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Adams operations on KO(X) ⊕ KSp (X) Allard, Jacques 1973

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ADAMS OPERATIONS ON KO(X) © KSp(X) by JACQUES ALLARD B.Sc, Universite 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1973 In presenting 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 requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying 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 gain s h a l l not be allowed without my w r i t t e n permission. Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada l i . Supervisor: U. Suter ABSTRACT Let KO(X) be the r e a l and KSP(X) be the quaternionic K-theory of a f i n i t e CW-complex X . The tensor product and the exterior powers of vector bundles induce on L(X) = KO(X) ® KSP(X) the structure of a -graded A-ring. In t h i s thesis i t i s shown, that the Adams operations ipk : L(X) + L(X) , k = l, 2, 3,..., which are associated to t h i s A-ring, are rin g homomorphisms and s a t i s f y the composition law ijjk o = \p^k = ijj^ o ipk t k,£ = 1, 2, 3,... F i n a l l y , the ring L(X) together with i t s ^-operations i s e x p l i c i t e l y determined for the quaternionic and complex projective" spaces. i i i . Acknowledgement The author i s pleased to acknowledge the very patient help and guidance of Dr. U l r i c h Suter during the preparation of t h i s thesis. He i s also thankful to Dr. Kee Lam for kindly reading the manuscript. i v . Table of Contents Page Introduction v. Chapter 1 §1.1. Definitions and preliminaries 1 §1.2. Main theorem 3 §1.3. A-semi-rings 4 §1.4. A-operations on V(X) 9 §1.5. Proof of the main theorem 16 §1.6. A natural transformation 16 §1.7. The reduced theory 17 Chapter 2. §2.1. Preliminaries 19 §2.2. A ring isomorphism 21 k k §2.3. Relation between <JJ and ip 24 Chapter 3 §3.1. Computation for HP n 27 §3.2. Computation for CP n 32 Bibliography 38 V. Introduction Let X be a f i n i t e connected CW-complex and l e t KO(X) , KU(X) and KSp(X) be respectively the r e a l , complex and quaternionic K-theory of X . This thesis i s concerned with the functor defined by L(X) = KO(X) © KSp(X) . Real and quaternionic vector bundles can. be viewed as complex vector bundles provided with some structure maps. This permits the d e f i n i t i o n of tensor product and exterior powers i n the category of re a l and quaternionic vector bundles, and, therefore, induces a structure of (Z2~graded) A-ring on L(X) . (A A-ring R i s a commutative ring with unit together with a set of maps A n : R -> R , n = 0, 1,..., having the formal properties of exterior powers). To any A-ring R , one can associate group homomorphisms k n tp : R -> R , k = 1, 2,..., given by universal polynomials i n A Our main theorem states that the homomorphisms tp associated to the A-ring L(X) (i) are ring homomorphisms ( i i ) s a t i s f y the r e l a t i o n <pN^ = = i|/ipk" k = 2., 2,.. Atiyah 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 's have properties ( i ) , ( i i ) . A A-ring s a t i s f y i n g those conditions i s cal l e d special A-ring. We prove that L(X) i s a special A-ring. Our proof i s based on the theorem that the representation ring of a compact Lie 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 related to the Adams operations i n many ways. Let LU(X) = KU(X) « KU(X) be given the Z 2-graded A-ring structure induced by the A-ring structure of KU(X) (essentially, define (a,b) • (a',b') = (aa' + bb', ab' + a'b) , A n(a,0) = (A n(a),0) A n(0,b) = (0,A n(b)) for n odd , A n(0,b) = (A n(b),0) for n even). There i s a natural Z^-graded ring