ADAMS OPERATIONS ON KO(X) © KSp(X) by JACQUES ALLARD B . S c , U n i v e r s i t e de Montreal, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the 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 J u l y , 1973 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s . study. c o p y i n g of t h i s be g r a n t e d by the Head of my thesis Department or I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t written permission. Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada my li. Supervisor: U. Suter ABSTRACT Let KO(X) K-theory of a f i n i t e be the r e a l and CW-complex KSP(X) be the quaternionic X . The tensor product and the e x t e r i o r powers of vector bundles induce on L(X) = KO(X) ® KSP(X) the s t r u c t u r e of a -graded A-ring. In t h i s t h e s i s i t i s shown, that the Adams operations ip : L(X) + L(X) k which are associated to t h i s , k = l, 2, 3,..., A-ring, are r i n g homomorphisms and s a t i s f y the composition law ijjk o = \p^ = ijj^ o ip F i n a l l y , the r i n g k k t k,£ = 1, 2, 3,... L(X) together with i t s ^-operations i s e x p l i c i t e l y determined f o r the quaternionic and complex p r o j e c t i v e " spaces. iii. Acknowledgement The author i s pleased to acknowledge the very p a t i e n t help and guidance of Dr. U l r i c h Suter during the preparation of t h i s t h e s i s . He i s also thankful to Dr. Kee Lam f o r k i n d l y reading the manuscript. iv. Table of Contents Page Introduction v. Chapter 1 §1.1. D e f i n i t i o n s and p r e l i m i n a r i e s 1 §1.2. Main theorem 3 §1.3. A-semi-rings 4 §1.4. A-operations on §1.5. Proof of the main theorem 16 §1.6. A n a t u r a l transformation 16 §1.7. The reduced theory 17 V(X) 9 Chapter 2. §2.1. Preliminaries 19 §2.2. A r i n g isomorphism 21 §2.3. R e l a t i o n between k <JJ and k ip 24 Chapter 3 §3.1. Computation f o r HP n 27 §3.2. Computation f o r CP n 32 Bibliography 38 V. Introduction Let KU(X) X be a f i n i t e connected and KSp(X) CW-complex and l e t KO(X) , be r e s p e c t i v e l y the r e a l , complex and quaternionic K-theory of X . This t h e s i s i s concerned w i t h the functor defined by L(X) = KO(X) © KSp(X) . Real and quaternionic vector bundles can. be viewed as complex vector bundles provided w i t h some s t r u c t u r e maps. This permits the d e f i n i t i o n of tensor product and e x t e r i o r powers i n the category of r e a l and quaternionic vector bundles, and, therefore, induces a s t r u c t u r e of (Z2~graded) A-ring on L(X) . (A A-ring commutative r i n g w i t h u n i t together w i t h a set of maps n = 0, 1,..., A n isa : R -> R , having the formal properties of e x t e r i o r powers). To any A-ring R , one can associate group homomorphisms k tp : R -> R , k = 1, 2,..., given by u n i v e r s a l polynomials Our main theorem states that the homomorphisms A-ring R tp in A n associated to the L(X) (i) (ii) are r i n g homomorphisms s a t i s f y the r e l a t i o n <pN^ = = i|/ipk" k = 2., 2,.. A t i y a h and T a l l [4] have given s u f f i c i e n t conditions on a A-ring i n order that i t s associated A 's have properties ( i ) , A-ring s a t i s f y i n g those conditions i s c a l l e d s p e c i a l prove that L(X) i s a s p e c i a l A-ring. (ii). A-ring. Our proof i s based on the We theorem that the representation r i n g of a compact L i e group i s a special A-ring. Adams [1] has introduced on the A-rings KO(X) and KU(X) the homomorphisms: i|£ : KO(X) -> KO(X) i|£ ; KU(X) -»• KU(X) The ip 's on L(X) are r e l a t e d t o the Adams operations i n many ways. Let LU(X) = KU(X) « KU(X) be given the Z -graded 2 s t r u c t u r e induced by the A-ring s t r u c t u r e of KU(X) define (essentially, (a,b) • (a',b') = (aa' + bb', ab' + a'b) , A (a,0) = (A (a),0) n A (0,b) = (0,A (b)) n n There i s a n a t u r a l for n odd , A (0,b) = (A (b),0) n n Z^-graded r i n g homomorphism n commuting w i t h the A 's L(X) A-ring i s torsion free, U n for n even). U : L(X) -> LU(X) k (and, hence, the ip ' s) . Moreover, when k i s a monomorphism, and the 's on L(X) are induced by the i|; . This gives an easy proof of our main theorem i n the t o r s i o n f r e e case, and was done i n [ 8 ] . In [8] a l s o , one can k f i n d an a p p l i c a t i o n of the p r o p e r t i e s of the The Bott isomorphism isomorphism KSp(X) = KO ^(X) L(X) = KO(X) © KO~^(X) 4 induces an (L( ) defined as u s u a l ) . isomorphism i s a c t u a l l y a r i n g isomorphism. KO~^(X) = KO(X/vS ) 's on L ( X ) . By d e f i n i t i o n , (XA.Y denotes the smashed product of X Hence the Adams operations ^ This and Y ) . are defined on KO(X) and on KO (X) vii. The i|) 's on L(X) r e s t r i c t to the \p ' s on KO(.X) . K Moreover, they induce the following commutative For k odd: diagrams: For