b ) , we defined a b i l i n e a r p a i r i n g (*,•) on Kq(M,3M), and made use of a quadratic form ip :Kq (M, 9M; Z 2)—>Z 2 . ^•1 Theorem: The obstruction 0 to thickening f:M—»A to a normal cobordism i s given by 0=(x',x') i f q i s even, 0=ty((xf)2) i f q i s odd, where (*):2 denotes reduction mod 2. Before proving Theorem 9.1, we s h a l l use i t to complete the proof of Theorem 7.1. Theorem 4.1 states that i f (f,b) i s normally cobordant r e l B to a homotopy equivalence, then a(f,b)=0. Thus, our intent i s to • assume that a(f,b)=0, and then to construct a normal cobordism of (f,b) to a homotopy equivalence. F i r s t , suppose q i s even. Then (f ,b)=-^-I(f) , so that i f o (f,b)=0, i t follows that 1 ( f ) , the signature of (•,•) on Kq(M,9M), i s zero. By Theorem 6.13 we may assume that (M^K"*"(M,9M)=0 for i<q, and i s free for i=q. By Proposition 5.3, there i s an x'eKq(M,9M) such that (x',x')=0, so by 9.1, [M]nx'=xeKq(M) can be represented by <f>:SqxDq—>int M, ( i . e . <JK(y)=x, y the generator of H ( S q * D q ) ) , * q such that the surgery based on § defines a normal cobordism of ( f , b ) . But we may choose x' to be i n d i v i s i b l e . ( f o r otherwise, x'=kx", where x" i s i n d i v i s i b l e , and (x',x,)=0=(kx",'kx")=k2 (x",x") , so (x",x")=0), so the generator of a d i r e c t summand of Kq(M,9M). Hence, by Corollary 7.12, rank K (M')<rank K (M), and f i s s t i l l q-connected, where q q f':(M',9M')—KA,B) r e s u l t s from normal surgery based on <(> (in f a c t , the rank decreases by 2: see Lemma 8.2). Since (f,b) and (f',b') are normally cobordant, I(f')=I(f)=0 (see Theorem 5.14), and we may repeat the procedure. In f a c t , i f we i t e r a t e the process u n t i l i s reduced to zero, the r e s u l t i n g map i s (q+1)-connected, as desired. 63 Now take q odd. Then a(f,b)=c(f,b) i s the Arf inva r i a n t of on K q(M,9M;Z 2). I f a(f,b)=0, then there i s c e r t a i n l y some yeK q(M,9M;Z 2) for which iKy)=0 (see for example Pr o p s i t i o n 5.8 or 5.10). I f f i s q-connected, then K q(M,9M;Z 2)=K q(M,9M)®Z 2, and y=(x') 2 for some i n d i v i s i b l e x'eK q(M,9M). By 9.1, x=[M]nx' i s represented by <j>:SqxDq—>-int M such that 9 defines a normal cobordism, and by Corollary 7.12, rank K q(M')<rank K^(M), with f' s t i l l q-connected. But a ( f ' ,b' )=a('f ,b)=0, since (f',b') i s normally cobordant to ( f , b ) , so we may proceed as above to.produce a (q+1)-connected map. This completes the proof of Theorem 7.1, and hence of the Fundamental Theorem. The balance of th i s section w i l l be taken up by the proof of 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. Choose an xeK q(M), and l e t i t be represented by an embedding a:S q—KLnt M. Let ? q denote the normal bundle of the image of a i n M, and set M='Mu Dq'+^. Then f may be extended to f:M—>A. Let Oeu (V, ) be the obstruction to thickening 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 k>q q - i connecting homomorphism i n the exact sequence of the f i b r e bundle p:S0(k+q)—>V, =S0(k+q)/S0(q), with f i b r e S0(q). k., q We define 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 follows: l e t E =(E,S ,TT) be a k-dimensional orientable vector bundle. If Sn={ (x_^)ei?n+"'"| x§+xf+ .. .+x2 = l} , then we may define two subsets and D_ such that D + (resp. D_) i s the hemisphere centred on the N (resp. S) pole of S n, i . e . D^={(x.)eS nix SO}, and s i m i l a r l y for D n. Cl e a r l y + i n — Sn=D^uDn, and i t i s easy to show that D^nD^-Sn (the 'equator' of S n) . Since the r e s t r i c t i o n s of ? to and are both t r i v i a l , we may choose t r i v i a l i s a t i o n s T + and T_ such that x +:E|D^—>D^xfl k ( s i m i l a r l y k k for T_) . Since T + and T are f i b r e isomorphic, the map s^ii? —hff defined for each xeS*1 "*"=D^ nD^  by T °T_|_^"(x,y) = (x,s (y)) i s i n fac t an orientation-preserving l i n e a r transformation of R , i . e . s^eSCKk). Thus, we have defined a map c(E):S n• —>-SO(k) given by c(£)(x)=s x. This i s c a l l e d the c h a r a c t e r i s t i c map of E , and although i t i s not