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UBC Theses and Dissertations

Cross-sections of the sphere and J-theory Mauro, David J. 1981

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CROSS-*-SECTIONS OF THE SPHERE AND J-THEORY ^ — B . S c , U.B.C., 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mathematics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1981 © David J. Mauro, 19 81 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 an advanced degree 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 and s t u d y . I f u r t h e r a gree 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 department o r by h i s o r h e r 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 . Department o f Mathematics  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date A p r i l 29, 1981 i i ABSTRACT Let. Fj denote the f i e l d R or C. Let 0 n ) k denote the man i f o l d of orthonormal k-frames i n F n, where 1'S k < n. We can f i b r e over ~ ^ n ^ by t ak i ng the l a s t v ec to r i n each k-frame. Here d = 1 or 2 , accord ing to whether F equals R or C. A c r o s s - s e c t i o n , S^ 1 1 -^ »- 0 n > k , ass igns to each po in t u e S^n- l an orthonormal k-frame (u-p U 2 , . . , u^..^, We wish to determine va lues f o r n and k which w i l l guarantee the ex i s t ence of such a c r o s s - s e c t i o n . In the r e a l case t h i s i s the c l a s s i c a l v e c t o r f i e l d s on spheres problem. A t i y a h and James prove that the c r o s s - s e c t i o n i n g problem i s equ i va len t to a problem i n J - t h e o r y . Let J (X) denote the set of equ iva lence c l a s se s of s t a b l e f i b r e homotopic orthogonal sphere bundles over a f i n t e CW-complex X. I t i s w e l l known that J (X) i s a f i n i t e a b e l i a n group. I f a i s an F |_ vec to r bundle over X, we can a s s o c i a t e w i t h i t a unique sphere bundle which we w i l l denote by (a ) . The c l a s s of (a) w i l l be denoted by J (a ) . Let £ denote the c anon i ca l r e a l d-dxmensional Hopf l i n e bundle over the F - p r o j e c t i v e space P^. The l i n k between c r o s s -s e c t i o n i n g and J - t heo r y can be s t a ted as f o l l o w s : The S t i e f e l f i b r i n g 0n^ >• S d n - 1 admits a c r o s s - s e c t i o n i f and on ly i f n i s a m u l t i p l e of the order of J (5) i n J ( P k ) . Thus the problem of f i n d i n g c r o s s - s e c t i o n s has been reduced to determin ing the J - o r d e r of E, i n J ( P k ) . i i i TABLE of CONTENTS 1. Vecto r Bundles 1 Induced Bundles 4 2. The Rings K p (X ) 6 Products 11 Bott P e r i o d i c i t y 13 The Adams Operat ions 14 P r o p e r t i e s of ¥ 17 Some Weil-Known Resu l t s 17 3. The J-groups 19 The J ' - and J " -groups 22 The J " -groups 23 The Adams Conjecture 23 The J* -groups 25 a The Chern Character 26 t in -c las ses and J ' - g r oup s 26 sh - c l a s se s and J R - g roups 27 Two Important Isomorphisms 29 4. Vecto r F i e l d s on Spheres 29 5. The At iyah-James Theorem 38 S-type and S - d u a l i t y 39 S - r e d u c i b i l i t y and S - c o r e d u c i b i l i t y 40 Thorn Spaces 42 A t i y a h ' s D u a l i t y Theorem 44 Thorn Spaces and J(X) 45 The Main Theorem 47 6. Complex C ro s s - sec t i on s 49 The J - o r de r of 5 i n J ( C P k ) 50 1 1. Vector Bundles A f i b r e bundle £ over a space X i s a t r i p l e (E , p , X) , where p : E >• X i s a cont inuous map, together w i t h the f o l l o w i n g p roper ty . There e x i s t s an open cover ing {U} of X and, f o r each po i n t x i n Ue {u} , a homeomorphism <J) : U x p "*"(x) *• p "^(U) such that the . f o l l o w i n g diagram commutes where fr i s p r o j e c t i o n to the f i r s t f a c t o r . We c a l l the space E , sometimes w r i t t e n as E(£) , the t o t a l space, the space X = X(£) the base space, the space p ^(x) = E^ the f i b r e and the map p the proj ect i on map . Let ? = (E , p , X) and £ ' = (E ' , p ' , X ' ) be two f i b r e bundles. A bundle morphism i s a p a i r (u , f ) : £ >• where u : E >• E' and f : X >• X ' such tha t the f o l l o w i n g diagram commutes U x p ( x ) u P X ( u ) u E u E' P P X f 2 I f £ and £' a re two bundles w i t h the same base space X , then a bundle morphism i s j u s t the p a i r (u , l x ) : Z, > £ ' . In t h i s case we c a l l the morphism an X-morphism and denote i t by u : £ • • A c r o s s - s e c t i o n - o f a! bundle- (E , p , X) i s a map s : X >• E such tha t ps = 1 . That i s , a c r o s s - s e c t i o n i s a map s : X > E such that s(x) e p ^"(x) f o r each x e X . Let F denote one o f the bas i c f i e l d s R o r C . R e c a l l that i f V i s the n-d imens iona l v e c t o r space F n , then the inner product f o r ' n ' by < x , y> = E x ^ y ^ • Here denotes the con jugat ion ope r a t i on . the elements x = ( x 2 , . . . , x^) and y = (y 1 , . . . , y^) of V i s g i ven £ x . y . i 1 1 Examples 1. Let 0 be the S t i e f e l v a r i e t y o f orthonormal k-frames i n F n . n,k We can t h i n k o f 0 : . as the subspace of (v , . . . , v , ) e ( S^ n ^) where n, K. K. < v ^ , v_.. >T.= 6\.. . The i n t ege r d equals 1 or 2 acco rd ing to whether F equals R or C . S ince 0 i s a c l o sed subset of a compact XX j AC space, i t i s a compact space. In f ace 0 admits the s t r u c t u r e of n,k a man i fo ld (compare Steenrod [20] ) . For m < k we have a f i b r e bundle (0 , , p , 0 ) where p n,k ' r ' n,m v ass igns t o each k-frame i n 0 , the l a s t m v e c t o r s . I f ° n,k v = (v, , . . . , v ) i s an element of 0 then p ^(v) w i l l c o n s i s t 1 ' ' m' n,m r of elements of the form (w, , . . . , w, , v, , . . . , v ) where. 1 k-m 1 m < w;; -, w.. > = <5. . ' a n d .-s^ w... v.-->=0 The vec to r s w , .. . , w, l i e i n i - ' • x j 1 - j k-m the o r thogona l complement of the space spanned by v^ , . . . , v . The 3 dimension of the o r thogona l complement i s n - m . Thus we can t h i n k of (w , . . . , w^ ) as an or thogona l set of vec to r s i n F n m . Therefore the f i b r e of the bundle can be i d e n t i f i e d w i t h 0 . n-m , k-m Where d i s t i n c t i o n i s necessary we denote 0 7 by V , i n the r e a l case and W , i n the complex case . n,k r 2. S ince S^ ^ c f , the sphere S^ n ^ " c p " has the s t r u c t u r e of an r.d-1 . „d - l „dn- l „dn- l . . fc. c , S - space where S x S > S i s the r e s t r i c t i o n of the v e c t o r space s t r u c t u r e o f F n over F to S^ n ^ . The quo t i en t space „dn—1 , , , . _ „d—1 . , __n—1 , . S modulo the a c t i o n of S i s the space FP , the r e a l o r complex p r o j e c t i v e space. Let q : S^ n > F P n ^ be the quot ient map. Then the t r i p l e (S^ n , q , F P n "*") i s a f i b r e bundle w i t h f i b r e S^ ^ . Th is i s the c l a s s i c a l Hopf f i b r a t i o n . An n -d imens iona l v e c t o r bundle £ over X i s a f i b r e bundle (E , p , X) where each f i b r e p "'"(x) has the s t r u c t u r e of an n -d imens iona l vec to r space F n over F and such that the r e s t r i c t i o n of $ to n —1 x x F — > p (x) i s a v ec to r space isomorphism f o r each x i n U e {U} . I f £ =' (E , p , X) and = (E 1 , p 1 , X ' ) a re two vec to r bundles then a v e c t o r bundle morphism i s a morphism (u , f ) of the unde r l y i n g bundles such that the r e s t r i c t i o n u : p "''(x) >• (p 1 ) ^"f(x) i s l i n e a r f o r each x e X . I f X = X ' then u : Z, > i s an isomorphism i f t he re e x i s t s a morphism v : £ > £ ' such that vu = 1 and uv = 1 , . 4 Induced Bundles I f a = (E , p , X) i s a bundle and f : Y *• X i s a map then the induced bundle f * ( a ) over Y has t o t a l space c o n s i s t i n g of the subspace of p a i r s (y , e) eY x E w i t h f ( y ) = p(e) , and p r o j e c t i o n map q w i t h q : (y , e) • y . We can de sc r i be the induced bundle as the p u l l b a c k of the f o l l o w i n g diagram E P Let £ and be two bundles over X . Then the Whitney sum, 5 © £' , of the bundles i s the p u l l b a c k of E X E ' p x p . X • X X X where A i s the d iagona l map. Example The Grassman v a r i e t y G^(F n ) i s the space of k -d imens iona l subspaces of F n w i t h the quo t i en t topology de f i ned by g : 0^ ^ > G^(F n) where g(v^ , . . . , v^) = <v^ , . . . , v ^ > , the subspace spanned by the k-frame (v, , . . . , v, ) . The c anon i c a l k -d imens iona l v e c t o r bundle 5 y£ = (E , p , G k ( F n ) ) on G k ( F n ) i s the bundle w i t h t o t a l space c o n s i s t i n g of the subspace of p a i r s ( V , x) £:G^(F n) x F n where x e V and p : (V , x) • > V . A s p e c i a l case of t h i s example occurs when k = 1 . Then G^(F n ) = F P n ^ and the c anon i ca l v e c t o r bundle y^ i s c a l l e d the c anon i c a l l i n e bundle. In t h i s case we w r i t e y\ = £ . 1 n The or thogona l complement bundle *y!J o v e r G , (F n ) i s the rv iC bundle w i t h t o t a l space the subspace of p a i r s (V , x) eG^ (F n ) x F n where <V , x > = 0 . That i s , x i s o r thogona l to V . A t r i v i a l bundle over a space X i s one w i t h t o t a l space equal to the product of the base space and the f i b r e . We denote an n-d imens iona l t r i v i a l v e c t o r bundle by 0 n . (1.1) I t can be shown that the bundle * y k » y k i s t r i v i a l . (cf . [16]) . ' n ' n (1.2) One of the main c l a s s i f i c a t i o n theorems f o r v ec to r bundles s t a t e s t ha t f o r any n -d imens iona l v e c t o r bundle £ over a paracompact space X , t h e r e - e x i s t s a map f : X;- »- G n ( F m ) ,' f o r .some m , such t ha t f * (y™) ' a nd £ ,-are X - i somorph ic . ( c f . [16 ] ) . (1.3) S ince the pu l l b a c k of a t r i v i a l bundle i s t r i v i a l , by combining (1.1) and (1.2) we see tha t f o r any vec to r bundle Z, over a paracompact space X , t he re e x i s t s a bundle x\ over X such tha t £ <& n i s t r i v i a l . 6 Another important theorem s t a te s the f o l l o w i n g . (1.4) I f f , g : X *• X ' a re homotopic maps where X i s paracompact, and £ i s a v e c t o r bundle over X ' , then f * ( 0 and g*(?) a re i somorph ic . Example The tangent bundle over S n _ 1 , denoted T ( S n ~ 1 ) = (T , p , S n 1 ) , i s the bundle w i t h t o t a l space c o n s i s t i n g of the subspace of p a i r s (b , x) e S n _ 1 x S.n where < b , x > = 0 and p : (b , x) » > b . The tangent bundle i s an (n -1 ) -d imens iona l v ec to r bundle w i t h f i b r e R n ^ . A c r o s s ~ s e c t i o n , s , of .f ( S n ~^ ) i s a tangent vec to r f i e l d on S n ^ . A set of k c r o s s - s e c t i o n s s^ , . . . , s^ of T ( S n ^) i s l i n e a r l y independent i f (s^(b) , . . . , s^(b)) i s a l i n e a r l y independent set of XI—1 Tl 1 v ec to r s i n ffi. f o r a l l b e S . By the Gram-Schmidt o r t h o g o n a l i z a t i o n process we can ask tha t the v e c t o r s s^(b) , . . . , s^(b) be orthonormal . Combining t h i s w i t h example 1 we see that the ex i s tence of k .- 1 n—1 l i n e a r l y independent vec to r f i e l d s on S i s equ i va len t to a c ro s s P n-1 s e c t i o n of the bundle V . , >• V , = S f o r k < n . n,k n , l One of our o b j e c t i v e s w i l l be to determine, f o r a g iven n , the maximum number k so that such a c ro s s s e c t i o n e x i s t s . 