homomorphism U : L(X) -> LU(X) n k commuting with the A 's (and, hence, the ip ' s) . Moreover, when k L(X) i s torsion free, U 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 torsion free case, and was done i n [8]. In [8] also, one can k find an application of the properties of the 's on L(X). The Bott isomorphism KSp(X) = KO ^(X) induces an isomorphism L(X) = KO(X) © KO~^(X) (L( ) defined as usual). This isomorphism i s actually a ring isomorphism. By d e f i n i t i o n , KO~^(X) = KO(X/vS4) (XA.Y denotes the smashed product of X and Y). Hence the Adams operations ^ are defined on KO(X) and on KO (X) v i i . 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 diagrams: For k odd: KSp (X) ,2 ,k k -\p KSp(X). Bott /v -A » KO (X) R Bott * , -* KO "(X) For k even: Bott - 4 >K0 (X) KSp(X) ; KO(X) I*2 * 2 e Bott S S p (X) > KO (X) R k - «%» k Hence, i n general, the ip 's on L(X) determine the i/> 's R on KO(X) and KO~4(X) . Also, ip k : L(X) -> L(X) i s a ring homomorphism, whereas ij£ © i|£ : KO(X) © KO~4(X) -> KO(X) © KO~\x) K, K i s not. We have actually computed the ring L(X) and the operations 4) for the quaternionic and complex projective spaces. The l a t e r case as torsion for odd dimensions. The content of the chapters i s as follows: Chapter 1 gives the d e f i n i t i o n of the ring L(X) and the maps A n on L(X) . The main theorem i s proved there. The d e f i n i t i o n of the natural homomorphism U and of the reduced theory L( ) i s given with the i r main properties. v i i i . Chapter 2 has two main theorems. In the f i r s t the isomorphism L(X) - KO(X) <& KO ^(X) i s established. In the second, we compare the k k ip ' s on L (X) with Adams' 1 s . Chapter 3 contains the computation of L(X) and the ip 's for the quaternionic and complex projective spaces. 1. Chapter 1 §1.1. Definitions and preliminaries For A = R, C, or H , and for a f i n i t e CW-complex X , l e t ^ e c t ^ 00 denote the class of A-vector bundles over X , and Vect^ (X) the set of isomorphism classes of A-vector bundles over X . The direct (or Whitney) sum of vector bundles induces a structure of abelian semi-group on Vect^ (X) and the tensor product induces a structure of commutative semi-ring on Vect (X) and Vect (X) . Let R C us write: V(X) = Vect,, (X) « Vect (X) K rl and l e t K ^ 0 0 denote the Grothendieck group of Vect^ (X) and L(X) the Grothendieck group of V(X) • We w i l l also use the alternative notations: KO (X) = K^(X) KU(X) = K C(X) 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 can be viewed as a complex vector bundle over X together with a structure map, i . e . as a pair (v,T) , where v e (/ect (X) and T : v v i s an isomorphism of the vector bundle v with i t s complex conjugate v , such that T 2 = Id (resp, T 2 = -Id ) v r v (Note, T can be considered as a conjugate-linear map from v to i t s e l f ) . The de 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 pairs (v,T) and (v',T') are said isomorphic i f V- and v' are isomorphic complex vector bundles v i a an isomorphism a : v -> v' which commutes with the structure maps, i . e . a o T = T'• » a . This d e f i n i t i o n coincides with 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 VCX) = U[u,S] , {v,T]) y, v, e l/ect (X) , S 2 = Id , T 2 = -Id } where S , T are structure maps and I M , S] and {v,Tj denote the isomorphism classes of ( y , S ) and (v,T) respectively. We define a semi-ring structure on V(X) as follows. Let [ y , S ] , I M ' , S»] e Vect R C X ) fv,T] , [v',T'] e Vect R (X) 3. and define: CIy,sj,o)-(ly',s'j,o) » Qy 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 class of the O-dimensional r e a l