Bott KSp (X) ,2 ,k k -\p Bott KSp(X). k /v -A » KO (X) KSp(X) R * , -* KO "(X) KO(X) I* even: Bott -4 ; >K0 (X) R 2 * 2 S S p (X) e Bott > KO (X) k«%» k Hence, i n general, the ip 's on L(X) determine the i/> 's R on KO(X) and KO~ (X) . A l s o , ip : L(X) -> L(X) 4 homomorphism, whereas k i s a ring ij£ © i|£ : KO(X) © KO~ (X) -> KO(X) © KO~\x) 4 K, K i s not. We have a c t u a l l y computed the r i n g 4) L(X) and the operations f o r the quaternionic and complex p r o j e c t i v e spaces. The l a t e r case as t o r s i o n f o r odd dimensions. The content of the chapters i s as f o l l o w s : Chapter 1 gives the d e f i n i t i o n of the r i n g maps A n on L(X) and the L(X) . The main theorem i s proved there. of the n a t u r a l homomorphism U The d e f i n i t i o n and of the reduced theory given w i t h t h e i r main p r o p e r t i e s . L( ) i s viii. Chapter 2 has two main theorems. L(X) - KO(X) <& KO ^(X) k ip ' s on L (X) i s established. w i t h Adams' k 1 In the f i r s t the isomorphism In the second, we compare the s . Chapter 3 contains the computation of L(X) f o r the quaternionic and complex p r o j e c t i v e spaces. and the ip 's 1. Chapter 1 §1.1. D e f i n i t i o n s and p r e l i m i n a r i e s For let ^ e c t ^ Vect^ (X) 00 A = R, C, or H , and f o r a f i n i t e denote the c l a s s of CW-complex A-vector bundles over the set of isomorphism classes of X , X , and A-vector bundles over X . The d i r e c t (or Whitney) sum of vector bundles induces a s t r u c t u r e of abelian semi-group on Vect^ (X) and the tensor product induces a s t r u c t u r e of commutative semi-ring on Vect (X) and Vect R (X) . Let C us w r i t e : V(X) = Vect,, (X) « Vect (X) K rl and l e t ^ 0 0 K denote the Grothendieck group of the Grothendieck group of Vect^ (X) and L(X) V(X) • We w i l l also use the a l t e r n a t i v e notations: KO (X) = K^(X) KU(X) = K (X) C KSp(X) = K^X) We have that L(X) * KO(X) ® KSp(X) . Following [ 3 ] , §1.5, we r e c a l l that a r e a l (resp. quaternionic) vector bundle over X X can be viewed as a complex vector bundle over together w i t h a structure map, i . e . as a p a i r (v,T) , where v e (/ect (X) and v T :v v w i t h i t s complex conjugate T (Note, T 2 = Id v i s an isomorphism of the vector bundle v , such that (resp, T r 2 = -Id ) v can be considered as a conjugate-linear map from v to itself). The d e t a i l s of t h i s construction are to be found i n 13]. However l e t us r e c a l l that two p a i r s isomorphic i f Van isomorphism and v' (v,T) and (v',T') are s a i d are isomorphic complex vector bundles v i a a : v -> v' which commutes with the s t r u c t u r e maps, i . e . a o T = T'• » a . This d e f i n i t i o n coincides w i t h the isomorphism of r e a l (resp. quaternionic) vector bundles under the present i d e n t i f i c a t i o n . Hence 1.1.1 where VCX) = U[u,S] , {v,T]) S, T y, v, e l/ect are s t r u c t u r e maps and isomorphism classes of ( y , S ) and (X) , S I M , S ] and IM',S»] Id , T {v,Tj denote the V(X) as f o l l o w s . Let , = (v,T) r e s p e c t i v e l y . We define a semi-ring structure on [y,S] 2 e Vect R CX) fv,T] , [v',T'] e V e c t (X) R 2 = -Id } 3. and define: CIy,sj,o)-(ly',s'j,o) » Q y M , s 8 s ],o) ([y,S],0)-(0,[v,T]) = (0,[y 8 v, S 8 T]) (OJv,T])-(0,[v' ,T' ]) = ([v 8 v» , S 8 f ] , 0 ) Here the tensor products are taken i n the category of complex vector bundles'. The 0 denotes the isomorphism c l a s s of the O-dimensional r e a l or quaternionic vector bundle. structure maps I t i s easy to check that the S 8 S* , S 8 T, I 0 T' have the appropriate so that these d e f i n i t i o n s make sense. squares, We extend the d e f i n i t i o n of the product f o r a r b i t r a r y elements of L(X) i n the obvious way. Hence V(X) becomes commutative semi-ring and L(X) therefore c a r r i e s the s t r u c t u r e of a commutative r i n g w i t h u n i t . Z2 graded. Actually _ of f i n i t e Notice that L(X) i s L(-) i s a contravariant functor from the category CW-complexes and continuous maps to the category of Z2~graded commutative r i n g s w i t h u n i t . CW-complex and f : X induced map Y If Y i s another f i n i t e i s a continuous map, we denote by f * the L(Y) -*• L(X) . I t i s the d i r e c t sum of the maps KO(Y) •> KO(X) and KSp(Y) -> KSp(X) §1.2. Main theorem: The r i n g induced by f . L(X) admits Adams operations, i . e . there are r i n g homomorphisms: ip k : L(X) •* L(X) k = 1,2,... such that (i) / (ii) ^ o / r (iii) = ^ = / r (x) = x^ (mod f o r a continuous k k,£ = 1, 2, p) p prime f : X -»- Y , * of o / * k = f o 4* The proof of t h i s theorem w i l l be completed i n 1.6. We need to introduce another concept. §1.3. A-semi-rings The f o l l o w i n g d e f i n i t i o n s are found i n [4] i n the case of rings. The extension to semi-rings was without problem. A A-semi-ring i s a commutative semi-ring together w i t h a set of maps A n :R R R , n = 0,1,..., with unit 1 such that x, y e R (1) A°(x) = 1 (2) A (x) = x (3) A (x+y) = 1 n I A (x) A ( y ) r=0 r n r If the semi-ring i s a r i n g we w i l l t a l k of a Let a....a and 1 q A-ring b,...b be indeterminates and l e t s. 1 r l and be the i elementary symmetric functions i n a^...a^ and b^.