2. The Rings ^ ( X ) For two vec to r bundles C , n over X the product bundle, ? H n , i s de f ined to be the bundle w i t h f i b r e over a po i n t x e X , the tensor 7 product of the corresponding f i b r e s E (?) and E (H) . The t o t a l space , E , i s the d i s j o i n t un ion of the f i b r e s and p : E >• X i s de f i ned by p(E ) = x . x Let Vect (X) be the set of isomorphism c l a s s e s of F - vec to r bundles over a space X . Vect (X) admits the s t r u c t u r e of a commutative semir ing where (£ , n ) »—>• Z, €> n i s the a d d i t i o n f u n c t i o n and (£ , n) »— Z,H n i s the m u l t i p l i c a t i o n f u n c t i o n . (We l e t £ denote both a vec to r bundle and i t s isomorphism c l a s s . ) For a po inted space X we have a semir ing morphism r k : Vectp(X) — 7L , which a s s i gns to Z, e Vect (X) the dimension of the vec t o r bundle on F the component of X c o n t a i n i n g the base p o i n t . The r i n g complet ion of a semir ing S i s a r i n g S* together w i t h a semir ing morphism 9 : S > S* w i t h the f o l l o w i n g u n i v e r s a l p rope r t y : i f f : S —•> R i s any morphism i n t o a r i n g R then there e x i s t s a unique r i n g morphism g : S* *• R such t ha t g6 = f . To con s t r uc t the r i n g S* cons ider p a i r s (a , b) e S x S w i t h the f o l l o w i n g equ iva lence r e l a t i o n : (a , b) and ( a ' , b ' ) a r e equ i va len t i f t he re e x i s t s a c e S such t h a t a + b ' + c = a ' + b + c . Let < a , b > denote the equ iva lence c l a s s of (a , b) . (We t h i n k of t h i s as < a , b > = a - b . ) Let S* denote the set of equ iva lence c l a s s e s <a , b > . Then we d e f i n e < a , b > + <c , d> = < a:!> c , b + d > -and < a , b > < c , d > = <ac + b d , b c + a d > . C l e a r l y < a , a > = < 0 , 0 > = 0 and the nega t i ve of < a , b > i s < b , a > . The map 9 : S > S* i s de f i ned by 0'(a) = < a , 0 > . I f R i s a r i n g and f : S > R i s a 8 morphism of semir ings then g : S* *• R i s the r i n g morphism de f i ned by g <a , b > = f ( a ) - f (b) . C l e a r l y ge = f and g i s un ique. For the uniqueness of S* cons ider another r i n g comp le t i on , S* , of S w i t h map 0^ : S • S* . Then we have the f o l l o w i n g commutative diagram w i t h f g = l g * and gf = l g ^ . We d e f i n e the r i n g K^CX) of a space X to be the r i n g complet ion of Vect^CX) . Vecty i s a co func to r from the category of spaces and maps to the category of semi r i ng s . Let f : Y > X be a map. Then _ Vect^Cf) : Vectp/X) - f * V e c t F ( Y ) i s de f i ned by V e c t y C f X ? ) equals the isomorphism c l a s s of f * ( ? ) over Y . We d e f i n e K^Cf) by the requirement tha t the f o l l o w i n g diagram commutes Vec V e c t p ( f ) V e c t F ( Y ) In o ther words, i f < Z, ,n > = Z, - n i s an element of ^ ( X ) then ( C - r i ) = f*(0 - f * ( n ) • I f g : Z >• Y i s a second map then K ^ f g ) = K F ( g ) K F ( f ) . A l so K p C y i s 1 ^ , the i d e n t i t y on ^ ( X ) Thus Kjp, i s a c o f u n c t o r . S ince r k : Vect^XX) > Z i s a semi r ing morphism there i s a map a l s o denoted r k , from Kp ( x ) t o z such t ha t rk0 = r k . That i s , r k ( C - n ) = r k ( £ ) - r k ( n ) . Denote the m u l t i p l i c a t i v e i d e n t i t y i n Kp ( x ) by 1 . .This i s represented by the t r i v i a l l i n e bundle 0 ^ . Def ine a morphism e :Z »-KpCX) by e ( l ) = 1 . Then f o r n > 0 , e(n) i s the c l a s s of 0 n and (rk) e = l z . Let ^ ( X ) , c a l l e d the reduced r i n g of Kp(X) , be the ke r ne l of the map r k : Kp(X) >- Z . Then we have a short exact sequenc of r i n g s o • i y x ) • K^CX) Z • 0 S ince ( rk) e = 1^ , t h i s sequence i s s p l i t exact and Kp(X) = (X)© TL Moreover, f o r a map f :Y >• X , K p ( f ) i s the r e s t r i c t i o n of KpCf) • i We w i l l denote Kp( f ) by f* . I f z, i s a .bundle over X and [r,] i s the c l a s s of t, i n K (X) then [ f * c ] = f ' [ c ] • Where d i s t i n c t i o n i s necessary we denote ^ ( X ) by KO(X) and K c (X ) by KU(X) . For now we l e t K = K^ , , F = R or C . We have the f o l l o w i n g p r o p o s i t i o n 10 (2.1) Let (X , A) be a r e l a t i v e CW-complex, X f i n i t e . Then a 8 ( i ) the maps A > X > X/A induce an exact sequence i i K ( X / A ) — ^ K ( X ) -^-y K(A) ( i i ) moreover, i f A i s c o n t r a c t i b l e then 3* : K(X/A) >• K(X) i s an isomorphism. (2.2) Let (X , A) be a r e l a t i v e CW-complex. Def ine the r e l a t i v e K-theory of (X , A) by K(X , A) = K(X/A) . The i n c l u s i o n A • X induces two morphisms K(X) *• K(A) and K(X) • K(A) w i t h the same ke r ne l and thus by (2.1) the f o l l o w i n g sequence i s exact K(X , A) K(X) y K(A) R e c a l l the Puppe sequence f o r a map f : X * Y , X y Y y C f y SX >• SY y where C^ i s the mapping cone of f and S i s the suspension o p e r a t i o n . By examining the homotopy p r o p e r t i e s of mapping cones and suspens ions, and app l y i ng (2.1) s e ve r a l t imes we f i n d tha t the f o l l o w i n g sequence i s exact K(SY) • K(SX) »• K (C f ) • K(Y) »• K(X) . I f (X , A) i s a r e l a t i v e CW-complex then the mapping cone, C^ , of the i n c l u s i o n f : A >• X i s homotopy equ i va len t to the quot ien t space X/A . Thus the f o l l o w i n g sequence i s exact K(SX) y K(SA) • K(X/A) • K'(X) • K(A) . As i n (2.2) the f o l l o w i n g sequence i s exact (2.3) K(SX) >• K(SA) • K(X , A) >• K(X) > K(A) . Def ine X + to be X/(j) = Xu {*} , the un ion of X p l u s an i s o l a t e d base p o i n t . We can d e f i n e a cohomology theory K* . For n > 0 d e f i n e K _ n ( X ) = K(S n X) and K~ n (X , A) = K~ n(X/A) . Then K~ n(X) = K _ n ( X , $) = K~ n (X + ) = K ( S n X + ) . Thus the exact sequence (2.3) extends to a long exact sequence (2.4) f K _ 2 ( X ) • K _ 2 ( A ) > K _ 1 ( X , A) > K _ 1 ( X ) • K _ 1 ( A ) • K°(X , A) • K°(X) • K°(A) . Products R e c a l l t ha t the tensor product of v e c t o r bundles induces a r i n g s t r u c t u r e on K(X) which i s a group morphism K(X) H K(X) > K(X) . We denote the image of a H b by ab . Let p : X x Y -—> X and p : X x y Y be the two K X F Y p r o j e c t i o n s from the product . The e x t e r n a l K-cup product i s a group I r morphism K(X) H K(Y) > K(X x y) de f i ned by a f i b >• p ' ( a ) p " ( b ) . X Y i i By abuse of n o t a t i o n we denote p"(a)p^.(b) by ab . A Y I t S ince rk (p^(a)pY(b) ) = r k ( a ) r k ( b ) , the e x t e r n a l K-cup product induces a w e l l - d e f i n e d K-cup product K ( X ) ® K ( Y ) • K(X x y) by r e s t r i c t i o n . Let q, : X > X ' Y and q. : Y • X w Y be the two n a t u r a l "iy V V i n c l u s i o n s . Then the sequence X ——> X y Y • X y Y/X = Y induces a s p l i t exact sequence T 0 > K(Y) • K(X v Y) - J L-> K(X) »• 0 Thus the group morphism (2-5) ( q ^ , q^) : K ( X v Y ) • K(X) + K(Y) i s an isomorphism. Using t h i s i t i s not hard to show tha t the sequence q P X Y >-X><Y • X Y V A induces the f o l l o w i n g s p l i t exact sequence (2.6) 0 • K ( X A Y)-£—> K ( X X Y ) > K(X A Y) > 0 w i t h s p l i t t i n g ( P X ® P Y % ? X ' <45 : ^ ( X v Y ) ** K(X x Y) . Note t ha t the compos i t ion t K(X)8 K(Y) K ( X X Y ) - 3—* K(X y Y) i s z e r o . For i f ab i s a cup product i n K ( X x Y ) then a •: p r o j e c t s to zero when p r o j e c ted i n K(Y) . S i m i l a r l y f o r the p r o j e c t i o n of b i n t o K(X) . Therefore the product i s zero when p r o j e c t e d i n t o K (X ' Y) ~ ~ cup ~ Thus the morphism K(X) HK(Y) • K ( X x y ) f a c t o r s un ique l y through K(X Y) . The unique group morphism K(X) H K(Y) >• K(X A Y) composed w i t h the monomorphism K(X Y) >• K(X x y) i s the K-cup product . From now on we w i l l r e f e r to K(X) H K(Y) • K ( X A Y) as the K-cup product . We w r i t e K ( X ) H K ( S k ) = (K(X) ®Z) S ( K ( S k ) » Z ) = (K(X) H K ( S k ) ) #K(X) ® K ( S k ) ®Z:. A l s o , by (2.5) and (2 .6 ) , we can w r i t e K ( X x S k ) = K ( X A S k ) ® K ( X W S k ) = K ( X A S k ) t K ( X ) » K ( S k ) . From t h i s we have the f o l l o w i n g commutative diagram K ( X ) H K ( S k ) = ( K ( X ) H K ( S k ) ) § K ( X ) l l ( S k ) # Z cup K (X x S^) = K(X S k ) « K ( X ) ® K ( S k ) ®Z A The cup product morphism i s the i d e n t i t y when r e s t r i c t e d to K(X) « K ( S k ) « Z . (2.7) Thus we see tha t the e x t e r na l cup product K ( X ) « K ( S ) > K(X x S i s an isomorphism i f and on l y i f the K-cup product K ( X ) 8 K ( S k ) >• K(X ^Sk) i s an isomorphism We now s t a t e one of the main theorems i n K- theory, known as the Bot t p e r i o d i c i t y theorem. Bott P e r i o d i c i t y Let X be a compact space. The e x t e r n a l cup products KU(X)HKU(S 2 ) v K U ( X x S 2 ) and KO(X) H KO(S^) )• K O ( X x S 8 ) a re isomorphisms. In v iew of (2.7) t h i s imp l i e s that KU(X) i s i somorphic to K U ( X A S 2 ) = KU(S 2X) = KU _ 2 ( X ) and KO(X) i s i somorphic to K O ( X A S 8 ) = KO(S 8X) = KO~ 8(X) . Now we can use these isomorphisms to i n d u c t i v e l y d e f i n e K (X) 14 f o r a l l i n t e ge r s n . The sequence (2.4) can now be extended to an exact sequence i n f i n i t e l y to the r i g h t . Thus we now have a g ene r a l i z ed cohomology theory . The Adams Operat ions Let V be a vec to r space over F = S or C . Let X 1 (V ) denote t h the i e x t e r i o r power of V . E x t e r i o r powers s a t i s f y the f o l l o w i n g r e l a t i o n (2.8) A n (V©W) = i + j ^ ^ O O B * J (W) We extend the X 1 opera t i on s to vec to r bundles i n the obvious way. I f t, i s a v e c t o r bundle over X w i t h f i b r e E at x e X , X 1 ( ? ) w i l l be the bundle whose f i b r e a t x i s X X ( E ) . We a l s o w r i t e X 1 ( ? ) f o r the isomorphism c l a s s 9 ( X 1 ( c ) ) i n K(X) . We may use (2.8) to d e f i n e n a t u r a l t r an s fo rmat ion s X 1 : K(X) »• K(X) i n the f o l l o w i n g manner. Let K ( X ) [ [ t ] ] be the fo rma l power s e r i e s ' i n t w i t h c o e f f i c i e n t s i n K(X) and l e t K^ (X ) [ [ t ] ] be the m u l t i p l i c a t i v e group of elements i n K ( X ) [ [ t ] ] which s t a r t w i t h 1 . For a v e c t o r bundle Z d e f i n e X (C) IspK^CX) [ [ t ] ] by X t (5 ) S - = ^ t 1 X : L ( ? ) . By (2.8) we have X = z t V a e K ' ) i = E t i ( e x m ( ? ) a x k ( c ' ) ) i m+k= i = E VE t m X m ( ? ) H t k X k ( ? ' ) i m+k=i Z t m X m ( c ) S E t k X k ( ? ' ) m k Hence X : Vect (X) >• K^ (X ) [ [ t ] ] i s a semigroup morphism. By t he u n i v e r s a l p roper ty of K(X) there i s a unique group morphism X : K(X) > itjCX) [ [ t ] ] . Thus we get a morphism X 1 : K(X) >• K(X) where X 1 ( ? ) i s de f i ned to be the c o e f f i c i e n t of t 1 i n X (?) . Examples From the p r o p e r t i e s of X 1 , i f V i s an n -d imens iona l v ec to r space, X (V) = 0 f o r k > n . Thus i f t, i s a l i n e bundle X (?) = 1 + tr, . Note that (1 - t ? ) ( E ( - t ) 1 X 1 ( - ? ) ) = 1 . i Thus X_ t ( - ? ) = E ( - t ) i X ± ( - ? ) 1 - tc. 2 2 1 + t ? + t C + We now d e f i n e the Adams opera t i on s i ^ 1 : K(X) > K(X) . To do t h i s set (2.9) i|i (x) = tTji(x) + tV(x) + = E tVW t 1 = 1 f o r x e K(X) , and d e f i n e if; by the formula dX (x) d(logX (x)) ^ - t ( x ) = " t " T t - - = _ t ~ • X t ( x ) The r i g h t hand s ide of (2.9) i s a w e l l - d e f i n e d element of K ( X ) [ [ t ] ] and so determines . We have (2.10) ij)_ t(x + y) = - t [ X f c ( x + y ) ] ' = - t . [ X t ( x ) X t ( y ) ] ' ^ t ( x + y) X t ( x ) X t ( y ) = - t [ X j ( x ) X t ( y ) + X t ( x )X^ (y ) ] X t ( x ) • X t ( y ) = - t r X j ( x ) ' X^(y) ' I x t (x> + x ^ y = * ._ t (x) + * _ t ( y ) I f we so l ve f o r the i); 1 i n (2.9) we ob ta i n the f o l l o w i n g formulas - . X 1 = 0 ' ' 