or quaternionic vector bundle. It i s easy to check that the structure maps S 8 S* , S 8 T, I 0 T' have the appropriate squares, so that these de f i n i t i o n s make sense. We extend the d e f i n i t i o n of the product for arbitrary elements of L(X) i n the obvious way. Hence V(X) becomes commutative semi-ring and L(X) therefore carries the structure of a commutative ring with unit. Notice that L(X) i s Z2 _graded. Actually L(-) i s a contravariant functor from the category of f i n i t e CW-complexes and continuous maps to the category of Z2~graded commutative rings with unit. I f Y i s another f i n i t e CW-complex and f : X Y i s a continuous map, we denote by f* the induced map L(Y) -*• L(X) . I t i s the direct sum of the maps KO(Y) •> KO(X) and KSp(Y) -> KSp(X) induced by f . §1.2. Main theorem: The ri n g L(X) admits Adams operations, i . e . there are ring homomorphisms: ip k : L(X) •* L(X) k = 1,2,... such that (i) / o / = ^ = / o / k,£ = 1, 2, r r ( i i ) ^ (x) = x^ (mod p) p prime ( i i i ) for a continuous f : X -»- Y , k * * k o f = 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 following 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 R with unit 1 together with a set of maps A n : R R , n = 0,1,..., such that x, y e R (1) A°(x) = 1 (2) A 1(x) = x (3) An(x+y) = I A r(x) A n r ( y ) r=0 If the semi-ring i s a rin g we w i l l t a l k of a A-ring Let a....a and b,...b be indeterminates and l e t s. 1 q 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 P (s. .. .s ; a- .. .a ) be the coe f f i c i e n t of t n i n II. . (1+a.b.t) n v 1 n' 1 n y 1 , 3 I j P (s. ...s ) be the coeffic i e n t of t n i n II (1+a. ...a. t) m,n l mn i . <...<i I , I 1 m 1 m P and P are uniquely defined polynomials with coef f i c i e n t s i n Z n m,n n J r J they s p l i t i n : P = P + - P n n n P = P + - P m,n m,n m,n where P + (resp. P + ) i s the sum of the terms of P (resp. P ) n m,n n r m,n having po s i t i v e coefficients and P (resp. P ) i s the sum of the n v r m,n ' terms of P^ (resp. P^ n ) having negative c o e f f i c i e n t s , multiplied by - 1 . Hence P + , P , P + , P have only positive c o e f f i c i e n t s n n m,n m,n and can be interpreted i n any semi-ring with unit. A A-semi-ring R i s a special A-semi-ring i f the following i d e n t i t i e s hold V x , y e R : (4) P n (A 1(x)..,A n(x) ; A 1(y)...A n(y)) + A n(xy) = P n + (A 1(x)...A n(x); A 1(y)...A n(y)) 6. (5) P~ > n (A1(.x) . ... A^ C x ) ) + A mA n C x ) = P m n + C A 1 C x ) . . . A m n ( x ) ) . m,n (the f i r s t i d e n t i t y relates A n(xy) and the A 1(x) , A 1(y) i l, . . . , n ; the second relates A mA n(x) and the A 1(x) , i = l,...,mn) . For convenience we w i l l refer to these i d e n t i t i e s i n the following form: (4') F n(x,y) = G n(x,y) C5») f (x) = G (x) m,n m,n i . e . we write F n Cx,y) for the l e f t hand side of (4), etc.... Moreover, even though we w i l l consider many A-semi-rings, we w i l l not use different symbols for the maps A n and the i d e n t i t i e s ( l ) - ( 5 ) , (4'), (5'). We hope t h i s w i l l not cause any d i f f i c u l t y . Let R and S be two A-semi-rings. A A-semi-ring  homomorphism g : R -> S i s a semi-ring homomorphism commuting with the maps A n , i . e . such that g ° A n = A n o g , n = 0 , 1,... The following two propositions motivate the introduction of the previous concepts. 