unique, i t i s well-defined up to homotopy. (Thus i t can be held that the c h a r a c t e r i s t i c 'map' i s not r e a l l y a map, but only an element of TT (SO(k)).) n — l With Z, and 0 defined as above, we have 9.2 Proposition: 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 q+k Proof: Choose a base point Jg eS0(q+k), a (q+k)-frame i n R . Let p:SO(q+k)—>V -SO(q+k)/SO(q) be the pr o j e c t i o n , given by s e l e c t i n g k., q the f i r s t k elements of a (k+q-)—frame. Let xgeS q be a base point such that, i f h:Sq—*-S0(q+k)/S0(q)=V represents 0, then h(x 0)=p(Jo) K. ,q Divide S q into two c e l l s , S q=D quD q, so that x 0eD qnD q=S q _ 1=3D q=3D q. Without loss of generality, we may assume that h(D q)=p (</Q ) , since D q i s c o n t r a c t i b l e . Let h:Dq^-*S0(q+k) be such that h ( x 0 ) = J 0 and p°h=h on D q. Then ph(S q 1)=h(S q 1)=p(J 0)» s o t h a t t h e f i r s t k elements of h(y) for yeS q ^ make up the base frame of V, . Let k,q i:S0(q)—>S0(q+k) be the representation of S0(q) acting on the subspace q+k of R orthogonal to the space spanned by p(J"o). Then there i s a map Y :-Sq-1—>S0(q) such that h(y)=J"0 (i°Y (y) ) • By the d e f i n i t i o n of 3, Y represents 30eiT q_ 1(S0(q)) (see [Steenrod 1951]). 65 Now ? i s the orhtogonal bundle to the t r i v i a l bundle spanned by h(x), for xeS q. Since h (Dq)=p (C7Q ) > the l a s t q vectors i n JQ give a t r i v i a l i s a t i o n of t, over D , and since p°h=h, the l a s t q vectors of h(x), for xeD q, give a t r i v i a l i s a t i o n of C over D q. Since y(y), for yeS q \ sends the l a s t part of JQ i n t o the l a s t part of h (y), i t follows that y i s c(£), the c h a r a c t e r i s t i c map of t, (see [Steenrod 1951, (18.1)]). QED From our discussion above of the homotopy properties of SO(n), we derive the following 9.3 Proposition: The boundary 3 : 7 r q ( ^ q ) — ^ ^(SO(q)) i s a monomorphism for q*l,3, or 7. Proof: By comparing various related f i b r e bundles, we produce the following commutative diagram: SO(q) — >-SO(q) —*SO(q)/SO(q-l)=S q - 1 1 3 SO (q+1). ^ >S0 (q+k)-'Pi s-q=v P2 P3 k+1,9"1 i , q k,q k,q where the p^ are the projections of f i b r e bundles, and i _ . are inclusions of f i b r e s . Let 9 be the connecting homomorphism i n the homotopy exact sequence of the bundle with pro j e c t i o n p^ . . By Lemma 6.9, i f q i s even, p ! , 3 i :TT (Sq)—*-TT , (S q "*") 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 . V q q-1 But by the commutativity of the diagram, P^°9]. =83 °j^ • Hence j ^ i s a monomorphism, and since by Theorem 6.12 TT (V, )=Z i f q i s even, i t q k,q follows that 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 Cor o l l a r y 6.11 ker i^=Z 2, where i ^ : T T q_ 1(SO(q) ) — ( S O (q+1)) . Hence 9 T i s onto Z 2 C T f q _ 1 ( S O ( q ) ) , 66 and since j ^ : i T q ( S q ) — ^ ^ 0 ^ ^) i s s u r j e c t i v e by Theorem 6.12, 3i =33°j^5 i t follows that 33(TT (V ))^ZZ. Since TT (V )=Z 2 for q odd (by 6.12), q k, q q k, q we have 83=8 a monomorphism for q*l,3, or 7. QED Thus for q*l,3, or 7, the obstruction 0 to doing normal surgery on a p a r t i c u l a r S q embedded i n M^q can be i d e n t i f i e d with the c h a r a c t e r i s t i c map of C , the normal bundle of the chosen S q i n M, Oeker i , , c T r ..(SO(q)), it q-1 and i s therefore zero i f t, i s t r i v i a l . Now ker i ^ i s generated by 3 j ( i ) , where i € T T q ( S q ) i s the class of the i d e n t i t y , so that 31 (t) i s the c h a r a c t e r i s t i c map for the tangent bundle T of S q. I t follows that 0=X(3 1(i)) for some XeZ. If q i s even, the Euler class x ( T)=2geH q(S q), where g i s the generator for which g [ S q ] = l . This follows from the general formula X ( T M ) = X ( M ) g J or may be deduced f o r M=Sq, q even, using the fact that i s equivalent to the normal bundle of the diagonal M i n M*M. For i f UeH q(E,E 0) i s the Thorn c l a s s , i t follows from Corollary 7.6 that [S qxS q]nri*U=[S q]®l+l®[S q] , the homology class of the diagonal, where ri:Sqxsq—>-E/Eg i s the natural c o l l a p s i n g