-.b^ respectively. Take q >_max(n,mn) , r >_ n and l e t i n II. . (1+a.b.t) 1,3 I j P (s. .. .s ; a- .. .a ) be the c o e f f i c i e n t of t n 1 n' 1 n n P II (1+a. ...a. t ) i . <...<i I, I 1 m 1 m v P y m,n (s. ...s ) be the c o e f f i c i e n t of t l mn n and P m,n n in are uniquely defined polynomials w i t h c o e f f i c i e n t s i n Z n J r J they s p l i t i n : P P where = P -P m,n m,n + m,n P (resp. P ) n m,n + = P -P n n + n i s the sum of the terms of P + having p o s i t i v e c o e f f i c i e n t s and P n terms of P^ (resp. P^ ) n -1 . Hence r (resp. P ) i s the sum of the m,n ' r having negative c o e f f i c i e n t s , m u l t i p l i e d by P ,P ,P ,P have only p o s i t i v e n n m,n m,n + coefficients + and can be interpreted A v (resp. P ) n m,n i n any semi-ring w i t h u n i t . A-semi-ring R i s a special A-semi-ring i f the f o l l o w i n g i d e n t i t i e s hold V x , y e R : (4) P ( A ( x ) . . , A ( x ) ; A ( y ) . . . A ( y ) ) + A (xy) 1 n n = P 1 + n n n (A (x)...A (x); A (y)...A (y)) 1 n 1 n 6. (5) P~ (A (.x) . ... A ^ C x ) = P (the A (y) i 1 ) + A A Cx) 1 >n m,n m m CA Cx)...A + 1 n f i r s t identity relates m n A (xy) n l , . . . , n ; the second r e l a t e s i = l,...,mn) . n (x)) . and the A A (x) m A (x) , 1 and the n A (x) , 1 For convenience we w i l l r e f e r to these i d e n t i t i e s i n the f o l l o w i n g form: i.e. (4') F (x,y) = G (x,y) C5») f n m,n n (x) = G F Cx,y) we w r i t e m,n (x) f o r the l e f t hand side of ( 4 ) , e t c . . . . n Moreover, even though we w i l l consider many A-semi-rings, we w i l l not use d i f f e r e n t symbols f o r the maps (5'). and the i d e n t i t i e s ( l ) - ( 5 ) , ( 4 ' ) , n We hope t h i s w i l l not cause any Let homomorphism maps A A n R and g : R -> S S be two difficulty. A-semi-rings. A A-semi-ring i s a semi-ring homomorphism commuting w i t h the , i . e . such that g ° A n = A n o g , n = 0 , 1,... The f o l l o w i n g two p r o p o s i t i o n s motivate the i n t r o d u c t i o n of the previous concepts. 1.3.1 Proposition Let group R of R R q be a (special) i s a Cspecial) A-semi-ring. A-ring. Then the Grothendieck Proof: I t i s w e l l known that extend to R 1.3). R i s a ring. The maps If R n : R. ->• R Q Q i n a unique way which s a t i s f i e s (3) ( c f . [ 7 ] , Chap. 12, The extension a l s o s a t i s f i e s (1) and ( 2 ) , hence A-ring. A i s special, q R R isa i s s p e c i a l , since (4) and (5) are compatible w i t h (3) by lemma 1.5 of 14]. j For k = 1, 2,..., l e t be the unique polynomial w i t h integer c o e f f i c i e n t s such that: Q (s ..s ) = a k r k k x +...+ a k q (with the n o t a t i o n formerly used and q >_ k ) . Then we have 1.3.2 Proposition Let R be a s p e c i a l A-ring and define / ( x ) = Q (A (x),...,A (x)) 1 k k for x e R , k = 1, 2,... . Then 4>N^ = 4^ <jj^ (x) = x^ c f [4] prop. 5.1, 5.2. Examples: (a) For A = R A n or are r i n g homomorphisms and = 1, 2,... = r Proof: \p r (mod p) p prime, x £ R j C , Vect^(X) i s a special A-semi-ring where the are induced by e x t e r i o r powers of vector bundles, i . e . : . A [v] = ,IA (v)] n By 1.3.1 v e |/ect n KO(X) and KU(X) are s p e c i a l A (X) A-rings, and the operations: ip : KO(X) •* KO(X) k tp : KU(X) -»• KU(X) k obtained v i a 1.3.2 are the c l a s s i c a l Adams operations. these f a c t s are found i n [1] or {4]. (b) Let LU(X) = KU(X) « and define a KU(X) Z2~graded r i n g s t r u c t u r e on LU(X) (a,b) o (a*,b ) = (aa' + bb', ab'+ f for (a,b) (a',b') e LU(X) . For k A^ (0,b) = ( A ( b ) , 0 ) K A 2 k + 1 ZK (0,b) = (0,A 2 k + 1 (b)) , and define , A (a,b) = K k I r=0 A Ca,0) A* (0,b) r a*b) k = 0, 1,.. set A (a,0) = (A (a),0) k by r The proof of 9. LU(X) w i t h these maps i s a s p e c i a l special (c) A-ring. This i s d i r e c t using the A-ring properties of KU(X) . Other examples can be found i n [ 4 ] . §1.4. A-operations on V(X) For n = 0, 1,..., we define: A ({y,S],0) = ( I A ( j i ) , A (S)],0) n n n A (0,{v,Tj) = ( I A ( v ) , A (T)],0) 2n 2 n 2n A 211+1,,, n \ /n r-, 2n+l / v , 2n+l ,„,. ,.. A (0,{v,Tj) = ( O J A (v), A (T)]) r m using the c h a r a c t e r i z a t i o n of V(X) given by 1.1.1 . Checking the squares of the s t r u c t u r e maps involved confirms the l e g i t i m i t y of this definition. We extend i t to V(X) by: A ( [ y , S ] , [v,T]) = n n I A ([y,S],0) A r=0 r n _ r (0,[v,Tj) . Theorem V(X) special Proof: together w i t h the A-operations j u s t defined, i s a A-semi-ring. Checking the conditions ( 1 ) , (2), (3) i s d i r e c t , using the p r o p e r t i e s of e x t e r i o r powers and, f o r (.3), v e r i f y i n g a simple identity. Hence V(X) i s a A-semi-ring. We now have to check (4) and (5). We w i l l do the work f o r (4) only since a s i m i l a r treatment deals w i t h (5). Using again the lemma 1.5 of 1.4], we remark that we need only to check these i d e n t i t i e s on a set of generators of the semi-group V(X) . The convenient set w i l l be: Vect where Vect R CX) U V e c t CX) . and Vect rl is. H (X) C V(X) (X) are i d e n t i f i e d w i t h t h e i r image under t h e i r canonical embedding i n V(X) . In X7J, Chap, 5, §6, Husemoller defines the notion of continuous f u n c t o r , and shows that a continuous functor: l/ect (0) x.. .xl/ect (0) -> l/ect (0) A where 0 A A denotes the 1-point space, induces a functor: l/ect A (X)x...xl/ect A (X) -> l/ect A (X) . The d e f i n i t i o n s and theorems of [7] can c l e a r l y be extended to the case: l/ect where A (0)x...x(/ t ec A, A', A" A (0) x (/ect , (0) x... xVect^, (0) + l/ect^, (0) A take the values R and H . For example, we have the f o l l o w i n g continuous functors: « : Vect (0) x V e c t A (.0) •> V&ct^ (0) 11. 0 : l/ect for A A (A, A', A") = (R, H, H) , (H, R, H), (H, H, R) \ for (0) x |/ect ,(0) - Vect „ (0) A : l/ect n (A,A') = (R, R) (0) -> V e c t or (H, H) and n , or (R, R, R) . (0) odd , or (H, R) and n even. These functors r e s p e c t i v e l y induce the d i r e c t sum of vector bundles, the tensor product of vector bundles and the e x t e r i o r powers of vector bundles. The functors F n and G n are compositions of these operations on vector bundles; they are induced by continuous functors F where n,0 ' n,0 G : " c c t A ( 0 ) X " e c t A' ( 0 ) * " e c t (A, A', A") = (R, R, R) , (R, H, H) , (H, R, H) for n odd, or (H, H, R) for n A» or ( 0 ) (H, H, H) even. By [ 7 ] , an equivalence of functors: u 0 n :F _ -»- G n,0 n,Q A induces an equivalence of functors: u :F n G' n and, passing to isomorphism c l a s s e s of vector bundles, one gets: F = G : Vect (X) x Vect , (X) + Vect ,, (X) . n n A A A 12. However, we are not i n t e r e s t e d i n the f u n c t o r i a l properties of and G , and an equivalence of the functors n n F -. and G „ f o r n,0 n,0 vector bundle isomorphisms w i l l be s u f f i c i e n t to give the r e s u l t wanted. (This can be seen i n greater d e t a i l by analyzing the proof of the theorem 6.2 i n [ 7 ] , Chapter 5 ) . Since Let us f i n d out such an equivalence. A-vector bundles over a point are e s s e n t i a l l y A-vector spaces, we can choose bases, and consider, f o r each F with n,0 ' n,0 G : G (A, A', A") z ( - h > ^ x k, k' e N : G W , A ) + G£(k",A") f as above and k" e N depending on k, k', n, A, A' . k k' k k' (We don't w r i t e the more accurate F ' , G '„ n,0 f o r fear of heaviness) ' n,0 Now, i n order to complete the proof, we need t o produce M e G£(k",A") such that; M F (A,B) = G^ (A,B) M n,U n,U n for a l l purpose. n (A,B) e G£(k,A) x G£(k',A') . The next lemma serves that Hence the proof of the theorem i s complete. J 1.4.1 Lemma There i s M e G£(k",A") M F for a l l n > Q (A,B) = G such that: n>Q (A,B) e Gil(k,A) x (A,B) M G£(k',A') 13. Proof; S i m i l a r l y as i n 1.1, r e a l and quaternionic representations of L i e groups can be viewed as complex representations together w i t h some structure maps. This i s c a r r i e d out i n [2], Chapter 3. The f o l l o w i n g i s a d i r e c t t r a n s l a t i o n of t h i s treatment i n term of matrix representations. ( I t can also be obtained w i t h the help of §1.1. f o r the 1-point space) • Let E = 1 or 2 and T e Gl (Em,C) such that: TT = Id i f E = 1 TT = -Id where T of conjugated. T if E = 2 denotes the complex matrix w i t h as c o e f f i c i e n t s , the c o e f f i c i e n t s Define: ©(Em, C ) = {A e Gl (Em,C) T TA = AT} . Then we have the f o l l o w i n g isomorphisms: G ; U m , R ) ~ Gl(m,<Z) if SS = Id G£(m,H) ~ Gn(2m,C) if S'S' = - I d s (The conjugate signs appearing here correspond to the conjugate l i n e a r i t y of the s t r u c t u r e maps of {2]). Examples of S, S' are: 14. S = Id f 0 -1 0 0 . . . 01 1 0 0 0 ... 0 0 0 0 -1 0 0 1 S' = 0 V. 0 oJ Using these i d e n t i f i c a t i o n s , the functors F - and G „ n,0 n,0 are as f o l l o w s : F G : GACEk,C) x 6 n E ' k , C ' ) , , n > 0 n,0 T : G £ ( E k ' > C X T T G ^ ' '. ) . E k G£(E"k",C) GA(E"k",C) c T G F ( t ) n,0 where E" i s determined by F _(T,T') F „(T,T') = G „(T,T') G „(T,T') n,0 n,0 n,0 n,0 (This e q u a l i t y i s easy to check). The existence of a matrix M e G£(k",A") lemma i s equivalent to the existence of a matrix such that: (i) V (ii) N F n>0 (U,V) = G (U,V> N (U,V) e 6*(Ek,C) x G£(E'k',C) , T T N F (T,T') = G(T, T') N . One can produce such a N i n the f o l l o w i n g way. s a t i s f y i n g the N e Gl(E" k",C) 15. F i r s t , the representation r i n g A-ring, where R(U(Ek) x U(E'k')) i s a s p e c i a l U(n) i s the u n i t a r y group of dimension n ( c f . [4] t h . 