1.3.1 Proposition Let R q be a (special) A-semi-ring. Then the Grothendieck g r o u p R of R i s a C s p e c i a l ) A - r i n g . Proof: I t i s well known that R i s a rin g . The maps A n : R.Q ->• R Q extend to R i n a unique way which s a t i s f i e s (3) (cf. [7], Chap. 12, 1.3). The extension also s a t i s f i e s (1) and (2), hence R i s a A-ring. I f R q i s spec i a l , R i s special, since (4) and (5) are compatible with (3) by lemma 1.5 of 14]. j For k = 1, 2,..., l e t be the unique polynomial with integer coe f f i c i e n t s such that: Q k ( s r . . s k ) = a x k +...+ a q k (with the notation formerly used and q >_ k ) . Then we have 1.3.2 Proposition Let R be a special A-ring and define / ( x ) = Q k(A 1(x),...,A k(x)) for x e R , k = 1, 2,... . Then \p are ring homomorphisms and 4>N^ = 4 ^ = = 1, 2,... r r <jj^  (x) = x^ (mod p) p prime, x £ R Proof: cf [4] prop. 5.1, 5.2. j Examples: (a) For A = R or C , Vect^(X) i s a special A-semi-ring where the A n are induced by exterior powers of vector bundles, i . e . : . A n[v] = ,IA n(v)] v e |/ectA (X) By 1.3.1 KO(X) and KU(X) are special A-rings, and the operations: ip k : KO(X) •* KO(X) tpk : KU(X) -»• KU(X) obtained v i a 1.3.2 are the c l a s s i c a l Adams operations. The proof of these facts are found i n [1] or {4]. (b) Let LU(X) = KU(X) « KU(X) and define a Z2~graded ring structure on LU(X) by (a,b) o (a*,b f) = (aa' + bb', ab'+ a*b) for (a,b) (a',b') e LU(X) . For k = 0, 1,.. set A k(a,0) = (A k(a),0) A^ K(0,b) = (A Z K(b),0) A 2 k + 1(0,b) = ( 0 , A 2 k + 1 ( b ) ) , and define , k A K(a,b) = I ArCa,0) A* r(0,b) r=0 9. LU(X) with these maps i s a special A-ring. This i s direct using the special A-ring properties of KU(X) . (c) Other examples can be found i n [4]. §1.4. A-operations on V(X) For n = 0, 1,..., we define: A n({y,S],0) = ( I A n ( j i ) , A n(S)],0) A 2 n(0,{v,Tj) = ( I A 2 n ( v ) , A 2 n(T)],0) A 211+1,,, r mn\ /n r-, 2n+l / v , 2n+l ,„,. ,.. A (0,{v,Tj) = (OJA (v), A (T)]) using the characterization of V(X) given by 1.1.1 . Checking the squares of the structure maps involved confirms the l e g i t i m i t y of t h i s d e f i n i t i o n . We extend i t to V(X) by: n A n([y,S], [v,T]) = I A r([y,S],0) A n _ r (0,[v,Tj) . r=0 Theorem V(X) together with the A-operations just defined, i s a special A-semi-ring. Proof: Checking the conditions (1), (2), (3) i s d i r e c t , using the properties of exterior powers and, for (.3), v e r i f y i n g a simple i d e n t i t y . 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 for (4) only since a similar treatment deals with (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 R CX) U Vect H (X) C V(X) where Vect CX) . and Vect (X) are i d e n t i f i e d with the i r image is. rl under thei r canonical embedding i n V(X) . In X7J, Chap, 5, §6, Husemoller defines the notion of continuous functor, and shows that a continuous functor: l/ect A(0) x.. .xl/ect A(0) -> l/ect A(0) where 0 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 A (0)x...x(/ e ct A (0) x (/ect A, (0) x... xVect^, (0) + l/ect^, (0) where A, A', A" take the values R and H . For example, we have the following continuous functors: « : Vect (0) x V e c t A (.0) •> V&ct^ (0) 11. 0 : l/ect A (0) x |/ectA,(0) - VectA„ (0) for (A, A', A") = (R, H, H) , (H, R, H), (H, H, R) or (R, R, R) . \ n : l/ect (0) -> V e c t , (0) for (A,A') = (R, R) or (H, H) and n odd , or (H, R) and n even. These functors respectively induce the direct sum of vector bundles, the tensor product of vector bundles and the exterior powers of vector bundles. The functors F and G are compositions of these n n operations on vector bundles; they are induced by continuous functors Fn,0 ' Gn,0 : " c c tA ( 0 ) X " e c tA' ( 0 ) * " e c tA» ( 0 ) where (A, A', A") = (R, R, R) , (R, H, H) , (H, R, H) or (H, H, H) for n odd, or (H, H, R) for n even. By [7], an equivalence of functors: u n : F _ -»- G A 0 n,0 n,Q induces an equivalence of functors: u : F G' n n and, passing to isomorphism classes of vector bundles, one gets: F = G : Vect (X) x Vect , (X) + Vect ,, (X) . n n A A A 12. However, we are not interested i n the fu n c t o r i a l properties of and G , and an equivalence of the functors F -. and G „ for n n 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 result 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). Let us find out such an equivalence. Since A-vector bundles over a point are es s e n t i a l l y A-vector spaces, we can choose bases, and consider, for each k, k' e N : Fn,0 ' Gn,0 : G z ( - h > ^ x G W , A f ) + G£(k",A") with (A, A', A") as above and k" e N depending on k, k', n, A, A' . k k' k k' (We don't write the more accurate F ' , G '„ for fear of heaviness) n,0 ' n,0 Now, i n order to complete the proof, we need to produce Me G£(k",A") such that; M F n(A,B) = G^  n(A,B) M n,U n,U for a l l (A,B) e G£(k,A) x G£(k',A') . The next lemma serves that purpose. Hence the proof of the theorem i s complete. J 1.4.1 Lemma There i s Me G£(k",A") such that: M F n > Q (A,B) = G n > Q(A,B) M for a l l (A,B) e Gil(k,A) x G£(k',A') 13. Proof; Si m i l a r l y as i n 1.1, r e a l and quaternionic representations of Lie groups can be viewed as complex representations together with some structure maps. This i s carried out i n [2], Chapter 3. The following i s a direct translation of th i s treatment i n term of matrix represen-tations. (It can also be obtained with the help of §1.1. for 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 i f E = 2 where T denotes the complex matrix with as c o e f f i c i e n t s , the coeff i c i e n t s of T conjugated. Define: ©(Em, C ) T = {A e Gl (Em,C) TA = AT} . Then we have the following isomorphisms: G ; U m,R) ~ Gl(m,<Z) i f SS = Id G£(m,H) ~ Gn(2m,C)s i f S'S' = -Id (The conjugate signs appearing here correspond to the conjugate l i n e a r i t y of the structure maps of {2]). Examples of S, S' are: 14. f 0 -1 0 0 . . . 01 1 0 0 0 . . . 0 S = Id S' = 0 V. 0 0 0 0 -1 0 0 1 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 follows: F n > 0 : GACEk,C)T x 6nE'k ,,C') T, G£(E"k",C)F Gn,0 : G £ ( E k ' C > T X G ^ E ' k ' . c ) T . GA(E"k",C)G ( 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 equality i s easy to check). The existence of a matrix M e G£(k",A") sa t i s f y i n g the lemma i s equivalent to the existence of a matrix N e Gl(E" k",C) such that: (i) N F n > 0(U,V) = G (U,V> N V (U,V) e 6*(Ek,C) T x G£(E'k',C)T, ( i i ) N F (T,T') = G(T, T') N . One can produce such a N i n the following way. 15. F i r s t , the representation ring R(U(Ek) x U(E'k')) i s a special A-ring, where U(n) i s the unitary group of dimension n (cf. [4] th. 1.5); hence the representations F , G ; UCEk) x'U(E'k') + G£(EMk",C) are equivalent. But two representations of G£(Ek, C) x G£(E'k', C) equivalent on U(Ek) x U(E'k') are equivalent. (cf. [1], proof 4.5). Therefore there exist P e G£(E"k", C) such that ( i i i ) P F n > Q(U,V) = G n > 0(U,V) P V (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 P by P . F i r s t start by writing ( i i i ) with U, V , and note that F n , G _ commute with the n,0 n,0 conjugation. Now l e t p e C be such that - p p ^  i s not an eigenvalue of P "*T , and define: N = p P + p"P = P(pp -"" 1 I + P - 1 P)p- . One sees that N e G£(E"k", C) and N s a t i s f i e s ( i i i ) , hence i t s a t i s f i e s ( i ) . But since N = N , i t s a t i s f i e s also ( i i ) . This proves the lemma. 