1 2 , 1 1 2 , i> X + 2X = 0 3' 2 1 1 ' 2 ' "3 , i , i - 1 , 1 , • 7,1 , i - 1 , . . i . - ijj X + +ip X ±xX = 0 These equat ions r e c u r s i v e l y d e f i n e the ty1 i n terms of the X 1 . I f we so lve aga in we get if/1 = s^(X , . . . , X 1 ) where i s the i ,-th Newton po lynomia l i n the X" P r o p e r t i e s of if k k I f t, i s a l i n e bundle then if (?) = S • For X t ( ' t ) = i-+ t? i m p l i e s X£( ? ) = s o * _ t ^ ) = -t \a) _ tx. X f c(C) = 1 + 2 2 3 3 Hence if (?) = tg = t ? + t £ + t ? + 1 - t? = Z tV(c) i2d Therefore i f k ( ? ) = ? k . I f we compare c o e f f i c i e n t s i n (2.10) we see tha t i f k ( x + y) = i f k ( x ) + i f k ( y ) . Fu r ther p r o p e r t i e s of if a re the f o l l o w i n g 1) / ( x y ) = i f k ( x ) / ( y ) ,Vk V P VP 2) if if (x) = if (x) , V k , I 3) V prime p i f P ( x ) E ' x P mod p Some We l l Known Re su l t s There are th ree n a t u r a l group morphisms r : KU(X) >• K0(X) c : K0(X) y KU(X) t : KU(X) y KU(X) 18 where r means t ak i ng the unde r l y i ng r e a l bundle, c means ten so r i ng w i t h C and t '.means complex con juga t i on . Furthermore the f u n c t i o n s c and t a re r i n g morphisms. Let t, denote the c a n o n i c a l l i n e bundle over FP . In the complex case l e t y = t, - 1 e KU(CP ) and i n the r e a l case l e t X = £ -1 £ KO ( l P k ) . Let to = ry e K0(CP k ) . (2.11) The Ring KU(CP k) (cf [ l ] and [4]) KU(CP k ) = Z [ y ] / y k + ^ . The Adams operat ions a re g iven by i A u r ) = ((1 + U ) n - l ) r • The p r o j e c t i o n C P k ->CPk/CPm maps KU(CP k /CP m ) i s o m o r p h i c a l l y onto the subgroup of KU(CP k ) ^ , , m+1 m+2 k generated by y , y y . The sequence CP V /CP W • CP U /CP W • CP U /CP Y leads to an exact sequence 0 • KU(CP /CP ) • KU(CP /CP ) »- KU(CP /CP ) — (2.12) The Ring K 0 ( C P k ) ( c f . [4]) Let a be an i n t e g e r . The r i n g K0(CP ) may be represented by the generator to= r u and the f o l l o w i n g r e l a t i o n s o 2a+l 2a+l n , _ . , . 2io = 0 , to = 0 i f k = 0 mod 4 2a+2 _ co = 0 i f k = 1 mod 4 2a +1 OJ = 0 i f k S 2 or 3 mod 4 . n n 2 The Adams ope ra t i on s a re g i ven by \\) (to) = k to (modulo h igher powers of to) . The sequence CP V CP W > CP U /CP W > CP U /CP V l ead s to an exact sequence 0 > K0(CP U /CP V ) —> K0(CP U /CP W ) > K0(CP V /CP W ) > 0 19 For X = CP k /CP m the morphism c : KO(X) • KU(X) i s a monomorphism i f k $ 1 mod 4 ; the morphism r : KU(X) • KO(X) i s an epimorphism i f m t 3 mod 4 and an isomorphism i f m i s even and k > 1 . (2.13) The Ring KO(RP k ) ( c f [1]) Let a(k) be the number of i n tege r s s w i t h o < s < k k-1 such t ha t s = 0 , 1 , 2 or 4 mod 8 . Then KO(EP ) may be de sc r i bed by the generator A and the two r e l a t i o n s X 2 = -2X , X a ( k ) + 1 - 0 (so tha t 2 a ^ k ^A = 0) . Let m be an i n t ege r and assume tha t m t 0 mod 4 . Let f = cr(k) - a(m) . Then K 0 ( R P k - 1 / K P m - 1 ) = Z 2 f . i f m = 0 mod 4 then K0(RP /RP C ) = Z0KO(RP /RP C ) . 3. The J-groups In t h i s s e c t i o n we w i l l assume t ha t X i s a f i n i t e CW-complex. Let £ = (E , p , X) and £' = ( E ' , p ' , X) be bundles over X . A f i b r e homotopy . h^ : E >• E ' i s a homotopy such tha t P ' h t = p f o r a l l t e [0 , 1] . 20 Bundle morphisms f , g : E > E ' a re f i b r e homotopic prov ided the re e x i s t s a f i b r e homotopy h : E »- E ' w i t h h Q = f and h^ = g . A bundle morphism f : E > E' i s a f i b r e homotopy equ iva lence prov ided the re e x i s t s a bundle morphism g : E ' > E w i t h f g and gf f i b r e homotopic to the i d e n t i t y . The bundles ? and ? ' have the same f i b r e homotopy type prov ided t he re e x i s t s a f i b r e homotopy equ iva lence f : E »- E ' . In t h i s case we w r i t e ? ~ ? ' . The bundles ? and ? ' have the same s t a b l e f i b r e homotopy type prov ided m m' the re e x i s t s i n t ege r s m and m' such tha t ? 0) 9 ~ ? ' # 9 . Here we w r i t e ? ~ ? ' and say t ha t ? and ?' a r e s t a b l e f i b r e homotopy equ i v a l en t . Example. We g i ve an a l t e r n a t e d e s c r i p t i o n of K (X) . C l e a r l y isomorphic vec to r bundles a r e s t a b l y e q u i v a l e n t . Thus s t a b l e equ iva lence can be thought of as an equ iva lence r e l a t i o n on Vectp(X) . Let TjXX) equal Vectp(X) modulo the r e l a t i o n of s t a b l e equ i va lence . View Z c Kp(x) u s ing e and de f i ne the f u n c t i o n a : Tp(x) *" Kj,(x) by a (?) = £ - r k ( ? ) . We c l a i m t h a t a i s a b i s e c t i o n f o r any paracompact space X . To show tha t a i s s u r j e c t i v e l e t ? - n be an element of KJJXX) w i t h r k ( ? ) = rk(n) . By (1.3) there e x i s t s a v e c t o r bundle n' such t h a t n©n' = 6 m • Then i n Kp(X) we have a(?®n') ='• ?® n* - rk( ?©n')= ?®n' - 0 m = ?® n' - n $ n' = £ - n • Thus a i s s u r j e c t i v e . To show i n j e c t i v i t y suppose tha t £^ and r\m are two • bundles such that £ - 8P = a (?) = a(n) = r\ - 8 m . S ince C - 8P i s equal to n - 6m i n Kp ( x ) > t he re e x i s t s a bundle E such tha t £ © 9 m ® 5 and n ® 8P $ E a re i somorph ic . Let £ ' be a bundle w i t h £ ® E' = 9q . Then £ # 0 m fl> £ 0 S' = £ © 9 m ® e q = e m + q and n # 8P $ E ® E' = n $ 8P ® 8q = 9 P + q a r e i somorph ic . Thus C, and ry a r e s t a b l y equ i va len t and a i s i n j e c t i v e . Hence we may v iew the r i n g K p ( x ) a s Vect^,(X) under the r e l a t i o n of s t a b l e equ i va lence . A sphere bundle over a space X i s a bundle w i t h a sphere as the f i b r e . Let £ be an F - vec to r bundle over X w i t h t o t a l space E = E(£) . The a s s o c i a t ed d i s k bundle, D(£) , i s the bundle having as a f i b r e over x e X , the set of a l l y e E w i t h || y || ^ 1 . The a s s o c i a t e d sphere bundle, S(£) , i s the bundle having as a f i b r e over x e X , the set of a l l y e E^ w i t h ||y|| = 1 • That i s , 8D(£) = S(0 , w i t h the boundary ope r a t i on be ing c a r r i e d out on each f i b r e . A l l sphere bundles can be regarded as the a s s oc i a t ed sphere bundle of some vec to r bundle. I f £ i s a complex vec to r bundle then S(£) can be regarded as the a s s o c i a t ed sphere bundle of the unde r l y i ng r e a l bundle of £ . Let J(X) denote the set of equ iva lence c l a s s e s of sphere bundles w i t h re spec t to s t a b l e f i b r e homotopy t ype . We denote the c l a s s of a vec to r bundle £ by J(C) . We have the f o l l o w i n g . The d i r e c t sum of vec to r bundles induces on J(X) the s t r u c t u r e of an a b e l i a n group. Inverses can be a r r i v e d a t v i a (1.3) . C l e a r l y , i somorphic vec to r bundles have i somorphic a s s o c i a t ed sphere bundles which a re equal under s t a b l e f i b r e homotopy equ i va lence . Thus s t a b l e f i b r e homotopy equ iva lence of sphere bundles can be thought of as an equ iva lence r e l a t i o n on Vectp(X) . Therefore by the u n i v e r s a l p roper ty of Kj.( x ) > the quot ient map Vectp(X) > J(X) l ead s to a group epimorphism KO(X) >• J (X) which we w i l l denote by J ; and a group homomorphism KU(X) * J(X) which we w i l l denote by J r . The . - . ' J ' - - and J " - groups. The groups J(X) a r e , i n gene ra l , ve ry d i f f i c u l t t o compute d i r e c t l y . We s h a l l compute J(X) by squeezing i t between a " lower bound", J ' ( X ) and an "upper bound" , J " (X ) . When we speak of a " lower bound" and an "upper bound" we mean we s h a l l c on s t r uc t a commutative diagram i n which t he v e r t i c a l arrows a r e epimorphisms as f o l l o w s . Th i s i s t he J ' of [ 4 ] and does not c o i n c i d e w i t h the J ' or the J ' of [ 2 ] . 23 K(X) The J " -g roups Let Z + be the non-negat ive i n t e g e r s . Given a f u n c t i o n f : Z y T+ , l e t W(f , X) denote the subgroup of K(X) generated by the elements t f ( t ) ( \ ( ; t - l ) x x e K ( X ) , t e Z Let W(X) = nW (f , X) where f runs over a l l such f unc t i o n s Z + > Z Then we d e f i n e J " (X ) = K(X)/W(X) . We w i l l c a l l the quot ien t map J " . t I f we a re g iven a map g : Y >- X then c l e a r l y g"W(f , X) c i W(f , Y) and hence g ' n W (f , x) c n W (f , Y) . Thus the map g : Y > X f f induces a map g " : J " (X ) • j " ( Y ) . The Adams Conjecture The Adams con j e c t u r e can be s ta ted as f o l l o w s (3.1) For any xe K(X) and any i n t ege r t , t he re e x i s t s a f u n c t i o n f : Z + •> Z such t h a t t f ^ t ^ ( ^ t - l ) x i s J - t r i v i a l . 24 From t h i s we can see tha t f o r such a f u n c t i o n f we have W(f , X) c ker J . Hence W(X) = ker J " c ker J . We have j u s t shown tha t the map J : K(X) • J(X) f a c t o r s through J " (X ) g i v i n g the f o l l o w i n g diagram of epimorphisms. In f a c t , one of the main theorems of J - t heo ry s t a t e s that 8" i s an isomorphism. (c f [13]) In [2 , I I ] Adams shows t h a t f o r a f i n i t e CW-complex X , the groups J " (X ) a re f i n i t e . Therefore J(X) i s a f i n i t e group. As an example of t h e i r u se fu lnes s we w i l l show how the groups J " (X ) a l l o w us to f i n d a (non-minimal) set of generators of the groups J(X) . The r e s u l t was po in ted out by K.Y. Lam. Let X be a f i n i t e CW-complex and x e K(X) . Then any element of the form J(<Jj P qx), where p and q are r e l a t i v e l y pr ime, can be w r i t t e n as J ( i j j p q x ) = J ( ^ P x ) + J(i/j qx) - J ( x ) . To prove t h i s l e t ( i ^ - l ) ( i| j q - l )x = ( i p P - l ) (ij; qx-x) = i j j p q x - I| J P X - t|jqx + x = y , say . By (3.1) the re e x i s t s an i n tege r N such t ha t P N ( ^ P - 1 ) [ ( ^ q - l ) x ] = p N y i s J " - t r i v i a l . S i m i l a r l y the re i s an i n t ege r M such that -q^( ' ^ P ' - l ) . ; ( i i j q - l )x = q M ( i| ; q - l ) [ ( I J J P - 1 ) X ] = q^y i s j " - t r i v i a l . S ince P^ and q^ are r e l a t i v e l y pr ime, t h i s i m p l i e s t h a t y i s J " - t r i v i a l . Hence y i s J - t r i v i a l . Therefore J(4> P qx) = J ( i A ) + K ^ q x ) - J (x ) . 25 As a c o r o l l a r y we have the f o l l o w i n g . Let E, denote the c a n o n i c a l l i n e bundle over CP . Then p i i the set of elements J(£ ) > w i t h p a prime and p < k , i s a set of generators f o r J(CP ) . The ^ - g r o u p s We f i r s t d e f i n e the groups J ' ( X ) f o r a t o p o l o g i c a l space I* X . Let us r e c a l l some f a c t s about the Chern c l a s s e s of an n -d imens iona l complex v e c t o r bundle £ over X . The i * " * 1 Chern c l a s s of ? , denoted c_^(?) , i s an element of the cohomology group H X ( X , Z) . For i > n the Chern c l a s s c^(0 i s de f i ned to be z e r o . The t o t a l Chern c l a s s , denoted by c ( 0 , i s the fo rmal sum 1 + c 1(C) + c 2 ( ? ) + : + c n ( 0 . The Chern c l a s s e s have the f o l l o w i n g p r o p e r t i e s . 1) ( N a t u r a l i t y ) I f (u , f ) : £ ' • £ i s a v ec to r bundle morphism then f * c ( 0 = c(C') . 1, v_ 2) I f 6 i s a t r i v i a l bundle over X then c(£ © 0 ) = c ( 0 . 3) Let £ and ? ' be two vec to r bundles over a paracompact space X Then c(C €> = c ( ? ) c ( C ) . 4) The cohomology r i n g H*(Cp n , Z) i s a t runcated po lynomia l r i n g t e rm ina t i n g i n dimension 2n and generated by the Chern c l a s s 26 5) I f £ and £ ' a re isomorphic vec to r bundles over X then c ( ? ) = c ( C ' ) . 