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Contribution to the theory of stably trivial vector bundles Allard, Jacques 1977

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CONTRIBUTION TO THE THEORY OF STABLY TRIVIAL VECTOR BUNDLES by Jacques A l l a r d B . S c , U n i v e r s i t e de Montreal, 1971 M.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 CE) Jacques A l l a r d , 1977 DOCTOR OF PHILOSOPHY i n In present ing th is thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f inanc ia l gain shal l not be allowed without my wri t ten permission. 5 r t m . „ _ f Mathematics Uepartment or The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date August 19th, 1977 Research Supervisor: Professor Kee Yuen Lam. A b s t r a c t A v e c t o r bundle £ over a CW-complex X i s s a i d to be s t a b l y t r i v i a l of type (n,k) i f E, © ke = ne, where e denotes the t r i v i a l l i n e bundle. Let V , be the S t i e f e l manifold of orthonormal k-n,k frame i n e u c l i d i a n n-space R n and l e t n , be the r e a l ( n - k ) -n,k dimensional v e c t o r bundle over V , whose f i b e r over a k—frame x n,k i s the subspace of R n orthogonal to the span of the v e c t o r s i n x . The ve c t o r bundle ri , i s "weakly u n i v e r s a l " f o r s t a b l y t r i v i a l v e c t o r bundles of type (n,k), i . e . f o r any s t a b l y t r i v i a l v e c t o r bundle of type (n,k), there i s a map f: X >• V , not n e c e s s a r i l y unique II y K. up to homotopy, such that f ri , - E, . n j K We study the f o l l o w i n g questions: (a) f o r which values of r i s the r - f o l d Whitney sum rri , t r i v i a l , and (b) what i s the maximum n, K number of l i n e a r l y independent c r o s s - s e c t i o n s of n @,se (0 < s < k - 1) . Among the r e s u l t s obtained are: (1) 2n „ i s — — n,2 t r i v i a l i f f n i s even or n = 3; (2) 3n „ i s t r i v i a l i f n i s n,2 even; (3) rn , i s not t r i v i a l i f r i s odd and < (n-2)/(n-k): n, K. (4) n , © ( k - l ) e i s not t r i v i a l i f n ^ 2,4,8 and 1 < k < n - 3; n,lc — — (5) n , © se admits e x a c t l y s l i n e a r l y independent c r o s s - s e c t i o n s n j K. i f n and k are odd; (6) n , © (k-2)e admits at most (k-1) n, k l i n e a r l y independent s e c t i o n s i f 2 < _ k < _ n - 3 . These r e s u l t s are used to construct examples of s t a b l y f r e e modules and unimodular matrices over commutative noetherian r i n g s . The techniques used are those of homotopy theory, i n c l u d i n g Postnikov systems, K-theory and, s p e c i a l l y , Spin operations on ve c t o r bundles. A chapter of the t h e s i s i s devoted to d e f i n i n g the Spin operations f o r m a l l y as a type of K - t h e o r e t i c c h a r a c t e r i s t i c c l a s s e s f o r a c e r t a i n type of r e a l v e c t o r bundles. Formulae to compute the Spin operations on a Whitney sum of v e c t o r bundles are given. i v . Table of Contents I n t r o d u c t i o n 1. Chapter I. P r e l i m i n a r i e s . 6. §1. S t i e f e l manifolds and s t a b l y t r i v i a l v e c t o r bundles. 7. §2. Cross-sections of n . . 9. n,k §3. T r i v i a l i t y of l a r g e m u l t i p l e s of n , . 9. II) K Chapter I I . Spin Operations on Vector Bundles. 12. §1. The a- c o n s t r u c t i o n and Spin r e d u c t i o n s . 13. §2. Complex Spin operations. 17. §3. An equivalence of r e p r e s e n t a t i o n s . 20. §4. Complex Spin operations and Whitney sums. 22. §5. Real and qua t e r n i o n i c Spin operations. 24. Chapter I I I . Whitney Sums of Stably T r i v i a l Vector Bundles. 35. * m §1. Sections of a (TS ) . 36. §2. The block map. 39. §3. The vect o r bundle n „ © n „ . 43. n, z n, / §4. The vect o r bundle r ) o ® T ) 0 ® T ) o . 52. n,3 n,3 n,3 §5. Odd m u l t i p l e s of n , . 59. n,k Chapter IV. Cross-sections of n , © ( k - l ) e . 62. n,k §1. S e c t i o n i n g of n , © ( k - l ) e . 63. n ^  K §2. N o n - t r i v i a l i t y of n . © ( k - l ) e . 66. n,k Chapter V. Examples of Stably Free Modules and Unimodular M a t r i c e s . 69. §1. Stably f r e e modules. 70. §2. Unimodular matrices. 73. Bi b l i o g r a p h y 76. V . L i s t of Tables I . Real and q u a t e r n i o n i c Spin r e p r e s e n t a t i o n s . 28. I I . Product formulae f o r r e a l Spin operations. 31. I I I . Product formulae f o r quaternionic Spin operations. 32. Acknowledgement The author wishes to express h i s deep g r a t i t u d e toward h i s t h e s i s s u p e r v i s o r , Dr. Kee Yuen Lam. Without h i s generous help both as a mathematician and as a f r i e n d , t h i s work could not have been completed. The author i s a l s o pleased to acknowledge h i s indebtedness to the numerous people who have a l s o helped him i n many d i r e c t and i n d i r e c t ways. These should i n c l u d e Dr. A. F r e i and Dr. E. L u f t who accepted to be members of the author's Ph.D. committee and Dr. S. G i t l e r and Dr. D. Sjerve who read and commented on an e a r l i e r d r a f t of t h i s t h e s i s . F i n a l l y , thanks are a l s o due to Miss Cathy Agnew who e f f i c i e n t l y typed t h i s manuscript from a very d i s o r g a n i z e d d r a f t . To Miyako 1. I n t r o d u c t i o n A r e a l v e c t o r bundle E over a CW-complex X i s s a i d to be s t a b l y t r i v i a l of type (n,k), o r, simply, of type (n,k) i f £ © ke = ne where e denotes a t r i v i a l l i n e bundle. The object of t h i s t h e s i s i s to study such v e c t o r bundles. For 1 < k < n - 1, l e t V , denote the S t i e f e l manifold of ortho — — n,k normal k-frames i n e u c l i d i a n n-space R n, and l e t n , be the (n-k)-dimensional r e a l v e c t o r bundle over V , whose f i b e r over a k-frame n ,k x = ,(x^,...,x^) c o n s i s t s of the vec t o r space x = {u: u € R n and u_Lx., i = l,...,k} . The vec t o r bundle n , i s s t a b l y t r i v i a l l n,k J of type (n,k) . In f a c t , f o r any vector bundle £ of type (n,k) over a CW-complex X, there i s a map f : X —•> V , such that £ = f n , . n,k n,k The map f i s not n e c e s s a r i l y unique up to homotopy. Therefore, we say that n , i s "weakly u n i v e r s a l " f o r v e c t o r bundles of type (n,k) . I t i s c l e a r that a general study of s t a b l y t r i v i a l v e c t o r bundles should begin w i t h a study of n , . n, i£ In t h i s t h e s i s , we have concentrated our a t t e n t i o n on the f o l l o w i n g questions. (1) For which values of r i s the r - f o l d Whitney sum rn n, k t r i v i a l ? (2) What i s the maximum number of l i n e a r l y independent c r o s s -s e c t i o n s of ri , © se, f o r 0 < s < k - l ? n,k — — Concerning the f i r s t question,.a purely a l g e b r a i c r e s u l t of T.Y. Lam i m p l i e s that rn i s t r i v i a l f o r r >^  k + k/(n-k) . In chapter I I I , n, & we prove the f o l l o w i n g : ( i ) 2n 9 i s t r i v i a l i f and only i f n i s n, z even or n = 3 ; ( i i ) 2n , i s not t r i v i a l i f n - k i s odd and > 3 n, K — 3. ( i i i ) 3n n ^ i s t r i v i a l i f n i s even. We a l s o o b t a i n some r e s u l t s f o r l a r g e values of k ( i . e . k > %n) . In p a r t i c u l a r , we prove: ( i v ) rn , i s not t r i v i a l i f r i s odd and < (n-2)/(n-k) . There n, k. i s a gap of approximately k u n i t s between the " p o s i t i v e " r e s u l t deduced from the theorem of T.Y. Lam and our "negative" r e s u l t ( i v ) . However, our r e s u l t suggests that even and odd m u l t i p l e s of ri may n, k behave i n very d i f f e r e n t ways. We study the second question i n Chapter IV. Assume that l < _ s < k < _ n - 3 i f n i s even and l < _ s < k < ^ n - 2 i f k i s odd. I f n 4- 2,4,8, we show that n , © ( k - l ) e i s not t r i v i a l . n, k Moreover, i f n i s odd, we show that ri , © ( k - l ) e admits e x a c t l y Tl y K. k - 1 l i n e a r l y independent c r o s s - s e c t i o n s except p o s s i b l y i f k i s even and smaller or equal to the Radon-Hurwitz number p( n - k - l ) . Using elementary arguments, these r e s u l t s already imply very strong r e s u l t s about the v e c t o r bundles n , © se . For i n s t a n c e , i t f o l l o w s that n , © (k-2)e never admits k l i n e a r l y independent c r o s s -Tl y K. s e c t i o n s . The complete s o l u t i o n of question (2) w i l l r e q u i r e i n f o r -mation l y i n g o u t side the truncated p r o j e c t i v e space RP n "^/RP11 ^ V , n, K and w i l l t h e r e f o r e r e q u i r e f u r t h e r study. In chapter V, we use the r e s u l t s of chapter I I I and IV to o b t a i n examples of modules w i t h s p e c i a l p r o p e r t i e s . We c o n s t r u c t a f a m i l y of commutative noetherian r i n g s A = A(n,k), n = 2,3,..., 1 <_ k <_ n - 1, such that there i s a f i n i t e l y generated p r o j e c t i v e A-module P = P(n,k) k k-1 w i t h the f o l l o w i n g p r o p e r t i e s : ( i ) P © A i s f r e e ; ( i i ) P © A i s not f r e e (with some p o s s i b l e exceptions); ( i i i ) P(n,2) © P(n,2) i s not f r e e i f n i s odd and >_ 5 . The f a m i l y of modules P(n,k) 4. g e n e r a l i z e s an example of [Swan 1962]. Moreover, ( i i i ) shows that a theorem of T.Y. Lam (mentioned above) i s best p o s s i b l e i n some cases. F i n a l l y , we use the modules P(n,k) to give examples of unimodular matrices which are ^.-stable but not (£+1)-stable i n the sense of [Gabel-Geramita 1974], f o r v a r i o u s values of Z . Our main t e c h n i c a l r e s u l t , to be found i n chapter I I of the t h e s i s , i s concerned w i t h Spin operations on vector bundles. A r e a l -^.-dimensional vector bundle £ over a f i n i t e CW complex X i s s a i d to have a Spin r e d u c t i o n i f the s t r u c t u r e group of £ can be taken to be the spinor group Spin(£) . When t h i s can be done i n an e s s e n t i a l l y unique way, we say that £ has a unique Spin r e d u c t i o n . I f £ has a unique Spin r e d u c t i o n , the s o - c a l l e d Spin rep r e s e n t a t i o n s of the spinor group can be used to construct elements A(E) and, f o r I even, A~(£), i n KU(X) (by the a - c o n s t r u c t i o n ) . We view A(E) and A~(£) as a kind of c h a r a c t e r i s t i c c l a s s f o r the vect o r bundle £ . These c l a s s e s are not n e c e s s a r i l y t r i v i a l even i f (a) £ admits a non-zero s e c t i o n , or (b) £ i s s t a b l y t r i v i a l . These p r o p e r t i e s are c l e a r l y an advantage f o r the study of s e c t i o n i n g problems f o r s t a b l y t r i v i a l v e c t o r bundles. In order to make e f f i c i e n t use of the Spin operations A(—) and A ~ ( — ) , we develop formulae r e l a t i n g A(£^ © E,^) t o A(E ) and AC^) and s i m i l a r l y f o r A (£ © E,^) • We a l s o complete t h i s program f o r the r e a l and q u a t e r n i o n i c Spin operations A (—) and A (—) which take K H values i n KO-theory and KSp-theory r e s p e c t i v e l y . Another t e c h n i c a l aspect of the t h e s i s which we would l i k e to point out concerns the use of Postnikov systems to analyze maps having c e r t a i n symmetry p r o p e r t i e s . Let Y = X x ... x X, and l e t T : Y —> Y 5. be the map permuting the t f a c t o r s of Y according to a permutation a € S . Suppose that we have a map f: Y >• Z such that f o T - f t a f o r any a € S . Then the ob s t r u c t i o n s to l i f t i n g f i n t o the Postnikov tower over Z must s a t i s f y c e r t a i n i n v a r i a n c e p r o p e r t i e s r e l a t i v e l y to some S t ~ a c t i o n s . These observations (and, i n one case, the Spin operations) are e s s e n t i a l i n our e v a l u a t i o n of some k - i n v a r i a n t s . These symmetry p r o p e r t i e s p l a y an important r o l e i n the study of the vecto r bundles rn , • n,k Chapter I contains p r e l i m i n a r y r e s u l t s . Chapter I Preliminaries 7. §1. S t i e f e l manifolds and s t a b l y t r i v i a l vector bundles. Let SO(n) be the L i e group of a l l n x n r e a l orthogonal matrices w i t h determinant +1 and BSO(n) the c l a s s i f y i n g space f o r p r i n c i p a l SO(n)-bundles. V denotes the u n i v e r s a l n-dimensional r e a l v e c t o r n bundle over BSO(n) . An orthogonal k-frame i n R n can be thought of as a k x n matrix x w i t h r e a l e n t r i e s s a t i s f y i n g the equation xx*" = I . Therefore, l e t p: SO(n) —>• V , be the map f o r g e t t i n g the l a s t n - k v e c t o r s of an n, K orthogonal m a t r i x . The map p gives r i s e to the well-known i d e n t i f i c a t i o n of V as a homogeneous space: n, K (1.1) SO(n-k) SO(n) SO(n)/SO(n-k) : V . . n,k. Consequently, we can consider the sequence (1.2) SO(n-k) * SO(n) -2—>• V . * BSO(n-k) B l > BSO(n) n, k where any two consecutive maps form a f i b e r bundle, and the map j i s a c l a s s i f y i n g map f o r the p r i n c i p a l SO(n-k)-bundle (1.1). Since the n-k vect o r bundle n , i s isomorphic to the f i b e r bundle w i t h f i b e r R n, k * a s s o c i a t e d to (1.1), we have that n , - j r . Therefore, we can n j K. n _ K. prove the f o l l o w i n g theorem. Theorem (1.3). Let £ be a r e a l (n-k)-dimensional v e c t o r bundle over a CW-complex X . Assume that £ © ke = ne . Then there i s a map * f: X *• V , such that £ = f n , . The map f i s not n e c e s s a r i l y n,k n,k ^ •L-unique up to homotopy. Proof. Since £ i s s t a b l y t r i v i a l , i t i s o r i e n t a b l e . Let f^: X *• BSO(n-k) be a c l a s s i f y i n g map f o r £ . Since £ © ke i s t r i v i a l , the composition B i o f i s null-homotopic, and s i n c e i B i V - J — B S O ( n - k ) >• BSO(n) i s a f i b r a t i o n , i t f o l l o w s that there n 5 K. i s a map f: X — * V , such that i o f - f„ . Moreover, v n,k J 0 ' £ = f . T , = i i T , - f n , as d e s i r e d . 