map. Hence n U=g®l+l®g, and n (U 2)=(ri U) 2=(g®l+l®g) 2=2g®g, 2Q 9 i f q i s even. Since n i s an isomorphism on H , i t follows that U =2gU, so x ( T) =2g, since by d e f i n i t i o n x(£)U^=(U^) 2 for a bundle E . The Euler c l a s s i s represented by the u n i v e r s a l Euler class XeH q(BSO(q)), where BSO(q) i s the c l a s s i f y i n g space for oriented q-plane bundles (see [Husemoller 1966] or [Steenrod 1951]). That i s , i f c:X—>BS0(q) i s the c l a s s i f y i n g map of a q-plane bundle E over X, c (Y)= E » where y i s the univ e r s a l q-plane bundle over BSO(q), then x ( C ) = c ( x ) • q * If c:S —>-BS0(q) represents T q, then c ( x ) = 2 g as above, but i f c' *-BS0(q) represents X (T q) i n the homotopy group TT (S0(q)),then Xc and c' are homotopic, i . e . [Xc] = [c'] i n TT q(BS0(q)). Hence c' =Xc- , 67 so we have: 9.4 Lemma: If q i s even and 320=A3;L ( i ) , then x(?) =2Ag, where £ i s the normal bundle of a(S q) i n M 2 q, representing an element i n (M) , 0 the obstruction to doing a normal surgery on th i s S q, 9.5 Lemma: y ( O [ S q ] = ( x ' , x ' ) , where [M]nx'=x, a:Sq—>MZq i s an embedding representing xeK q(M), t, the normal bundle of ot(S q), as above. Proof: x(?)U=U 2 by d e f i n i t i o n of x » where UeH q(E(?)/E 0 ( O ) i s the Thorn c l a s s . C l e a r l y ( x ( ? ) ) [ S q ] =( x(C)U)[E]=U 2[E]=(n*U) 2[M], where [E]eH 2 (E(?)/E 0(?)) i s the o r i e n t a t i o n c l a s s , so [E]=n*[M], where TI :M/3M—>E/EQ i s the natural c o l l a p s i n g map. By Corollary 7.6, [M]nn U=x, so that n U=x'. Hence X(?)[S q]=(n*U) 2[M]=(x') 2[M]=(x',x'). QED By 9.4 and 9.5 for q even, (x*,x')=2A where 3 20=A3 1(i). By 9.3 3 2 i s a monomorphism for q even, so we may i d e n t i f y 0 with (x',x'), which proves Theorem 9.1 for q even. Finally., we turn our- attention to the case of q odd. Let a_^:Sq—>-M.Zq, i=l,2, be 'embe'da'ings representing x^eK q(M), where, as usual, K^(M) i s defined using a normal map ( f , b ) , f:(M,3M)—>-(A,B), (f | 3M) A :HA (3M)—>-H^(B) an isomorphism. Suppose the have d i s j o i n t images, and l e t 0\ and c92 be the obstructions to doing normal surgery on ai(Sq) and a 2 ( S q ) r e s p e c t i v e l y . Join (S q) to a 2 ( S q ) by an arc, d i s j o i n t (except, of course, at i t s endpoints) from both images. By thickening t h i s to a tube T-D qx[l,2] we may take ((*! ( S q ) \ (D qxl) ) U30TU ( < x 2 (S q) \ (D qx2) ) , where 30T=3Dqx [1,2] , D qxi=Tna ( S q ) . This subset of M i s homeomorphic to S q, and so gives us an embedding a:Sq—>-M representing x^+x2, which can be made d i f f e r e n t i a b l e by 'rounding the corners'. 9.6 Lemma: c9=c9i+t9o i n ir (V, ), where 0 i s the obstruction to doing • 1 q k,q & a surgery on a(S ). Proof: Since TcM, we may multiply T by [0 ,e] to obtain Tx [0 ,e ]<=MxI. If we have McD m + k, then M x l < = D m + k x l t and by composing embeddings we produce T x [ 0 , e ] c D m + k x i . Choose D q + 1 c D m + k x i such that a (S q)=9D q + 1, and D q + " ^ meets D m + k x 0 t r a n s v e r s a l l y i n a ^ ( S q ) . Then we may assume that a neighbourhood of a (S q) i n D q + 1 i s given by a ( S q ) x [ 0 , e j . Set D q + 1 = { D q + 1 \ ( D q x l x [ 0 , e ] ) } u { ( 8 D q x [ l , 2 ] x [ 0 , e ] ) u ( D q x [ l , 2 ] x e ) } u { D q + 1 \ ( D q x 2 x [ 0 , e ] ) } . This i s a (q+l)-rcell meeting D m + k x 0 t r a n s v e r s a l l y i n a ( S q ) , and we may smooth t h i s D q + " ' " , together with a ( S q ) , by 'rounding corners'. The smoothed D q + " ' " i s the union of three c e l l s , D q +^"=AiuBuA 2, which correspond to the three expressions i n braces, i n the expression f o r D q + ^ " above, a f t e r closure and smoothing. Assume A . <=D q + ' ' " . Then C = D . \ i n t A J i s a (q+1)-cell, 3C.n9D =FJ , F a q - c e l l i n 3D., i l i 1 l i i i l BnA^g^nA^gB and 3B\((3C 1nA 1)u(9C 2nA 2)) = S q - 1 x I . Since the d e f i n i t i o n of the obstruction 0 doesn't depend on the choice of the framing of the normal bundle y of D q + \ we may assume that the framings over D q + \ D q + " ' " , and D 2 + " ' " have been chosen