6) By p r o p e r t i e s 2) and 5) i f £ and £ ' a re s t a b l y equ i va len t vec to r bundles over X then c(?) = c ( ? ' ) . The Chern c h a r a c t e r By the s p l i t t i n g p r i n c i p l e i n K-theory we may regard an n-d imens iona l v ec to r bundle, t, , as a sum of l i n e bundles a,# Ba . Let A. = c , ( a . ) . The Chern c h a r a c t e r , 1 n i 1 i ch : KU(X) »- H * (X,Q) , i s a r i n g morphism de f i ned by ch(C) = e^- + + e^ 1 1 where e ^ i s the formal e xponen t i a l power s e r i e s i n X. . Let (C , 6 ) be an element of KU(X) = KU(X) #Z . We de f i ne the Chern cha rac te r on KU(X) by l e t t i n g ch(£ , 0m) = n - m + ch(£) . We remark tha t ch i s a n a t u r a l t r an s fo rmat i on of f u n c t o r s . bh -c l a s se s and J ' - g r oup s For a bundle £ over a f i n i t e CW-complex X , we w i l l d e f i ne 2 " a c h a r a c t e r i s t i c c l a s s bh(?) e l + £ H (X , Q) where bh i s an i " e x p o n e n t i a l " morphism KU(X) > 1 + Z H 2 l ( X , Q) . Let a±9 ®a n be the s p l i t t i n g of £ . Def ine bh(?) by n c, (a.) bh(g) = n e x 1 - 1 i = l c X c u ) 27 By the n a t u r a l i t y of the Chern c l a s s e s i t f o l l o w s t ha t the f o l l o w i n g diagram commutes f o r maps f : Y *• X KUCX) ™ > H * ( X « t f • f A KU(Y) — > H*(Y , Q) . Let t, be a l i n e bundle and l e t y = c£(g) Then bh( 5) = 1 +_y_ tL , A 2! + 3! + 4 ! f Let V (X) be the set of elements aeKU(X ) such t ha t bh(a) = c h ( l + 3) f o r some 3eKU(X ) . - S ince ch : 1 + KU(X) > 1 + EH2^"(X , Q) i s a r i n g morphism and bh i s " e x p o n e n t i a l " , V^(X) i s a subgroup. Thus we de f i ne J£(X) = KU(X)/V C (X) . We denote the quo t i en t map KU(X) • J C ^ X ^ b y J C * T h e n a t u r a l i t y of bh and ch imply t ha t maps f : Y > X induce group morphisms f ' : JJ,(X) • J£(Y) . s h - c l a s se s and J^-groups We d e f i n e the groups J.L(X) i n a s i m i l a r f a s h i o n as we d i d f o r the groups J ' ( X ) . Def ine the c h a r a c t e r i s t i c c l a s s 41 sh(£) e 1 + EH (X , Q) f o r a bundle £ over a space X . Let £ be i a complex v e c t o r bundle over X w i t h s p l i t t i n g a^® j • fflkx . Def ine sh : KO(X) • 1 + E H 4 l ( X , Q) by sh(rO rO = S e ^ i a ± ) - e " * c i ( a i > i=.l c ^ o ^ ) Let V (X) be the set of elements aeKO(X) such that K. sh(a) = ch c ( l + 3) f o r some 3e KO(X) . As be fo re V R ( X ) i s a subgroup. Thus we d e f i n e j£ (X) = KO(X)/V R(X) and denote the quo t i en t map KO(X) • J£(X) by . Maps f : Y induce group morphisms f ' : JJL(X) > J™ 0 0 . Ne i t he r of the f unc t o r s J ' or J ' i s the J ' f unc to r of C ^ Adams. However, i n p rov ing tha t h i s J ' f a c t o r ed through h i s group J(X) Adams proved the f o l l o w i n g . Le t £ and ? ' be two vec to r bundles over a f i n i t e CW-complex X . I f the a s s o c i a t e d sphere bundl of £ and ? ' a r e f i b r e homotopy equ i va len t then the re e x i s t s an element 3 e Kj,(X) such t ha t ( bh(£) = b h ( C ' ) c h ( l + 3) , F = C sh(£) = sh(c3')ch c ( l + 3) , F = K. . S ince the bh c l a s s of a t r i v i a l bundle i s 1 , t h i s i m p l i e s t ha t i f £ i s a v e c t o r bundle w i t h a s s o c i a t ed sphere bundle f i b r e homotopy t r i v i a l then £ e V_(X) . That i s , f o r a f i n i t e CW-complex X , we w have ker J c ker J ' and ker J r c ker J ' . Hence we have the f o l l o w i n g commutative diagrams 29 Two Important Isomorphisms ~ k k (3.2) The group KO(RP ) i s isomorphic to J(KP ).(cf. [9] ) Thus, i f % i s the c anon i c a l l i n e bundle over RP then k o"(k) J(RP ) i s c y c l i c of order 2 , generated by J (£) . k k (3.3) The groups J ' ( C P ) and J ' ( C P ) a re i somorphic, ( c f . [13]) JR. C 4. Vector f i e l d s on spheres The vec to r f i e l d on spheres problem has generated a great dea l of i n t e r e s t du r i ng t h i s c en tu r y . Seve ra l people i n c l u d i n g A t i y a h , Adams, James, Steenrod and Whitehead, Toda, Hurw i t z , Radon and Eckmann, have c o n t r i b u t e d to the s o l u t i o n by g i v i n g r e s u l t s i n s p e c i a l ca se s . A s u f f i c i e n t c o n d i t i o n f o r the ex i s t ence of k l i n e a r l y independent v e c t o r f i e l d s on the (n-1) - sphere i s a t t r i b u t e d to Hurw i t z , Radon and Eckmann. Us ing l i n e a r a l g e b r a i c methods they found tha t the (n - 1 ) - sphere admitted p ( n ) - l v e c to r f i e l d s where p(n) = 2 + 8d , 0 < c < 3 and n = (2a + l ) 2 C + 4 d . Adams [ 1 ] , employed the methods o f K-theory to prove tha t t h i s was the best p o s s i b l e r e s u l t , i ' . e . , S n ^ does not admit p(n) v ec to r f i e l d s . 30 xi X R e c a l l t ha t a v e c t o r f i e l d on S i s a s e c t i o n of the tangent bundle x ( S n ^) . The problem i s to f i n d the maximum va l ue f o r the i n t ege r k f o r which there e x i s t s k s ec t i on s s 1 , , s f c : S • T(S ) such tha t the v e c t o r s s^(x) , , s ^ ( x ) a r e l i n e a r l y independent f o r a l l x e S n . As before we may use the Gram-Schmidt o r t h o g o n a l i z a t i o n process to assure t h a t the k v e c t o r s s^(x) , , s^Cx) a r e or thonormal . We re fo rmu la te the vec to r f i e l d problem as f o l l o w s . For 1 < m 2 k the S t i e f e l man i fo ld V . f i b r e s over V , n,k n,m w i t h f i b r e V , , by t a k i n g the l a s t m ve c t o r s from each k-frame. n-m,k-m J ° In p a r t i c u l a r , V , f i b r e s over V .. = S n ^ by t a k i n g the l a s t v ec to r from each k-frame. A c r o s s - s e c t i o n S n ^ > V , a s s i gns to each XI y K. XI 1 po i n t v e S an orthonormal k-frame (v^ , , v^_^ > v) . Thus the vec to r f i e l d problem can be regarded as f o l l o w s : f o r a g i ven i n tege r n , determine the maximum va lue of an i n t ege r k such that the S t i e f e l f i b r i n g V . > S n ^ admits a c r o s s - s e c t i o n . XI j K. The ve c t o r f i e l d problem i s then a p a r t i c u l a r case of the f o l l o w i n g problem. Let d = 1 o r 2 accord ing to whether fF = 1 o r C . For what va lues of n and k does the S t i e f e l f i b r i n g 0 , > S^ n ^ XI ^ K. admit a c r o s s - s e c t i o n ? We beg in w i t h the r e a l case by o u t l i n i n g some s o l u t i o n s f o r s p e c i a l v a l ue s of k . 31 Let k = 2 . We w i l l show that a c r o s s - s e c t i o n of the f i b r i n g 2 *• S n ^ e x i s t s i f and on l y i f n i s even. An orthonormal 2-frame i s a p a i r (u , v) £ R n x R n such tha t <u , v > = 0 and || u|| = || v|| = 1 . Thus a c r o s s - s e c t i o n S n ^ > V n 2 i s a se l f -map, g , of S n ^ such that g(v) i s o r thogona l to v f o r a l l v e S n ^ . Suppose n i s even, n = 2m say. Then we can regard v as a complex m-vector and de f i ne g by m u l t i p l i c a t i o n by i . I f V = ( V 0 ' V l ' V 2 m - 2 > V 2 m - 1 ) t h e n g(v) = , v Q , , ~^2m-i , v 2 m _ 2 ) and < v , g(v) > = 0 . Now suppose that g e x i s t s w i t h <v , g(v) > = 0 f o r a l l „n - l _ _. , , n-1 ,. „n—1 , v e S . Def ine a homotopy h : S x i y s by h- t(v) = v cos Trt + g(v) s in ? r t , 0 < t < 1 . Th i s i s a homotopy between the i d e n t i t y on S n ^ and the a n t i p o d a l map. The degree of the a n t i p o d a l map i s ( - l ) n and so i t f o l l o w s that n—1 n must be even. Thus the f i b r i n g V „ > S admits a c r o s s - s e c t i o n ° n, 2 i f and on l y i f n i s even. In the above example n o t i c e t ha t the s u f f i c i e n c y par t was e s t a b l i s h e d i n a s t r a i g h t f o r w a r d manner. The n e c e s s i t y p a r t , however, r equ i r ed more s ub t l e methods. Th is was a l s o the case i n the genera l problem; the s u f f i c i e n c y being e s t a b l i s h e d a t a much e a r l i e r date by Eckmann, by u s i n g C l i f f o r d a l g e b r a s . B r i e f l y , we d e f i n e an orthogonal m u l t i p l i c a t i o n to be a b i l i n e a r f u n c t i o n U : R k x R n y R n such tha t ||y(y , x) || = ||y|:| ||x|| f o r a l l y e R k and x e R n . Let e ± = (0 , , 0 , 1 , 0 , . 0) w i t h 1 i n the i t n p o s i t i o n . We say t ha t u i s normal i zed prov ided Ute^ > x) = x . Now f o r each x e S the vec to r s u(e^ , x) , V(^2 ' x ^ » » ^ e k ' x ^ n—1 a re or thonormal . Thus we can d e f i n e a c r o s s - s e c t i o n f : S *• V , n, k by f (x) = ( y ( e 1 , x) , l - K e ^ » x) , x ) . The set of normal i zed or thogona l m u l t i p l i c a t i o n s y : R k x R n y R n maps b i j e c t i v e l y onto the set of 2 u, , u_ , u, , e 0(n) such tha t u. = 1 and u.u. + u .u . = 0 f o r 1 2 ' k-1 l x j j x i ^ j . Th i s correspondence i s de f i ned by s e t t i n g u^(x) = u(4^ > x) f o r a g iven u . Such a s e t , u^ , ' u k 1 ' de f i ne s a C l i f f o r d a l geb ra C^ . ^ w i t h ba s i s c o n s i s t i n g of elements of the form u. * u . , s , i ( l ) < < i(m) . So the ex i s tence of an x ( l ) x(m) or thogona l m u l t i p l i c a t i o n y : l k x R n y R n i s equ i va len t to the ex i s tence of a C, n -module s t r u c t u r e on R n . Hence i f R n admits k-1 the s t r u c t u r e of C, .. -module then V , y S n ^ has a c r o s s - s e c t i o n k-1 n,k Let F(n) be the r i n g of n x n mat r i ce s w i t h e n t r i e s i n F . From the genera l p r o p e r t i e s of ma t r i x a l gebras we can determine the i r r e d u c i b l e modules over F(n) and F(n) © F(n) . For F(n) there i s on l y one, namely, the a c t i o n of F(n) on F n and i t s dimension over F i s n . In the case of F(n) © F(n) there a re two, namely, the two p r o j e c t i o n s F(n) <& F(n) y F(n) f o l l owed by t he a c t i o n of F(n) on F n , and both have dimension n . Let M denote the quate rn ions . Consider the f o l l o w i n g t a b l e . Table of I r r e d u c i b l e Modules over C, , — _ _ _ k - l C, , l R C H ffl»M H(2) C(4) R(8) R(8) ®R(8.) R(16) k-1 33 There i s a l s o an isomorphism C k + g C f c 8R(16) . Let a(k) be the number of s w i t h 0 < s < k and s = 0 , 1 , 2 or 4 mod 8 Then R n admits a ^-module s t r u c t u r e i f and on l y i f n = 0 mod 2°^^ . Thus i f n = 0 mod 2 ^ ^ then V , admits a c r o s s - s e c i t o n over S U ^ . n, k Now i f we a re g iven n , we wish to f i n d the maximal number k We f i r s t make some observat ions about a . Let 0 < c < 3 and l e t b be a p o s i t i v e i n t ege r such that a(b) = c . Then the p o s s i b l e va lues f o r b and c a re such t ha t n = 0 mod o r , e q u i v a l e n t l y , n = ( o d d ) 2 0 ^ c b 0 1 1 2 2 3 or 4 3 5,6,7 or Note a l s o tha t f o r an i n tege r d we have a(b + 8d) = a(b) + 4d . c+4d Now l e t n = (odd)2 , 0 < c < 3 . Then a(k) = c + 4d = a(b) + 4d = a(b + 8d) . By the above observat ions we w i l l o b t a i n a maximal k i f we take b to be maximal. From the above t a b l e we see t ha t f o r a g i ven c we o b t a i n a maximal b i f we c c take b = 2 . Hence k = 2 + 8d = p(n) . No t i ce tha t a c r o s s - s e c t i o n n-1 , „n - l S > V , a s s o c i a t e s to each v e S n, k an orthonormal k-frame v, n ) as a (k -1) - f rame ( v 1 , , v ^ , v) . We regard (y± n—1 n 1 1 - 1 of tangents to S a t v . Thus f o r a g i ven n the sphere S admits k-1 = p ( n ) - l l i n e a r l y independent v e c t o r f i e l d s as de s i r ed 34 In f a c t , V . admits a c r o s s - s e c t i o n over S n ^ i f and ' n,:k on l y i f n = 0 mod m ^he proof of n e c e s s i t y , however, turned out t o be con s i de rab l y more d i f f i c u l t . As w i t h many c l a s s i c a l problems, t h i s one was a l s o sub ject to the law of d im i n i s h i n g r e t u r n s . More and more work was done i n order to get a r e s u l t out of the dwind l i ng supply of s p e c i a l cases . Toda [23] remarked, " . . . our problem i s s t i l l open i n ques t ion on the sphere S ^ 4 ^ . " But a f i n a l r e s u l t i n a s p e c i a l case was i n i t s e l f not as important as the techniques used to o b t a i n i t . To i l l u s t r a t e t h i s we i n d i c a t e a proof f o r the cases when k-1 i s a power of two and when k £ ± 3 mod 8 . Let P^ k be the quot ien t space R p m "*"/RPm k . The f o l l o w i n g r e s u l t i s due to James [13]. (4.1) I f V , has a c r o s s - s e c t i o n over S N ^ then S n P , n,k m,k and P . , have the same homotopy type f o r a l l m S k . m+n,k r j J * Let f : S n > V , be a c r o s s - s e c t i o n and f o r v e S D ^ n,k k n l e t f : K. > R be the norm-preserv ing t r an s f o rmat i on corresponding to f (v) . Cons ider the map 0 : B n x R m - k x R k > Rm+n-k x R k ? m > k ^ 2 1/2 g iven by 9 ( t v , y , z) = (y , t f ^ ( z ) , ( 1 - t ) z) where rx~l in lv k 0 < t < l , v e S , y e R and z e R . S ince f i s norm p re se r v i ng and f (-z) = - f (z) we have 0 ( t v , -y , - z ) = - 9 ( t v , y , z) and ||Q(tv , y , z) || = || (y , z) || . I t f o l l o w s that 0 induces a map B n x R p m > R p m + n which c a r r i e s the subspace B n x R P ^ " 1 . S ^ X R P 1 1 1 " 1 of B n x R P m - 1 to the subspace R P ^ 1 1 - ^ 1 u i- «„ni+n—1 „, . of RP . Thus we have a map / T n „ m-1 „n „„m-k- l ^ n - l _ m-1, ,__m+n-l ^ i r r i -n-k- l , . (B x RP , B x RP u S x Rp ) y (RP , RP ) which i n t u rn induces a map * : ( B n / S n - 1 ) A ( R P M - V R P M _ K ~ 1 ) . ( R P ^ - V r P 1 " ^ ^ " 1 ) . That i s , —KP " . Now <j> induces an isomorphism i n homology and m,k m+n, k. hence S NP , and P , , have the same homotopy type, by Whitehead m,k m+n,k theorem. Next we r e c a l l some f a c t s about the e f f e c t of the Steenrod squares on the mod 2 cohomology of the p r o j e c t i v e spaces. We have H * ( R P m _ 1 ) = Z 2 [w ]/ (w n ) where w generates H ^ R P ™ 1 " 1 ) ; and Sq ' V 3 = (^ ) w1"*"-1 by the Cartan fo rmu la . For m - k < r < m H (P . , ) i s generated by an element w and S w. = (,)w. . f o r m,k' ° J r q ] 1 i+j j > m - k and i + j < m. The next two r e s u l t s are due to James [13] (4.2) Let n be an i n t e g e r and suppose that there i s a c r o s s -s e c t i o n of the f i b r i n g V , • S n where k = 2 S + 1 n, tc s+1 f o r some s. Then n = 0 mod 2 - . s+1 Choose m < k so that m = k mod 2 Then by the above remarks S q 1 H m _ k ( P , ) = 0 m, k r f o r a l l i > 0 . Suppose n i s an odd m u l t i p l e of 2 where r < s T h £ n S ^ W m+n-k = ( " T ^ m + n - k + i ^ 0 f o r 1 = ? • T h u s gqi^m+n k^p ) ^ 0 . S ince S q 1 commutes w i t h suspens ion, t h i s m+n,k i m p l i e s t ha t S n P , and P , . are not of the same homotopy type, m,k m+n,k J J which i s a c o n t r a c t i o n by (4 .1 ) . Thus n must be an even m u l t i p l e s of 2 and we are done. Let x(k) be the number of s with o < s < k such that s = 0 , 1 » 3 or 5 mod 8 . Note that x(k) = a(k) - 1 for k f ± 3 mod 8 and T(k)= a(k) otherwise. (4.3) Let k be an integer and l e t n = 0 mod 8 . Suppose that and P . have the same homotopy type for a l l m,k nrrn,k T He") m > k . Then n i s d i v i s i b l e by 2 v ' . Choose m > k so that m t k mod 4 . For n = 0 mod 8 the following diagram commutes (4.4) K 0 ( Pm 0 (S*> > KO(SnP , ) m,k ko(p ) — j z + k o ( s n p , ) m,k tn/2^ S A^n m.k7 for a l l integers t , where if denotes the Adams operations. By (2.13) we know that KO(RPm ^) i s c y c l i c of order a , say, with generator \ = 5 - 1 . Now m 2 t t K = 5 K C i s t r i v i a l and since if (5) = 5 we have if CX = 0 or X according to whether t i s even or odd. For m $ k mod 4 , KO(P , ) can be i d e n t i f i e d with the m,K subgroup of KO(RPm 1) generated by a . X and m—K. if*" = 0 or 1 according to whether t i s even or odd. t t n / 2 Thus ib = 1 i n KO(P . ) and s i nce (4.4) commutes ib = t i n m, K. KO(S nP ) . But / = 1 i n KO(P ) "s. KO(S nP ) . Let m,K. m+n,k m,K. f = a(m) - a(m-k) . R e c a l l that a l l these groups are c y c l i c of order 2^ and so we have (4.5) n/2 _ f t = 1 mod 2 . To complete the proof of (4.3) we need a lemma by Adams [ l ] . (4.6) Let f be a p o s i t i v e i n t e g e r . I f n i s an odd m u l t i p l e of 2^ , n = (2a+l)2^ , say then 3 n - l = 2 ^ + 2 mod 2 ^ + 3 . 2 To show t h i s f i r s t note tha t s i nce 3 = 1 mod 8 then 3^ n + 1 = 2 mod 8-. We now-proceed, by i n d u c t i o n over f , to prove that 3 -1=2 mod 2 . For f = 1 the r e s u l t i s t r ue s i nce 2 3 - 1 = 8 . Suppose the r e s u l t i s t r ue f o r some f > 1 . Then ? ( f + D , f , f 3 - 1 = (3 - 1)(3 + 1) = ( 2 f + 2 + x 2 f + 4 ) ( 2 + y 2 3 ) by i n d u c t i o n „f+3 . _f+5 - 2 mod 2 which completes the i n d u c t i o n . M • 0 2 f + ^ _ ^ - of+3 T, 0 ( 2 a ) 2 ^ _ . , „f+3 Now s i n ce 3 = 1 mod 2 we have 3 = 1 mod 2 , „ ( 2 a + l ) 2 f _ „ 2 f f+3 , , and 3 - 1 = 3 - 1 mod 2 and we are done. n/ 2 f We have found that t = 1 mod 2 . I f n i s an odd e-2 n / 2 _ e-1 e m u l t i p l e of 2 f o r any e > 3 then 3 - 1 = 2 mod 2 by (4.6). S e t t i n g t = 3 i n (4.5) we ob ta i n a c o n t r a d i c t i o n . Thus n f - 2 must be an even m u l t i p l e of 2 . However, we can choose an m w i t h m t k mod 4 so that f '- 1 = T ( k ) . Hence n i s a multiple of T ( k ) 2 . This completes the proof of (4.3). We may sum up our results so far by the following. (4.7) Let k < 10 or k = + 3 mod 8 . Then the S t i e f e l manifold :s a i ,o(k) V , admit cross-section over S n ^ i f and only i f n,k n = 0 mod 2 We use (4.2) for the cases k < 4 and to show that for k > 5 , n must be at least a multiple of = 8 . Now we use this together with (4.3) and the fact that for k 4 ± 3 mod 8 , x(k) = a(k) , to get the result for 6 < k < 10 and k i ± 3 mod 8 . Before completing the vector f i e l d on spheres problem i t should be mentioned that the general problem of finding cross-sections of the S t i e f e l f i b r i n g V , • V , 1 < £ < k < n has been completely i i y K. n j x. solved. The solution can be stated as follows. Let 1 < H < k < n . Then V , admits a cross-section over n,k V „ i n the following cases: (i\ V »• V , n,£ 6 W n,n n,n-l <14> V7,3 > V7,2 ( i i i ) v M >v 8 > 3. 5. The Atiyah-James Theorem As one might have expected, the complete solution of the vector f i e l d problem was not found by exhausting a l l the cases i n the proof of necessity but by the development and application of new methods. 39 In a d d i t i o n to c o n t r i b u t i n g to the s o l u t i o n of the v e c t o r f i e l d problem, these methods shed cons ide rab le l i g h t on the complex (and qua te rn i on i c ) c r o s s - s e c t i o n i n g problems. In an e legant paper by A t i y a h [5] he " u n i f i e d " the c r o s s - s e c t i o n i n g problems by showing that determin ing c r o s s - s e c t i o n s of 0 , was equ i va len t to determin ing n, K k-1 the J - o r de r of the c anon i c a l l i n e bundle i n J (FP ) . S-type and S - d u a l i t y Assume a l l spaces have base p o i n t . Let [X,Y] denote the homotopy c l a s se s of maps from X to Y . The suspension f u n c t o r , S , de f i ne s a map S:[X,Y] • [SX,SY]... The se t [SX,SY] i s a group and the f u n c t i o n S ^ s V s V l ^ [ S k + 1 X , S k + 1 Y ] i s a group morphism f o r k ^ 1 . Denote the l i m i t of the sequence of a b e l i a n groups [ S 2 X , S 2 Y ] >-[s 3X , S 3 Y] • [ S ^ . S 1 ^ ] »• by {X,Y} . An element of {X,Y} i s c a l l e d an S-map from X to Y . C l e a r l y the suspension f u n c t i o n S:{X,Y} —>• {SX,SY} i s an isomorphism. The f o l l o w i n g r e s u l t can be found i n Spanier [19 ] . (5.1) The n a t u r a l f u n c t i o n [X,Y] 4- {X,Y} i s a b i j e c t i o n f o r X , a CW-complex of dimension n , and Y , an r -connected space, r > 1 , where n - 1 < 2r - 1 . Now l e t f : X — » Y be a map and form the Puppe sequence X Y >- C f > SX SY . For a space Z we get the f o l l o w i n g sequence [X,Z] « [Y,Z] < [ C f ,Z] < [SX,Z] + [SY,Z] . I f we apply the suspension f unc t o r S to t h i s sequence and take the d i r e c t l i m i t then we get the f o l l o w i n g exact sequence (5.2) { X , Z } « — {Y,Z} < {Cf,Z}< {SX,Z}^ {SY,Z> C a l l maps u : X X ' >• S n , n - p a i r i n g s . These maps de f i ne A group morphisms u : {Z,X '} — — y {X Z , S n} and Z A -u Z : {Z..X} K Z X ' , S n} by the r e l a t i o n s u i f } = { u ( l f)> and A Z A Z k k u {,gl = { U ( S A 1 ) ^ • I f Z = S we w r i t e u^ and u r e s p e c t i v e l y . An n - p a i r i n g u i X ^ X ' —> S n i s c a l l e d an n - d u a l i t y i f u k : { S k , X ' } — ^ { X A S k , S n } and u k : { S k , X} — M S ^ X ' , S n } are group isomorphisms. I f an n - d u a l i t y map u : X A X ' >• S n e x i s t s then X ' i s c a l l e d an n -dua l of X . We c a l l X ' an S -dua l of X i f some suspension of X ' i s n -dua l to some suspension of X f o r some n . Spaces X ' and X are s a i d to have the same S-type p rov ided there e x i s t s a homotopy equ iva lence between some suspension of X ' and some suspension of X . S - r e d u c i b i l i t y and S - c o r e d u c i b i l i t y A space X i s r e d u c i b l e prov ided there e x i s t s a map f : S n * X such that f . : H.(S n ) >• H.(X) i s an isomorphism * x 1 f o r i > n . A space X i s S - r educ i b l e i f S X i s r e d u c i b l e f o r some k . A space X i s co reduc i b l e prov ided there e x i s t s a map g : X • S n such that g* : H 1 ( S n ) H 1 (X) i s an isomorphism f o r i < n . A space X i s S - co reduc ib le i f S X i s co reduc i b l e f o r some k . Le t u : X X ' *• S n be a map . For two spaces W and Z de f i ne a f u n c t i o n [ W , Z A X ] >• .[w X ' , S n z ] where the image of [ f ] :W > Z A X i s [ (1 u ) ( f 1) ] . Taking the l i m i t we get a morphism 6 : {-W;, 'Z "X} • {W X ' ,-SnZ} . I f W and Z are f i n i t e CW-complexes and u : X A X ' >• S n i s an n - d u a l i t y map then 6 i s an isomorphism. Let u : X X 1 -—> S n and v : Y Y ' > S n be two n - d u a l i t y A A J maps. (5.3) Le t f : Y • X and g : X ' • Y ' be two maps such tha t v ^ | u Y { f } = {g> , where v~) u ' Y :{Y:,X} —> {X ' , Y ' } . That i s , {u( f 1)} = { v ( l g)} i n {Y X ' , S n} . I f W and Z A A A are f i n i t e CW-complexes then the f o l l o w i n g diagram commutes (5.4) . {W,Z Y} »• {W Y* , S nZ} A A {w,z x} > {w x ' , s n z > A A Cons ider the Ei lenberg-MacLane space K(G , i ) and r e c a l l tha t H 1 (X,G) i s i somorphic to [ X ,K (G , i ) ] and ^[^^(X.G) i s i somorphic to TT. . (K(G,k) X) f o r l a r g e k . Whitehead [25] K.H"i A showed that K (G , i ) and K(G,k) can be rep laced by s u i t a b l e ske le tons K ( G , i ) * and K (G ,k ) * which w i l l be f i n i t e CW-complexes f o r f i n i t e l y generated G . 42 The r - p a i r i n g v : S n S™ >• S r , n+m = r , i s an r - d u a l i t y s i n ce S : {X,Y} — » - {SX,SY} i s an isomorphism. Thus by (5.4) we have the f o l l o w i n g commutative diagram \{Sn) = { S k + q , K ( Z , q ) * A S n } ^ - > { S k + q A S m , S r K ( Z , q ) * } = ST^s" 1) H k (X) = { S k + q , K ( Z , q ) * A X } ^ { S k + q A X ' , S r K ( Z , q ) * } = 5 r _ k ( X ' ) where f : S n > X and g : X ' • S™ are as i n (5.3) and X X 1 • S r i s an r - d u a l i t y . Therefore f A : H ± ( S n ) • H ± (X) i s an isomorphism f o r i < n i f and on ly i f g* : H (S ) >• H J ( X ' ) i s an isomorphism f o r m < j . Thus we get the f o l l o w i n g . (5.5) Let X and X ' be f i n i t e CW-complexes that are S-dual to each o the r . Then X i s S - r educ i b l e i f and only i f X ' i s S - c o r educ i b l e . Thorn Spaces. Let £ be a r e a l v e c t o r bundle over a compact CW-complex, w i t h a reimannian m e t r i c . Let D(£) be the a s soc i a ted d i s k bundle and S (£) the a s soc i a ted sphere bundle. The Thorn space of E, , denoted T(£) , i s the quo t i en t space D(£)/S(£) . Note tha t T(£); can a l s o be regarded as the one po i n t c o m p a c t i f i c a t i o n of the t o t a l space, E(5) . 