0 n-k J n-k n,k The f o l l o w i n g example shows that the map f i s not n e c e s s a r i l y unique. Let X = V .. ~ S n ^ and £ = n -, - x S n . Assume that n , l n , l * n i s even . Then £ = f n . f o r any map f: V ., V , of n , l n , l n , l odd degree. (In g e n e r a l , the non-uniqueness of f i s measured by the image of p„: [X,SO(n)] >- [X,V ,]) .• if n, K D e f i n i t i o n (1.4). A s t a b l y t r i v i a l v ector bundle £ over a CW-complex X i s s a i d to be of type (n,k) i f i t s a t i s f i e s the equation £ © ke = ne . We w i l l r e f e r to the s i t u a t i o n described i n theorem (1.3) by saying that "V , i s a weak c l a s s i f y i n g space f o r v e c t o r bundles n j K. of type (n,k) 1 1 and that ! l r i , i s a weakly u n i v e r s a l v e c t o r bundle rij K. — "• " 1 • — -of type (n , k ) . " We now give some n o t a t i o n and s t a t e elementary p r o p e r t i e s of the vector bundle n n,k Assume that l < £ < k < n - l . Let p: V . >- V „ be the — — — n,k n,JC map f o r g e t t i n g the l a s t k - Z vectors of a k-frame, and l e t i : V p , „ *• V , be the map transforming a (k-£)-frame i n n--c, K--C n, k R n ^ i n t o a k-frame i n R n by adding to i t the l a s t £ v e c t o r s of the standard b a s i s of R n, i . e . , i n matrix n o t a t i o n , i ( x ) = (Q j ) • k Throughout the t h e s i s , we w i l l use the l e t t e r s p and i to r e f e r to the above maps. The sequence V 0,0 ~^>- V , — V n i s a n--c, K--C n, K n, X, 9. f i b r a t i o n . P r o p o s i t i o n ( 1 . 5 ) . i n , = n » . B and p n „ = n , © (k-£)e . n, k n--t, K--C. n, -t n, k Proof. Omitted. • R e c a l l a l s o that there are n a t u r a l i d e n t i f i c a t i o n s V „ ~ S n , l and V - ~ SO(n) . Moreover, r| , i s isomorphic to the tangent n, n - l n , l b bundle t S n of the u n i t sphere S n ^. n ^ i s a t r i v i a l l i n e bundle. n,n-l §2. Cross-sections of n , . n,k The f o l l o w i n g theorem i s due to G.W. Whitehead. Theorem (1.6). Let 2 < k < n - 2 . The vec t o r bundle n , does _ _ n,k not admit a non-zero c r o s s - s e c t i o n unless (n,k) = (7,2) or_ (8,3) . The vec t o r bundles n „ and n 0 o each admit e x a c t l y one l i n e a r l y /,z • — - O,J independent c r o s s - s e c t i o n . Proof. See [Whitehead 1963]. • Of course, i f k = 1, the vector bundle n , = xS admits n , l e x a c t l y p(n) - 1 l i n e a r l y independent s e c t i o n s , where p(n) i s the Hurwitz-Radon number [Adams 1962]. §3. T r i v i a l i t y of l a r g e m u l t i p l e s of ti , . A f i n i t e l y generated module P over a commutative r i n g R i s s a i d to be s t a b l y f r e e i f there are i n t e g e r s n and k such that k n P © R = R . The f o l l o w i n g theorem i s due to T.Y. Lam. Theorem (1.7). Let R be a commutative r i n g and P be a non-zero k n s t a b l y f r e e R-module such that P ffi R = R . Then the r - f o l d d i r e c t  sum rP i s f r e e f o r r >^  k + k/(n-k) . Proof. We give an o u t l i n e of the proof which can be found i n [Lam, T.Y. 1976]. Let 1 <_ k <^  n and assume that r >_ k + k/ (n-k) . Since k n P © R = R , there i s a k * n matrix a w i t h e n t r i e s i n R such Ti k n k that a: R >- R i s an epimorphism and P - Ker {a: R >• R } . Let 6>r a 0 0 0. . . . . 0 . . 0 .0 a ^^ XT XT Tl XT lc Then rP = Ker {a : R > R } . We w i l l show that there i s a ©r sequence of elementary row and column operations transforming a i n t o a matrix (a )' such that rP = Ker(a )' i s o b v i o u s l y f r e e . Write a = (M,V) where M i s a k x k m a t r i x and V i s a k x (n-k) matrix. Using elementary row and column o p e r a t i o n s , one ©r can show that a i s equivalent to the matrix r r—1 M M V MV V 0 0 I ( r - l ) k (Try w i t h r = 2! Use the f a c t that a has a r i g h t i n v e r s e a') . Now, by the Cayley-Hamilton theorem, the k x k m a t r i x M s a t i s f i e s i t s c h a r a c t e r i s t i c polynomial, which i s monic of degree k . I t follows e a s i l y that there i s a new sequence of elementary row and column operations transforming the l a s t matrix into the matrix M r 0 ... 0 M k _ 1V MV V 0 0 0 ... 0 0 1,.... ( r - l ) k and f i n a l l y into a matrix - ©r ' r 0 W 0 ( a > = ( 0 0 I, _ J ( r - l ) k where W i s a k x (k(n-k) + k) matrix. Let Q = Ker W . Then R k@ Q = R k( n- k> + k . A l s 0 j q u i t e c l e a r l V ) KerCa®17)' = R ( r " k ) ( n - k ) 0 Q Since r > k + k/(n-k), (r-k)(n-k) >_ k . I t follows that rP ^ Ker(a® r)' s R ( r - k ) ( n - k ) 0 Q s R r ( n - k ) _ . Cor o l l a r y (1.8). Let E be a r e a l vector bundle of type (n,k) over a f i n i t e CW-complex X . Then the r - f o l d Whitney sum r£ i s t r i v i a l f o r r > k + k/(n-k) . In p a r t i c u l a r , the vector bundle rn , jLs_ n j K. t r i v i a l f o r r >_ k + k/ (n-k) . Proof. Let T(E) denote the set of continuous sections of £ . T(£) i s a f i n i t e l y generated module over the r i n g C(X) of continuous r e a l -valued functions on X . Since E i s of type (n,k), we have that T ( E ) ® C ( X ) k = C ( X ) n . By theorem (1.7), i t follows that rT ( E ) = T(r£) i s fr e e . This, i n turn, implies that r£ i s t r i v i a l [Swan 1962, cor. 4]. a Chapter I I Spin Operations on Vector Bundl 13. §1. The a - c o n s t r u c t i o n and Spin-reductions. Let A denote the f i e l d of r e a l numbers R, the f i e l d of complex numbers C or the s k e w - f i e l d of q u a t e r n i o n i c numbers H . Denote by 0(k,A) the L i e group of k x k matrices A w i t h c o e f f i c i e n t s i n A which s a t i s f y the equation AA*" = I . Let G be a compact L i e group and 6: G —> 0(k,A), a m a t r i x r e p r e s e n t a t i o n of G . I f q: E —*• X i s a p r i n c i p a l G-bundle over a f i n i t e CW-complex X, we can d e f i n e a k-dimensional A-vector bundle 6(E) over X i n the f o l l o w i n g way. Using the r i g h t a c t i o n of G on k -1 E, d e f i n e a r i g h t a c t i o n of G on E x A by (e,\)g = (eg, 8(g) X) (e U , 1 ( A , g £ G) . The t o t a l space of 9(E) i s taken to be the k k quotient space (E x A )/G and the p r o j e c t i o n map (E x A )/G —> X i s defined by [e,A] — y q(e) . The isomorphism c l a s s of 6(E) . depends only on the equivalence c l a s s of the r e p r e s e n t a t i o n 0 and on the i s o -morphism c l a s s of the p r i n c i p a l G-bundle E . Let K A(X) denote KO(X), K(X) = KU(X) or KSp(X) r e s p e c t i v e l y . The v e c t o r bundle 0(E) defined above determines an element (also denoted 8(E)) i n K^(X) . This c o n s t r u c t i o n i s r e f e r r e d to as the a - c o n s t r u c t i o n and the element 8(E) € K^(X) i s denoted sometimes by a E(8) . For A = R, C or H, l e t R^(G) denote the r e a l r e p r e s e n t a t i o n r i n g RO(G), the complex r e p r e s e n t a t i o n r i n g R(G) = RU(G) or the q u a t e r n i o n i c r e p r e s e n t a t i o n group RSp(G), r e s p e c t i v e l y . The f o l l o w i n g two p r o p e r t i e s of the a - c o n s t r u c t i o n w i l l be used l a t e r . (2.1). For f i x e d E, the f u n c t i o n 0 I—*- 8(E) d e f i n e s by l i n e a r extension a group homomorphism R^(G) — K A ® ' ^ A = R or C, t h i s homomorphism i s a l s o a r i n g homomorphism. (2.2). The a - c o n s t r u c t i o n i s n a t u r a l i n X and G ( i n a s u i t a b l e sense). The reader i s r e f e r r e d to [Bott 1969, p.52] f o r more d e t a i l e d statements. Let p : Spin(£.) — > S0(£.) be the standard r e a l r e p r e s e n t a t i o n of the ^-dimensional spinor group. The map p : Spin(£) —*• S0(£) a l s o e x h i b i t s Spin(£) as double covering of S0(£) . D e f i n i t i o n (2.3). A r e a l ^-dimensional v e c t o r bundle E, over a f i n i t e  CW-complex X i s s a i d to have a Spin r e d u c t i o n i f there e x i s t a  p r i n c i p a l Spin (I)-bundle E over X such that E, = p ( E ) . Moreover, g i s s a i d to have a unique Spin r e d u c t i o n i f the p r i n c i p a l Spin,(£)- bundle E i s uniquely determined by E, (up to isomorphism). I t i s w e l l known that a vec t o r bundle E, has a Spin r e d u c t i o n i f and only i f the f i r s t two St i e f e l - W h i t n e y c l a s s e s w^(£) and w 2(£) are zero. Let f : X — y BSO(£) be a c l a s s i f y i n g map f o r an o r i e n t a b l e v e c t o r bundle E, . Of course, E, has a Spin r e d u c t i o n i f and only i f the map f admits a l i f t i n g f to the c l a s s i f y i n g space BSpin(£): K(Z 2,1) = BZ 2 BSpinGO —* BSO(£) _ v f v f X Moreover, E, has a unique Spin r e d u c t i o n i f and only i f the l i f t i n g f i s unique (up to homotopy). I Example (2.4). Let E, be a s t a b l y t r i v i a l v e c t o r bundle over a simply-connected CW-complex X . Then E, has a unique Spin r e d u c t i o n . Proof: Since £ i s s t a b l y t r i v i a l , w^(£) and w 2(£) are 0 . Hence £ has a Spin r e d u c t i o n , and there i s a l i f t i n g f : X —> BSpin(£) of the c l a s s i f y i n g map f : X —* BSO(£) of £ . Since [X, K ( Z 2 , 1 ) ] = H 1(X; Z 2) = 0, t h i s l i f t i n g i s unique i n view of the f i b r a t i o n K(Z ,1) -»• BSpin(£) BS0(£) SL m P r o p o s i t i o n (2.5). Let £ and n be r e a l v e c t o r bundles over f i n i t e  CW-complexes X and Y r e s p e c t i v e l y . Assume that £ and n have  unique Spin r e d u c t i o n s . Then the v e c t o r bundle £ * n over X x Y has  a unique Spin r e d u c t i o n . I f X = Y, the vector bundle £ © n a l s o  has a unique Spin r e d u c t i o n . Proof. Let b: S0(£) x SO(m) —> S0(£+m) and b: Spin(£) x Spin(m) —> m £+m Spin(.£+m) be the maps induced by the n a t u r a l i d e n t i f i c a t i o n R x R = R (see §3 f o r d e t a i l s ) . The f o l l o w i n g diagram i s commutative ( i n the category of groups): Z2 * Z 2 *" sP i n(£) x Spin(m) >- S0(£) x S0(m) i I5 l " Z 2 y Spin(£+m) > S0(£+m) Applying the c l a s s i f y i n g space f u n c t o r , we o b t a i n again a commutative diagram: BZ 2 x BZ 2 y BSpin(£) x BSpin(m) > BS0(£) x BSO(m) l» | B b | B b BZ 0 > BSpin(£+m) y BS0(£+m) The map u: BZ 2 x BZ,, *• BZ 2 i s the m u l t i p l i c a t i o n and the two h o r i z o n t a l sequences of maps are p r i n c i p a l bundles. The map Bb i s a p r i n c i p a l bundle map. Let f and g be c l a s s i f y i n g maps f o r E and n r e s p e c t i v e l y . Assuming that E and n have Spin r e d u c t i o n s , there are l i f t i n g s f : X — y BSpin(£) and g: Y —>• BSpin(m) of f and g r e s p e c t i v e l y . The map Bb o ( f x g ) i s obviously a l i f t i n g of Bb o ( f x g ) . Since the l a t t e r map c l a s s i f i e s E x 1, we have shown that t h i s v e c t o r bundle has a Spin r e d u c t i o n . Now assume that E and ri have unique Spin r e d u c t i o n s , and l e t F: X x Y y BSpin(£rm) be any l i f t i n g of Bb o (fxg) . Let v^: B S p i n ( i ) x K(Z 2,1) —*• B S p i n ( i ) denote the a c t i o n of K(Z 2 >1) on the t o t a l space of the p r i n c i p a l f i b r a t i o n K(Z 2,1) —• BS p i n ( i ) —>• BSO(i). Then F - v^^o- (Bb o (f xg) > a ) f o r some map a: X x Y —*- K(Z 2,1) . Now [XxY, K(Z 2,1)] = ^ ( X x Y ; Z^ = H^X; Z 2) + H^Y; Z^ = [X, K(Z 2,1)] x [Y, K(Z 2,1)] . Thus, i f a^ denotes the r e s t r i c t i o n of a to XC X x y and a v denotes the r e s t r i c t i o n of a to YC X x Y, we have a - u o ( a x x a^ .) . Consequently, F - v ^ ^ o (Bb o (fxg),a) -o (Bb o (fxg) ^ 0 ( 3 ^ 3 ^ ) ) - B b ( v ^ o ( f , a x ) x V m o ( g , a y ) ) . The l a s t - i s due to the f a c t that Bb i s a p r i n c i p a l bundle map. Since we assume that E and n have unique Spin r e d u c t i o n s , we must have v»o ( f , a ) - f and v o (g,a ) - g . We deduce that F - Bb o (fxg) . A m x This shows that E x q has a unique Spin r e d u c t i o n . The proof of the statement f o r E © n i f X = Y i s s i m i l a r . • §2. Complex Spin operations. In t h i s s e c t i o n , the complex Spin r e p r e s e n t a t i o n s are used to d e f i n e an operation on r e a l v e c t o r bundles which admit a unique Spin r e d u c t i o n . We f i r s t r e c a l l some f a c t s about the r e p r e s e n t a t i o n r i n g of the spinor groups. I Let C^ be the C l i f f o r d algebra over R (given i t s usual negative d e f i n i t e quadratic form). There are n a t u r a l i n c l u s i o n s t-1 t S C R £ C^ . The group Spin(£) can be taken as the subgroup of the group of i n v e r t i b l e elements of C^ c o n s i s t i n g of those of the t-1 form u 1 ' - - u 2 p s ) • L E T R = U £ / 2 ] » T = RMTTZ and T r = Tx...xT . Consider the map j ^ : T r — • Spin(£) defined by: J£(8 ,...,6r) = ( c o s % e i - e 1 e 2 s i n % 9 1 ) . . . ( c o s % 0 r - e 2 r _ 1 e 2 r s i n % 0 r ) t ( 0 ± C [0,4ir) and {e : i=l,...,£} = Standard b a s i s of R ) . The image of i s a maximal torus of Spin(£). Let a^: T >• C be the r e p r e s e n t a t i o n of T defined by a (0 ,...,9 ) = exp(%i9,) (l<k<r) . The (complex) r e p r e s e n t a t i o n r i n g R(T ) i s isomorphic to the polynomial r i n g Z[a , ...,0^, , .. . ,a^~] . Let = i - t h elementary symmetric f u n c t i o n i n a^,...,a r A 2 r = I ai ••• a r 2 r J 1 v 1 I = ) a, ... a T T 1 r  r where I = { ( e ^ . . . ,£ r) ; e i = ±1 and e^.