1.5); hence the representations F are equivalent. equivalent on ,G M But two representations of G£(Ek, C) x G£(E'k', C) U(Ek) x U(E'k') are equivalent. P e G£(E"k", C) Therefore there e x i s t (iii) V ; UCEk) x'U(E'k') + G£(E k",C) P F n>Q (U,V) = G n>0 such that (U,V) P (U,V) e G£(Ek, C) x G£(E'k', C) . Now, ( i i i ) s t i l l holds i f one replaces writing ( i i i ) with P U, V , and note that by P . F i r s t s t a r t by F n ,G n,0 conjugation. of ( c f . [ 1 ] , proof 4.5). Now l e t p e C be such that -pp ^ _ commute w i t h the n,0 i s not an eigenvalue P "*T , and define: N = p P + p"P = P(pp "" - One sees that satisfies (i). N e G£(E"k", C) But since 1 I + P - 1 and N P)p- . s a t i s f i e s ( i i i ) , hence i t N = N , i t s a t i s f i e s also ( i i ) . proves the lemma. 1.4.2 C o r o l l a r y to the theorem L(X) Proof: i s a special A-ring. Theorem §1.4 and p r o p o s i t i o n 1.3.1. / This 16, §1.5. Proof of the main theorem By 1.4.2 and p r o p o s i t i o n 1.3.2, we obtain the r i n g homomor- phisms ip with t h e i r f i r s t two p r o p e r t i e s . since f isa §1.6. A natural A-ring homomorphism. The l a s t one i s c l e a r / transformation R e c a l l the canonical homomorphisms c : KO(X) -> KU(X) c r : KU(X) -> KOCX) c and q : KU(X) and c' are induced by the maps KSp(X) . feet (X) ->• feet R C (X) fect„ (X) -»- feet (X) f o r g e t t i n g the s t r u c t u r e maps ( c f 1.1); r H. L i s induced by the map l/ect (X) q ' : KSp(X) ->- KU(X) i s induced by the map where e i s the t r i v i a l feet (X) f o r g e t t i n g the complex s t r u c t u r e ; feet (X) ->- feet (X) defined by n ~+ n 8 e 1-dimensional quaternionic vector bundle over Define: U = c « c' : L(X) -> LU(X) U' = r # q : LU(X) -* L(X) . Proposition The map L(X) U : L(X) i s torsion free, U LU(X) i s a A-ring homomorphism. I f i s a monomorphism, i . e . L(X) i s a sub-A-ring of LU(X) . Proof; The f i r s t part i s obvious from the d e f i n i t i o n of the maps A n X . 17. on L(X) . For the second part we have that U' o U = 2 where 2 : LCX) ->• L(X) for instance. i s the map x ->• 2x x e L(X) . Cf [5] or [6] Hence, i f L(X) i s t o r s i o n f r e e , U' ° U is a monomorphism and so i s U . j §1.7. The reduced theory From now on we suppose that the spaces considered have base points: (X, Xq) s . t . X q e X . The i n c l u s i o n by X we mean a p a i r i : x« -*• X induces an exact sequence: 0 -> Ker i * L(X) i L(x_) -> 0 . Define L(X) = Ker i Then we have a n a t u r a l s p l i t t i n g L(X) - L(X) e L ( x ) . n Let L(xq) e be the 1-dimensional quaternionic vector bundle over i s the free abelian group on two generators product s t r u c t u r e i s given by the r e l a t i o n k = 1, 3 , . . . k = 2, 4, . . . Moreover, 1, e , and the 2 e = 4 . The maps defined by: Xq . k ip are 18. L(X) - KOCX) © KSp(X) and the k ip 's and U pass to the reduced functors (with LU(X) - KU(X) 9 KU(X)) . F i n a l l y , i f X* denotes the d i s j o i n t union of point which we take as canonical base p o i n t , we have that L(X) - L ( X ) . + X with a Chapter 2 §2.1. P r e l i m i n a r i e s Let (Y,Vq) S n be f i n i t e denote the n-dimensional sphere and CW-complexes w i t h base p o i n t . X A Y = X X Y X / X y Q xj ( X , X q ) and Define as usual: x Y X q Consider the obvious p r o j e c t i o n maps: /I 4 X x s X and denote by v 4 S X the generator of A S 4 KSp(S ) - Z . : K S p ( X ) + K 0 ( X * S ) C l"(X x s) 4 $ ( 3 ) = p*(g) • q (v) by 1 Define: 4 V g e KSp(X) KO(X) + KSp(X x S ) C L(X x S ) 4 by Since 4 $ (a) = p*(a) • q (v) v i s a quaternionic bundle, V a e KO(X) . p*(a) • q (v) and p * ( 3 ) • q*(\ are r e s p e c t i v e l y quaternionic and r e a l bundles as i n d i c a t e d , f* : KOCXAS )^ 4 KOCX x S ) 4 and f * : K S p ( X A S ) -> KSp(X x s ) 4 4 20. are monomorphisms, and we have the Bott isomorphism theorem: (i) Consider f* KSp(X) then $^ * K0(X i s an isomorphism onto (ii) K0(XAS ) x s ) «- Im f * Consider ~ 2 ~ L f* ~ L K0(X) -> KSp(X x S ) $• KSp(XAS ) $ then Proof: i- s isomorphism onto a n c f [5]. Im f * . J Next, we r e c a l l that one defines f o r n = 0, 1, 2,... K0 (X) = K0(XA S ) _n and that K0(X) # KO ^(X) f o l l o w i n g way. Let n can be given a r i n g s t r u c t u r e i n the a , a ' e lcO(X) , 8, 3 ' e K0~^(X)=K0(X A sS denote the product i n KO(X) « K0~^(X) F i r s t define a x a' = oca' . Now l e t X x s - ^ X x X x s by x . , and 21. be the maps defined by: A(x,s) = ( x , x , s ) Pj^Cx.yjS) = x 4 for x , y e X , s e S P .