43 (5.6) By a s imple argument, the Thorn space T(E <B 0 n ) i s homeomorphic to the n - f o l d suspension S n T(E) . k The r e a l p r o j e c t i v e space, BSP , can be thought of as k the quo t i en t space of S modulo the a c t i o n of TL^ . In o ther words k k RP equals S modulo the r e l a t i o n x i s equ i va l en t to - x , f o r k k x e S . Let denote the c a n o n i c a l l i n e bundle on RP . Then It TO the t o t a l space E(mE^) i s the quo t i en t space S x ffi. modulo k rn. Z 0 and D(mE. ) i s the quot ien t space S x D modulo Z 0 . Let < x ,y >. denote the c l a s s of (x,y) i n D(mE^) . Then < x,y > e S:(mEk) i f and only i f ||y|| = 1 . m _ k m—1 We de f i ne a homeomorphism h : T(mE, ) >• KP /RP • Def ine a map f : S k x D m • S m f k by f (x,y) = (y, (1 - || y|| 2 )x) . Then f ( S k x S m _ 1 ) = S ™ " 1 c . Now f ( - x , - y ) = - f ( x , y ) so f de f ines a map g : D(mE k) —>• RP11^"15" when we mod out the Z2~ac t i on and t^t ~ \ \ ™.^ ni—1 „ r „k . „m „m+k „m— 1 g ( S (m£ k ) ) = RP . Now f : S x i n t D • S - S i s a homeomorphism and hence g : D(mE^) - S(mE k) >• R P m f k - RP m i s a homeomorphism. Therefore g induces the r equ i r ed quo t i en t map h . Combining t h i s r e s u l t w i t h (5.6) y i e l d s a homeomorphism between the Thorn space T ( m E k & 0 n ) and the n - f o l d suspension s n ( k p k + m , M p m - l ) ^ We remark t h a t i f E i s the c anon i c a l 2 -d imens iona l l i n e bundle k k n over CP then we a l s o have T(mE,^ 9»0 ) i s homeomorphic to s n ( c p m f k / c p m " 1 ) . 44 In v iew of ( 5 .6 ) , the S-type of T(£) depends on ly on the c l a s s of £ i n KO(X) . S ince every element n i s e x p r e s s i b l e as £ - 0 n , f o r some i n t e ge r n , i t f o l l o w s that we can extend our n o t a t i o n and speak of the S-type of T(n) f o r any n e KO(X) . A t i y a h ' s D u a l i t y Theorem By u s ing the theorems of Whitney concerning embeddings of man i fo lds w i t h boundary i t i s not d i f f i c u l t to show that i f X i s a compact, d i f f e r e n t i a b l e man i fo ld w i t h boundary and x i s the tangent bundle of X , then T( -x) i s the S -dua l of X/6X . From t h i s we can deduce A t i y a h ' s d u a l i t y theorem f o r Thorn complexes. (5.7) Let X be a compact,, d i f f e r e n t i a b l e man i fo ld (without boundary), w i t h tangent bundle x . Let a be a r e a l v e c t o r bundle over X . Then the S-dual of T(a) i s T ( -a -x ) . To prove t h i s we f i r s t g i ve a a d i f f e r e n t i a b l e s t r u c t u r e by t a k i n g the c l a s s i f y i n g map f o r a and approximating i t w i t h a d i f f e r e n t i a b l e map g . The map g induces a d i f f e r e n t i a b l e bundle over X which i s equ i va l en t to a . Then D(a) i s a compact d i f f e r e n t i a b l e man i fo ld w i t h boundary S(a) . By the above remarks, the S-dual of T(a) = D(a)/S(a) i s the Thorn space T ( - t ) , where t i s the tangent bundle of D(a) . Let IT : D(a) >• X be the p r o j e c t i o n map. Then t i s i somorphic to x r ^ x f t a ) and s i n ce xrt i s a homotopy equ iva lence i t f o l l ows that T ( - t ) and T( -a -x ) are of the same S-type. Hence T( -a -x ) i s the S-dual of T(a) as r e q u i r e d . 45 Thom Spaces and J ( X ) In t h i s s e c t i o n we i n d i c a t e a proof of the f o l l o w i n g . (5.8) Let E be a r e a l v e c t o r bundle over a connected space X . Then T(E) i s S - co reduc ib l e i f and on ly i f J ( E ) = 0 . ) We beg in by p rov ing (5.9) I f E and n are two ve c t o r bundles over X w i t h J ( E ) = J ( n ) then T ( E ) r and T(n) are of the same S-type. Now J ( E ) = J ( n ) i m p l i e s that S ( E © 0 n ) and S(n<B0m) have the same homotopy type. Let E' = E ® and n ' = n <B 0 m .' Le t f : S (E ' ) > S ( n ' ) and g : S ( n ' ) »• S (E ' ) be f i b r e homotopy i nve r se s of each o the r . We can extend f and g r a d i a l l y to f : D(E ' ) H>(n') and. g : D(n') >• D(E ' ) . The homotopy between f g and the i d e n t i t y extends to a homotopy (D(E ' ) , S ( E ' ) ) K D ( E ' ) , S ( E ' ) ) between f ' g ' and the i d e n t i t y . S i m i l a r l y g ' f i s homotopic to the i d e n t i t y . By pass ing to quo t i en t s we get maps f : T ( E ' ) *-T(TI') and g : T ( n ' ) • T ( E ' ) which are homotopy i nve r se s of each o the r . Therefore S n T(E) = T ( E © 0 n ) = T ( E ' ) and S m T ( n ) = T ( n ® 0 m ) = T ( n ' ) have the same homotopy type. Thus T(E) and T ( n ) are of the same S-type. As a c o r o l l a r y suppose n = 0 , the t r i v i a l bundle of dimension 0 . Then T(0) = X + , the d i s j o i n t un ion of X and an i s o l a t e d base p o i n t , {«>.} . Thus i f J ( E ) = 0 then T(E) and T(0) have the same S-type. 46 To complete the proof of (5.8) we show tha t the f o l l o w i n g statements are e q u i v a l e n t . (1) J ( 0 = 0 i n J(X) (2) The space T(£) i s S - co reduc ib l e (3) The spaces T(£) and T(0) = X + have the same S-type. We have a l ready shown tha t (1) imp l i e s ( 3 ) . To show that (3) i m p l i e s (2) l e t g : X + • S^ be a map de f i ned by g(X) = -1 and g(°°) = 1 . Then g* : H i ( S ° ) y H i ( X + ) i s an isomorphism f o r i < 0 . Therefore T(0) i s co reduc i b l e and s i nce T(£) and T(0) have the same S-type, T(£) i s S - co reduc i b l e . To prove that (2) i m p l i e s (1) we take an i n t e ge r m l a r ge enough so that n = £ <B 6 m has the p roper ty that T(n) = T(£ © e m) = S mT(5) i s c o r educ i b l e and m > dim X . Let g : (D (n) , S ( n ) ) • ( S n , *) be the co reduc t i on map f o r T(n) -., ••. Then by (5.1) and (5.2) we have the f o l l o w i n g commutative diagram where the h o r i z o n t a l arrows are isomorphisms [ s ( n ) , s n _ 1 ] • [D(n)/s(n) , s n ] [s(n ) , s 1 1 " 1 ] • [ D ( n ) / s ( n ) , s n ] Thus the co reduc t i on map def ines a map f : S(TI) > S such that f| , v : S(n ) y S i s a homotopy equ iva lence f o r a l l x i n X x X 47 Now a theorem due to Dold s t a t e s tha t under t h i s c o n d i t i o n S ( n ) and X x S n must be f i b r e homotopic e q u i v a l e n t . Hence S(£) i s s t a b l e f i b r e homotopic t r i v i a l . Or, J (£ ) = 0 . Th i s completes the proof of (5 .8 ) . We now combine (5.7) and (5 .8 ) . (5.10) Le t X be a connected, compact, d i f f e r e n t i a b l e man i fo ld w i t h tangent bundle x . Let a be a r e a l v ec to r bundle over X . Then T(a) i s S - redud ib le i f and on ly i f J ( a ) = J ( - x ) . By (5.7) T(o) i s S -dual to T ( -a -x ) . By (5.5) T(a) i s S - r educ i b l e i f and on ly i f T ( -a -x ) i s S - co reduc i b l e . Now (5.8) - i m p l i e s that T ( -a -x ) i s S - co reduc ib l e i f and on ly i f J ( - a - x ) = 0 . Hence J (a ) = J ( - x ) . The Main Theorem In t h i s s e c t i o n l e t P be the r e a l or complex p r o j e c t i v e space and £ the c anon i c a l r e a l d -d imens iona l l i n e bundle over P . Le t x be the tangent bundle over P and l e t be the dua l bundle of £ . A t i y a h found t h a t n = 5 ®-r, K* had the proper ty that x <B n = k£ and tha t r) decomposes as n = 8^ © ? f o r some bundle t, . i Now James [12] de f ines a subspace Q of 0 which he L n n c a l l e d a q u a s i p r o j e c t i v e space. These spaces have n a t u r a l i n c l u s i o n s Q , c Q . Tak ing quo t i en t s we get the t runcated I 48 q u a s i p r o j e c t i v e spaces ^ = Q n/Q n_k • A t i y a h proved that the space Q , was homeomorphic to the Thorn space T( (n -k )£ <& £) . Th i s leads to the f o l l o w i n g r e s u l t . (5.11) Q i s S - r e d u c i b l e i f and only i f n i s a m u l t i p l e of ri j K. the order of J ( 0 . For Q i s S - r e d u c i b l e i f and on ly i f T ( ( n - k ) £ # C ) i s S - r e d u c i b l e i f and on ly i f J ( (n-k)E+ 5) = -J(T) , by (5 .10) , i f and on ly i f J ( (n-kK + C + T ) = 0 i f and on ly i f J ( ( n - k ) £ + t, + (k£ - ?)) = 0 , by the above remarks, i f and on ly i f J(n£) = 0 i f and on ly i f nJ(£) = 0 . (5.12) James [12] , by determin ing the c o n n e c t i v i t y of the p a i r (0 , , Q , ) , d i s covered that Q .. i s r e d u c i b l e i f and n, k. n, k n, k dn—1 on ly i f 0 , >• S has a c r o s s - s e c t i o n and f o r n,k F = R , n > 2 k or k = 1 . (5.13) Us ing a r e s u l t by Toda [24] , A t i y a h goes on to prove that Q , i s r e d u c i b l e i f and only i f i t i s S - r e d u c i b l e and n j K, n > 2k or k = 1 . Us ing (4 .1 ) , (4.2) and (4.3) we can show that i n the r e a l case n > 2k i f k> 9 and that V , > S N has a c r o s s - s e c t i o n n,k i f k <10 . I n . the complex case A t i y a h and Todd [7] showed that n > 2k f o r k > 1 . These remarks together w i t h (5 .11), (5.12) and (5.13) y i e l d : 49 dn~ 1 Theorem. The S t i e f e l f i b r i n g 0 , • S admits a c r o s s -n,k s e c t i o n i f and on ly i f n i s a m u l t i p l e of the order of J (E ) i n J ( P k ) . Th is theorem together w i t h the isomorphism KO(EP k) = J ( RP k ) , ( 3 .2 ) , and the de te rminat i on of KO(RP k) by Adams, (2.13) , complete ly so l ves the vec to r f i e l d on spheres problem. 6. Complex C ro s s - s ec t i on s The genera l problem of c r o s s - s e c t i o n s of S t i e f e l man i fo lds has been " r educed " to a problem i n J - t h e o r y . In the complex case, however, we do not have an isomorphism s i m i l a r to (3.2) f o r CP . Thus we must use other methods to determine the order of J(E) i n J(CP ) . The isomorphism (3.3) a l l ows us to a l t e r our n o t a t i o n and w r i t e J * (CP k ) f o r e i t h e r J R ( C P k ) or J£.