-e^. = 1} and J = {(e-,...,e ); e. = ±1 and e ...e = -1} . 1 r l 1 r 18. [£/2] Theorem (2.6). The I n c l u s i o n of the maximal torus j ^ : T • Spin(£) induces the f o l l o w i n g isomorphisms: R(Spin(2r)) = Z j A ^ . . . ,X^, A ^ , A ^ ] R(Spin(2r+l)) = Z[X±,. . ., A ^ , ^ 2 r + 1 ] • Moreover, i n R ( S p i n ( 2 r ) ) > (A^ ) 2 + (A" ) 2 = X + 2X n + 2X , + ... 2r 2r r r-2 r-4 and i n R(Spin(2r+l)) (A„ ^ ) 2 = X + X .. + ...+ A.. + 1 2r+l r r-1 1 Proof. See [Husemoller 1966]. • + .+ The rep r e s e n t a t i o n s A 2 r , A 2 r = A 2 r + A 2 r and A2r+1 a V & c a ^ e ^ ± r-1 the Spin r e p r e s e n t a t i o n s . N o t i c e that dim = 2 and dim A„ = dim A„ ... = 2 2r 2r+l Let £ be a r e a l ^.-dimensional v e c t o r bundle over a f i n i t e CW-complex X . Assume that £ has a unique Spin r e d u c t i o n , i . e . £ = p(E) f o r some p r i n c i p a l Spin(£)-bundle E over X and E i s unique up to isomorphism. Define the f o l l o w i n g element(s) of KU(X): A(£) = A £(E) i f I = 1,2,3,... A ±(£) = A*(E) i f I = 2,4,6,... . D e f i n i t i o n (2.7). A(—) and A~(—) are c a l l e d the (complex) Spin  operations. They are defined f o r any r e a l v e c t o r bundle £ over a  f i n i t e CW-complex X and such t h a t £ has a unique Spin r e d u c t i o n , and they take value i n KU(X). Remark (2.8). The Spin operations have been p r e v i o u s l y used i n v a r i o u s context without being f o r m a l l y defined. For i n s t a n c e , see [Feder 1966]. In t h i s t h e s i s , we view the Spin operations as a type of K U - t h e o r i t i c c h a r a c t e r i s t i c c l a s s e s defined f o r c e r t a i n r e a l vector bundles. To our best knowledge, t h i s p o i n t of view has not been taken before. Example (2.9). Let E = ne be the n-dimensional t r i v i a l v e c t o r bundle r /on over a simply-connected CW-complex X . Then A(E) = 2 . I f n = 2s ± s-1 i s even, then A (E) = 2 Proof. The c o n d i t i o n that X i s simply connected insures that E admits a unique Spin r e d u c t i o n . C l e a r l y , E = p(X x Spin(n)) and A(X x Spin(n)) = X x C k, k = 2 [ n / 2 - 1 . S i m i l a r l y f o r A ±(E) i f n = 2s. • I Example (2.10). Let E be a r e a l v e c t o r bundle over X and l e t f: Y —* X be a continuous map. Assume that E and f E have unique * i Spin r e d u c t i o n s . Then Af (E) = f'A(E) and, i f £ i s even, A ±f*(E) = f ! A ± ( E ) , where f ! : KU(X) —> KU(Y) i s the homomorphism induced by f . Proof. N a t u r a l i t y of the a - c o n s t r u c t i o n . • Example (2.11). Let n _> 1 and l e t Y 2 n denote a generator of K U(S 2 n) = Z . Then A + ( x S 2 n ) = 2 n _ 1 ± y 2 n and A ~ ( T S 2 N ) = - A + ( x S 2 n ) . 2n 2n Proof. Since xS i s s t a b l y t r i v i a l and S i s simply connected, TS has a unique Spin r e d u c t i o n and A~(xS ) are defined. The p r i n c i p a l Spin(2n)-bundle E: Spin(2n) —>• Spin(2n+1) —> Spin(2n+1)/Spin(2n) s a t i s f i e s p(E) = x S 2 n . I t i s w e l l known that ± Y 2 n = ^ 2 n ^ ^ - 2 n are generators of KU(S ) [Bott 1969, thm. I l l , p.75].B 20. Example (2.12). Let 1 <_ k <_ n - 2 and l e t n = \ ^ b e t h e v e c t o r bundles over V , described i n 1.1. Then A(n ,) = T , + 2 ^ n - k ^ 2 ^ n,k n,k n,k >u t where T i s a t o r s i o n element of KU(V , ) w i t h order 2 and n,k n,k where t = %k - 1 i f n and k are even and t = [k/2] otherwise. I f n - k i s even, A _(n) = ±y , + 2 2 ^ n ^ where Y I generates n-k n-k an i n f i n i t e c y c l i c summand of KU(V , ) . n j K Proof. R e c a l l that V . i s (n-k-1)-connected. Hence, f o r k <. n - 2, n, k. V , i s simply connected. Since 11 = 1 1 , i s s t a b l y t r i v i a l , we deduce n,k v J n,k that ri has a unique Spin r e d u c t i o n . Let E be the p r i n c i p a l Spin(n-k)-bundle Spin (n-k) — > Spin(n) >• V We have p(E) = n , . G i t l e r n j K n j K and Lam [ G i t l e r and Lam 1970, pp.45-46] have shown that T , = A , (E) - 2 ^ n " k ^ 2 ^ i s a t o r s i o n element of KU(V ,) w i t h n,k n-k n,k order as described above. I f n - k i s even, they have shown that Y , = A - , (E) - 2 2 ^ n k ) a r e the generators of an i n f i n i t e c y c l i c n-k n-k summand of KU(V . ) . • n,k §3. An equivalence of r e p r e s e n t a t i o n s . The r e s u l t of t h i s s e c t i o n i s used to prove theorem (2.14) and (2.25) below. Let b: S0(£) x S0(m) —*• S0(£+m) and b: Spin(£) x Spin(m) —*• Spin(£+m) be the group homomorphisms induced by the n a t u r a l i d e n t i f i c a t i o n R^ x R m = R^ 4™ . For matrices A i n S0(£) and B i n S0(m), we have b(A,B) = (Q g) • F o r u i , , , u 2 p € Spin(£) and v 1 ' - ' v 2 q t Spin(m) (u±e S £ - 1 C R^ 3 R £ x ,0 and v ± € S m _ 1 C R™ = 0 x R m ) , we have b(u,...u„ , v, . . .v_ ) = u... ..u_ v., . . .v„ . Moreover, the homomorphism 1 2p 1 zq I zp 1 Zq b i s the l i f t i n g of b to the covering spaces, i . e . the f o l l o w i n g diagram commutes:. Spin(£) x Spin(m) > Spin(£+m) SO(l) x SO(m) • SO(£+m) . R e c a l l that f o r compact L i e groups G and H, there i s a n a t u r a l isomorphism R(GxH) = R(G) 0 R(H) induced by the tensor product of re p r e s e n t a t i o n s . Theorem (2.13). We have the f o l l o w i n g e q u a l i t i e s i n R(Spin(£) x Spin(m)) (a) f o r l,m even: b * ( A » . ) = A^ ® A + + A* ® A~ ; £+m I m I m -* (b) f o r I even and m odd: b ( A „ , ) = A ( ? ® A ; -c+m JL m (c) f o r £,m odd: b ( A ^ ) = A^ ® ^ m Proof. We prove only ( a ) . The proofs of (b) and (c) are s i m i l a r . We use the n o t a t i o n of §2. Assume that £ = 2r and m = 2s and consider r 3 = r4"S the map c: T x T *• T induced by the n a t u r a l isomorphism r s r~t*s R x R y R m Xhe f o l l o w i n g diagram commutes: T r x T S 5 • T r + S l j £ X j m lj£-hn b Spin(£) x Spin(m) • Spin(£+m) . I d e n t i f y R ( T r + S ) s Z[a±,. . . ,a r + s> o ^ 1 , . . . ,a r* g] = Z [ a 1 , . . . , a r , a ^ , . . . , a r 1 ] ® Z [ a r + 1 , . . . , a r + g , a ^ , . . . . a ^ ] = R(T r x T S) . We have: * ^ * ^l+n> = ^ 4+m ^l+x? Er+s 1 r+s 1 r+s . . . e =1 1 r 1 r 1 r ve ......E , =1 r+1 r+s r+1 r+s a ... a r+1 r+s + ''£.....£ =-1 1 r 1 r 1 ' ' ' r £ r + l " " ' e r + s 1 r+1 *r+l • r+s r+s = a * AJ) ® a * A+> + a * A") ® ( J * A;> = ( j , x j J ( A » ® A + A T ® A J m m * -* + Since ( j ^ x j ) i s a monomorphism, we deduce that b (A^.^) = + + -* -A„ ® A + A 0 ® A as d e s i r e d . S i m i l a r l y f o r b (A„, ) . • X, m L m -L+m §4. Complex Spin operations and Whitney sums. R e c a l l that the e x t e r n a l tensor product of compact vector bundles induces a b i l i n e a r p a i r i n g ®: KU(X) ® KU(Y) —-> KU(XxY) . We prove the f o l l o w i n g theorem. Theorem (2.14). Let £^ and n™ be v e c t o r bundles over f i n i t e CW- complexes X and Y r e s p e c t i v e l y . Assume that £ and n have unique  Spin r e d u c t i o n s . The Spin operations s a t i s f y the f o l l o w i n g e q u a l i t i e s : (a) f o r l,m even: A ± ( ? x n ) = A ±(£) ® A + ( n ) + A*(£) ® A~(n); (b) f o r I even and m odd: A(£xn) = A(£) ® A ( n ) ; (c) f o r l,m odd: A ^ x n ) = A ( £ ) ® A ( n ) . Proof. The v e c t o r bundle E, x r\ has a unique Spin r e d u c t i o n by p r o p o s i t i o n (2.5). Consequently, the Spin operations are defined on E, x n . Let E be a p r i n c i p a l Spin(£)-bundle over X such that E, = p (E) and s i m i l a r l y f o r F over X w i t h n = p(F) . Then E x F i s a p r i n c i p a l Spin(£) x Spin(m)-bundle over X x y . Let b: Spin G O x Spin(m) > Spin(£+m) be as i n the previous s e c t i o n and l e t b(ExF) denote the f i b e r bundle w i t h f i b e r Spin(£+m) (considered as Spin(£) x Spin(m) - space through b) ass o c i a t e d to E x F . The f i b e r bundle b(ExF) i s a p r i n c i p a l Spin(£+m)-bundle and p(b(ExF)) = £ x n . Now, assume that £ and m are even. By n a t u r a l i t y of the co n s t r u c t i o n s i n v o l v e d , we have: A ±(?xn) = A | + m ( b ( E x F ) ) = ( A ^ . b)(ExF) = ^  W(EXF) • Since the a - c o n s t r u c t i o n induces a r i n g homomorphism R(—) —> KU(—) , we a l s o have the f o l l o w i n g e q u a l i t i e s : A ± ( ? ) ® A +(n) + A+(?) ® A"(n) = A | (E) ® A*(F) + A*(E) ® A"(F) = (Ap 0 A + + At <g> A") (EXF) . •C m JL m -* ± ± + £ '-By theorem (2.13), b A ^ ^ = A^ ® ^ + A^ ® A^ . Consequently b* Ap, (ExF) = (Ap ® A + + Ap ® A~)(ExF) . The e q u a l i t i e s above then -c+m Z. m -c m imply that A ±(Sxn) = A ±(5) ® A +(n) + A +(?) ® A~(n), as d e s i r e d . The proof of (b) and (c) i s s i m i l a r . • The f o l l o w i n g c o r o l l a r y i s immediate. 24. C o r o l l a r y (2.15). Let E^ and be r e a l v e c t o r bundles over a f i n i t e CW-complex X . Assume that E and n have unique Spin  r e d u c t i o n s . The Spin operations s a t i s f y the f o l l o w i n g i d e n t i t i e s : (a) f o r l,m even: A± (E©n) = A ± ( E)A +(n) + A* (E)A~(n); (b) f o r I even, m odd: A(E®n) = A(E)A(n); (c) f o r £,m odd: A ± (E©n) = A(E)A(n) . • ' §5. Real and qua t e r n i o n i c Spin operations. In t h i s s e c t i o n , we d e f i n e r e a l and qua t e r n i o n i c Spin operations, i . e . Spin operations defined on r e a l v e c t o r bundles having a unique Spin r e d u c t i o n and ta k i n g values i n KO-theory and KSp-theory. Formulae to compute these operations on Whitney sums of vector bundles are a l s o given. This s e c t i o n i s included f o r the sake of completeness, s i n c e the t h e o r e t i c a l r e s u l t s obtained are not used i n the r e s t of t h i s t h e s i s . However, the author hopes that example (2.27) w i l l convince the reader that they are of some value. We begin by r e c a l l i n g some f a c t s about r e a l and quaternionic r e p r e s e n t a t i o n r i n g s and r e a l and quat e r n i o n i c K-theory. Most d e t a i l s are omitted. The reader who i s not f a m i l i a r w i t h the approach taken should consult [Adams 1967] and [Atiyah 1969, §1.5]. R e c a l l that a r e a l (resp.: q u a t e r n i o n i c ) r e p r e s e n t a t i o n of a compact L i e group G can be regarded as a p a i r (V,j) where V i s a complex r e p r e s e n t a t i o n and j : V > V i s a co n j u g a t e - l i n e a r G-map 2 2 such that j = 1 (resp.: j = - 1 ) . The map j i s c a l l e d a s t r u c t u r e  map. F o r g e t t i n g the s t r u c t u r e maps induces the " c o m p l e x i f i c a t i o n " 25. homomorphisms: c: RO(G) y R(G) c': RSp(G) y R(G) . The homomorphisms c and c 1 are monomorphisms [Adams 1969, 3.27]. There are a l s o " r e a l i f i c a t i o n " and " q u a t e r n i o n i f i c a t i o n " homomorphisms: r : R(G) y RO(G) q: R(G) y RSp(G) and a "conjugation" homomorphism: t : R(G) y R(G) [Adams 1969, §3.5 ( i ) , ( i v ) , ( v ) ] . The f o l l o w i n g i d e n t i t i e s are s a t i s f i e d : r c = 2 (2.16) c r = 1 + t qc' = 2 c'q = 1 + t Let a = (V,j) and a' = (V',j') be r e a l r e p r e s e n t a t i o n s of G 2 2 ( i . e . j = l v and j ' = 1 ,) . Let* a l s o 3 = (W,k) and 3 ' = (W',k') 2 2 be quaternionic r e p r e s e n t a t i o n s of G ( i . e . k = -1 and k 1 = -1,.,) . w w Then (j®j') 2 = 1 V 0 V , and (k®k') 2 = - 1 ^ , • Thus a ® a' = (V®V\ j®j') arid 3 ® 3 ' = (W®W', k®k') are r e a l r e p r e s e n t a t i o n s of G . S i m i l a r l y , a ® 3 ' = (V0W', j®k') and 3 ® a' = (W®V , k®j ') are qua t e r n i o n i c r e p r e s e n t a t i o n s of G . Therefore, we can d e f i n e a m u l t i p l i c a t i o n i n RO(G) © RSp(G) by s e t t i n g ( [ a ] , [ 3 ] ) • ( [ a ' ] , [ 3 ' ] ) = ([a®a'] + [ 3 ® 3 ' ] , [a® 3 ' ] + [ 3®a ' D and extending l i n e a r l y to RO(G) © RSp(G) . The group RO(G) © RSp(G) i s endowed i n t h i s way of a n a t u r a l s t r u c t u r e of Z 2~graded r i n g . Moreover, we can give a s t r u c t u r e of Z,,-graded r i n g to R(G) © R(G) by d e f i n i n g (u,v)(u',v') = (uu' + w', uv' + v u ! ) f o r u,u',v,v' € R(G) . Then the map (2.17) c" = c © c': RO(G) © RSp(G) > RU(G) © RU(G) i s a n a t u r a l monomorphisn of Z 2-graded r i n g s . I f G and H are compact L i e groups, l e t ir^,: G x H —> G and i t ^ : G x H >• H be the p r o j e c t i o n maps. The reader w i l l e a s i l y see that vn and IT can be used i n the usual way to d e f i n e an e x t e r n a l G H product: (2.18) [RO(G) © RSp(G)] <8> [RO(H) © RSp(H)] y [RO(GxH) ©RSp(GxH)] where ® denotes the tensor product of Z 2~graded r i n g s . I t should b n o t i c e d that (2.18) i s not an isomorphism. In a very s i m i l a r way, r e a l and q u a t e r n i o n i c v e c t o r bundles over f i n i t e CW-complexes can be regarded as complex ve c t o r bundles w i t h s t r u c t u r e maps. A r e a l (Resp.: quater n i o n i c ) v e c t o r bundle over a f i n i t e CW-complex X can be thought of as a p a i r (E,T) where E i s complex ve c t o r bundle over X and T: E —> E i s a c o n j u g a t e - l i n e a r 2 2 bundle map such that T = 1 (resp.: T = -1). One can proceed as we d i d f o r group r e p r e s e n t a t i o n s and d e f i n e maps c: KO(X) — y K(X), c': KSp(X) >-K(X), e t c . . . . The i d e n t i t i e s (2.16) a l s o hold. The tensor product of ve c t o r bundles and s t r u c t u r e maps induces a n a t u r a l s t r u c t u r e of Z„-graded r i n g on KO(X) © KSp(X) . I f Y i s an other CW-complex, the e x t e r i o r tensor product induces a b i l i n e a r p a i r i n g (2.19) ®: [KO(X) © KSp(X)] ® [KO(Y) © KSp(Y)] > KO(XxY) + KSp(XxY) . Remark (2.20). The existence of t h i s n a t u r a l Z 2-graded r i n g s t r