(x,y,s) = (y,s) 2 Then y = A*[ define P ; L • p£p*CB)l *(v) e Im(f* : K O ( X A S ) KO(XH S )-) 4 4 and a x 3 = (f ) ~ ( ) e K*0(XA S ) . S i m i l a r l y , l e t Y 4 4 x s X 4 X A A' x S S > X 4 A x x x s \, *• 4 S A 4 x s 4 2 * X x s 4 X x s be the obvious p r o j e c t i o n s plus the map defined by A ' ( x , s , t ) = (x,s,x,t) 4 x - e X , s, t e S for Y ' . Again = A ' * [ q * p * ( 3 ) - q * p * ( B ' ) ] e Im(g*:K0(XA S ^ S ) •+ K0(X x S 4 and define 3 x 3 ' = ($ $ ) 2 _ 1 (g*) - 1 (y») 4 4 x S )) 4 e K0(X) . (These d e f i n i t i o n s are equivalent to those of [3] where the d e t a i l s can be obtained). §2.2. A r i n g isomorphism From the Bott isomorphism theorem, we obtain a group isomorphism: B = Id 9 $ 1 : L(X) K0(X) 0 K0~ (X) 4 22. We have the following Theorem; B : L(X) -»• KO(X) 6 KO~ (X) 4 i s a r i n g isomorphism. Proof: E s s e n t i a l l y , the proof consists of checking the commutativity of the two f o l l o w i n g diagrams: KO(X) 8 KSp(X) Id 8 $]_ » KO(X) 8 KO(XA S ) KO(X) 8 KO(X x S ) PI<->-P*C-) (I) KO(X x X x s ) A KO(X x S ) f •RSp (X) 1 VKOCX A S ) 4 KSp(X) 8 KSp(X) = ±-*KO(X f S ) 0 KO(X S ) A 8 f KO(X x s ) 0 KO(X x S ) 4 & 4 it % (~)-q (-) 0 (ID KO(X x S KO(X x s KO(X)' 1 2 ~> , -» KO (X A 4 4 x X x S) 4 x s ) 4 4 4 S A S ) In each case, the l e f t hand v e r t i c a l arrow i s the product L(X) and the r i g h t hand v e r t i c a l sequence i s the product i n KO(X) « KO~ (X) . Let us check (I) : V a e KO(X) , V g e KSp(X) , 4 A*[p*(a) • p* = A*[p*(»-p* (p*(0) • q*(y))] = f * [ p * ( a - g ) q* ( v ) J = f *,(a-g) We used mainly that continuous maps induce r i n g homomorphisms on L(-) and that p 2 6 A = Id xx g 4 and p ][ 0 A = p ., 24. Similarly for (II) §2.3. R e l a t i o n between k and ip k We need the f o l l o w i n g Lemma: (i) 4 L(S ) = ev Z « v Z «~ where v 4 i s the generator of KSp(S ) - Z quaternionic (ii) and e i s the t r i v i a l 1-dimensional vector bundle; the r i n g homomorphisms ip are given by: k v 2 /(v) = , 2 k ev k = 2, 4, 6,. . ~ ,4 Proof: of of KO(S ) - Z , and l e t a ^ ( S ) . By p r o p o s i t i o n §1.6, U : *L(S ) -> LU(S ) 4 phism. of Let p be generator 4 By [1] or [ 7 ] , U(p) = (2a,0) and 4 4 hence i s given by i s a monomor- U(v) = (0,a) . LU(X) - KU(X) © KU(X) w i l l be w r i t t e n as p a i r s U(e) e LU(S ) be generator (x,y)) . Since U(e) = (0,2) , one has that U(u) = U(e)-U(v) , p = ev . Using [ 1 ] , c o r r . 5.2, one computes that: i> k (0,a) k. "~ 1 j 3 ^ • • * { k (a,0) lc * 2 j A ^ • • • (0,a) = — (Elements 25. Hence: k v k = 1, 3, 2 k 2 k = 2, 4,, E V k 4 ip 's on L(S ) k know the value of ip (e) • Remark: / 2 the are now completely determined since we R e c a l l (1.3., example (a)) that we have the c l a s s i c a l Adams operations \£ K : KO(X) The r e s t r i c t i o n of and -> K O ( X ) \fc : R i p : L ( X ) -> L ( X ) 4 , with ^ : KO(XA 4 S) . 4 i s ij£ . In the next R <p : 'L(X) -»- T ( X ) to to k theorem, we compare the r e s t r i c t i o n of KSp(X) - K O ( X A S ) KO(XA S ) KO(X) k -> KO(XAS ) 4 K KO(XA S) . 4 Theorem The f o l l o w i n g diagrams are commutative. For *KSp(X) — KSp(X) - For Bott —z ^ > KO(XAS Bott * 1 A ) R KO(XAS") k = 2, 4,, KSp(X) • KO(X) Bott —z ^ > KO(XAS "R A ) k = 1, 3,.. k where -r- e 2 Proof: k a -> -y e«a 2 denotes the map For f o r a e KO(X) k = 2, 4,... ^(3) l£ / = <l£(p*(6) • q*(v)) p*(g) • / q*(v) p* / ( g ) • q* / ( v ) = p* / ( g ) • q * ( | ev) = p | (*) because R (**) = * because ^ C3) 4 -J q (> e v (e./(B)) 1 k ip <y i s a r i n g homomorphism on L(-) and for a l l k = 1, 2, KO(-) ip Similarly for k are n a t u r a l odd. J (Main theorem) . Chapter 3 §3.1. Computation f o r HP n F i r s t we r e c a l l some r e s u l t s . Let and l e t £ r\ be the canonical complex l i n e bundle over be the canonical quaternionic l i n e bundle over Also c f . §1.6. f o r C 3.1.1 : HP n n . KSp(X) ->- KU(X) . Theorem: KU(CP ) n generator i s a truncated polynomial r i n g over u = n - 1 and one r e l a t i o n ]s ^~ = n+ i|£(y) = ( l + y ) - l Z w i t h one 0 . Moreover: k = 1, 2,... k Proof: CP c f . [1] For a l l k = 1, 2,..., there i s a unique polynomial T k e Z[X] such that: 1 V 1 T, (z + — - 2) = z + —, - 2 k z k 3.1.2 Theorem KU(HP ) n generator v = c f i s a truncated polynomial r i n g over £ - 2 and one r e l a t i o n ^ ( v ) = T (v) k v Z w i t h one = 0 . Moreover: k = 1, 2,, Proof; The structure of KU(HP ) i s w e l l known. n operations, the canonical map f : CP "^ -»- HP 211 For the Adams induces a n monomorphism: f * n ; KU(HP ) KU(CP ?n+1 ) given by f*(.c'?) = n + n . By [ l ] , t h . 