so that the framings over D q + ^ and D q + ^ coincide over A.. Further we may l l assume that the framings of v, the normal bundle of M i n D ™ ^ , over a ( S q ) , a i ( S q ) , and a 2 ( S q ) , induced by b, have been chosen so that over F^ .they are a l l the same, coming from a framing of v|T (note that T i s a c e l l ) , and the framings of y, Y l J a n d Y2 m a y be assumed to extend that of v over Tna (S q) , Tna_^(Sq) (as i s appropriate). Thus the three maps 3,3., i = l , 2, 3 :a (S q)—*V. , 3 . :a, (S q)—•V, l k,q l i k,q def i n i n g 0 and 0^, may be taken to be the base k-frame over Tna(S q), T n a i ( S q ) , and 3|(a ( S q ) n a ( S q ) ) = 3 ± | ( a ( S q ) n a ( S q ) ) . It follows that for the homotopy class e s , [3] = [ 3 i ]+[621 ± n 7 7 ( v v )» o r 0=0i+02 . QED q k,q 9.7 Lemma: If 0=0, then I|J ((x T ) 2)=0 , with notation as above. Proof: Since 0=0, we can perform normal surgery based on a:Sq—>-M2q, so that the trace i s a normal cobordism W2q+"'", W=Mu (3M*I)uM' , and i f i:3W—*W and k:M—>3W are i n c l u s i o n s , i.k.x=0. I t follows from * * elementary r e s u l t s about and K (see p. 21 above), that x"=i z, zeK q(W), where x"ekq(3W) i s defined by [3W]nx"=k.,.x, and Kq(W) comes from the map F:W—>AxI extending f on M. If K q(3W;Z 2) i s defined f o r the map 3F:3W—*Ax0uBxIuAxl, and T|JQ i s the quadratic form Kq(3W;Z2)—>-Z2 used i n the d e f i n i t i o n of the Kervaire i n v a r i a n t , i t follows from a lemma i n [Browder 1972, III.4.13] that i>oX{i z) 2)=iK(x") 2)=0. Now 3F i s c l e a r l y the sum of (f,b) on M and (f',b') on M' (the r e s u l t of surgery). By an intermediate r e s u l t i n the proof of Theorem 5.12, i|>0 (n* (x' ) 2 )=iji ( (x* ) 2 ) , x'eKq(M,3M), so i t remains to show that n (x') 2=(x") 2 (where n:3W—*M/3M). Consider k^x=k^ ( [M] nx' )=k^(ri^ [3W] nx' ) = [3W] nri x', using i d e n t i t i e s of the cap product ( c f . Corollary 7.6), so that since [SWjnx'^k^x, i t follows that x"=n x', and hence i K(x') 2)=0. QED Now we prove that 0=iK(x') 2).. If 0=0, then iK(x') 2)=0 by 9.7, so i t remains to show that i f 0=1 then 4 )((x') 2)=l. By taking the connected sum with the map S qxS q—*S 2 q, or a l t e r n a t e l y doing a normal surgery on. a S q " * " c D / q c M 2 q , we may add to K^(M) the free module on two generators a x and a 2 , corresponding to [S q]®l and l®[S q] i n H q ( S q x S q ) , and add to Kq(M,3M) the elements g :,g 2 such that [M#(S qxs q)]ng i=a ±, with ( g 1 , g 2 ) = l , (g i,g ±)=0, 1=1,2, orthogonal to the o r i g i n a l Kq(M,3M), and such that T/J (g x )=ij> (g 2)=0. Hence if) (gi+g 2)=^ (gi)+4<(82)+(gl >82)=1-If g r S ^ M i (S^xS^) represents the diagonal class ai+a 2, i t follows from 9.7 that the obstruction c9 to surgery on $ i s 1, since i f i t were zero, then i(>(gT+g2) would be zero.. Then on the sum embedding a+3 representing x+(ai+a 2), the obstruction 0"=0+0' by Lemma 9.6, so that 0"=1+1=O.. Hence if) ((x' ) 2+(gl+ g 2 ) )=0 by 9.7. But since ((x' ) 2 , (gi+g 2 ) )=0, * ( (x') 2-Kgl +g 2 ):) (• (x' ) 2 ) n ( gi +82 ) ( (x' ) 2 ) +1=0 > we see that ip ( ( x ' ) 2 ) = l . QED ThiV completes the proof of Theorems 9.1 and 7.1, and thus of the Fundamental Theorem. 71 Chapter I I I . Plumbing and the C l a s s i f i c a t i o n of Manifolds. §10. Intersection and Plumbing. Let Ni and N 2 be smooth submanifolds of dimension p (resp. q) of a smooth m-manifold M, such that p+q=m. A point xeNjnN 2 w i l l be c a l l e d d i s c r e t e i f there i s an open neighbourhood V of x i n M such that VnNi nN2={'x} . Note that i f every point i n N^  nN2 i s d i s c r e t e , then N'f"nN2 "is a d i s c r e t e subset of M. I f xeNinN 2 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 and VnNinN2={x}) then (V\Ni)u(V\N 2)=V\{x}. Thus we have a p a i r i n g H q(V,V\N 1)®H P(V,V\N 2)—>H P + q(V,V\{.x}) given by the r e l a t i v e cup product. Suppose that M, Ni and N 2 are oriented, and l e t [M] xeHm(M,M\{x}), [Nj] yeH p(N!,Ni\{y}) and [N 2] zeH (N 2,N 2\{z}) be the generators compatible with the o r i e n t a t i o n s . Let E_^ , i=l,2, be a