u c t u r e on K0(X) © KSp(X) was made obvious by the work of A t i y a h , B o t t , . . . . However, to our best knowledge, i t was f i r s t used e x p l i c i t l y i n [ S i g r i s t and Suter 1972]. See a l s o [ A l l a r d 1974]. Let q: E —> X and 0: G —»- 0(k,A) (A = R or H) be as i n §1. The element 0(E) obtained by the a - c o n s t r u c t i o n can be regarded as an element of K0(X) © KSp(X) v i a the n a t u r a l i n c l u s i o n K0(X) y K0(X) © KSp(X) or KSp(X) > K0(X) © KSp(X) . The reader w i l l e a s i l y prove the f o l l o w i n g property of the a - c o n s t r u c t i o n (compare (2.1)) (2.21) For f i x e d E, the a - c o n s t r u c t i o n 0 i—»- 0(E) induces a homomorphism of Z^-graded r i n g R0(G) © RSp(G) v K0(X) © KSp(X) . We now d e s c r i b e the r e a l and q u a t e r n i o n i c Spin r e p r e s e n t a t i o n s of the group Spin(£). In view of the monomorphism (2.17), i t s u f f i c e s to g i v e t h e i r images under c and c' r e s p e c t i v e l y . Table ( I ) contains t h i s l i s t . £(8) Real Spin r e p r e s e n t a t i o n s Quaternionic Spin r e p r e s e n t a t i o n s + + 0 2 A £" 1 A £ 2 A £ 2 A £ A £ 3 2 A £ A £ + + 4 2 A £" 5 2 A £ A £ 6 A £ A £ 7 A £ 2 A £ TABLE I : Real and qua t e r n i o n i c Spin operations The l i s t of r e a l Spin r e p r e s e n t a t i o n s i s taken from [Bott 1969, p.66]. Knowing the r e a l Spin r e p r e s e n t a t i o n s , one can deduce the l i s t of q u a t e r n i o n i c Spin r e p r e s e n t a t i o n s using elementary r e p r e s e n t a t i o n theory. For example, i t i s a theorem that a s e l f - c o n j u g a t e i r r e d u c i b l e complex r e p r e s e n t a t i o n i s e i t h e r r e a l or qua t e r n i o n i c ( i . e . i n Im c or i n Im c ' ) , but not both (see [Adams 1969, 3.56]). I f £ = 4( 8 ) , the i r r e d u c i b l e complex r e p r e s e n t a t i o n s A^ are s e l f - c o n j u g a t e , but not r e a l ( i . e . not i n Bott's l i s t ) . Hence they must be qua t e r n i o n i c .. We are now ready to de f i n e r e a l and quaternionic Spin operations. £ Let E be a £-dimensional r e a l v e c t o r bundle over a compact connected £ CW complex X, and assume that E has a unique Spin r e d u c t i o n , i . e . E = p(E) f o r some Spin(£)-bundle E over X and E i s unique up to isomorphism. Denote the r e a l Spin r e p r e s e n t a t i o n ( s ) of Spin(£) by cf>£ i f £ f 0(4) and by cj>£ i f £ = 0(4) . In the l a t t e r case d e f i n e a l s o <j)« = ^"t + §B . Define the f o l l o w i n g element(s) of K0(X): A R ( ? ) = <f)£(E) f o r £ = 1,2,... A R ( 5 ) = < f > £ ( E ) f o r 1 = 4' 8'* * S i m i l a r l y , denote the q u a t e r n i o n i c Spin r e p r e s e n t a t i o n s of Spin(£) by i f £ £ 0(4) and by ^ i f £ = 0(4) . In the l a t t e r case a l s o d e f i n e ^£ = ^£ + ^£ • Define the f o l l o w i n g element(s) of KSp(X) A H(£) = if^(E) i f £ = 1,2,... A*(£) = ^ ( E ) i f £ = 4,8,... . + D e f i n i t i o n (2.22). A (—) and A (—) are c a l l e d the r e a l Spin K K operations and A (—) and A~(—) are c a l l e d the q u a t e r n i o n i c Spin  operations. These operations are defined f o r any r e a l v e c t o r bundle £ over a f i n i t e CW-complex X and such that £ has a unique Spin  r e d u c t i o n , and they take values i n K0(X) and KSp(X) r e s p e c t i v e l y . They can a l s o be regarded as t a k i n g values i n the Z^-graded r i n g KO(X) © KSp(X) v i a the n a t u r a l i n c l u s i o n s K0(X) >• KO(X) © KSp(X) and KSp(X) >• K0(X) © KSp(X) . Example (2.23). Let £ = £e be a t r i v i a l £-dimensional vect o r bundle over a simply connected CW-complex X . Then A D (C) = 2 V J l £ K0(X) w i t h r = 0,-1,0,1,2,1,0,-1 f o r £ = 0,1,...,7(8) r e s p e c t i v e l y . I f £ = 0 ( 8 ) , A*(?) = 2 ( £ " 2 ) / 2 and i f £ = 4 ( 8 ) , A*(S) = 2 £ / 2 . Example (2.24). R e c a l l that K0(S n) = Z i f n = 0(4), K0(S n) = Z 2 i f n = 1,2(8) and K0(S ) = 0 otherwise. Let 3 n denote a generator of K0(S n) i f t h i s group i s not t r i v i a l . Then A^(xS n) = ±6 + 2 ( n _ 2 ) / 2 i f n E 0(8) and A*(TS n) = 3 + 2 n / 2 i f n = 4(8). n K n In both cases, A f x S 1 1 ) = -A^CxS11) . I f n 5 1 ( 8 ) , A^xS 1 1) = 6 + 2 ( n - 1 ) / 2 and i f n = 2 ( 8 ) , A_(TS n) = 0 + 2 n / 2 . The proof n K n of these f a c t s i s contained i n [ A t i y a h , Bott and Shapiro 1963]. Theorem (2.25). Let X,Y, and n m be as i n theorem (2.14). Let A D(—) and ATT ( ) take values i n the Z 0-graded r i n g K0(-) © KSp(-) . Then A t )(£x n) (A*(£xn) i f £ + m E 0 ( 4 ) ) i s equal to the expression given i n Table ( I I ) , and A^(£xn) (A~(£xn) i f I + m = 0(4)) i s equal to the expression given i n Table ( I I I ) . The product o should be taken to be the e x t e r n a l product ® (2.19). Proof. We give the proof of the theorem f o r A~(—) and £,m = 0(8). The other cases are s i m i l a r . Moreover, the proof i s s i m i l a r to that of theorem (2.14). Consequently, l e t b, E, F and b(ExF) be as i n the proof of (2.14). By the n a t u r a l i t y of the c o n s t r u c t i o n s i n v o l v e d , we have that A^(C*n) = ^ ^ ( b C E x F ) ) = ( f ^ o b) (E*F) = b* ^ ^ ( E x F ) , where b now stands f o r the group homomorphism induced by b on the qua t e r n i o n i c r e p r e s e n t a t i o n r i n g s . On the other hand, using (2.21), we deduce that A*U) ® A*(n) + A*(£) ® A~(n) = ip^(E)-® <|>*(F) + i|/t(E) ® <J>~(F) = (ij)» ® <|>+ + ifp ® <|>~)(ExF) • (Given groups A and B we w i l l w r i t e a f o r (a,0) €• A x B and b f o r (0,b) g A x B i f there i s no danger of co n f u s i o n ) . We have the f o l l o w i n g e q u a l i t i e s i n R0(G) © RSp(G), G = Spin(£) x Spin(m) . F i r s t l y , from the n a t u r a l i t y of the homo-- * ± _ * ± ~* ± morphism c , we have that c b = b c = ^ ^' 2 A£+m^ * Using theorem (2.13) and the Z 2~graded r i n g s t r u c t u r e given to R(-) © R ( ~ ) , we have that b*(0,2A| ) = (0,2A^) ® (A*,0) + (0,2At) ® (A ,0) . F i n a l l y , s i n c e c" i s a homomorphism of Z -Z. m z m< 8) 0 1 2 3 4 5 6 7 I (8) 0 A R ( S ) o A R ( n ) +A ? Ra)*A^(n) A R a)oA R(n) A RU)°A R(n) A R ( c)oA R( n) A*(?)oA R(n) +A^(OoA R(n) A R(?)oA R(n) A R ( 5)oA R(n) A R U ) o A R(n) 1 A R(£)oA R(n) 2A R(C)oA R(n) 2A R(C)oA R(n) A R(?)oA R(n) A R ( 5)oA R(n) A R ( 5)oA R(n) A R(?)oA R (n) A R ( 5)o A R(n) 2 A R(OoA R(n) 2A R ( 5)oA R(n) 2 r ( A ± ( U p A + ( n ) ) A R(?)oA R(n) A H(?)oA H(n) A H(C)oA H(n) r ( A ± ( O o A + ( n ) ) A R ( 5)o A R(n) 3 A R(?)oA R(n) A R(s)oA R(n) A R ( ? ) o A R ( n ) 2A H ( 5)oA H(n) A H(C)oA H(n) A H ( 5)«A H(n) A H(DoA H(n) A H ( O o A H (n) 4 A R(S)oA R(n) +A+(5) A R ( n ) A R(?)oA R (n) A H(?)oA H(n) A H(S)oA H(n) A * ( 5)oA R(n) +^a)»A~(n) A H ( 5)oA R(n) AH(c)oAH(n) A H(?)o A R(n) 5 A R(?)oA R (n) A R(£)oA R(n) AH(?)-»AH(n) A H ( ? ) o A H ( n ) A H ( 5 ) ° A H ( n ) 2A H (5)oA H(n) 2A R ( 5)oA H(n) A H ( 5 ) . A R(n) 6 A R(?M R(n) A R(?)oA R(n) r(A ±(?)oA + (n ) ) AH(?)«»AH(n) A H ( ? ) o A H ( n ) 2A HU)°A H(n) 2r(A ±(OoA + (n ) ) 2A R(5) o A R ( n ) 7 A R ( 5)oA R(n) A R(?)oA R ( n ) A R a)oA R (n) A H ( c)oA H( n) A R ( 5)oA H(n) A H ( 5)oA R(n) 2A R(£)oA R(n) 2A R(5) oA R(n) TABLE I I : Product formulae f o r Real Spin Operations 1 (8) 0 m(8) 0 1 2 3 4 5 6 7 A*(S)oA+(n) A R ( 5)'oA H ( n ) A R ( c ) o A H ( n ) A R ( 0 » A H ( n ) A R(?)e A H ( n ) +A+(5)oA^( n) AR(5)»A I I(n) A R U > A R ( n ) A R ( 5)°A H ( n ) +A+(5)oA~(n) 1 A H a ) o A R ( r i ) A H ( 5)oA R ( n ) A R ( 5)oA R ( n ) A R(?)oA H ( n ) A R ( 5)oA H ( n ) A R(?)oA R (n) 2A R(?)o A R ( n ) A R(?)oA H ( n ) 2 A H(?)°A R ( n ) V s ) o V n ) q ( A± ( O o A + ( n ) ) A R U ) » A H ( n ) 2A R(c:)oAH(n) A R(?)oA R ( n ) q(A ±(5)oA(n).) A R a ) o A H ( n ) 3 A H(^)oA R ( n ) A h ( 5 )O A R ( n ) A H(OoA R (n) A H(?)oA R ( n ) A R(C)oA H ( n ) A R ( 5)«A H ( n ) A R U ) o A R ( n ) 2A r ( 5 )OA r(TI) 4 A*(e;)«>A+(n) +A^(5)oA R ( n ) A H ( 5)oA R ( n ) 2A H ( 5)oA R ( n ) A H ( 5)oA R ( n ) A * ( O o A R ( n ) +A+(?)bA R ( n ) A H a ) ° A R ( n ) A R ( 5)oA R ( n ) A R(?)oA R ( n ) 5 A H ( 5)oA R ( n ) A H ( 0 » A R ( n ) A H (5)oA R(n) A H (5)oA R(n) A R ( 5)oA H ( n ) A r ( 5 )OA r(TI) A H ( 5)oA R ( n ) A H(?)oA R ( n ) 6 A H ( 5)oA R(n) 2A H ( 5)°A R ( n ) q(A ±(c:)oA ( n ) , ) A R U)o AR(n) A R ( 5)oA H ( n ) A R ( 5)oA H ( n ) q ( A . ± ( 5)«A + ( n ) ) A R ( 5)oA R(n) 7 A H ( ? ) o A R ( n ) A H ( 5)°A R ( n ) A H ( ? ) o A R ( n ) 2A R(C)oA H ( n ) A R ( 5 ) o A H ( n ) A R ( 5)»A H ( n ) A R(S> A H ( n ) A R ( D°A H ( n ) TABLE I I I : Product formulae f o r Quaternionic Spin Operations graded r i n g , the l a s t expression i s equal to c"(lp„ ® <))+ + tyt ® cf> ) . -c. m -c m - * + i , ± + r -Thus we have shown that c"b ib„, = c (iK ® d> + ii't ® <b ) . Since -c-rm -c m X, m -* ± • ± + r -c" i s a monomorphism, we deduce that b i K . = \pe ® <j> + \pt ® d> -c-rm -c m X, m I t f o l l o w s that b % o , (ExF) = ( i ^ ® <j>+ + ® <j>~) (ExF) . In view JL+m X. m X. m of the l a s t paragraph, t h i s i m p l i e s the r e s u l t wanted, i . e . A^(Exn) = A*(£) ® A R(n) + A + ( 0 ® A~(n) . • C o r o l l a r y (2.26); Let X, and n m be as i n c o r o l l a r y (2.15). Let A (—) and A (—) take values i n the Z 0-graded r i n g R H / K0(~) © KSp(-) . Then AR(?©n) (AR(£©n) i f £ + m E 0(4)) i s equal to the expression given i n Table ( I I ) and A^(§®ri) (A~(£©n) _if_ £ + m E 0(4)) i s equal to the expression given i n Table ( I I I ) • The product o should be taken to be the Z^-graded product i n K0(X) © KSp(X) . Proof. Obvious. • Example (2.27). Let x n = x S n and l e t £^ k be a r e a l vector bundle 0 8 k + l . . 8k _ ,. 8k+l m u 8k 8£ . over S such that £ © e = x . Then £ x T i s not . . , „8k+l e8£ . t r i v x a l over S x s Proof. The r e s u l t i s t r i v i a l i f k = 0 . Therefore, we assume k > ( R e c a l l examples (2.23) and (2.24). By c o r o l l a r y (2.26), we have that W i • V T 8 k + 1 ) - 2 4 k - v.- E> - * 4 K = ( A R ( ^ + V ^ V e > -2 4 k = (A R ( ? ) - 2 4 k _ 1 ) + ( A ~ ( 0 - 2 4 k _ 1 ) . Since K 0 ( S 8 k + 1 ) = Z^ i t f o l l o w s that A R ( ? ) - 2 4 k _ 1 = 0 and A~(c) - 2 4 k _ 1 = B g k + 1 , or 4- £V=1 - 4k-l Al(C) - 2 = 3 0 1 J , and A_(?) - 2 4 = 0 . Without l o s s of R o K+1 R g e n e r a l i t y , assume the l a t t e r . Then, by theorem (2.25), we have that . + , 8k 8-t\ . + / s . + / 8£ x , ,-, . „ — f 8£. A R ( ? x T ) = A R(5) ® A R ( x ) + A R ( ? ) ® A R ( x ) = B Q 1 ... ® g o P + 2^k +"^ ^ Applying Bott's p e r i o d i c i t y theorem i n d u c t i v e l y , we have that 0 + $ 8 k + 1 ® P g£ G K O ( S 8 k x S 8 £ ) . There f o r e , 2 4 ( £ + k > " 1 * A+<,8k * x 8 £ ) 6 K O ( S 8 k + 1 x S*1) . The r e s u l t K f o l l o w s . • Chapter I I I Whitney Sums of Stably T r i v i a l Vector Bundl This chapter i s concerned w i t h the r - f o l d Whitney sums of the ve c t o r bundle n , defined i n Chapter I . The n o t a t i o n i s that of n, K that chapter unless otherwise i n d i c a t e d . Moreover, the truncated r e a l p r o j e c t i v e space RP n/RP m ^ i s denoted by P n . Sections 1 and 2 m co n t a i n a u x i l i a r y r e s u l t s . §1. Sections of a (TS ) . P r o p o s i t i o n (3.1). Let n represent the generator of IT ,. (S m) = Z, m " m+1 i Assume that m i s even and m > 6 . Then *k m ( i ) n (TS ) i s not t r i v i a l ; m * TI * m ( i i ) i f m a 0 ( 4 ) , n (TS ) and (n on .,) (TS ) admit — m m TB.+X. e x a c t l y one l i n e a r l y independent s e c t i o n ; ( i i i ) ±f_ m = 2(8) and m >^  18, h (TS ) admits at most 5 l i n e a r l y independent s e c t i o n s . ^ m — Proof. ( i ) I f n TS i s t r i v i a l , there i s a l i f t i n g r\ of n m m m as shown i n the f o l l o w i n g diagram: * v j.1 = SO (m+1) • n , ^  m+l,m I p sm+l' D ^ sm Consequently, 3(n ) = 0 where 9 i s the boundary homomorphism i n the exact homotopy sequence as s o c i a t e d to the f i b e r i n g SO(m+1)/SO(m) However, t h i s i s the case only i f m = 3(4) [Kervaire 1960, thm. 1 ] . ( i i ) Let a: S M + 1 —• S™ be a continuous map. The vec t o r bundle % m a (TS ) admits k . l i n e a r l y independent s e c t i o n s i f and only i f there i s a l i f t i n g a of a as- shown i n diagram (1). V a P m .