5.1, Hence n = n 1 f*(v) = f*(c'£ - 2) = n. + - - 2 , and: * f k k * u£(v) = U£ k >'c f (v) = ./£ f - 2) = i|£(n + r f - 2) = ^ ( n ) + / ( r f ) - 2 1 1 n Hence by n a t u r a l i t y , k ^( ) v = -^k^) ' / We w i l l now compute L(HP ) n w i t h the help of the theorem §1.6. / and the ij^'s on L(HP ) n We proceed through a s e r i e s of lemmas. 3.1.3 Lemma The c o f i b r a t i o n HP n g •> HP n g -»• S sequences: ~,„4n. l ~, 2 ~, n-1. 0 + LCS"" ) L(HP ) S LCHP""") •* 0 g (1) 1 g 11 n induces the exact 29. # (2) 0 -> LUCS Proof: ) + # LU(HP ) V* LU(HP ). + 0 . This r e s u l t s from a study of the long exact sequences of the c o f i b r a t i o n i n KO- and KU-theory. For example, we show that g^ i s a monomorphism. One has the f o l l o w i n g part of exact sequence: + 1cocs 4 n + 1 ) 5 KO-.VP ) fir 11 on? - ) ^ K O ( S ) - 1 11 F i r s t one shows i n d u c t i v e l y that Indeed 1 KO 4 n (HP ) K 0 ( H P ) - KO" (S ) = 0 . Also _1 1 Suppose that 1 KO "''(HP ) ->• KO(S ) - Z 4n i s t o r s i o n or 0 , as wanted. is 0 or From t h a t , homomorphism, hence g^ : K 0 ( S ) morphism. The same considerations about 4n * ~ —4 g^ : KO 4 n + 5 n - 5 4n (S n ~* —4 ) -»- KO n» (HP ) 11 1 1 K0(HP ) ) + K 0 ( H P ) -KO-^HP - ) - K 0 ( S Z n 4 n + 4 i s also a monomorphism. 2 . : Hence K0~ (HP ~ ) 1 0 . or , so that i s the 0 ^K0(S ) =0 i s the 0 homomorphism. But Ker 8 = Im a epimorphism. 4 n + 1 0 . Then y i s t o r s i o n or 111 111 KO(S 4 KO ''"(HP ) i s t o r s i o n or 8 KO i s an (HP ) K0(S ) - Z 4n i s a mono- )+ show, that Hence g j : L ( S ) - L(HP ) 4 n i s a monomorphism. 3.1.4- n (We use the n a t u r a l isomorphism of §2.2). J Corollary L(HP ) n monomorphism. i s torsion-free and U : L(HP ) -> LU(HP ) n n isa 30. 1 Proof: 4 L(HP ) - LCS ) i s t o r s i o n f r e e ( c f . §2.3 lemma). Then use induction on -n w i t h the exact sequence (1) of lemma 3.1.3. The second a s s e r t i o n of the c o r o l l a r y i s due to §1.6. J Let us denote again the elements of LU(X) as p a i r s ( a , 3 ) a , g e KU(X) . Also define: l even 5. = \ l 3.1.5. i odd Lemma Im (U : 1!(HP ) +'LU(HP )) n i s the free abelian group 2 (2v,0) , (v ,0) generated by: (0,6 n n 2 (6 v ,0) , (0,v) , (0,2v ),, v ) . n n + 1 Proof: Lemma 3.1.3 and the n a t u r a l transformation U give the f o l l o w i n g commutative diagram w i t h exact rows: 0 - L(S ) I 4 n L(HP ) ! n 1 2 LOIP - ) - K g K g 0 + L U ( S ) + LU(HP ) + 11 # 4n Let a 1 n 1 3 LU(HP ) 0 # i l J 2 n 1 •+ 0 . be a generator of KU(S ) - Z . Since: 8 2 ^ , 0 ) = ( v \ o ) g^CO.v ) = ( O ^ ) 1 g ^ C O ) = (0,0) 1 i = 1, 2,...n-1 g^CO.v ) = (0,0) 11 one gets that: gj(a,0) = (v ,0) g*(0,a) = (0,v ) n We also have that (6^01,0) and (°» Im (5 i s the f r e e abelian group generated by i) • a n+ n ( * t ] ) ' Y definition, cf 1 B (0,v) = (0, '?-2) = l i a - e ) e Im U c £ (2v,0) = (0,2)-(0,v) = U ( e ) ' U ( ? - e ) E Im U and also . Hence 2 (2v,0), ( v , 0 ) , . . . , ( 6 v , 0 ) , (0,v), ( 0 , 2 v ) , . . . , ( 0 , 6 , v ) e Im U , 2 n 2 n n n+1 2 1 Now we can prove the lemma by induction on n . For HP c f . §2.3, lemma. Assuming the r e s u l t f o r HP n 1 4 ~ S , , look at the s p l i t exact sequence: 0 -> Im U -> Im U x 2 -> Im U We can define a s p l i t t i n g map H : Im H(6 v ,0) = (6 v ,0) and H ( 0 , 6 1 ^ 0 3 x i Im U" 2 v ) = (0,6 j V ) f o r 1 i+1 by 1 i = l , . . . , n - l . Then we see that Im U 2 - g (Im V ) ± 2 « H(Im Uj) i s as described i n the lemma Theorem: The r i n g w i t h the r e l a t i o n s L(HP ) n e 2 =4 i s generated by and 1, e , and n 11 T = 0 . Moreover: x = £ - e Proof: I m ( U : L ( H P ) •+ L U ( H P ) ) By 3.1.5, (0,2), (0,v) . n U ( T ) = (0,v) Since s t r u c t u r e of L(HP ) . Now use 3.1.2 i s generated by n and (1,0), U ( e ) = (0,2) we have the k n n to compute the ip 's LU(HP ) . on The obvious r e s u l t i s : (0, T (v)) k = 1, 3, 5... ( T ( v ) , 0) k = 2, 4, fc /(0,v) = \ k x (v ,0) Noting that U(e) » ^ i (0,v ) and X X TT and we use the f a c t that can be w r i t t e n r e s p e c t i v e l y as , . U(e) • V(T) U(e) • ——- , V(T) U commutes w i t h stated i n the theorem. j~ Examples 2 2 (T) = 3 ^ (x) = + 2ex T 3 T 2 + 3ex + 9x 2 Proof: T (x) = x + 4x 2 T (x) = x 3 3 + 6x 2 6... + 9x . the , . for if ' S J w e , l = 1, 2,...n , g e t the r e s u l t 33. §3.2. Computation f o r CP n We s t i l l denote by p a generator of 1cU(CP ) . Let n a generator of 'KUCS ) - Z . Let y = r ( y g ) e £o~ (CP ) , where 2 2 Let v 3.2.1. 4 n be and i s as i n §1.6. = $ ~ ( y ) e KSp(CP ) . n 2 Theorem Let by = r ( y ) e £b(CP ) Q r : KU(X) -> KO(X) n 1 2 p g n be even. ^(CP ) i s the free abelian group generated H -1 n -1 p , PQ,...,P2 , v , v p ,...,v p2 . The m u l t i p l i c a t i v e 11 Q 2 2 Q 2 2 s t r u c t u r e i s completed by the r e l a t i o n 2 = P Q • Moreover: / ( y ) = T (y ) Q k k = 1, 2, Q * C v ) = -I k 2 v 2 —y Proof; T.k(p0 ) v r Q 16] gives the structure of have the s t r u c t u r e of we remark that T C C P . ) 11 U : L(CP ) -> LU(CP ) n n Let L(CP ) n k = 1, 3,, y n KO(CP ) « KO~ (CP ) . Hence we n by §2.2. 