tubular neighbourhood of N ± i n M, E°=E_L\N_L. Then the i n c l u s i o n (E i,E°)c(M,M\N i> i s an e x c i s i o n , so H (M,M\NJ)=H (E.,E°). If the E. are oriented, and r . denotes the i l l 1 i i n c l u s i o n (V,V\N^)c(E i,E°), then by the Thorn Isomorphism Theorem there i s an element Ui eHq(E]_ ,E? ) such that r*UieH q(V,V ) i s a generator, and ' U T J i , • nUj are isomorphisms ( s i m i l a r l y for N 2). We s h a l l also assume that the orientations are compatible, i . e . so that [M] nr.U =[N.] x i i I x for xeN.. l Under the preceding conditions we may define the sign or o r i e n t a t i o n * * of a d i s c r e t e point xeN^nN2 by sgn(x)=(riUiUr 2U 2)[M]^, using the p a i r i n g above. We s h a l l c a l l x a (homologically) transverse point of i n t e r s e c t i o n i f sgn(x)=±l. Note that geometrically transverse points are also homologically transverse. (A point xeNjnN 2 i s geometrically transverse i f x has an open neighbourhood V i n M such that there i s a diffeomorphism (V.VnNi ,VnN 2)—>(i? m,i? PxO,Oxi? q) .) If N} i s compact and N}nN2n3M i s empty, i t has been shown that given an e>0 there i s a diffeomorphism 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 to 1^, such that h(Ni)nN 2 consists s o l e l y of (geometrically) transverse points. On p.50 above we defined a p a i r i n g • : H^ (M) ®H^ (M)—>Z by x*y=(x'uy') [M.]., where x'eHq(M,3M), y'eHP(M) are defined by [M]nx'=x, [M]ny'=j Ay, and j i s i n c l u s i o n . Let NP,- N 2 be compact oriented submanifolds of a compact ' oriented manifold with boundary, m=p+q, and suppose Ni_ i s closed i n M, 3MnNj=cf>, and . 8MnN2 = 8N 2. Assume further that Nj and N 2 i n t e r s e c t (homologically) transver s a l l y . Let i :Nj—»-M denote . the i n c l u s i o n s . We state without proof the following theorem from [Browder 1972], 10.1 Theorem: ( i i ^ t N i ] ) • ( i 2 ^ [ N 2 ] ) = Z s g n ( x ) , where the sum i s taken over a l l points xeN^ nN 2. Thus, the i n t e r s e c t i o n of the o r i e n t a t i o n classes counts the number of i n t e r s e c t i o n points, with sign. If N q i s a closed submanifold l y i n g i n the i n t e r i o r of M 2 q, with normal bundle ? q, then we may consider how N in t e r s e c t s i t s e l f . I t i s possible (see above) to change N by an e-isotopy so that i t i n t e r s e c t s i t s e l f t r a n s v e r s a l l y . Then Theorem 10.1 gives us: i^[N]•i^[N]=Isgn(x), the sum running over the points of s e l f - i n t e r s e c t i o n . However, we can also i n t e r p r e t t h i s r e s u l t using the normal bundle £: 10.2 Proposition: i^[N]•i^[N ] = x(?)[N], where x ( ? ) 'is the Euler class of We are now prepared to describe the construction known as plumbing dis c bundles. Let t,. be a q-plane bundle over a smooth q-manifold , and l e t E be the t o t a l space of the closed d i s c bundle associated to ?.. 1 Suppose that and are oriented compatibly for 1=1,2. Choose x.eN. and B.cN. a q - c e l l with x.eint B_, . Since B. 1 1 1 1 i i l i s c o n t r a c t i b l e , t,. B. i s t r i v i a l , and that part of E, l y i n g i i i over B. i s diffeomorphic to B,xD., where D. i s a q-disc, such l i l l that the f i b r e s are mapped to x*D^. We may choose diffeomorphisms h_,h+:B1->D2, k_,k+:D!->B2, where a subscripted + indicates orientation-preserving, and a -indicates o r i e n t a t i o n - r e v e r s i n g . We plumb E]_ with E 2 at x^ and x 2 by i d e n t i f y i n g the subsets of the d i s j o i n t union E i u E 2 given by B^xD^ and B 2xD 2 using the map I +(x,.y) = (k y ,h +x) or the map I_(x,y)=(k_y ,h_x) . We s h a l l say that the plumbing i s with sign +1 i f .1 i s used, and with sign -1 i f I_ i s used. The r e s u l t i n g manifold i s denoted by ET_DE 2, and i t can be smoothed i n a canonical way. Since both of I and I preserve o r i e n t a t i o n i f q i s even, and reverse i t i f q i s odd, Ei_DE2 can be oriented compatibly with N j . ^ i , N 2, and C 2 If 1 l s even, and with Ni,r,n ,-N2, and c;2 i f q i s odd. Note.that ,N^cE^cE 1DE 2, where the inclusions are obvious, and that