* m+l,k+l m-k c sm+i a y sm sm+i a y gm (1) (2) I f 2k < m - i - 1, by c e l l u l a r approximation, t h i s i s e q u i v a l e n t , „m to having the homotopy commutative diagram ( 2 ) , where c: P i n i s the map c o l l a p s i n g the (m-1)-skeleton of t o a p o i n t m To prove that n (TS ) admits one non-zero s e c t i o n f o r m even, m i t s u f f i c e s to consider the c o f i b r a t i o n sequence sm-l _ 2 ^ sm-l _ ^ pm c sm ^ gm m-1 „ \m \ ' ' v s m + l / * n / 2on -m/ m Since 2ori i s homotopically t r i v i a l , the map. n f a c t o r s through m m c as d e s i r e d , and the r e s u l t f o l l o w s . Of course, that * m * * m (n o n ) (TS ) = n ,-• (n ( TS )) admits at l e a s t one non-zero s e c t i o n m m+1 m+1 m a l s o f o l l o w s . To prove that (n o n ,n) (TS ) does not admit 2 l i n e a r l y m m+± independent s e c t i o n s i f m = 0( 4 ) , consider the c o f i b r a t i o n sequence sm-l pm-l ^  sm-l v sm-2 _ ^ pm c gm £A_^ . pm-l m-2 m-z • m-z sm+2 The a t t a c h i n g map A: S *• ?m_2 ~ ^ V ^  1 S homotopic to the map n _v 2: S m > S m 2 v S m (consider the a c t i o n . of the Steenrod m-z * m z algebra on H (P ; Z ) ) . Consequently, the composition m—z (ZA) o (n on - ((En J o n on ) V * i s not homotopically t r i v i a l , m m+l m-z m m+l and the map n ° n . -, does not l i f t through c . We deduce that m m+l & m (n o n ,, ) (TS ) does not admit 2 l i n e a r l y independent s e c t i o n s m m+l & m fo r m = 0(4). This i m p l i e s that n ( TS ) does not admit 2 l i n e a r l y m independent s e c t i o n s e i t h e r . ( i i i ) (The f o l l o w i n g proof was suggested by S. G i t l e r , thus avoiding an unnecessary reference to the l i t e r a t u r e ) . Assume that m = 2(8) & m and suppose that n (TS ) admits 6 l i n e a r l y independent s e c t i o n s . Since we assume that m __ 18, we deduce that there e x i s t s a f a c t o r i z a t i o n n m of n m through the map c as i n the f o l l o w i n g diagram: _m _m m+2 n . P , > P , u- e m „ * m-6 m-6 n I m c sm+l m ^ sm ________ ^ sm J em+2 m^ Let X = P™_£ u - e m + 2 . For i = m-6,m-5,...,m, denote the generator i m m+2 of H (X;Z 2) = Z 2 by x_^  . Denote a l s o the generator of H™ (X;Z 2) = Z 2 2 3 by y . By n a t u r a l i t y , Sq x = y . Since Sq x „ = x we deduce that m m—j m 2 3 2 3 4 1 1 4 Sq Sq x m _ 2 = Y • Using the Adem r e l a t i o n Sq Sq + Sq Sq + Sq Sq and the f a c t that H m + 1 ( X ; Z 2 ) = 0 , we have that Sc^Km_2 = y " S i n c e 4 4 4 1 Sq x , = x O J we deduce that Sq Sq x , = y . However, Sq x , = 0 m-6 m-2 m-6 n m-6 2 4 4 7 1 6 2 and Sq x m _ ^ = 0, and using the Adem r e l a t i o n Sq Sq + Sq Sq + Sq Sq , 4 4 we deduce that Sq Sq x , = 0, a c o n t r a d i c t i o n . • m-6 * m Remark (3.2). I t has been shown that n (TS ) admits e x a c t l y 5 l i n e a r l y m independent s e c t i o n s i f m = 2(8) [Hoo 1964]. The author does not know the corresponding r e s u l t f o r m = 6(8). §2. The block map. Let 1 <_ k < n ( i = l , 2 , . . . , t ) and l e t N = l n ± and K = Y,k± . We d e f i n e the map b: V , n l ' k l * x V V k t -y V by: N,K y b(A l 5...,A t) = A1 0 0 0 0 0 0 where A. i s a k. x n. matrix such A.A. = I, l x i l x k. We c a l l the map b the block map and we w i l l g e n e r a l l y assume that the i n d i c e s (n^,k^) are c l e a r l y i n d i c a t e d by the context. P r o p o s i t i o n (3.3). Let b: V x . . . x V , >• V V 1 V t ' K be the block map_. Then b = * Proof. Omitted. x n V k t P r o p o s i t i o n (3.4). Assume that f o r some i ^ , 1 < i n < t , we have 0^ k. < Z (n.-k.) . Then the r e s t r i c t i o n of the block map b to the x„ — . x x -0 subproduct V , x ... x V , 1 1 x 0 - l x 0 - l i s homotopically t r i v i a l . x * x V x V 1 ' V 1 x V V k t Proof. For the sake of t e c h n i c a l c l e a r n e s s , we give the proof f o r t = 2 . The reader w i l l see immediately the m o d i f i c a t i o n s necessary f o r the general case. For t = 2, assume that i ^ = 1 . We have to prove that the map V , >- V , , defined by A -n^, K.^  n ^ n ^ , K^+K^ homotopically t r i v i a l . Since - _^ k^, the map ,A 0 0 N . * (0 I, o ) 1 3 k l (A,s) |7 l-s 2 A 0 s i , 0 k l 0 0 0 (0 <_ s <_ 1) d e f i n e s a homotopy between the above map and the constant map. • We w i l l only use the map b i n cases where the i n d i c e s (n_^,k^) are a l l equal, i . e . (n^,k^.) = (n,k) i = l , . . . , t . The f o l l o w i n g two p r o p o s i t i o n s are r e l a t e d to t h i s case. For a space X and a permutation ... x X >• X x ... x X the map which permutes the t f a c t o r s of X x ... x X according to the permutation a . a € S^, we denote by T : X x t a P r o p o s i t i o n (3.5). (Symmetry property of b) . Consider the block map b: V . x n,k X Vv v , >- V . . Assume that e i t h e r k i s even, or n,k t n , t k — n i s even and k and t are odd. Then b o T - b f o r any permutation a € S_ . Proof. There i s an obvious map T': V , >• V , permuting the r a t n , t k t n , t k ^ 6 elements of the tk-frames i n V f c n ^ i n such a way that the f o l l o w i n g diagram ( s t r i c t l y ) commutes: V , x ... x V . n,k n,k V , x ... x V , n,k n,k -y V t n , t k T* a -> V t n , t k I f k i s even, T^ i s a "row operation of even order" [James 1958]. In t h i s case, i t i s known that T^ i s homotopic to the i d e n t i t y map on V , [Ibidem]. The p r o p o s i t i o n f o r k even f o l l o w s immediately. UT1 j UK I f n i s even and k and t are odd, TV i s al s o a row op e r a t i o n , but p o s s i b l y of odd order. However, under these assumptions, t n i s even and t k i s odd, and a l l row operations on V , are homotopic LL1 5 T-K. to the i d e n t i t y map [Ibidem, cor. 1.2]. The p r o p o s i t i o n f o l l o w s e a s i l y i n t h i s case a l s o . • Remark (3 .6) . Let f : Xfc = X x ... x x > Y be a based point preserving map such that f o T - f f o r a l l a € S^ . . Of course, S ac t s on H (X x ... x X; Z^) (on the r i g h t ) v i a the homomorphisms T . Let y€H*(Y;Z 2) . Then T* f * ( y ) = (f e T_)*(y) = f*( y ) f o r a l l o 6 S . Thus, the c l a s s f (y) € H (X x ... x X; Z ) i s f i x e d under the Sfc-a c t i o n . Now assume that Y i s h i g h l y connected. I f H (f%Z^) i s the zero homomorphism, the map f l i f t s to the f i r s t stage of the Postnikov system of Y: X x ... x X • Y Suppose that cj> € H^(E ;Z 2) i s a k - i n v a r i a n t defined by a r e l a t i o n S q ^ _ 1y = 0 f o r some y € H 1 + 1 ( Y ; Z 2 ) . Then, the set {_*<{>}, where f runs through a l l the p o s s i b l e l i f t i n g s of f , determines an element E i n the quotient group H J (X x ... x X; Z 2)/Sq : , _ : LH : L(X x ... x X; Z ) . Using the n a t u r a l i t y of the Steenrod operations, we have that T* Sq^~ 1H 1(X x ... x X; Z 2) = Sq- 5" 1 T* H^X x ... x X; Z £) f o r any a € S . Consequently, Sq"3 1 H"L(X x ... x X; Z 2) i s an S t ~ i n v a r i a n t subgroup of H J(X x ... x X; Z 2 ) , and acts n a t u r a l l y on H J(X x ... x X ; Z 2 ) / S q J ~ 1 H^X x ... x X; Z2> . We wish to point out that E i s f i x e d under t h i s S t ~ a c t i o n . Indeed, i f X x ... x X —-> i s a l i f t i n g of f , then f o i s a l s o a l i f t i n g of f - f «T , up _ * A —A to homotopy. Since (f » T ) (<}>) = T f (cf>) , t h i s i m p l i e s that the set -* i {f ((J))} i s i n v a r i a n t under the a c t i o n of on H J (X x . . . x X; Z^) and that E i s f i x e d under the a c t i o n of S^ . on H J(X x ... x X ; Z 2 ) / S q 3 - 1 H X(X x ... x X; Z 2) . The reader i s asked to n o t i c e that these remarks g e n e r a l i z e to k - i n v a r i a n t s <|> defined by more complicated r e l a t i o n s and to k - i n v a r i a n t s i n higher stages of the Postnikov system f o r Y, even though the d e t a i l s are more complicated as u s u a l . In § § 3 , 4 , we w i l l apply these c o n s i d e r a t i o n s to the map b: V , x ... x v , >• V , w i t h n,k and t s a t i s f y i n g the I I 5 K II j K. t i l j L K c o n d i t i o n s of p r o p o s i t i o n ( 3 . 5 ) . • Let 1 <^  k < n . R e c a l l [Steenrod-Epstein 1962] that the cohomology r i n g H (V t;Z„) i s a commutative a s s o c i a t i v e algebra over Z„ w i t h generators x n_] c» • • • ' x n_i "*"n d l m e n s i ° n n-k,. . . , n - l r e s p e c t i v e l y , and 2 2 r e l a t i o n s x. = x„. f o r n - k < i < % ( n - l ) and x. = 0 f o r i 2 i — — I % ( n - l ) < i < n - 1 . Moreover, the n a t u r a l i n c l u s i o n i : V „ , „ —*• V , — n-Z,k-£ n,k induces the cohomology homomorphism defined by i (x.) = x. f o r n - k < j < n - £ - l and i x = 0 f o r n - £ < ^ j £ n - l . The n a t u r a l p r o j e c t i o n p: V , *• V » induces the homomorphism defined by p (x ) = x f o r n-Z^_j<_n-l . P r o p o s i t i o n ( 3 . 7 ) . Assume that 1 <_ k <_ ( t - l ) n / t . Then the block map b: V , x ... x v , >• V , induces the zero homomorphism i n reduced n,k n,k t n , t k -cohomology w i t h Z ^ - c o e f f i c i e n t s . Proof. Let B: SO(n) x ... x SO(n) • SO(tn) be the map defined by B(A 1,...,A t). = A1 0 . . . 0 0 A, and l e t y: SO(n) x ... x SO(n) —'SO(n) be the m u l t i p l i c a t i o n map y(A l 5...,A t) = A 1- .A . Let i : SO(n) —* SO(tn) denote the n a t u r a l i n c l u s i o n . I t i s w e l l known that i o y - B . Consequently, the f o l l o w i n g diagram i s homotopy commutative: SO(n) x ... x SO(n) |px...xp V , x ... x V , n,k n,k SO(n) I 1 -y SO(tn) t n , t k Since k <_ ( t - l ) n / t , i t f o l l o w s that t n - t k >_ n and that the map * p o i induces the t r i v i a l homomorphism (p o i ) i n reduced cohomology * . . >v w i t h Z2~coef f i c i e n t s . Thus (p»iop) = (b o (p x . . . x p)) = (p x ... x p) o b i s a l s o the zero homomorphism. Since (p x ... x i s a monomorphism, the p r o p o s i t i o n f o l l o w s . §3. The v e c t o r bundle n o © n n,2 n,2 In t h i s s e c t i o n we prove the f o l l o w i n g theorem. Theorem (3.8) . Let n >_ 4 . The vect o r bundle n „ © n „ over V n,2 n,2 n i s t r i v i a l i f and only i f n i s even. The f o l l o w i n g c o r o l l a r i e s are almost immediate. 44. C o r o l l a r y (3.9). Let E, be an even dimensional r e a l v e c t o r bundle over a f i n i t e CW-complex such that £ © 2e i s t r i v i a l . Then E, © E, ijs t r i v i a l . Proof. The v e c t o r bundle n 0 i s "weakly u n i v e r s a l " f o r such E, n,z ( c f . chapter I) . • C o r o l l a r y (3.10). Lf n - k >_ 3 i s odd, then n , © ri , i s not n ) K. Tl j K. t r i v i a l over V , . n,k Proof. Consider the i n c l u s i o n map i : V , ,„ „ >• V , . We have n-k+2,2 n,k * that i ( n , © n I ) = TI , , „ „ © n 1 | 0 0 . The l a t t e r vector bundle n,k n,k n-k+2,2 n-k+2,2 i s not t r i v i a l by Theorem (3.8). Thus n , © n , i s not t r i v i a l . • n,k n,k Proof of Theorem (3.8). For n > 4, l e t d: V „ • V „ x V „ be the diagonal map, — n,2 n,2 n,2 ° and b: V „ x V „ > V„ . the block map. By (3.3), we have n,2 n,2 2n,4 b ( n 0 , ) - n o x n o • Hence, d b ( n 0 ,) = n „ © n „ . The 2n,4 n,2 n,2 2n,4 n,2 n,2 f i r s t step of the proof w i l l be to study the map b o d . I f n i s even, we show that b o d l i f t s a r b i t r a r i l y high i n t o the (modified) Postnikov tower over 4 * Consequently, b o d i s homotopically t r i v i a l , and the theorem f o l l o w s f o r t h i s case. I f n i s odd, we f i n d that b o d cannot be l i f t e d past the f i r s t stage of the Postnikov tower over V^ n ^ . I t f o l l o w s that b o d i s not homotopically t r i v i a l . We deduce that n, 0 © n „ = (b » d) rio / i s not t r i v i a l by an n, I n, z zn, 4-argument i n v o l v i n g p r o p o s i t i o n (3.1). Let K. denote the Eilenberg-Maclam space K(i,Z„) and l e t i . be the fundamental c l a s s , x w i l l denote the cohomology t r a n s g r e s s i o n . Let V . - q 2 _ q l . (3.11) —-> E. • ... E 0 • E- — ^ V 0 . x 2 1 2n,4 A be the (modified) Postnikov tower over ^ ( ). Case n even. The f o l l o w i n g i s a l i s t of a l l k - i n v a r i a n t s i n dimension <_ 2n - 3 oc c u r r i n g i n the Postnikov tower over ^, n = 0(2): (0) k - i n v a r i a n t s i n H (V„ .; Z„) zn, 4 z X2n-4 ' X2n-3 A (1) k - i n v a r i a n t s i n H (E^; Z^) *2n-4 : S ^ X 2 n - 4 = ° *2n-3 : S q 2 X2n-4 = ° ( i ) k - i n v a r i a n t s H ( E ± ; Z^) i __ 2 , ( i ) . 1 (1-1) <|)2n-4 ' S q * = ° * Due to p r o p o s i t i o n (3.7), the map b admits a l i f t i n g b^ to the f i r s t stage of (3.11). P r o p o s i t i o n (3.5) a p p l i e s to b . Consequently, * we can apply remark (3.6) to the c l a s s bn iL>_ „ . Since 1 in—_> 2 2n-5 * Sq H (V „ x V Z_) = 0, b.. ib„ „ does not depend on the choice n,2 n,2 2 1 2n-3 * of the l i f t i n g b^ of b . Thus, by remark (3.6), b^ ^2n-3 ^S a c l a s s of H 2 n ^(V „ x v „; Z„) i n v a r i a n t under the a c t i o n of S„ . n,2 n,2 2 2 ( ) We w i l l use the Postnikov towers a s s o c i a t e d w i t h the path-loop f i b r a t i o n s . This i m p l i e s that b.. ili„ _ = a(x „x n ® 1 + 1 ® x „x , ) + 1 2n-3 n-2 n-1 n-2 n-1 B(x ~ ® x , + x , @ x .) f o r some a, 3 € Z„ . Hence n-2 n-1 n-1 n-2 2 d*b* . = 0 H 2 n _ 3 ( V Z_) . Since H^V Z_) = 0 f o r I z n - j n,z z n,z z i = 2n - 4 and f o r i __ 2n - 2, we deduce that b o d l i f t s a r b i t r a r i l y high i n the Postnikov tower (3.11). Therefore b o d i s homotopically t r i v i a l . I t f o l l o w s that n r, © n _ - (b o d) n_ , n,2 n,2 2n,4 i s t r i v i a l . Case n odd. The f o l l o w i n g i s a l i s t of a l l k - i n v a r i a n t s i n dimension <_ 2n o c c u r r i n g i n the Postnikov tower over ^, n = 1(2): (0) k - i n v a r i a n t s i n H (V„ .; Z„) 2n,4 2 X2n-4 ' X2n-3 (1) k - i n v a r i a n t s i n H (E^; Z^) *2n-4 = S q l x 2 n - 4 = ° *2n-3 = S q 2 X2n-4 + S q l X2n-3 = ° A ( i ) k - i n v a r i a n t s i n H ( E ^ Z^) i >_ 2 ,(i) . 1 .(i-D _ n *2n-4 : S q *2n-4 " ° As i n the case f o r n even, we o b t a i n a l i f t i n g b^ of b tc the f i r s t stage of the Postnikov tower (3.11). Again, we study A b^ ^2n-3 us-*-n§ remark (3.6). In t h i s case, however, we have that I = Sq 2 H 2 n 5 ( V x V ; Z ) + Sq 1 H 2 n 4 ( V ^ x V ; Z ) i s the n,z n,z z n,z n,2 Z subgroup of H 2 n ^(V „ * V 9 ; Z . ) generated by * x n ® x 0 + x „ ® x _ . Consequently, the set {b- (JJ„ „}, where n-1 n-2 n-2 n-1 1 T2n-3 b^ runs through a l l the p o s s i b l e l i f t i n g s of b to E^, determines an element E i n H (V _ x v ; Z 9 ) / I , and, by remark (3.6), E i s i n v a r i a n t under the induced a c t i o n of • A f t e r examining the a c t i o n of S 2 on the group H 2 n ~^(Vn 2 * V n 2' Z2^*> i t : i s e a s y t o see that E = a x „ ® x _ + g ( x „x 1 ® l + l ® x „x ,) + I f o r n-2 n-1 n-2 n-1 n-2 n-1 some a,B £• Z 2 • P r o p o s i t i o n (3.4) i m p l i e s immediately that the composition V „ v V „ V 0 x V „ — V „ . i s homotopically n,z n,2 n,2 n,2 2n,4 t r i v i a l . I t i s then a r o u t i n e matter to check that we must have j b 1 2^n-3 = ^ ^ o r a n y l i f t i n S ^1 o f h • This i m p l i e s that & = 0 I t i s more d i f f i c u l t to determine a . We postpone the proof of the f o l l o w i n g c l a i m to the end of the proof of theorem (3.8). Claim (3.12): a + 0 . Assuming that we have proved t h i s c l a i m , we continue the proof. * Thus, E = x „ ® x , + I , i . e . , b., I(J0 0 = x „ ® x . or n-2 n-1 1 r2n-3 n-2 n-1 * b n „ = x ® x „ depending on the choice of the l i f t i n g b, of 1 2n-3 n-1 n-2 1 b . I t f o l l o w s that d*b* ipn , = x „x . f 0 € H 2 n _ 3 ( V Z „ ) . 