1.4.2 Corollary to the theorem L(X) i s a special A-ring. Proof: Theorem §1.4 and proposition 1.3.1. / 16, §1.5. Proof of the main theorem By 1.4.2 and proposition 1.3.2, we obtain the ring homomor-phisms ip with th e i r f i r s t two properties. The l a s t one i s clear since f i s a A-ring homomorphism. §1.6. A natural transformation Recall the canonical homomorphisms c : KO(X) -> KU(X) c' : KSp(X) ->- KU(X) r : KU(X) -> KOCX) q : KU(X) KSp(X) . c and c' are induced by the maps feet (X) ->• feet (X) R C and fect„ (X) -»- feet (X) forgetting the structure maps (cf 1.1); r H. L i s induced by the map l/ect (X) feet (X) forgetting the complex structure; q i s induced by the map feet (X) ->- feet (X) defined by n ~+ n 8 e where e i s the t r i v i a l 1-dimensional quaternionic vector bundle over X . Define: U = c « c' : L(X) -> LU(X) U' = r # q : LU(X) -* L(X) . Proposition The map U : L(X) LU(X) i s a A-ring homomorphism. I f L(X) i s torsion free, U 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 / 17. on L(X) . For the second part we have that U' o U = 2 where 2 : LCX) ->• L(X) i s the map x ->• 2x x e L(X) . Cf [5] or [6] for instance. Hence, i f L(X) i s torsion free, U' ° U i s 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: by X we mean a pair (X, Xq) s.t. Xq e X . The inclusion i : x« -*• X induces an exact sequence: Let e be the 1-dimensional quaternionic vector bundle over Xq . L(xq) i s the free abelian group on two generators 1, e , and the 2 k product structure i s given by the r e l a t i o n e = 4 . The maps ip are defined by: 0 -> Ker i * L(X) i L(x_) -> 0 . Define L(X) = Ker i Then we have a natural s p l i t t i n g L(X) - L(X) e L(x n) . k = 1, 3 k = 2, 4 , . . . , . . . Moreover, 18. L(X) - KOCX) © KSp(X) k and the 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 X with a point which we take as canonical base point, we have that L(X) - L(X +) . Chapter 2 §2.1. Preliminaries Let S n denote the n-dimensional sphere and (X , Xq) and ( Y , V q ) be f i n i t e CW-complexes with base point. Define as usual: X A Y = X X Y / X X y Q xj X q x Y Consider the obvious projection maps: 4 X x s / I 4 4 X S X A S and denote by v the generator of KSp(S ) - Z . Define: : KSp(X) + K0 ( X*S 4) C l"(X x s 4) by $ 1 ( 3 ) = p*(g) • q (v) V g e KSp(X) KO(X) + KSp(X x S 4) C L(X x S 4) by $ (a) = p*(a) • q (v) V a e KO(X) . Since v i s a quaternionic bundle, p*(a) • q (v) and p*(3) • q*(\ are respectively quaternionic and re a l bundles as indicated, f* : K O C X A S 4 ) ^ KOCX x S 4) and f* : KSp(XAS 4) -> KSp(X x s 4) 20. are monomorphisms, and we have the Bott isomorphism theorem: (i) Consider f * KSp (X) * K0 ( X x s ) «- K 0 ( X A S ) then $^ i s an isomorphism onto Im f* ( i i ) Consider ~ $2 ~ L f * ~ L K0(X) -> KSp(X x S ) $• KSp(XAS ) then i - s a n isomorphism onto Im f* . Proof: cf [5]. J Next, we r e c a l l that one defines for n = 0, 1, 2,... K0 _ n(X) = K0(XA S n) and that K0(X) # KO ^(X) can be given a ring structure i n the following way. Let a , a ' e lcO(X) , 8, 3 ' e K0~^(X)=K0(X A sS , and denote the product i n KO(X) « K0~^(X) by x . F i r s t define a x a ' = o c a ' . Now l e t X x s - ^ X x X x s 21. be the maps defined by: A(x,s) = ( x , x,s) Pj^Cx.yjS) = x 4 P2.