N}nN2={x^}={x2} ( i n E^DE 2), which i s a transversal i n t e r s e c t i o n , and that the sign of x i s the same as the sign of the plumbing. (Of course, a l l of t h i s discussion can be applied to the case of plumbing one manifold with i t s e l f , i f we choose two d i s t i n c t points i n i t and take E}=E2.) If we choose several p a i r s / o f points i n and N 2, we may plumb Ei and E 2 together repeatedly, choosing the sign of each plumbing. We w i l l s t i l l denote the r e s u l t by EiDE 2, and we see from 10.1 that i-l * [Ni ] * i-z-k t^2 ] i s determined by the way we choose the sign of the plumbings. Thus, i f we choose a number n\2, and plumb- Ei with E 2 at n X2 points, always with sign +1, then we have i ^ [NT_ ] • ±2^[N2 ]=nj2 • We may go on to plumb with other d i s c bundles, by making sure that the points i n NT_UN2 we choose to plumb at are we l l away from the f i n i t e number of points i n Ni_nN2, and by choosing the signs of the plumbings, we may cause i . [N. ] • i . [N. ]=n M , j*k, to take on any value * 3 * k j k we l i k e . (Note that we must have n =(-l) qn., .) The s e l f - i n t e r s e c t i o n s kj J k are determined by the Euler class x ( £ ^ ) > according to Proposition 10.2. Thus, we a r r i v e at the remarkable 10.3 Theorem: Let M be a symmetric n><n matrix with integer e n t r i e s , 4k and with even diagonal e n t r i e s . Then for k>l there i s a manifold W with boundary such that W i s (2k-l)-connected, 8W i s (2k-2)-connected, H 2 k(W) i s free abelian, the matrix of the i n t e r s e c t i o n p a i r i n g * * 2 k 8 ^ 2 k — ^ ^ S S^- V e n by M (or equivalently, M i s the matrix of the b i l i n e a r form (•,•) on H.(W,8W)), and there i s a normal map ( f , b ) , with 4k 4k-1 f: (W, 3W)—KD ,S ) f o r which M i s the i n t e r s e c t i o n matrix on ^ ^(W) . The proof i s provided i n d e t a i l i n [Browder 1972]. We have from the same source the 10.4 Lemma: In the construction of 10.3, 3W i s a homotopy sphere i f and only i f the determinant of M i s ±1. Consider the following 8x8 matrix due to Hirzebruch: M0 = 0 2 1 1 2 1 1 2 1 1 2 1 1 2 1 0 1 1 2 1 0 0 1 2 0 1 0 0 2 0 This matrix i s , as required, symmetric and even on the diagonal. Simple computation shows that |M Q|=1 and that the signature of Mg i s 8. We may quickly prove the following theorem of Milnor. 10.5_ Theorem: Let k>l. There i s a manifold W and a normal map ( f , b ) , 4k 4 k-l j f:(W,8W):=^(D ,S ) such that (f|3W) i s a homotopy equivalence, and a(f,b)=l. Proof: Let W be the 4k-manifold with boundary constructed i n Theorem 10.3 using the matrix Mg. Since f M Q[=1, we have by 10.4 that 3W i s a 2k homotopy sphere. By 10.3, the b i l i n e a r form (*,*) on K (W,3W) has matrix Mg, and sgn Mg=8. Thus, i f (f,b) i s the normal map of 10.3, i t follows, that a (f ,b)=~rl (f )=isgn M 0=l. QED o o A somewhat.different construction i n dimensions congruent to 2 mod 4 gives us the following theorem of Kervaire. 10.6 Theorem: For q odd there i s a manifold U and a normal map (g,c) such that g:(U,3U)—>(D 2 q,S 2 q _ 1) with a(g,c)=l. Taking Theorems 10.5 and 10.6 together with Proposition 5.35 (the Addition Property of a), we derive immediately the Plumbing Theorem: 10.7 Theorem: I f m=2k>4, then there i s an m-manifold M with boundary, and a normal map (g,c), g:(M,3M)—>-(D ,S ), c:v —>-e (where e i s the t r i v i a l bundle over D™), with g|9M a homotopy equivalence and with a(g,c) taking on any desired value. ; \1. The HomotoT-v Types of Sicocth xl&nifoHn and Class!i' : cation.. It: has si-iows by [Browdar 1-962]. a-i'': [Ncvikov 1964] •• th* ;, - ': 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 of Smooth Manifolds 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 c e r t a i n necessary conditions for a space to be of the homotopy type of a smooth manifold are sometimes also s u f f i c i e n t . In the theorem we w i l l use the following notation: h:Tr.—>-H. i s the i i Hurewicz homomorphism, E i s an oriented 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 Pontrjagin classes, and L, are i k the Hirzebruch polynomials. 