1 z n - j n-2 n-1 n,2 2 Since S q 2 H 2 n " 5 ( V £ ; Z£ + S q 1 H 2 n _ 4 ( V n ; Z ) = 0, we deduce that * f ^2n-3 = X n - 2 X n - l a n y l i f t ^ S f °f the map d o b to the f i r s t stage of the Postnikov tower (3.11). Now, r e c a l l that the S t i e f e l manifold V „ can be obtained by n,2 at t a c h i n g a ( 2 n - 3 ) - c e l l to the truncated p r o j e c t i v e space P n ^ . n-2 Thus, V „ - P n }. u e 2 n 3 and we have a map c: V „ > S 2 n 3 n,2 n-2 A n,2 c o l l a p s i n g P n_2 t o a P°i n t. By c e l l u l a r approximation, we have that // 2n—3 c : [S , V„ .] y- [V 0 , V„ .] i s an isomorphism. Therefore, the zn,4 n,z Zn,4 2n 3 map b o d f a c t o r s through the space S as f o l l o w s . Using our previous c o n s i d e r a t i o n s about the map b o d , i t i s easy 2n 3 to deduce that the map g admits a l i f t i n g g^: S >• to the f i r s t stage of the Postnikov tower (3.11), and that g^ ^2n-3 ^ 0 ^ ( S 2 n ^ ; Z^) f o r . any l i f t i n g chosen. Comparing the Postnikov tower over S 2 n 4 —^ > V„ , and the Postnikov tower 2n,4 (3.11), one can deduce e a s i l y that the map g i s homotopic to the . . „2n-3 n „2n-4 i „ , composxtion S >• S y ^ where n represents the 2n-4 generator of ^ n - S ^ ^ " Summing up these c o n s i d e r a t i o n s , we now * * * have that b o d - i r n i o c . Therefore, n 0 ffi n „ - c n i (n„ .) n, Z n, z zn, 4 * 2n-4 * * 2n-4 Since i n „ . = xS , t h i s means that n „ © n „ = c n (xS ) 2n,4 n , 2 n , 2 We wish to deduce that n ^ © n „ i s not t r i v i a l . We suppose n,z n,z that • n „ © n „ i s t r i v i a l and we w i l l get a c o n t r a d i c t i o n . Let n,2 n,2 r „ , denote the u n i v e r s a l v e c t o r bundle over BSO(2n-4) and l e t 2n-4 2n-4 2n-4 t : S y BSO(2n-4) be a c l a s s i f y i n g map f o r xS . Consider the f o l l o w i n g diagram: s2n-4 p n - l _ ^ c g2n-3 _ZA^ E p n - 1 n-2 n,2 I n-2 n s 2 n " 4 / h BSO(2n-4) The h o r i z o n t a l sequence of maps i s a c o f i b r a t i o n sequence. I f n » © n ~ i s t r i v i a l , then the composition t o no c must be n, z n, z homotopically t r i v i a l . Therefore, there i s a map h: E P N \ > BSO(2n-4) n-z such that h o E A - t o n . F i r s t l e t n = 3(8) . N o t i c e that, the map h f a c t o r s through BSO(n) C BSO(2n-4) f o r dimensional reasons. I t f o l l o w s that the v e c t o r bundle n (TS n _ ) s n t (T„ .) = ( E A ) h (T„ ,) zn-4 zn-4 admits at l e a s t n - 4 s e c t i o n s . But n = 3(8) and n >_ 4, so that 2n — 4 E 2(8) and n - 4 >_ 7 . Thus we have obtained a c o n t r a d i c t i o n w i t h p r o p o s i t i o n ( 3 . 1 ) i i i . This proves the theorem f o r n = 3(8) . For n E 1,5 or 7 ( 8 ) , f i r s t r e c a l l that the S t i e f e l manifolds are s t a b l y p a r a l l e l i z a b l e . I t f o l l o w s by a standard argument that the a t t a c h i n g map A : S^ n 4 > P N }. of the top c e l l e 2 n 3 of V „ n-z n,z must be s t a b l y homotopically t r i v i a l . Hence the composition r,2n-3 ,. ^ „2n-4 E A ^ n-1 c 1 „n , , , . . . , S - E S *• ^ p n_2 S must be s t a b l y homotopxcally t r x v x a l a l s o ( c ' : ^P^_2 *" ^ n denotes the map c o l l a p s i n g the bottom c e l l of E P ^ _ 2 to a p o i n t ) . By the Freudenthal suspension theorem, t h i s i m p l i e s that the composition c' o ( E A ) must be homotopically t r i v i a l i t s e l f . Considering the cof i b r a t i o n S*""1 - U » E P n " i S n . we deduce that n-2 E A f a c t o r s through j ' : S >• £ p n_2 • Since n >_ 5, [ E P N " J , BSO(2n-4)] 3 [ E P N ~ J , BSO] (= K O ( E P n ~ b . Using t h i s f a c t n-Z n-Z n-z together w i t h Bott's computation of n\(BSO) and the c o f i b r a t i o n S N _ 1 S N - 1 -J-U E P n " i , one sees that n-2 I m { j , # : [ S ] Pn—2' B S ° ( 2 n _ 4 ) ] * [ S n _ 1 5 BS0(2n-4)]} = 0 . Thus, the composition h o E A must be homotopically t r i v i a l . Since * 2n-4 * * * 2n-4 n (xS ) = ( E A ) h (^2"n-4^' w e O D t a i n t n a t n (TS ) i s a t r i v i a l v e c t o r bundle. This i s a c o n t r a d i c t i o n w i t h p r o p o s i t i o n ( 3 . 1 ) i . This proves the theorem f o r n E 1,5 or 7 (8) . This concludes the proof of theorem (3.8), assuming c l a i m (3.12). Proof of Claim (3.12) Let us make the s u p p o s i t i o n that a = 0 . R e c a l l that n i s odd. Then there i s a l i f t i n g of b to the f i r s t stage of the Postnikov tower (3.11) such that b^ ^ 2n-3 = ^ " Consider the i n c l u s i o n j : P n ~ i > V „ and l e t b' = b o ( j x j ) . Then b' = b, o ( j x j ) i s J n-2 n,2 J J 1 1 V J J / * a l i f t i n g of b' to . Our hypothesis i m p l i e s that b| 2^n-3 = ^ * 1 (1) 1 * (1) Moreover, si n c e Sq (l>2n-4 = ^' w e m u s t bave that Sq (bj 2^n-4^  = ^ ' * (1) This i m p l i e s that b' <f>„ . = 0 . Thus, b' admits a l i f t i n g b' to 1 T2n-4 1 2 * (2) E2 • Again, we must have b^ (<f>2n-4^ = 0 by the same argument as above. In t h i s way, the map b' l i f t s a r b i t r a r i l y high i n the Postnikov tower (3.11). Therefore, under the hypotheses that a = 0, we have that b' - * . We now o b t a i n a c o n t r a d i c t i o n by showing t h a t , i n f a c t , b' ? * . S p e c i f i c a l l y , we prove that ( b ' ) ! : KU(V 0 .) — K U ( P n ~ * x P n ~ b i s 2n,4 n-2 n-2 not the zero homomorphism. Let £ denote the r e s t r i c t i o n of n _ to n,2 P n \ . R e c a l l that the composition p n \ — V ~ — S n ^ i s homotopic n-2 n-2 n,2 r -n-1 „n-l ,, . „n-2 , _ n - l to the map c: P „ > S c o l l a p s i n g S C P „ to a p o i n t . We n-2 r 0 n-2 can compute A(£) (see chapter I I ) as f o l l o w s : HO = A ( j (n , ) ) n, z A = j A ( n „) (by n a t u r a l i t y of A) n, z. = j * ( A ( n 7 ) A ( e ) ) ( A(e) = 1) n, z = j * ( A + ( n _ © e)) ( c o r o l l a r y (2.15)) n, z A + A n-1 * n-1 = j (A p ( x S n X ) ) (p ( x S n X ) = n „ © e) n, z A A + N _ _ + = j p (A (xS )) (by n a t u r a l i t y of A ) = C * ( Y n - l ) + 2 * ( n _ 3 ) (by example (2.11)) = v + 2 * < n " 3 > where v denotes the generator of KEJCP"^) = Z^ and the l a s t e q u a l i t y comes from the f a c t that c': K U ( S n 1 ) K6(?^_^) . Now, consider Y2n-4 " A + ( r i 2 n , 4 } " ^ 6 ™(V2n,S ' ^ ^ b , , ^ 2 n - 4 > - b , , ^ 2 n , 4 » " ^ = A + b ,*^2n,4> " 2 n " 3 = _ + u*5> - 2 n " 3 = A(5) ® A(S) - 2 n " 3 (by theorem (2.14)) = (v + 2 % ( n " 3 ) ) ® (v + 2 % ( n - 3 ) ) - 2 n " 3 = v ® v (since n > 5) . Since the e x t e r i o r tensor product induces a monomorphism ®: KU(X) ® KU(X) >- KU(XxX) [Atiyah 1962], we deduce that 0 / v ® v = b , ! ( Y o ,) € KU(Pn~i x P n"i) . Therefore, b 1 zn-4 n-z n—z and t h i s concludes the proof of the c l a i m (3.12). This completes the proof of theorem (3.8). • §4. The v e c t o r bundle n ~ © n ~ © n n, 3 n , i n, 3 We prove the f o l l o w i n g theorem. Theorem (3.13). Let n ^ 4 be even. Then the v e c t o r bundle n 0 © n © n 0 over V „ i s t r i v i a l . n,3 n,3 n,3 n,3 The f o l l o w i n g c o r o l l a r y i s immediate. C o r o l l a r y (3.14). Let B, be an odd dimensional v e c t o r bundle over a  f i n i t e CW complex. Assume that £ ffi 3e i s t r i v i a l . Then £ © £ © E, i s t r i v i a l . • Proof of theorem (3.13). Let d: V „ —-> V 0 x V 0 x V , be the diagonal map and l e t n,3 n,3 n,3 n,3 b : V 0 X V 0 x V _ — V „ _ be the block map. We have that n , i n,3 n,3 3n,9 A (b o d) (n Q) = 3n „ . We w i l l show that b o d - * by l i f t i n g t h i s map a r b i t r a r i l y h i g h l y i n the Postnikov tower over V^ n g . This i m p l i e s of course that 3 r i n ^ i s t r i v i a l . We have to separate the proof i n two p a r t s according to the cases n = 0(4) and n = 2(4) . Case n = 0(4) Since ^ i s t r i v i a l , we can assume that n >_ 8 . We f i r s t prove that 3n „ © e i s t r i v i a l . n,3 We can construct a non-zero s e c t i o n of 3n „ as f o l l o w s . Let n, 3 (x, ,x.,x 0) be a 3-frame i n R n ,and l e t p: V „ — > S n be the 1 2 3 n,3 n a t u r a l p r o j e c t i o n map, p(x^,x 2,x^) = x^ . Then ( nn ?)<v v V N = {u: u € R n and [u,x ] = 0 , i = 1,2,3} and * n—1 n p (TS ) . . = {u: u € R and [u,x ] = 0} . There i s an ( . X ^ y X ^ j X ^ ^ X * n-1 obvious ve c t o r bundle map P: p (xS ) *• n n 3 defined by P(u) = u - [ u , x 2 ] x 2 - [u,x^]x^ . Since n = 0( 4 ) , we can f i n d 3 l i n e a r l y independent s e c t i o n s s^,s 2,s- 3 of x S n ^  . Then we de f i n e a s e c t i o n s of 3n _ by s e t t i n g n, 3 s ( x 1 , x 2 , x 3 ) = ( P ( s 1 ( x 1 ) ) , P ( s 2 ( x ) ) , P ( s 3 ( x ) ) ) . I t i s easy to check that s ( x 1 , x 0 , x 0 ) 4 0 f o r a l l (x.,x„,x„) £ V „ . X Z J X Z J n,j Since 3 r i n 3 admits a non-zero s e c t i o n , there i s a (3n-10)-dimensional v e c t o r bundle ? over V „ such that 3n _ = £ © £ . n,3 n,3 We have that £ © 3e = 3n 0 © 2e = n . © p 1 (2n „) where n,o n,J n,Z p': V ~ > V „ . The l a s t v e c t o r bundle i s t r i v i a l s i n c e 2n n,3 n,2 'n,2 i s t r i v i a l by theorem (3.8) and 2(n-2) __ 3 . Thus £ © 3e i s t r i v i a l and by theorem (1.3), there i s a map f : V „ >• V„ _ „ such that n,3 3n-7,3 f n3 n_7 3 s ? • Therefore, we have that 3n_ -3 © e = Z © 2e = f * ( n 3 n _ 7 3 © 2e) = (p" o f ) * ( t S 3 n " 8 ) where 3n—8 p": V 3 n _ 7 3 *" s i s the p r o j e c t i o n map. We cl a i m that p" o f - * Indeed, by c e l l u l a r approximation, we have the f o l l o w i n g commutative diagram where the v e r t i c a l maps are isomorphisms. P" n,3 3n-/,3 n,3 3n-6 3n-8 °# r_3n-6 Q3n-8 n [ b ' F3n-10 J * [ S , S ] By the proof of p r o p o s i t i o n ( 3 . 1 ) i i , one sees e a s i l y that the arrow forming the bottom s i d e of the square i s the zero homomorphism. I t f o l l o w s that the arrow at the top of the square i s a l s o the zero homomorphism. This i m p l i e s that p"o f = P^(f) - * as claimed. Since ^ g we have shown that 3r) n ^ © £ = (p" ° f ) (xS ), we o b t a i n that (3.14) 3n o © e i s t r i v i a l . n,3 We w i l l use t h i s r e s u l t l a t e r . We now proceed to show that the map b o d i s homotopically t r i v i a l . Let (3.15) ... > E. • ... • E_ • E. • V_ n l / 1 3n, 9 be the Postnikov tower over V_ _ . Fol l o w i n g are l i s t s of the k-3n, 9 ° i n v a r i a n t s o c c u r r i n g i n t h i s tower i n dimension <_ 3n - 6 . For n = 0(8): A (0) k - i n v a r i a n t i n H (V„ „; Z„) 3n, y I X3n-9 A (1) k - i n v a r i a n t i n H (E^; Z^) *3n-7 : S ^ ' l x 3 n - 9 = ° A (2) k - i n v a r i a n t i n H ( E 2 ; Z^) *3n-6 = ^ *3n-7 = ° For n = 4(8): A (0) k - i n v a r i a n t i n H (V. n; Z 0) '3n,9' V X3n-9 (1) k - i n v a r i a n t s i n H (E^; Z^) >3n-7 : S q 2 > 1 X3n-9 " ° *3n-6 : X3n-9 = ° (2) k - i n v a r i a n t s i n H (E,,; Z^) *3n-6 : ^ *3n-7 = ° By p r o p o s i t i o n (3.7), H (b;Z 2) = 0 . • Hence there e x i s t s a l i f t i n g of b to E^ . We now study the composition b^o d . Since H (V „; Z.) = 0, ( b . o d ) <b0 n = 0 . For n = 4 ( 8 ) , n,3 2 1 T3n-7 A we a l s o want to evaluate (b, o d) i i i , . . In t h i s case, 1 on—o 4 3n-10 * Sq H (V Q x V x V Z.) = 0 . Hence the c l a s s (b, o d) iji- , n,3 n,3 n,3 2 1 r3n-6 i s independent of the choice of the l i f t i n g b^ . Notice that p r o p o s i t i o n (3.5) a p p l i e s to the map b . Therefore, by remark (3.6), b* ip. , must be an element of H 3 n _ 6 ( V x V ~ x V „; Z„) f i x e d 1 3n-6 n,3 n,3 n,3 2 by the S^-action. I t i s easy to check any such element i s p u l l e d back A A t r i v i a l l y by d . Thus ( b o d ) ^ 3 n _ 6 = 0 l f n = 4(8) . This shows A that H ( b ^ o d ; Z 2) = 0 f o r n = 0 or 4(8) . Consequently, there i s a l i f t i n g g: V „ > E„ of the composition b o d . n, j J z x Claim (3.16). g* tyl , = 0 . jn-b Let us assume that we have proved t h i s c l a i m . Then d and, hence, b o d , l i f t to E^ • However, E^ i s already (3n-5)-connected. Since dim V _ = 3n - 6, we deduce that b o d - * . This completes n,3 the proof of the theorem f o r n = 0 ( 4 ) , assuming (3.16). Proof of c l a i m (3.16). * 3n-6 Suppose that 0 ^ g , € H (V Z.) . We w i l l o b t a i n a jn-o N5-J z c o n t r a d i c t i o n . Let (3.17) >• E! — • »• E l • E' y V. Q x 2 1 3n,8 be a (modified) Postnikov tower over V„ „ . The f o l l o w i n g i s a l i s t 3n,o of the k - i n v a r i a n t s o c c u r r i n g i n that tower i n dimension <^  3n - 6 . (0) k - i n v a r i a n t s i n H (V^ Q ; Z„) i n , o / X2n-8 ' X3n-7 (1) k - i n v a r i a n t s i n H ( E ^ Z ^ *3n-8 = ^ X3n-8 = 0 *3n-7 : X2n-8 = ° 2 K„ : Sq x„ .. = 0 3n-6 3n-7 * (2) k - i n v a r i a n t s i n H ( E 2 ; Z 2 ) *3n-6 : S <> 3 +3^-8 + S " 2 *3n-7 " 0 ' ( i ) k - i n v a r i a n t s i n H (E ;Z 2) i >_ 3 - ( i ) _ 1 T ( i - D n *3n-8 : S q *3n-8 = ° ' The p r o j e c t i o n map p: g ^ '> V ^ n g induces a map between the Postnikov towers. Let p.: E. — y E| denote the maps induced. r x x x A r o u t i n e computation shows that p„ ibl r - tyl r • Now, n o t i c e 2 3n-6 3n-6 that p 2 o g i s a l i f t i n g of the composition p o b o d , and that under our hypothesis, (p 2° g) ^3 n_6 ^ 0 • BY c e l l u l a r approximation, we can consider p o b o d : V „ • V „ as mapping i n t o P 3 n „ . Since n,J jn,o 3n—o n _ i ^3n—5 _ „3n-8 _3n—5 _, 3n - 8 = 0 ( 4 ) , we have P„ „ - S v P- -, • Then, xt xs easy to 3n-8 3n-7 * ~ see that the c o n d i t i o n (P 2° g) ^ ^ i m p l i e s that the map po bo d f a c t o r s as shown i n the f o l l o w i n g diagram where c : ^ n 3 ^ 1 S t n e degree 1 map. We deduce that c * ( n 2 ) * ( T S 3 N " 8 ) = c * ( n 2 ) V ( n 3 n _ g ) = (bo d)*p * ( n 3 _ g ) * = ( b o d ) (ru Q © e) = 3ri © e . We have already shown (3.14) that the Jn,y n,J l a s t v e c t o r bundle i s t r i v i a l . Therefore, under the hypothesis that * * 2 * 3n-8 g ^ 3 n _ 5 r 0 , we deduce that c (n ) (TS ) i s t r i v i a l . I s i s easy 2 * 3n-8 to see that t h i s i m p l i e s that (n ) (TS ) admits more than 1 s e c t i o n ( c f . proof of theorem (3.8), case n = 3 ( 8 ) ) . However, t h i s i s a c o n t r a d i c t i o n w i t h p r o p o s i t i o n ( 3 . 