4 In order to compute the i s torsion-free (for n i s a monomorphism. $ : KU(X) •> KU (X) n even). ip s , Therefore, We f i r s t determine t h i s map. be the complex Bott isomorphism as described i n £3J §2.2 f o r instance. k, The f o l l o w i n g diagram i s 34. commutative (compare {5J) t o " ( X ) $ KU~ (X) 4 4 KSp(X) where c, c' KU(X) as i n §1.6. Moreover c » r = 1 + (~) : KU(X) KU(X) (cf [6] f o r instance). With the help of these remarks, we f i n d out that: U(u ) = (c o r (u), 0) = (y+y, 0) Q U ( v ) = (0, $ 2 = (0, $ 2 2 c ' $ (v )) x 2 c'(y )) 2 = (0, $ " ( g y + g y ) ) 2 2 2 since = (0, * ~ ( g ( y + y ) ) ) 2 2 g 2 = g 2 = (0, y+y)" By theorem 3.1.1 we can compute if : LU(CP ) -> LU(CP ) n k / ( y , 0 ) = ( ( l + y ) - l , 0) k (0, ( l + y ) - l ) K /(0,y) = 4 { ((1+y) -1, 0) n lc 1j 2y • lc X^3^ • • lc 2 ^ 4,. (cf [6]) 35. From t h i s , we get: / ( y + y , 0) = CT (y+y), 0) k = 1, 2, k (0, T (y+y)) k = 1, 3, ( T ( + i i ) , 0) k = 2, 4,, k / ( 0 , y+y) = \ k M ( D e t a i l s as i n proof of 3.1.2). ( ( y + y ) , 0) = U ( y ) 1 F i n a l l y , since 1 0 (0, (y+7) ) = U(y ) 1 for i _ 1 0 U(v ) 2 i = 1, 2,...— , we get the r e s u l t as stated i n the theorem, 3.2.2. j Theorem (i) For n = 4t + 1 , "L(C7 ) i s the d i r e c t sum of the free U abelian group generated by 2 2t y , y ,...,y ,v Q Q 0 2 > 2t-l v y ,...,y y 2 Q 2 Q 2t+l y^ . The 2 2 m u l t i p l i c a t i v e structure i s completed by the r e l a t i o n 2 ^0 " and the c y c l i c group of order two generated by V (ii) = For n = 4t + 3 , L(CP ) i s the d i r e c t sum of the free abelian group generated by 2 2t+l y , y ,...,y , Q Q 0 v^, ^ Q''''> 2 0 V v y ' ^ e 2t a n d 2 2t+l the c y c l i c group of order two generated by 2 2 s t r u c t u r e i s completed by = y^ . multiplicative 36. Moreover k = 1, 2, * CP ) = T ( y ) 0 k Q k = 2, 4, /(v ) 2 k = 1, 3, ~0 W where the s u i t a b l e c o e f f i c i e n t has to be taken modulo 2 i n each case. Proof; [6] and our §2.2 give again the structure of lT(CP ) . n , < . For the i|> 's , the n a t u r a l i n c l u s i o n k CP ,n+l CP . induces an epimorphism: iT(cp ) n+1 n + 1 i s even. iT(cp ) n Hence, using the r e s u l t j u s t found f o r L(CP "'") , i t < n+ i s c l e a r out to get the r e s u l t f o r L(CP ) , by n a t u r a l i t y of the \p s I 3.2.3. Remark The r e s u l t f o r the truncated complex p r o j e c t i v e spaces are £ now also obvious. Looking at the c o f i b r a t i o n one observes that 77 induces an embedding; 77 * : ~-„„n+£. IACP / C „. P £ ) CP ~, ii+£. •> L C C P ) -> CP n+£ 77 n+£ -> CP /' £ , -5 £ /v/ —1 £ KO (CP ) = KO (CP ) = 0 . I t i s then c l e a r how to get the n+£ r e s u l t s for CP / £. M since Bibliography J . F. Adams, Vector f i e l d s on Spheres, Annals of Math., V o l . 75 (1962), pp. 603-632. J . F. Adams, Lectures on L i e Groups, W. A. Benjamin, i n c . , New York (1969). M. A t i y a h , K-Theory, W. A. Benjamin, i n c . , New York (1967). M. A. A t i y a h and D. 0. T a l l , Group Representations, the X-Rings and J-Homomorphism, Topology, V o l . 8 (1969) pp. 253-297. R. B o t t , Quelques remar ques sur l e s theoremes de p£riodicite, B u l l . Soc. Math. France V o l 87 (1959), pp. 293-310. M. F u j i i , K^-Groups of P r o j e c t i v e Spaces, Osaka J . Math., V o l . 4 (1967), pp. 141-149. D. Husemoller, Fiber Bundles, McGraw-Hill, New York (1966). F. S i g r i s t and U. Suter, Eine Anwendung der K-Theorie i n der Theorie der H-Raume, Commentarii Math. H e l v e t i c i , V o l . 47 (1972), pp. 36-52.
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Adams operations on KO(X) ⊕ KSp (X) Allard, Jacques 1973
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Title | Adams operations on KO(X) ⊕ KSp (X) |
Creator |
Allard, Jacques |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | Let KO(X) be the real and KSP(X) be the quaternionic K-theory of a finite CW-complex X . The tensor product and the exterior powers of vector bundles induce on L(X) = KO(X) ⊕ KSP(X) the structure of Z₂ -graded λ-ring. In this thesis it is shown, that the Adams operations Ѱk : L(X) → L(X) , k = l, 2, 3,..., which are associated to this λ-ring, are ring homomorphisms and satisfy the composition law Ѱk ₀ Ѱℓ = Ѱℓk = Ѱℓ ₀ Ѱk , k, ℓ = 1, 2, 3,... Finally, the ring L(X) together with its Ѱ-operations is explicitely determined for the quaternionic and complex projective spaces. |
Subject |
K-theory. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080120 |
URI | http://hdl.handle.net/2429/32701 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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