11.1 Theorem: Let X be a simply-connected Poincard complex of dimension m>5, E an oriented k-plane bundle over X, k>m+l, aeir ,. (T(E )) such that m+k. h(a)nU=[X]. If (1) m i s odd, or (2) m=4k and index X=(L^(pi,P2,...,p^))[X], then there i s a homotopy equivalence f :M—*-X, for some smooth m-manifold * m+k M, such that v=f (E) i s the normal bundle of an embedding McS , and f can be found i n the normal cobordism class represented by a. Outline of Proof: A representative f:S m + k—>T(E) of a i s chosen, and the manifold.M i s defined by p u l l i n g X back to a submanifold of S m + k v i a f (after some modifications). The map f induces a normal map (f,b) with f:M—*X, b:v—>E. Then by the Fundamental Theorem of Surgery (4.2), (f,b) i s normally cobordant to a homotopy equivalence i f m i s odd, and i f m=2q then (f,b) i s normally cobordant to a homotopy equivalence i f and only i f a(f,b)=0. But i f m=4k, then by the Index Property (Proposition 5.35), a(f,b)=(L (p :,...,p ))[X]-index X, which i s zerotwhen (2) holds. QED K. K. Remark: If m=6,14,30, or 62 (none of which are covered by 11.1), then with the above hypotheses there i s a homotopy equivalence f:M—>X with" f (E)=v, but f may not be normally cobordant to a map representing a. ,77 We have defined ab ove the connected sum of Poincare complexes for the purpose o f the Addition Property. Given Poincare pa i r s ( X _ ^ , Y ^ ) , k-plane bundles over X_^ , smooth manifolds (M^,3M_^), and normal maps ( f i , b i ) such that f ± : (M,SM^)—+ ( X ± , Y ± ) , we have the Poincar£ p a i r ( X i # X 2 J Y I U Y J ) , the smooth manifold M^#M2 with boundary 3 M T U 3 M 2 , and the normal map (f j #f 2 .bj #b2 ) such that f i #f 2 : (Mx #M2 , 3MT_IJ 3M 2 )—»-(X! #X2 , Y T U Y 2 ) and bj#b2 :vj~Ki#?2 » where i s the normal bundle of MjZ/M^  i n D™"^. If and Y ^ are a l l nonempty, we may define the connected sum along (components o f ) the boundary. See [Browder 1972] for d e t a i l s . We produce analogous constructs: M^iLM2 , X]UX2, and maps f i J l f 2 , b|JLb2 • Note that 3 ( M T O I I ^ ) = 3M} #8M2 , and that .(XiliX 2 JY\#Y2) form a Poincarg p a i r . Then (fjlLf2 ,b]rJLb2) i s a normal map. 11.2 Proposition: Let (f,b),(g,c) be normal maps with f : (M,3M)—>-(X,Y) , g: (N,3N)—KD m,S m - 1) • . Then ( f i l g , b i l c ) i s normally cobordant t o ( f , b ) . This proposition together with previous r e s u l t s leads to the 11.3 Theorem: Let ( X , Y ) be a m-dimensional Poincar£ p a i r with X simply-connecetd and Y nonempty, mfc5, and l e t (f,b) be a normal map with f: (M,3M)-*(X ,Y) and (f|3M)^ an isomorphism. Then there i s a normal map (g,c), g:(U,3U)—»-(D ,S ) with g|3U a homotopy equivalence, such that (fjLg,hu.c) i s normally cobordant r e l Y to a homotopy equivalence. In p a r t i c u l a r , (f ,b) i s normally cobordant to a homotopy equivalence. Proof: By the Plumbing Theorem (10.7) there i s a (g,c) as above with a(g,c)=-c(f ,b) . By the Addition Property, Proposition 5.35, a ( f j i g,bjLc) =a(f,b)+a(g,c)=0, s o - b y the Fundamental Theorem (4.2) (fjLg,b!Lc) i s normally cobordant r e l Y to ( f ' , b ' ) , where f':M'—>X i s a homotopy equivalence. (Note that ( X A D M , Y # S m _ 1 ) = ( X , Y ) ) . Then 11.2 shows that (f,b) i s normally cobordant t o ( f ' , b ' ) . QED R e c a l l that a cobordism W between M and M' ( i . e . 9W=MuUuM', 3McTJ, 3M'cTj) i s an h-cobordism i f the incl u s i o n s McW, M'cW, 9McTJ, and 9 M ' C T J are a l l homotopy equivalences. With t h i s d e f i n i t i o n we can state the c l a s s i f i c a t i o n theorem of Novikov, and i t s c o r o l l a r y . 