(CP k) . To compute the order of J (E) i n J(CP ) , we use the k k groups J ' ( C P ) and J " (CP ) and the f o l l o w i n g commutative diagram 50 I t i s i n t e r e s t i n g to note that a l though the methods and procedure are the same, the qua te r n i on i c c r o s s - s e c t i o n problem was not so l ved u n t i l n ine years a f t e r the complex one. The reason f o r t h i s i s tha t i n order to a r r i v e a t a s o l u t i o n we f i r s t show t h a t , modulo p a t ho l o g i e s , the map 0 ' i s an isomorphism; then we c a l c u l a t e the J' -o rder of E . Th i s i s done by showing t h a t , modulo p a t ho l o g i e s , the compos i t ion O ' 0 " i s an isomorphism. As we have seen, the map 6 : J((EP K) —*• J' (CP K) was de r i ved a t i n a d i r e c t manner. D e r i v i n g the map 6" : J"(CP K) J(CP K) , however, depended upon showing that the space C P K was one of the s p e c i a l cases which s a t i s f i e s the Adams con jec tu re , (S ince the space MP was apparent l y not one of these s p e c i a l cases, i t was not u n t i l the Adams con jec tu re was proved, by Q u i l l a n [17 ] , f o r k k f i n i t e CW-complexes, that the map 0 " : J"(HP ) • J(HP ) cou ld be w e l l - d e f i n e d . S h o r t l y t h e r e a f t e r S i g r i s t and Suter [18] so lved the qua te rn i on i c c r o s s - s e c t i o n problem.) The J - o r de r of E i n J(CP K) The r e s u l t s of t h i s s e c t i o n are due p r i m a r i l y to Adams [ 1 , 2 ] , A t i y a h and Todd [7] and Adams and Walker [ 4 ] . r l r 2 Le t 0 # x = p^ p 2 be the prime f a c t o r i z a t i o n of x and de f i ne the number - theoret i c f u n c t i o n v by v (x) = r . P • P • X For a r a t i o n a l number x/y , v^(x/y) = Vp (x) - v^(y) . Let [x] denote the g rea te s t i n t e g e r not exceeding x , f o r a r a t i o n a l number x . Def ine the i n t e g e r M, as f o l l o w s 51 v max(r + v ( r ) ) , 1 < r < [^-7] i f p < k p P - l r 0 i f p > k k-1 Theorem. The order of J (E) i n J(CP ) i s R e c a l l tha t u = E - 1 and r : KU(CP k ) > KO(CP k) i s the ' r e a l i z a t i o n ' map. The theorem w i l l be proved by p rov ing the f o l l o w i n g statements. (6.1) The order of J ' ( E ) i n J ' ( C P k _ 1 ) i s ^ . (6.2) I f k f 1 mod 4 then the map k k 9 ' : J(CP ) y J ' ( C P ) i s an isomorphism, and i f k = 1 mod 4 then ker 0 ' = Z^ and i s generated by J r u ^ W + ^ " where k = 4w+l . k k A l so , the map 0 " : J " (CP ) • J(CP ) i s an isomorphism f o r a l l k . (6.3) I f w>o then the order of J r E i n J ( C P 4 w + 1 ) i s M 4w+2 * Proof of (6.1) We say that an i n t e ge r n s a t i s f i e s c o n d i t i o n (C^) i f f o r each i n t e g e r 1 < q < k - 1 , the c o e f f i c i e n t of z q i n / l ° g ( l + z ) \ i s an i n t e g e r . Here, ( ± 0 g ( ± + z ) ) denotes the u sua l power z z s e r i e s expans ion. Now we w i l l prove the f o l l o w i n g . 52 (6.4) An i n t e ge r n s a t i s f i e s c o n d i t i o n (C^) i f a n c * on ly i f J ' ( n S ) = 0 i n J ' ( C P k - 1 ) . S ince J ' i s a homomorphism we may r ep l ace J ' ( n E ) w i t h J ' ( - n E ) . By d e f i n i t i o n , J ' ( - n § ) = 0 i m p l i e s that there e x i s t s a ~ k-1 B e KU(CP .) such that bh(-nE) = ch(l+B) . The cohomology c l a s se s k-1 k-1 1 , ch(y) , , ch(y ) f r e e l y generate H*(CP , Q) and so there e x i s t unique r a t i o n a l s a , , a, ., such that o k-1 k-1 . ^ bh(-nE) = E a . ch ( y X ) and a = 1 . But B e KU(CP ) i m p l i e s 1 o i=o tha t there e x i s t unique i n t ege r s b Q , , b^_^ such that k-1 . k-1 . k-1 . B = E b . y 1 . Thus ch(l+B) = 1 + E b . ch ( y X ) = E b . ch ( y 1 ) i = l i = l i=o where b = 1 . Hence such a B e x i s t s i f and on ly i f a l l the a. are o 1 i n t e g e r s . Let z = ch(y) and r e c a l l tha t ch(y) = e^ - 1 where y = c^(E) . , Then y = l og ( l +z ) where : l og ( l+z ) i s de f i ned i n k-1 H*(CP , flj) by i t s u sua l power s e r i e s expans ion. Note that i i k ch(y ) = z , o < i < k-1 , and z = 0 . Thus a^ i s the c o e f f i c i e n t of z i n bh(-nE) = bh(E) n = ( 4 - ^ ) n = ( - ^ - ) n y e y - l = ( l o g ( l + z ) n z Therefore J ' ( n E ) = 0 i s equ i va l en t to the c o n d i t i o n that the c o e f f i c i e n t of z~ i n ( x ° g ( ± + z ) ) should be an i n t e ge r f o r o ^ i < k - l . 53 We have reduced the problem of f i n d i n g the J ' - o r d e r of £ , to determin ing the c o e f f i c i e n t s of a c e r t a i n power s e r i e s . So we f o l l o w [7] and prove the f o l l o w i n g . (6.5) An i n t e ge r n s a t i s f i e s c o n d i t i o n (C^) i f and on ly i f n i s a m u l t i p l e of . F i r s t we need a couple of lemmas. (6.6) Let p be a pr ime, s a p o s i t i v e i n t e ge r and l e t k-1 = s ( p - l ) . Suppose that v p ( n ) - v p ( M ] c _ i ^ • Then v ( n ) = v (n) - v ( r ) f o r 1 < r < s p r p p P roo f . Let 1 < m < s . Then v^(m) < s - 1 because p - 1 * . > A l s o , v p ( n ) * y M ^ ) * C f E f ] ' t-Zt^-i -[s - —^r] = s - 1 . Thus f o r 1 < m < s , v ( m ) = v (n-m) . Now L p-1 P P V r > r ! ( n - r ) ! r ^ r-1 n-m Therefore i t f o l l o w s that v p ( ^ = v p0 = V p ^ - V p ^ a n d w e a r e done. (6.7) Let a i » CT2 ' ' a h ^ e P o s l t l v e i n t e g e r s . Let a = £c. , v = v ( a ) and p = min v (a.) . 1 P ± P 1 Then v ( : — ~ r ) > v - p . p ffl!a2!....ah! 5 4 Proo f . Without l o s s of g e n e r a l i t y suppose that p = V p ( ° q ) • We have the f o l l o w i n g i d e n t i t y a ! a _ a - 1 . ( C T~ g l ) ! a l ! A 2 ! ' " - a h ! ° 1 V 1 ° 2 ! V ' The l a s t two f a c t o r s are i n t e ge r s and v (—) = v (a) - v ( a j = v - p P o"1 p p i The lemma f o l l o w s immediate ly. . ^ l og ( l +z ) Now the power s e r i e s expansion of — i s CO "1 "f ( - 1 ) z E — :—-— . Def ine the c o e f f i c i e n t s v by the formula i = 0 1 + 1 ( " ( - D V V : I E,. . J = E y z • \ i = 0 1 + 1 / i = 0 N ' R Using the mu l t i n om ia l expansion-we f i n d i r ( 6 . 8 ) y n _ = S ( ~ D r —TTT T T n „ = T ( n , r , S ) , n ' r S s 0 ! s 1 ! . . . . . s r ! ± = 0 ( . + 1 ) s . say, where the summation extends over a l l ordered se t s S = ( S Q , s . . . . . . . , s^) of non-negat ive i n t e ge r s such that Es^ = n and E i s ^ = r . ( 6 . 5 ) , and hence ( 6 . 1 ) , w i l l be a consequence of the f o l l o w i n g p r o p o s i t i o n . ( 6 . 9 ) Le t p be a pr ime, n and k p o s i t i v e i n t e g e r s . Then the f o l l o w i n g statements are equ i v a l en t . (C, ) : v (Y ) > 0 f o r 1 < r < k - 1 k,p p ' n , r (Mk,P):Vn) * w (T, ) : v (T (n , r ,S ) ) > 0 f o r 1 < r < k - 1 and k,p p ' ' a l l sequences S o c cu r r i n g i n ( 6 . 8 ) . 55 The proof of (6.9) i s by i n d u c t i o n on k . The r e s u l t i s t r i v i a l when k = 1 . The i n d u c t i v e step i s prov ided by (6.10) In ( 6 .9 ) , w i t h k > 1 , assume that p ) and (T, n ) are s a t i s f i e d . Then v (T(n,k,S)) > 0 f o r a l l k - l , p p ' sequences S o c c u r r i n g i n (6.8) w i t h the f o l l o w i n g p o s s i b l e e x cep t i on : i f k - 1 = s ( p - l ) w i t h s i n t e g r a l , and i f S i s the sequence i n which s„ = n - s , s , = s and a l l 0 p-1 other s. are zero then we have l - v (T(n,k,S)) = v (n) - v (s) - s . p P P Before p rov ing (6.10) we show that i t does, i n f a c t , y i e l d the de s i r ed equ i va lences . F i r s t we d i v i d e the sum (6.8) i n t o two summands, Si c c Y , , = T + T * where -r i s the summand corresponding to the above ' n , k - l b d d sequence, S , and i s the sum of a l l o ther summands. Now (6.10) a. i m p l i e s that V p ( ^ ) - 0 . A s imple c a l c u l a t i o n then shows that v (Y . T ) > 0 , i f . and o n l y i f v (-j) > 0 , which y i e l d s the p ' n , k - l ' ' J p d ' ' equ iva lence between (C, ) and (T, ) . To see the equ iva lence K.,p K ,p between (T, ) and (M, ) , r e c a l l that (M, ) i m p l i e s that K , p K , p K l , p v (n) > max(r+v ( r ) ) , 1 < r < [r-rr] . Now (6.10) i m p l i e s that f o r every P r P P-1 summand — e -f- , v (—) > 0 . Then (6.6) i m p l i e s that v b p v v p(^-) = v p ( n ) - v p ( s ) - s > 0 i f and on ly i f v p ( n ) > v p ( s ) + s = v (k j") + ^ j" and we are done, p p - 1 ' p-1 Proof of (6.10) We proceed by an enumeration of cases of the sequences S k-1 Without l o s s of g e n e r a l i t y we may omit the f a c t o r (-1) (a) Suppose that some s.. i s p o s i t i v e , j > 0 , and j + 1 i 0 mod Cons ider the ordered set S' de f i ned by s^ = s Q + s^ , s^ = 0 and s! = s. , i = 0 or j . Then i i k' - 1 = E i s ! = E i s . - i s . < k - 1 . Now I i 3 i k-1 n T(n ,k ' , S ' ) = . ^ x , a n ; . . , . . . a r n ( S 0 + S j ) ! s l ! S k - r i=0 ( i + l ) S i ' Thus T(n,k,S) = T ( n , k ' , S ' ) ( J ) - . S ince ( J ) i s an so ( j + i ) s J S 0 i n tege r and v (j+1) = 0 we have v (T(n,k,S)) > v ( T (n , k JS ' ) ) > 0 P P P by the i n d u c t i o n hypothes i s . (b) Suppose that some s.. i s p o s i t i v e w i t h j = tp^ 1 - 1 where t > 1 and t ^ 0 mod p . Cons ider the ordered set S' de f i ned by s' = s , + s. , s! = 0 and s| = s. , i ^ i or p -1 . h. , h 3 J x i p ; - l p -1 J J Then k' - 1 = E i s ! = E i s . - ( t - l ) p s. < k - 1 . Thus i i J S h + s • T(n,k,S) = T ( n , k ' , S ' ) ( p - 1 ' J ) ^ — • S i n c e S . S - ; j t J v p ( t ) = 0 , v (T(n,k,S)) > V p ( T ( n , k ' , S ' ) ) > 0 by hypothes i s . (c) The terms not so f a r cons idered are those f o r which the on ly non-zero s^ , i > 0 , a r i s e when i has the form p -1 . For s i m p l i c i t y we make a change of n o t a t i o n w r i t i n g a, f o r s n n . P "I 57 h 2 Then E a h = n and E (p —l)o"^ = k-1 . Let s = + (p_+l)a 2 + (p +p+l)a 3+ Then we can w r i t e k-1 = s ( p - l ) . Let a = n - and a ' = + 2a^ + 3a^+ . Suppose that f o r some h > 1 we have a, > 0 . Cons ider the ordered set S' de f ined by s ' = a ' , s ' = n - a ' h p—J- u and sl_ = 0 , i + 0 or p-1 . By hypothes i s v (n) > v p ( M k _ 1 ) > s-1 s-1 so that n s p > s S a ' and hence s^ > 0 . We have k ' - l = E is ' . = ( p - l ) o ' = ( p - l ) s - E [ ( p X - l ) - i ( p - l ) ] a . < k-1 because i a, > 0 f o r some h > 1 . A f t e r computing T ( n , k ' , S ' ) we f i n d that h (!) T(n,k,S) = T d i . k ' . S ' ) ° a ! ( n , ) a l ! 0 2 ! a Now k-1 = s ( p - l ) and v (n) > v (M, .) by hypothes i s . Thus, s i n ce p p k-1 a and a ' < s , we have by (6.6), v p ^ ) = V p ^ n ) ~ V p^° ) a n < * v ( n , ) = v (n) - v ( a ' ) . A l s o , by (6.7) we have p a p p v (—-1—: ) > v (a) - min v (a .) > si (a) - v ( a ' ) . Hence P a 1 ! a 2 ! p P i P P > 0 . Therefore v (T(n,k,S)) > v ( T ( n , k ' , S ' ) ) > 0 by hypothes i s . P P (d) The on ly term not so f a r cons idered i s the one corresponding to the set S where S Q = n-s , s p _ n = s a n & a 1 1 o ther s^ are n X ze ro . So k-1 = E i s . = s ( p - l ) . We have T(n,k,S) = ( ) — • 1 s t > p App l y i ng (6.6) once more y i e l d s V p ( T ( n , k , S ) ) = v p ( < ° ) - ^ ) = v p ( n ) - v p ( s ) - s . This completes the induction step, the proof of (6.9) and hence (6.1) i s proved. Proof of (6.2) To prove (6.2) we w i l l need several lemmas. Most of the proofs are taken from [2] and [4] . (6.11) If v > 0 then the injection i:CP 2 v • C P 2 v + 1 induces an isomorphism from J' (CP2v+"*") onto J'(CP 2 v) . Proof. Since the map i*:KO(CP 2 v + 1) • KO(CP 2 v) i s an epimorphism i t suffices to show that ( i * ) - 1 V a ( C P 2 v ) c V R ( C P 2 v + 1 ) . Suppose then that a e KO(CP 2 v + 1) i s such that shi*a = i*sha = ch c(l+B) for some g e KO(CP2v) . Since i * is an epimorphism we can write g = ±*y for some y • Then i*sha = ch c(l+i*y) = i*ch c(l+y) . So we have shot = ch c(l+y) mod H 4 v + 2 ( C P 2 v + 1 , «8) . But sha and ch C(1+Y) have components equal to zero in dimension 2 4v + 2 . Therefore sha = ch C(1+Y) and so a e V_(CP ) . This IK. completes the proof of (6.11). i j (6.12) If X Y • Y/X is a cofibring such that ch:KU(X) • H*(X,Q) and j*:H*(Y/X,Q) v H*(Y,Q) are monomorphisms then j*:J'(Y/X) *• J'(Y) i s a monomorphism. 59 P roo f . Let a e KU(Y/X) be an element such that J ' ( j * a ) = 0 . By d e f i n i t i o n t h i s means that bhj*a^= ch(l+g) where $ e KU(Y) . Then we have ch i * ( l+P) = i * ch(l+B) . = i * bh j * a = bh i * j * a = 1 . S ince ch i s a monomorphism we have i*(l+3) = 1 . By exactness •1+13 = j * ( l + y ) where y e KU(Y/X) . I t f o l l ows that j * bha = b h j * a = ch(l+B) = ch j * ( l + y ) = j * ch(l+Y) • S ince j * i s a monomorphism we have bha = ch(l+y) . Thus J ' ( a ) = 0 , by d e f i n i t i o n . This completes the proof of (6.12). Before s t a t i n g the next lemma we need to de f i ne the number-t h e o r e t i c f u n c t i o n m(t) as f o l l o w s . For an odd prime p , 0 i f t t 0 mod (p-1) 1+v ( t ) i f t = 0 mod (p-1) . v <m(t)) = For p := 2 , v2(m(t)) = { 1 i f t ^ 0 mod 2 2 + v 2 ( t ) i f t = 0 mod 2 Note that m(2v+l) = 2 . M i l n o r and Keva i re [15] proved the f o l l o w i n g , which can be used as an a l t e r n a t i v e d e f i n i t i o n of m(2v) . Let and g f c be x , oo t ' e —1 x de f i ned by the power s e r i e s expansion l o g — - — = E a — and 00 t = E 3^ - r • F ° r t > 1 i t i s not d i f f i c u l t to show that e X - l t - 0 fc t ! 