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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The fundamental surgery theorem and the classification of manifolds Cameron, Richard Bruce 1980
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Title | The fundamental surgery theorem and the classification of manifolds |
Creator |
Cameron, Richard Bruce |
Date Issued | 1980 |
Description | The purpose of this paper is to present a survey of some important results in the classification of differentiable manifolds. We begin with the Poincaré conjecture and its partial solution 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 their work, and we present its solution in the simply-connected case, by Browder. This solution amounts to the surgery invariant theorem, the fundamental surgery theorem and associated results. We end our discussion with the plumbing theorem, and several important classification theorems of Browder, Novikov and Wall. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080354 |
URI | http://hdl.handle.net/2429/22469 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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