11.4 Theorem; Let X be a simply-connected Poincare complex of dimension m>4, and (f-£>D^) for i=0,l , be normal maps with f_^ :M_j—->-X, where i s a smooth m-manifold. Suppose that fg and f^ are homotopy equivalences. I f fg i s normally cobordant to fi, then there i s a normal map (g,c) with g:(U,9U)—>(T>m+^~,Sm) , where g| 9U i s a homotopy equivalence, such that (fg,bg) i s h-cobordant to (f\ g| 3 U,bi c | 9 U ) . In p a r t i c u l a r , Mg i s h-cobordant to Mj_ i f m i s even, and to Mj#(9U) i f m i s odd. 11.5 C o r o l l a r y : Let M and M' be closed smooth simply-connected manifolds of dimension not less that 5. A homotopy equivalence f :M—*-M' i s homotopic to a d i f f eomorphism f':M#E—>-M' for 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 only i f there i s a bundle map b:v—*v ' covering f such that T(b)^(a)=a', where a,a' are the natural c o l l a p s i n g maps a £ % + k ( T ( v ) ) > a ' - m + k ( T ( V , ) ) -F i n a l l y we have a theorem of Wall and i t s c o r o l l a r y . 11.6 Theorem; Let (X,Y) be a Poincare p a i r of dimension m>6, with both.X and Y simply-connected, Y nonempty. Let E be a k-plane bundle over X, and choose aeir (T(E) ,T(E IY)) such that h(a)nU=[X]. Then m+k the normal map represented by a i s normally cobordant to a homotopy equivalence ( f , b ) , f:(M,9M)—>(X,Y), which i s unique up to h-cobordism. In p a r t i c u l a r , (X,Y) has the homotopy type of a d i f f e r e n t i a b l e manifold, 79 unique up to h-cobordism i n the given normal cobordism c l a s s . We w i l l prove the existence part of t h i s theorem. The proof of uniqueness (as w e l l as the other proofs omitted from t h i s section) i s to be found i n [Browder 1972, I I . 3 ] . Proof: Let (f',b') with f':(Mr,9M')—>(X,Y) be a normal map representing a. By the Cobordism Property, 5.36, a(f'|9M',b'|9M')=0, so that by the Fundamental Theorem (4.2) (f'|9M',b'[9M') i s normally cobordant to a homotopy equivalence. This normal cobordism extends to a normal cobordism of (f',b') to some (f",b") such that f"|9M" i s a homotopy equivalence. By Theorem 11.3, (f",b") i s normally cobordant to a homotopy equivalence, ( f , b ) . 11.7 C o r o l l a r y : Let M and M' be compact smooth simply-connected manifolds of dimension m>6, with 9M and 9M' simply-connected and nonempty. Then a homotopy equivalence f : (M, 9M)—• (M' , 9M1) i s i s o t o p i c to a d i f f eomorphism f' :M—>-M' i f and only i f there i s a bundle map b:v—>-v' covering f such that T(b)^(a)=a', where v ,v ' are the normal bundles, and aeir t 1 (T(v) ,T(v I 9M) ) , a'eiT (T (v ' ) ,T(v ' | 9M' ) ) are the m+k m+k co l l a p s i n g maps. Bibliography Browder,W.: Homotopy type of d i f f e r e n t i a b l e manifolds. Proceedings of the Aarhus Symposium, 1962, 42-46. Surgery on simply-connected manifolds. Berlin-Heidelberg-New York: Springer 1972. Hirzebruch,F.: New to p o l o g i c a l methods i n algebraic geometry. 3rd Ed. Berlin-Heidelberg-New York: Springer 1966. Husemoller,D.: Fibre bundles. New York: McGraw H i l l 1966. Kervaire,M. : An i n t e r p r e t a t i o n of G. Whitehead's generalisation of 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. Milnor,J.: On manifolds homeomorphic to the 7-sphere. Ann. Math. 64 (1956), 399-405. A procedure for k i l l i n g the homotopy groups of d i f f e r e n t i a b l e manifolds. Symposia i n Pure Math., Amer. Math. Soc. 3 (1961), 39-55. Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow. Princeton: University Press 1965. C h a r a c t e r i s t i c classes. Princeton: U n i v e r s i t y Press 1974. Morse,M. : Relations between the numbers of c r i t i c a l points of a r e a l function of n independent v a r i a b l e s . Trans. Amer. Math. Soc. 27 (1925), 345-396. Novikov,S.P.: Homotopy equivalent smooth manifolds 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 topologiques des variet£s feuillet£es, Actual, s c i . industr. 1183, P a r i s , 1952, 91-154. Serre,J.-P.: Homologie s i n g u l i e r e des espaces f i b r e s . A pplications, Ann. Math. 54 (1951), 425-505. Smale,S.: Generalized.Poincare conjecture i n dimensions greater than four, Ann. Math. 74 (1961), 391-406. Steenrod,N.: The topology of f i b r e bundles. Princeton Math. Series 14. Princeton: University Press 1951. Epstein,D.B.A.: Cohomology operations. Annals of Math. Studies No. 50, Princeton. Univ. Press 1962. Thorn,R.: Quelques propri£t£s globales des varietds d i f f e r e n t i a b l e s . Comment. Math. Helv. 28 (1954), 17-86. 

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