1 ) i i . Hence we must deduce that A g ty\ r ~ 0 • This completes the proof of c l a i m (3.16). jn-o This completes the proof of theorem (3.13) f o r the case n = 0(4) . Case n = 2(4). In t h i s case i t i s easy to prove that the map b o d: V „ n, 3 i s homotopically t r i v i a l . Again, l e t (3.18) ... >• E. • ... • E. • E. • V„ . l z l i n , y be the Postnikov tower over V„ _ . The k - i n v a r i a n t s o c c u r r i n g 3n,9 & dimensions <_ 3n - 6 are as f o l l o w s : For n = 2(8): A (0) k - i n v a r i a n t s i n H (V „; Z„) n, J z X3n-9 ' X3n-7 A (1) k - i n v a r i a n t s i n H (E^; Z^) *3n-6 = S q 4 X3n-9 + S q 2 *3n-7 = ° For n = 6(8) (0) k - i n v a r i a n t s i n H (V „; Z„) n, 5 z X3n-9 ' X3n-7 (1) k - i n v a r i a n t s i n H (E^; Z^) hn-6 1 S q 4 X3n-9 = ° By p r o p o s i t i o n (3.7), the map b admits a l i f t i n g b^ to E A A We w i l l show that d b n , = 0 . The theorem f o l l o w s . 1 3n-6 Notice that the map b s a t i s f i e s the c o n d i t i o n s of p r o p o s i t i o n (3.5). Consequently, we can apply remark (3.6). I f n = 2( 8 ) , l e t I = S q V n ~ 1 0 ( V , x V x V ; Z,) + S q 2 H 3 n _ 8 ( V , x V x V Z_) . n,3 n,3 n,3 2 ^ n,3 n,3 n,3 2 * The set {b. to. ,}, where b.. runs through a l l the p o s s i b l e l i f t i n g s 1 jn-o 1 3n—6 of b, determines an element E i n H (V „ x V „ x V „)/I . As n,3 n,3 n,3 explained i n remark (3.6)-, E must be f i x e d under the S^-action. Let a = x • , ® x „ ® x T + X „ ® x „ ® x T + X ~ ® x „ ® x „ and n-1 n-2 n-3 n-3 n-2 n-1 n-2 n-2 n-2 b = x „x „ ® x _ ® l + x „x . ® x _ ® 1 . I t i s easy to compute n-2 n-3 n-1 n-2 n-1 n-3 3n—6 that I i s the subgroup o f H ( V 0 x V „ x V „ ; Z„) generated by n,J n,o n,J Z the elements i n the o r b i t s of a and b under the S^-action. One 3n—6 computes q u i c k l y that the only elements of H (V 0 X V 0 X V „; Z„)/I n,3 n,3 n,3 I f i x e d under the S„-action are [0] and [x _ ® x „ ® x „] . I t 3 n-2 n-2 n-2 * * f o l l o w s that d b, , = 0 as d e s i r e d . 1 3n-6 I f n s 6(8), the same argument a p p l i e s , but the computations are e a s i e r . This concludes the proof of the case n = 2(4) . • §5. Odd m u l t i p l e s of n , . n , K The f o l l o w i n g theorem i s meaningful only i f k i s approximately equal to or l a r g e r than 2n/3 . Theorem (3.18). The vector bundle (2s+l )n , i s not t r i v i a l i f 2s + 1 < (n+<5) / (n-k), where 6 = 6(n,k) i s equal to 0 jLf n and k are even, 1 i f n and k are odd, and -2 otherwise. Before proving theorem (3.12), we e s t a b l i s h the f o l l o w i n g lemma. I P r o p o s i t i o n (3.13). Let E, be a s t a b l y t r i v i a l r e a l v e c t o r bundle over a f i n i t e CW-complex X . Assume that E, has a unique Spin + l - l r e d u c t i o n . Then A (£ © £) = 2 i n KU(X) . Proof. Let us assume that £ = 2r + 1 i s odd. The proof f o r L even i s s i m i l a r . We use the n o t a t i o n of chapter I I . Let E be a p r i n c i p a l Spin(£)-bundle over X such that E, = p ( E ) . Applying (2.15), we have that A ± ( ? © O = A ( £ ) A ( £ ) • Hence the element A ± ( ? © E,) 6 KU(X) i s represented by the vec t o r bundle ^ ( E ) <S> ^ ( E ) . Using the property (2.1) of the a - c o n s t r u c t i o n , we have that A ^ ( E ) ® ^ ( E ) ~ '. A ^ ) ( E ) . Since A ^ • A ^ = + >^r_1 + ••• + ^ + 1 (theorem ( 2 . 6 ) ) , we deduce that A - ( 5 © K) i s represented by the vec t o r bundle ^ r (C ) + xr_i^ + ... + A 1(?) + e . I t i s w e l l known that A (£) i s s t a b l y t r i v i a l f o r any s t a b l y t r i v i a l v e c t o r bundle E, . Thus, + A (E, © E,) i s represented i n KU(X) by a s t a b l y t r i v i a l v e c t o r bundle. The p r o p o s i t i o n f o l l o w s . • Proof of theorem (3.18) We give the proof f o r n and k even. The other cases are t r e a t e d s i m i l a r l y . To show that ( 2 s + l ) n , i s not t r i v i a l , i t i s n j K. s u f f i c i e n t to show that A + ( ( 2 s + l ) n . ) 4 z = Im{KU(*) >- KU(V .)} n, K • n, k. (see example (2.9)). Using s u c c e s s i v e l y (2.15), (3.13) and (2.12), we have: A + ( ( 2 s + l ) n . ) = A + ( 2 s n . ) A + ( n ,) + A~(2s n , ) A ~ ( n , ) n,K. n,k n,k. n,k. n,k = 2 s ( n - k ) " 1 A(n t ) = 2 s ( n " k ) - 1 x „ + 2 ( s + ^ ( n - k ) - 1 . Since the n,k n,k % k - l order of T , i s 2 , we deduce that ( 2 s + l ) n , i s not t r i v i a l n,k n,k i f s(n-k) - 1 < %k - 1, i . e . (2s+l) < k/(n-k) . • Remark ( 3 . 2 0 ) ( i ) . The proof of theorem" (3.18) a c t u a l l y gives somewhat stronger i n f o r m a t i o n . Let 0 < j <_ k - 1 . There i s an embedding P n _ k + j y P 1 1" 1 C V , . The element x , € KU(V , ) r e s t r i c t s to n-k n-k n,k n,k n,k / v / n—k-t~i the generator of the t o r s i o n subgroup of KU(P ) [ G i t l e r and Lam 1970]. Using the n a t u r a l i t y of the Spin operations, i t i s easy to see that the v e c t o r bundle (2s+l)n , i s not t r i v i a l over P n k + 3 n,k n-k i f j i s s u f f i c i e n t l y l a r g e . For i n s t a n c e , i f n - k = j + l E 0(2), t h i s i s the case i f j > 2s(n-k) - 1 . ( i i ) I t i s i n t e r e s t i n g to n o t i c e that i f rn , i s t r i v i a l , one can n, k. deduce that (r+l)n i s a l s o t r i v i a l as long as r £ n/(n-k) . The XI y K, r e s u l t s of theorem (3.18) are e x a c t l y i n the complementary range. Thus, theorem (3.18) gives r i s e to the i n t r i g u i n g p o s s i b i l i t y that even and odd m u l t i p l e s of n , behave i n very d i f f e r e n t ways. n, re ( i i i ) S l i g h t l y b e t t e r r e s u l t s can be obtained by the use of r e a l and qua t e r n i o n i c Spin operations (§11-5). For instance one can show that 7n, r c , i s not t r i v i a l i n t h i s way. 65,56 62. Chapter IV Cross-sections of ri . © ( k - l ) e . n, K §1. S e c t i o n i n g of n © ( k - l ) e n j K Theorem (4.1). If. n,k are odd and 1 <_ k <_ n - 2, or i f n i s  odd, k i s even and min(p ( n - k - l ) + l , % ( n - l ) ) <_ k <_ n - 3, then the v e c t o r bundle n , © (k-1)e admits e x a c t l y k - 1 l i n e a r l y n j K. independent s e c t i o n s . I f n i s even and 1 <_ k <_ p ( n ) , then n , © ( k - l ) e admits e x a c t l y p(n) - 1 l i n e a r l y independent s e c t i o n s . Remark (4.2). The f o l l o w i n g f a c t s are elementary consequences of (1.5). (A) I f n , © ( k - l ) e admits at most r l . i . s e c t i o n s , then Tl y tc n . , © (k-2)e admits at most r l . i . s e c t i o n s a l s o . (B) Let n j tc -L l < s ' < s < k - l . I f n . © se admits at most s + d l . i . — — n,k s e c t i o n s , then n , © s'e admits at most s' + d l . i . s e c t i o n s . n,k (C) I f n , © se admits at most r l . i . s e c t i o n s , then n j ic n_ +£ ic+£ ® s e admits at most r l . i . s e c t i o n s a l s o . Using these f a c t s , the reader w i l l e a s i l y convince himself that theorem (4.1) gives extensive i n f o r m a t i o n about the s e c t i o n i n g problem f o r n , © s e , 1 <_ s <_ k - 1, and t h a t , i n many cases, the r e s u l t s are best p o s s i b l e . We must prove the f o l l o w i n g w e l l known lemma before g i v i n g the proof of theorem (4.1). Lemma (4.3). Let Y,Z be CW-complexes and f : Z —> Y a continuous map. Assume that we have cohomology c l a s s e s y^ € B^iYiZ^) and z. € H^ZjZ,,) f o r 2a + 1 < i < 2b and that Sq Jy. = ( 1 ) y . and l 2 - - • ' i J l+j Sq Jz. = (.)z f o r j <_min(2i,2b) - i . Then, i f f y. = z. i 3 i J i Q i Q f o r some i ^ , 2a + 1 <_ i ^ <_ 2b, we have that f y^ = z^ f o r a l l i , 2a + 1 < i < 2b . Proof. Use the Steenrod operations Sq and Sq i n d u c t i v e l y . • Proof of theorem (4.1). I f n i s even and k <_ p (n), then the p r o j e c t i o n map p., : V , • S n ^ admits a c r o s s - s e c t i o n , i . e . there i s a map * I n,k n—1 s: S > V . such that p ° s - Id , . Then we have that n,k *1 m-1 p * ( T S n _ 1 ) = n , © ( k - l ) e and TS 1 1" 1 = s*(n , © ( k - l ) e ) . Since 1 n,k n,k x S n ^ admits e x a c t l y p(n) - 1 l . i . s e c t i o n s , we deduce that n . © ( k - l ) e admits e x a c t l y p(n) - 1 l . i . s e c t i o n s a l s o . This n, k proves the l a s t a s s e r t i o n of the theorem. To prove the other cases of the theorem, l e t us suppose that n , © ( k - l ) e admits k l . i . s e c t i o n s . Then, there i s a (n-k-1)-n,k dimensional v e c t o r bundle C over V , such that n,k ? © ke = n . © ( k - l ) e . The sum ? © (k+l)e = n , © ke i s t r i v i a l , n, K n, k Therefore, by theorem (1.3) there i s a continuous map f: V , —> V II j K Tl j K.T"_L * n-1 such that f (n ,.,) = £ . Moreover, i f p: V , ,n • S i s the n,k+l n,k+l n a t u r a l p r o j e c t i o n , then we have that (p » f ) * ( x S n 1 ) = f*p*xS n 1 = f ' V ( n , , n © ke) = n, k+1 n , © ( k - l ) e . n, K r\j n—1 Now, l e t us assume that n and k are odd. Let y € KU(S ) n-1 and x , € KU(V , ) be as i n example 2.11 and 2.12 r e s p e c t i v e l y . Tl y K. Tl y K. R e c a l l that we can choose y . such that v -. = A + ( x S n - 2^U n-1 n-1 65. and T 1 = A(n ) - 2^ n k)/2 . Using c o r o l l a r y 2.15 and the II y K. II y K. 1 n a t u r a l i t y of the Spin operatio n s , we have that (p o f ) ' ( y ) = A + ( f * P * CxS 1 1" 1)) - 2 ( n " 3 ) / 2 = A + ( , t © ( k - l ) s ) - 2 ( n " 3 ) / 2 n = n,k _(k-3)/2 . 9(n-3)/2 „(k-3)/2 , 2 A(n . ) - 2 = 2 T , . O n the other hand, n,k n,k ! (k-1)/2 we have that p*(y ,) = 2 T , (by computing as above, f o r n-1 n,k+1 (k-3)/2 1 1 i n s t a n c e ) . Therefore, we must have 2 T , = f ' p * ( v ,) = n,k c 'n-1 2(k 1)/2 ^ . H D w e v e r t h i s i s impossible because the order of n, k+1 T , i s 2^ k x ) / 2 > This i s a c o n t r a d i c t i o n , and we must deduce n,k that n , © ( k - l ) e admits only (k-1) l . i . s e c t i o n s f o r n and XI y K. k odd, as s t a t e d . i I f n i s odd and k i s even, we f i n d that (p o f ) * (Y ,) = n-1 (k 2)/2 n 1 2 T , by the same computation as above. Let i : P , —> V , n,k r J n-k n,k t f\j n—1 be the u s u a l embedding. Since j ' ( T ) generates KU(P ) = Z„t n, k n-k z and t = ( k - l ) / 2 [ G i t l e r and Lam 1970], we deduce that i (p o f o j ) ' ( Y n _ j ) ^ 0 • Therefore, the map p o f o j i s not homo-t o p i c a l l y t r i v i a l . By the Hopf c l a s s i f i c a t i o n theorem, we deduce ^ 2 ^ 2^  that p o f o j i s homotopic to the map c o l l a p s i n g P C P to n—k n—k a p o i n t . Therefore, (p o f o j ) x , = x n where x. i s as i n "n—1 n—± l * A §111-2. Since p x , = x , , we deduce that (f o i ) x -,=x „ . n-1 n-1 J n-1 n-1 * Using lemma (4.3), i t f o l l o w s that (f « j ) x_^  = x. f o r n - k < _ i < _ n - l . n-k-1 F i r s t l y , assume that % ( n - l ) < k < n - 3 . Then Sq x , , = x„ „, „ — — n-k-1 2n-2k-2 ± n H * ( V n , k + l ; V • S l n c e H n " k _ 1 ( P n - i ; V = °' ^ " ^ n^-k-^  = 0 • * n-k-1 * Therefore ( f o j ) x2 n-2k-2 = S q ( f°j) ( x n _ k _ i ^ = 0 • However, n - k j < 2 n - 2 k - 2 < n - 1, and t h i s i s a c o n t r a d i c t i o n w i t h our previous statement. Secondly, assume that p(n-k-1) <_ k - 1 . We can a l s o assume that k < % ( n - l ) i n view of the previous case. Since the (n-1)-skeleton of V , ,.. i s P n ?" , , we can homotope the map f « i n,k+l n-k-1 r t- J . . . ,_n—1. ^ _n—1 T —n—1 —n—1 so that f © j (P , ) C P , , . Let q: P , , —*• P . be the map n-k n-k-1 n-k-1 n-k c o l l a p s i n g the bottom c e l l of t o a p o i n t . Then, of course, ( q o f ©j) x n_2 = x n _ _ a§ ain. Therefore, by applying lemma (4.3), * * n-1 * n-1 we o b t a i n that (q o f © j ) : H (P , ; Z„) —> H (P . ; Z„) i s an i s o -n—K. z n-k z morphism. By the u n i v e r s a l c o e f f i c i e n t theorem, we deduce that * ft n _ l ft n__ ( q o f o j ) H (P , ; Z) —• H (P , ; Z) i s an isomorphism. I t f o l l o w s n-k n-k r by Whitehead's theorem, that q 6 f o j i s a homotopy equivalence, i . e we have the f o l l o w i n g homotopy commutative diagram: p n - l _ f _ _ ^ p n - l p n - l n-k n-k-1 n-k Id n—1 Therefore, ^ n - ^ - i ^ s c o - r e d u c i b l e . However, t h i s i s impossible by [Adams 1962, Theorem 1.2]. This c o n t r a d i c t i o n i m p l i e s the theorem i n t h i s case. This completes the proof of (4.1). • §2. N o n - t r i v i a l i t y of ri , © ( k - l ) e Theorem (4.4). I f n i s odd and 2 <_ k <_ n - 2, or i f n ^ 4,8 i s  even and 2 <^  k <_ n - 3, then n , © ( k - l ) e i s not t r i v i a l . _1 j K. Proof. Under the p r o j e c t i o n map V » y V , the vec t o r bundle n ) K. i A1- n J K. \ k ® ( k - 1 ) e P u l l s back to n n k + £ © (k+£-l)e . Therefore, i t i s s u f f i c i e n t to prove theorem (4.4) f o r l a r g e values of k . I f n i s odd, n 0 © (n-3)e i s not t r i v i a l by theorem (4.1). We deduce n,n-2 J that n , © ( k - l ) e i s not t r i v i a l f o r l < k < n - 2 i n t h i s case n,k. — — For n even, n 4- 4,8 n , © ( k - l ) e i s not t r i v i a l f o r k <_ p(n) by theorem (4.1) a l s o . The only case remaining i s n even and p(n) < k <_ n - 3 . Assume t h i s . By the remark at the beginning of the proof, we can a l s o assume that k i s odd. By theorem (1.3), i f n , © ( k - l ) e = ( n - l ) e , we have a map f: V . > V such that f n -, , -, - n . . Let n,k n - l , k - l n - l , k - l n,k * i : V , , y V , be the i n c l u s i o n map. Then ( i o f ) (n , ) = n , . n - l , k - l n,k c n,k n,k *~ V n - l , k - l V n,k Since x , = A(n . ) - 2^ n k l ) / z ^ by n a t u r a l i t y of the Spin operations, n j K n j K. we have that ( i o f ) ' ( t ) = x . . Because k < n - 3, P n JC+2 i s the n, k. n, k. — *" (n-k+2)-skeleton of V n,k , ^n-k+2 „n-k+2 map and g: P n > P . n-k n-k n-k n-k-2 Let i : P , — • V , denote the i n c l u s i o n n-k n,k the r e s t r i c t i o n of i o f to P n-k-2 n-k Using [ G i t l e r and Lam 1970] again, we have that j ' ( x , ) i s the n j K n-k+2 1 generator of KU(P , ) = Z„ . I t f o l l o w s e a s i l y that g' i s an n K isomorphism. A f t e r studying the homotopy commutative diagram 3n-k+2 " n-k c ,n-k+2 -> P n-k+2 n-k c n-k+2 f i r s t i n KU-theory, and then i n Z2~cohomology, one deduces that ,n-k, •.n-k, H (g;Z„) must be an isomorphism. Since H (V ;Z 0) = H' ' V(V , .) = Z„ z n,k z n - l , k - l Z n—k n k i t f o l l o w s that H ( i o f ; Z^) and H ( f ; Z^) are isomorphisms. Now l e t j 3 n - l n-k V . and j " : Pn ? n,k J n-k V , , , be the usu a l i n c l u s i o n s . n - l , k - l Let N be a very large power of 2 . R e c a l l from [James 1959, thm. 2.5] , . • . N N n-2 that there i s a map h: E V , , , —**• E P , such that the n-1,k-1 n-k Nn-2 F . E N p n - 2 E P , n-k n-k E N p n - l E ^ E N V J * f ^ E N V n-k n,k n-1,k-1 N N n-2 composition h o (E j") i s homotopic to the i d e n t i t y map on E ?n_-^ • „. ^N+n-k^N^ „ . „ ^N+n-k., , . Since H (E V , . ,; Z„) = Z„, H (h;Z„) must be an isomorphism, n—1,K-1 z z z Let i , : P n 2 — P n f be the i n c l u s i o n and F = ho E^ff o i ' o i . ) . J l n-k n-k J J l N+n-k H (F;Z2) i s an isomorphism. By lemma (4.3) i t follows that * H (F;Z 2) i s an isomorphism also. Using the univer s a l c o e f f i c i e n t s theorem and Whitehead's theorem, we deduce that F i s a homotopy equivalence N N AT O ^ 3 T XT , h E (f j ' ) „ „ „N n-2 J l Nn-1 J „N n-2 F: E P , E P , • E P , n-k n-k n-k Because N i s large, t h i s implies that E^P n ^  i s reducible. However, since N i s also a power of 2, E N p n ?" - P?."!^ 1 } [Atiyah 1961]. n-k N+n-k Therefore, we must have k <_ p(N+n) . But t h i s i s a cont r a d i c t i o n with our hypothesis. Thus, we have proven the theorem for n even and p (n) < k <_ n - 3 . 'This completes the proof of theorem (4.4). • Chapter V Examples of Stably Free Modules and Unimodular M a t r i c e s . In t h i s chapter, we use the r e s u l t s of chapter I I I and IV to construct examples of s t a b l y f r e e modules and of unimodular matrices over commutative noetherian r i n g s . §1. Stably f r e e modules. For a compact Hausdorff space X and a r e a l v e c t o r bundle £ over X, r e c a l l that C(X) denotes the r i n g of r e a l valued continuous f u n c t i o n s on X and T(X) denotes the C(X)-module of continuous s e c t i o n s of £ . The b a s i c p r o p e r t i e s of the func t o r F(—) are w e l l explained i n [Swan 1962]. The most important one i s given i n the f o l l o w i n g theorem. Theorem (5.1). (Swan). The correspondence £ — r ( £ ) d e f i n e s a  b i s e c t i o n between the set of isomorphism c l a s s e s of r e a l v e c t o r bundles  over X and the set of isomorphism c l a s s e s of f i n i t e l y generated  p r o j e c t i v e C(X)-modules. Proof. [Swan 1962]. • We now wish to use the v e c t o r bundle n , over V . to n,k n,k co n s t r u c t examples of s t a b l y f r e e modules. However, the r i n g C(V , ) n 5 K-i s g e n e r a l l y too l a r g e to be a l g e b r a i c a l l y i n t e r e s t i n g . Therefore, f o l l o w i n g [Swan 1962, example 1], we proceed as f o l l o w s . Denote the p o i n t s of R n by column v e c t o r s and l e t {e^; i = l,...,n} be the standard b a s i s . R e c a l l that the manifold V . c o n s i s t s of the k x n matrices x = (x..) w i t h r e a l e n t r i e s n,k I J s a t i s f y i n g the equation xx' = I, . The t r i v i a l n-dimensional vector bundle e n = ne over V c o n s i s t s of the p a i r s (x,u) where n j K x 6 V , and u € R n • Let y k be the subbundle of e n c o n s i s t i n g n,k of the p a i r s (x,u) where u i s a l i n e a r combination of the v e c t o r s t k i n the k-frame x, i . e . u = x A f o r some A £ R . The k column t k v e c t o r s of the m a t r i x x d e f i n e a t r i v i a l i z a t i o n of u . Let f: e n y e 1 1 be the v e c t o r bundle map defined by f(x,u) = ( x , x t x u) . k Then, Im f = u and Ker f = n , . n,k The s e c t i o n s s^: x y (x,e_^), i = l , . . . , n , form a b a s i s of the f r e e C(V .)-module T(e n) . R e l a t i v e l y to the b a s i s {s.}, the n, K. x C(V )-module homomorphism T ( f ) : r ( e n ) > T(e n) i s represented by the matrix x*"x where x = ( x ^ j ) n o w stands f o r the matrix of coordinate f u n c t i o n s on V , . n,k For 1 <_ k <_ n - 1, d e f i n e the r i n g A = A(n,k) to be the quotient of the polynomial r i n g on nk unknown R[x...,...,x n ,x01,...,x„ ,...,x, ] 11 In 2.L Zn kxi by the i d e a l generated by the polynomials E x. »x. » - 6 . ., 1 £ i £ j £ k (6^^ i s the Kronecker symbol). Notice that A(n,k) i s a commutative noetherian r i n g w i t h u n i t . Moreover, A(n,k) C C(V ,) . Let F be n j K the f r e e A(n,k)-module over the set ^s-^: ^ = l , . . . 9 n } . There i s an n k obvious isomorphism F ®. C(V , ) = T(e ) . Assume that A i s given iV II y K. k k i t s standard b a s i s , and l e t h: F *• A and h': A > F be the module homomorphisms corresponding to the matrices x = ( x ^ j ) a n <^ x t r e s p e c t i v e l y . Since hh' i s represented by the matrix xx', hh' = I 1 . Let g = h'h: F y F . The homomorphism g i s represented by the m a t r i x x t x . Thus Ker g ®. C(V , ) = r(n ) . Obviously, iV n y K. XI y K. Ker g 5 Ker h . Since h = (hh')h = h(h'h) = hg, Ker h ) Ker g, i . e . Ker g = Ker h . Therefore, we have an exact sequence 0 — y Ker g >• F — A y 0 . Hence Ker g i s a p r o j e c t i v e module k n and ker g © A z A . Let P = P(n,k) = Ker g . Notice that P ~ F/Im h' . These r e s u l t s are included i n the f o l l o w i n g theorem. Theorem (5.2). Let 1 < k < n - 1 and l e t R be the f i e l d of r e a l numbers. Let A = A(n,k) be the quotient r i n g of the polynomial r i n g i n nk v a r i a b l e s R[x, _ ,. . . ,x_ ,. . . ,x„ ,. . . ,x, ] by the i d e a l — 11 l n 21 Zn kn generated by the polynomials E x.„x.» - 6.., 1 _ i _ j <_ k . Let P = P(n,k) be the A(n,k)-module w i t h generators s_^, i = l , . . . , n and r e l a t i o n s E x..s., i = l , . . . , k . Then: ( i ) P © A i s f r e e of rank n; k-1 ( i i ) I f n i s odd and 1 <_ k <_ n - 2, then P © A i s not k-1 f r e e ; i f k i s a l s o odd, then P © A does not c o n t a i n a f r e e submodule of rank > k - 1 ; k-1 ( i i i ) I f n 4 2,4,8 i s even and 1 <_ k <_ n - 3, then P © A i s not f r e e . Proof. ( i ) was proven above. To prove ( i i ) , n o t i c e that (p © A k _ 1 ) ®. c ( v ) = (p ® c ( v _ ) ) © c ( v _ ) k _ 1 a r ( n , © ( k - i ) e ) . A n,K A n,k. n,K n,k The v e c t o r bundle n , © ( k - l ) e i s not t r i v i a l by theorem (4.4). Therefore, by theorem (5.1), r ( n , © ( k - l ) e ) i s not f r e e . I t f o l l o w s tl j K. k-1 that P © A i s not f r e e . I f k i s odd, Theorem (4.1) i m p l i e s that the v e c t o r bundle n , © ( k - l ) e does not c o n t a i n a t r i v i a l subbundle XI 5 r C of dimension > k - 1 . Therefore, r ( n , © ( k - l ) e ) does not c o n t a i n n, K k-1 f r e e submodule of rank > k - 1 . Hence P © A does not c o n t a i n a f r e e submodule of rank > k - 1 e i t h e r . ( i i i ) i s proved i n a s i m i l a r way. • Remark (5.3). For k = 1, theorem (5.2) was proven i n [Swan 1962]. Theorem (5.4). Let A = A(n,k) and P = P(n,k) be as i n theorem (5.2). Then: ( i ) Let 2 <_ k <_ n - 3 . Lf n - k i s odd then the module P © P i s not f r e e ; ( i i ) Let 6 be as i n theorem (3.18) and l e t r be odd. Then the module rP = P © ... © P i s not f r e e i f r < (n+6)/(n-k). Proof. Same as f o r theorem (5.2), using c o r o l l a r y (3.10) and theorem (3.18) i n s t e a d . • Remark (5.6). I f n >_ 5 i s odd and k = 2, theorem (5.4) shows that T.Y. Lam's theorem (1.7). i s best p o s s i b l e . For l a r g e values of k, n o t i c e that theorem (5.4) s t a t e s that rP i s not t r i v i a l i f r i s odd and r <_ k/(n-k) approximately. Thus, there i s a gap of approximately k u n i t s between the " p o s i t i v e r e s u l t " (theorem (1.7)) and the "negative r e s u l t " (theorem (5.4)). §2. Unimodular matrices. Let R be a commutative r i n g w i t h u n i t . R e c a l l that a k x n matrix a w i t h e n t r i e s i n R i s s a i d to be unimodular i f the map n k a: R y R i s an onto mapping. The f o l l o w i n g d e f i n i t i o n i s adapted from [Gabel-Geramita 1974, def. 2.1]. D e f i n i t i o n (5.7). Let a be a k x n unimodular matrix over R and  l e t 1 <_ I <_ n . We say that a i s £-stable i f there e x i s t s a Z x (n-Z) m a t r i x y w i t h e n t r i e s i n R such that the ma t r i x a i s a l s o unimodular. I Y J The f o l l o w i n g theorem r e l a t e s unimodular matrices to s t a b l y f r e e modules. I t i s adapted from [Gabel-Geramita 1974, thm. 2.3]. Theorem (5.8). Let R be a commutative r i n g w i t h u n i t and l e t a be_ a k x n unimodular matrix over R . Assume that a £.-stable. Then: £ (1) Ker a - R © Q, f o r some p r o j e c t i v e R-module Q . (2) I f 1 <_ Z < k, then Ker a © R k _ £ = Rn~l . (3) I f Z _> k, then Ker 'a i s f r e e . Proof. See [Gabel-Geramita 1974, p. 101]. • We apply theorem (5.8) to prove the f o l l o w i n g theorem. Theorem (5.9). Let A = A(n,k) be as i n theorem (5.2). Assume that n ^ 2,4,8, and that 0 <_> s < k <_ n - 3 ±i_ n i s even and 0 < _ s < k < ^ n - 2 i ^ n i s odd. Let a(n,k,s) be the k x (n+s) matrix ( ( x . . ) , 0 ) where 0 denotes the k x s 0-matrix. Then the 13" s y s matrix a(n,k,s) i s £-stable i f and only i f 1 <^  Z ^_ s . Proof. The matrix a(n,k,s) i s obviously ^ - s t a b l e f o r 1 < £ < s . Notice that Ker a(n,k,s) = P(n,k) © A s . By theorem (5.2), t h i s module i s not f r e e f o r 0 <_ s <_ k - 1 . Hence a(n,k,s) i s not Z-s t a b l e f o r Z >_ k by theorem (5.8)(3). Therefore, assume that 1 _< £ <_ k and suppose that a(n,k,s) i s ^ .-stable. Then Ker a(n,k,s) © A i s f r e e by theorem (5.8)(2). By theorem (5.2), i t f o l l o w s that k - s <_ k - £, i . e . , t <_ s . This concludes the proof of the theorem. • B i b l i o g r a p h y J.F. Adams. Vector fields on spheres, Ann, of Math., 75 (1962), 603-32. J.F. Adams. Lectures on L i e Groups, W.A. Benjamin, i n c . , New York 1969. J . A l l a r d . Adams Operations on K0(X) + KSp(X), B o l . Soc. B r a s i l e i r a  Mat., 5 (1974), 85-96. M.F. A t i y a h . Thorn complexes, Proc. London Math. S o c , 11 (1961), 291-310. M.F. A t i y a h . Vector bundles and the Runneth formula. Topology, 1 (1962), 245-48. M.F. A t i y a h , R. Bott and A. Shapiro. Clifford modules, Topology, 3 (Suppl. 1) (1964), 3-38. H. Bass. K-theory and s t a b l e a l g e b r a, I n s t . Hautes Etudes S c i . , P ubl. Math., 22 (1969). R. B o t t . Lectures on K(X), W.A. Benjamin, New York, 1969. M.R. Gabel and A.V. Geramita. Stable ranges for matrices, J . Pure  Appl. Algebra, 5 (1974), 97-112. S. G i t l e r and K.Y. Lam. The K-theory of S t i e f e l Manifolds, The Steenrod  Algebra and i t s a p p l i c a t i o n s , Lecture notes i n mathematics, 168, Springer 1970. C. S. Hoo. Homotopy groups of S t i e f e l m anifolds, Ph.D. t h e s i s , Syracuse U n i v e r s i t y , Syracuse, N.Y., 1964. D. Husemoller. F i b e r Bundles, McGraw H i l l Book Co., New York, N.Y., 1966. I. M. James. The intrinsic join, Proc. London Math. S o c , 8 (1958), 507-35. I.M. James. Spaces associated with S t i e f e l manifolds, Proc. London Math. Soc., 9 (1959), 115-40. M. K e r v a i r e . Some nonstable homotopy groups of Lie groups, 111. J . of  Math., 4 (1960), 161-9. T.Y. Lam. Series summation of stably free modules, Quat. J . Math. Oxford  (2) , 27 (1976), 37-46. N.E. Steenrod and D.B.A. Ep s t e i n . Cohomology Operations, Ann. of Math., Study 50, P r i n c e t o n , N.J., 1962. F. S i g r i s t and U. Suter. Eine Anwendung der K-Theorie in der Theorie der H-Raume, Comment. Math. Helv., 47 (1972), 36-52. 77. R. Swan. Vector bundles and p r o j e c t i v e modules3 Trans. Am. Math. S o c , 105 (1962), 264-77. G.W. Whitehead. Note on cross-sections in S t i e f e l manifolds , Comment. Math. Helv., 37 (1963), 239-40. 

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