3 f c = t a f c . The r e l a t i o n between the c o e f f i c i e n t s 8 and the s-1 c l a s s i c a l B e r n o u l l i numbers B i s 8„•• = (-1) B . M i l n o r and s 2s s Keva i re proved that m(2v) i s the denominator of 2v = 2v = v-1 _ v 2 4v ^ ; 4v when t h i s f r a c t i o n i s expressed i n i t s lowest terms. The f o l l o w i n g i s due to [ 4 ] . (proof omitted) (6.13) The group J 1 ( C P 2 v / C P 2 v ~ 2 ) i s c y c l i c of order m(2v) , generated by J ' (ooV) , where ••«.'= ry = r(E - l ) . The group J " ( d P 2 v / C P 2 v 2 ) i s c y c l i c of order m(2v) , generated by J M ( u V ) . In [2] , Adams showed that i f KO(X) KO(Y) * KO(Z) i s exact where each of the groups i s f i n i t e l y generated and such that J".(X) i s f i n i t e , then the sequence 61 J " ( X ) • J " ( Y ) • J " ( Z ) -> 0 i s exact . There fo re , s i n ce J " ( X ) i s always f i n i t e , by (2.12) the sequence J "0CP k /CP m ) J " ( C P k ) j"(c;pm) -y 0 , corresponding to the c o f i b r i n g CP ,^--y CP y CP /CP , i s exact. (6.14) The map G ' 0 M : J " ( C P 2 v ) • J 1 1 ( C P 2 v ) i s an isomorphism. P roo f . The proof i s by i n d u c t i o n on v , the case v = 0 being 2v-2 t r i v i a l . Suppose the statement ho lds f o r (CP . Consider the c o f i b r i n g ,CP 2 - 2 _ i _ ffip2v _ J _ C p 2 v / C p 2 v - 2 Then the f o l l o w i n g diagram commutes J " ( C P 2 ^ / C P 2 v 2 ) ~^-y J " ( C P 2 v ) J " ( C P 2 v 2 ) y 0 J ' ( C P 2 v / C P 2 v - 2 ) ~^-y J ' (2v) .2vl i (2v-2) 2v-2 s (CP ) •*• J ' (CP ) °y 0 The l e f t - h a n d arrow i s an isomorphism by (6.13); the upper row i s exact by the above remarks; the r i gh t -hand arrow i s an isomorphism by i n d u c t i o n ; the map j ' i s a monomorphism by (6.12); and the c e n t r a l arrow i s always an epimorphism. Now i f ® '® " ( 2 v ) ^ a ^ = ^ then i " ( a ) = 0 . So there e x i s t s a g i n J " ( C P 2 v / ( E P 2 v - 2 ) such that 62 j " ( 8 ) = a . S ince the l e f t - h a n d arrow i s an isomorphism t h i s i m p l i e s tha t j ' ( 3 ) = 0 . Therefore 3 = 0 because j ' i s a monomorphism. Thus a = 0 and ® '®"(2v) ^ S a m o n o m o r P n : L S m * Th is completes the proof of (6.14). We have j u s t proved that e'e" : J " ( C P k ) • J ' ( C P k ) i s an isomorphism i f k = 0 or 2 mod 4 . Before d e a l i n g w i t h the case k = 1 or 3 mod 4 we need the f o l l o w i n g lemma. 4w+l 4w+l , (6.15) In J(CP ) we have J ry ± 0 . P roo f . Let f : RP 8 v ^ - 2 • C P 4 W + 1 be the p r o j e c t i o n w i t h f ( E P 8 w + 1 ) c <GP4w Le t . j : C P 4 w + 1 > C P 4 w + 1 / C P 4 w be the quo t i en t map.. S ince C P 4 w f l / C P 4 w = S 8 w f 2 we have K U ( ( C P 4 w f l / C P 4 w ) = Z generated by y , say, and K 0 ( C P 4 w + 1 / C P 4 w ) = Z £ generated by ry . Now y 4 w + 1 = j*Y . S ince f has degree 1 , f * r y i s a generator of K0(ffiP /RP ) . In a prev ious r e s u l t due to Adams [ 1 ] , i t i s shown that the image ~ 8w+2 of the generator i n K0(RP ) , under the map induced by i : R P 8 w f 2 • jgp8w+2 / R p8w+l ^ ± & n o n _ z e r o > S o ± * f * ^ ± 0 . Now we have the f o l l o w i n g commutative diagram. 63 K O ( R P 8 w f 2 ) O — K O ( C P 4 w H > r- K U ( C P 4 w f l ) i * K O ( R P 8 W F 2 / R P 8 W F L ) < K O ( C P 4 W F L / C P 4 W ) ^ K U ( C P 4 W F L / C P 4 W ) Thus i * f * r y = f * r j * y 4 0 and hence f * r y 4 w + 1 4 0 . By (3.2) the -~ 8wH~2 8w 12 map. J :KO(RP ) > J(RP ) i s an isomorphism and so J f * r y 4 w + 1 4 0 . Therefore f * J r y 4 w + 1 4 0 . Whence J r y 4 w + 1 ^ 0. We are done. Next we cons ide r the cases when k = 1 or 3 mod 4 . (6.16) Cons ider the epimorphisms j»<CP 2 v f l) ^ j ^ M l j . e ^ j . ^ v f l j Then ( i ) i f v = 2w+l then 0'0" i s an isomorphism ( i i ) i f v = 2w , w 4- 0 , 0" i s an isomorphism and ker 0' = Z generated by J r y 4 w + 1 . P roo f . Cons ider the f o l l o w i n g commutative diagram J " ( C P 2 V + 1 / C P 2 V ) ^ > J » ( C P 2 V + 1 ) -^U J " ( C P 2 V ) 0 '0' (2v+l) 0'0' (2v) J ' ( C P 2 V + 1 ) - i U J ' ( C P 2 V ) We know that 0'©' i s an epimorphism; by (6.14) , 0'0",o i s (2v+l) _ r ~ r ' " J " « ( 2 v ) an isomorphism; the top row i s exact by (2.12); and i ' i s an 64 isomorphism by (6.11). Therefore ker ( ® ' ® " ) 2 v + T c l i n J " * ( i ) I f v = 2w+l then 2v+l = 4w+3 and K O ( C P 4 w f 3 / C P 4 w f 2 ) = K O ( S 8 ^ 6 ) = 0 . Thus im j " •= 0 and ® 'Q " (2v+ l ) 1 S a m o n o m o r P h i s m . Hence i t i s an isomorphism. ( i i ) I f v = 2w then 2v+l = 4w+l and TOCCP^V4*) = K O ( S 8 w f 2 ) = Z 2 , generated by r y 4 w + ± . Thus, e i t h e r ker ® '® " (2v+ l ) 1 S ^ ° r l t : 1 S , generated by J " r y 4 w + " ' " . From the c o n s t r u c t i o n of 0 " , n „ T „ 4w+l _ 4w+l° 0 J " r y = J r y and t h i s i s non-zero by (6.15) . Thus 0" i s a monomorphism. Hence -i i »^ t 4w+l • 4w+l i t i s an xsomorphism. I t remains to c a l c u l a t e 0 J ry = J ry 4w+l , 4w+l We have shry = 1 and so J ry = 0 . This completes the proof of (6.16) and hence (6.2) i s proved. Proof of (6 .3 ) . 1 Aw By (6.1) and (6.2) we see that orders of JrE i n J(CP ) and J ( C P 4 w + 2 ) are M, ,.. and. M. ,„ r e s p e c t i v e l y . Let 4w+l " 4w+3 a = J r (M, K) e J « E P 4 w + 2 ) . Then the order of a i s M .,/M. . 4w+l 4w+3 4w+l Now ^ 2 ( M 4 w + 3 ) = m a x ( s 4 v 2 ( s ) ) » 1 - s - 4 w + 2 a n d V 2 ( M 4 w + l ) = max ( s +v„ ( s ) ) , 1 < s < 4w . Note that i n the l a t t e r exp re s s i on we s ^ can have s = 4w which i m p l i e s that v„(M. ) > 4w+2 . The on ly 65 terms which en te r i n t o the f i r s t maximum but not the second are those w i t h s = 4w+l and s = 4w+2 . These terms have s+v^Cs) = 4w+l and s+v 0 (s ) = 4w+3 . Therefore v.(M. < v.(M. + 1 and 2 2 4w+3 2 4w+l v„(M. . 0 /M, ) < 1 . Thus the order of a conta ins a t most one power 2 4w+3 4w+l of 2. We r e f e r to the f o l l o w i n g commutative diagram. — — J(CP 4 V^ 2/CP 4 w ) j ( C p 4 w f 2 ) - ^ L J ( C P 4 W ) i * 0 ' 0* J ' ( C P 4 w f l ) J ' ( C P 4 W ) -> 0 J (CP 4 W + 1/CP 4 w ) — i ^ U j ( C p 4 w f l ) -2—> J (CP 4 w ) • o -*• 0 The rows are exact by (6.2) and the remarks preced ing (6.14); i * i s induced by i n c l u s i o n ; j * i s induced by the quot ien t map; the r i g h t -hand map 0 ' i s an isomorphism by ( 6 . 2 ) ; and i s an isomorphism by (6.11). S ince i * a = 0 , a l i e s i n the subgroup j * J ( C P 4 w + 2 / C P 4 w ) 4w+2 of J(CP ) . By ( 6 .2 ) , (6.12) and (6.13) t h i s subgroup i s c y c l i c of order m(4w+2) . From the d e f i n i t i o n we noted that m(2w+l) = 2 . I t i s a l s o c l e a r from the d e f i n i t i o n that m(2(2w+l)) = m(4w+2)= 2 . Aw I 2 Aw Therefore a i s d i v i s i b l e by 4 i n the subgroup j * J ( C P /CP ) . Now apply i * to a to get i * a = J r ( M ^ w + 1 ? ) e J ( C P 4 W + 1 ) . We see tha t J r ( M ^ w + 1 C ) i s d i v i s i b l e by 4 i n the subgroup j * J (<EP 4 W + 1/(EP 4 W) 66 S ince the bottom row i s exac t , we have ker i * = j * J ( C P 4 W + 1 / C P 4 W ) . By commutat iv i ty ker i * = ker G' and ker 0 ' = by (6 .2 ) . Thus j * J ( C P 4 W + 1 / C P 4 W ) = Z 0 and hence J r (M. _ E ) = 0 i n J ( C P 4 W + 1 ) . Let 2 ^4w+l the order of JrE i n J ( £ P 4 V ^ X ) be M . Then we have shown that M i s a m u l t i p l e of M . A l s o , M i s a m u l t i p l e of M „ s i nce 4w+l 4w+Z M. ,„ i s the order J ' E i n J ' ( C P 4 w + ~ S . Now s i nce M. ,_ i s a 4w+2 4w+2 m u l t i p l e of M, , we have M = M, , = M, „ . Th is completes the 4w+l 4w+l 4w+2 proof of (6 .3 ) . Our theorem i s now complete ly proved and hence we f i n a l l y have the e n t i r e s o l u t i o n to the complex c r o s s - s e c t i o n problem. That i s , the complex S t i e f e l man i fo ld W , admits a c r o s s - s e c t i o n over n,k S n X i f and on ly i f n i s a m u l t i p l e of . For completeness we g i ve the s o l u t i o n to the qua te rn i on i c c r o s s - s e c t i o n problem. The s ymp lec t i c S t i e f e l man i fo ld X , admits IT j rC 4 n - l a c r o s s - s e c t i o n over S i f and on ly i f n i s a m u l t i p l e of C , K. which i s de f i ned as f o l l o w s . v 2 ( C k ) = max ( 2 k - l , 2s+v 2 ( s ) ) , 1 < s < k-1 s max ( t + v p ( t ) ) , 1 < t < [^pj] > p odd < 2k 0 , p odd > 2k Th i s r e s u l t i s due to S i g r i s t and Suter [18 ] . 67 REFERENCES [ I ] J . F . Adams, Vecto r f i e l d s on spheres, Ann. of Math., 75(1962), 603-32 [2]'-J.F.-.rAdams, On the groups J(X) I-IV, Topology, 2(1963), 181-95; 3(1965), 137-72, 193-222; 5(1966), 21-71 [3] J . F . Adams, A l g e b r a i c Topology i n the l a s t Decade, Proc of Symposia i n Pure Mathematics, V o l XXII [4] J . F . Adams and G. Walker, On complex S t i e f e l man i f o l d s , P roc . Camb. P h i l . S o c , 61(1965), 81-103 [5] M.F. A t i y a h , Thorn Complexes, P r o c . Lond. Math. Soc. (3) , 11(1961), 291-310 [6 ] M.F. A t i y a h , K - theory , Benjamin, New York, 1967 [7] M.F. A t i y a h and J .A . Todd, On complex S t i e f e l man i f o l d s , P r oc . Camb. P h i l . S o c , 56(1960), 342-53 [8] R. B o t t , Lec tu re s on K (X ) , Benjamin, New York, 1969 [9] D. Husemol ler , F i b r e Bundles, S p r i n ge r -Ve r l a g , 19 75 [10] I. M. James, The i n t r i n s i c j o i n : A study o f the homotopy of S t i e f e l man i f o l d s , P roc . Lond. teth. Soc. (3 ) , 8(1958), 507-35 [ I I ] I.E. James, C ro s s - sec t i on s of S t i e f e l man i f o l d s , P roc . Lond. teth. Soc. (3 ) , 8(1958), 536-47 [12] I. K James, Spaces a s soc i a ted w i t h S t i e f e l man i f o l d s , P roc . Lond. Mith.:-Soev (3 ) , 9 (1959), 115-40 [ 13]. I. MrrrJames, The Topology of S t i e f e l man i fo ld s , Camb. Un iv . P re s s , 19 76 [14] K Ka roub i , K-Theory, An I n t r o d u c t i o n , S p r i n ge r -Ve r l a g , 1978 [15] J . M i lnor and M. K e v a i r e , B e r n o u l l i numbers, homotopy groups and a theorem of R o h l i n , P r oc . I n t . Cong. Math. Ed inburgh, 1958, Camb. Un i v . P re s s , London, 1960 [ 1 6 J . J . M i lnor and J.D. S t a she f f , C h a r a c t e r i s t i c C l a s se s , P r i n c e t o n Un iv . P re s s , 1974 [17] D. Q u i l l a n , The Adams Conjectu re , Topology, 10(19 71), 67-80 68 [18] F. S i g r i s t and U. Suter , C ro s s - sec t i on s of s ymp lec t i c S t i e f e l man i f o l d s , Trans. Amer. Math. S o c , 184 (19 73) , 247-59 [19] E.H. Span ier , A l g e b r a i c Topology, McGraw-Hi l l , 1966 [20] N.E. Steenrod, The topology of f i b r e bund les , P r i n c e t o n Un iv . P re s s , 1951 [21] N.E. Steenrod, Cohomology Operat ions , P r i n c e t o n Un iv . P res s , 1962 [22] N.E. Steenrod and J.H.C. Whitehead, Vector f i e l d s on the n-sphere, P r oc . Nat. Acad. S c i . , 37(1951), 58-63 [23] H. Toda, Vecto r f i e l d s on spheres, B u l l . Amer. Math. S o c , 67(1961), 408-12 [24].HioToda, Genera l ized Whitehead products and homotopy groups of spheres, J . I n s t . P o l y t e c h . Osaka, 3(1952), 43-82 [25] J.H.C. Whitehead, General ized Cohomology Theor ie s , Trans. Amer. Math. S o c , 102(1962), 227-83 

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