(x,y,s) = (y,s) for x , y e X , s e S Then y = A * [ P ; L * ( v ) • p£p*CB)l e Im(f* : K O ( X A S 4) KO(XH S4)-) and define a x 3 = (f )~ ( Y) e K*0(X A S ) . S i m i l a r l y , l e t 4 4 A ' 4 4 X x s x S > X x s x x x s A \, 2 4 4 *• 4 * 4 X A S A S X x s X x s be the obvious projections plus the map defined by A'(x,s,t) = (x,s,x,t) 4 for x -e X , s, t e S . Again Y ' = A'*[q* p* ( 3)-q* p * ( B ' ) ] e Im(g*:K0(XA S 4^ S 4) •+ K0(X x S 4 x S 4)) and define 3 x 3 ' = ($ $ 2 ) _ 1 ( g * ) - 1 ( y » ) e K0(X) . (These def i n i t i o n s are equivalent to those of [3] where the de t a i l s can be obtained). §2.2. A ring isomorphism From the Bott isomorphism theorem, we obtain a group isomorphism: B = Id 9 $ 1 : L(X) K0(X) 0 K0~ 4(X) 22. We have the following Theorem; B : L(X) -»• KO(X) 6 KO~4(X) i s a ring isomorphism. Proof: E s s e n t i a l l y , the proof consists of checking the commutativity of the two following diagrams: KO(X) 8 KSp(X) Id 8 $]_ » K O ( X ) 8 K O ( X A S ) KO(X) 8 KO(X x S ) (I) PI<->-P*C-) KO(X x X x s ) A KO(X x S ) f •RSp (X) 1 V K O C X A S4) KSp(X) 8 KSp(X) = ±-*KO(X S ) 0 KO(X AS ) f 8 f (ID KO(X x s 4) 0 KO(X x S 4) & it % (~)-q 0(-) KO(X x S 4 x X x S 4) KO(X x s 4 x s 4) KO(X)' ~> , 4 4 -» KO (X A S A S ) 1 2 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 right hand v e r t i c a l sequence i s the product i n KO(X) « KO~4(X) . Let us check (I) : V a e KO(X) , V g e KSp(X) , 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 ring homomorphisms on L(-) and that p 2 6 A = I d x x g4 and p][ 0 A = p . , 24. Simi l a r l y for (II) k k §2.3. Relation between and ip We need the following Lemma: 4 (i ) L(S ) = ev Z « v Z «~ 4 where v i s the generator of KSp(S ) - Z and e i s the t r i v i a l quaternionic 1-dimensional vector bundle; ( i i ) the ring homomorphisms ip are given by: / ( v ) = k 2v , 2 k ev k = 2, 4, 6,. . ~ , 4 Proof: Let p be generator of KO(S ) - Z , and l e t a be generator of ^ ( S 4 ) . By proposition §1.6, U : *L(S 4) -> LU(S 4) i s a monomor-phism. By [1] or [7], U(p) = (2a,0) and U(v) = (0,a) . (Elements of LU(X) - KU(X) © KU(X) w i l l be written as pairs (x,y)) . Since U(e) e LU(S 4) i s given by U(e) = (0,2) , one has that U(u) = U(e)-U(v) , hence p = ev . Using [1], corr. 5.2, one computes that: i> (0,a) = k (0,a) { k (a,0) k. "~ 1 j 3 ^  • • * lc —* 2 j A ^  • • • 25. Hence: k 2v k 2 2 E V k = 1, 3, k = 2, 4,, / k 4 Remark: the ip 's on L(S ) are now completely determined since we k know the value of ip (e) • Recall (1.3., example (a)) that we have the c l a s s i c a l Adams operations \£ : K O ( X ) -> K O ( X ) and \fc : K O ( X A S 4 ) K O ( X A S 4) . K R The r e s t r i c t i o n of i p k : L ( X ) -> L ( X ) to KO(X) i s ij£ . In the next R theorem, we compare the r e s t r i c t i o n of <pk : 'L(X) -»- T ( X ) to KSp (X) - K O ( X A S 4 ) , with ^ : K O ( X A S 4 ) -> K O ( X A S 4) . K Theorem The following diagrams are commutative. For k = 1, 3,.. *KSp(X) — KSp(X) -Bott ^ A —z > K O ( X A S ) 1 R Bott * K O ( X A S " ) For k = 2, 4,, KSp(X) • KO(X) Bott ^ A —z > K O ( X A S ) "R k 2 k 2 where -r- e denotes the map a -> -y e«a for a e KO(X) Proof: For k = 2, 4,... l£ ^ ( 3 ) = <l£(p*(6) • q*(v)) / p*(g) • / q*(v) p* / ( g ) • q* / ( v ) = p* / ( g ) • q*(| ev) = p ^ C3) -J e q (v> | 4 1 (e./(B)) k <y (*) because ip i s a ring homomorphism on L(-) and R = * KO(-) for a l l k = 1, 2, (**) because ip are natural (Main theorem) . Simi l a r l y for k odd. J Chapter 3 §3.1. Computation for HP n F i r s t we r e c a l l some results. Let r\ be the canonical complex l i n e bundle over CP n and l e t £ be the canonical quaternionic l i n e bundle over HP n . Also cf. §1.6. for C : KSp(X) ->- KU(X) . 3.1.1 Theorem: KU(CPn) i s a truncated polynomial ring over Z with one generator u = n - 1 and one r e l a t i o n ]sn+^~ = 0 . Moreover: i|£(y) = ( l + y ) k - l k = 1, 2,... Proof: 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(HPn) i s a truncated polynomial ring over Z with one generator v = c f £ - 2 and one r e l a t i o n v = 0 . Moreover: ^ ( v ) = T k(v) k = 1, 2,, Proof; The structure of KU(HPn) i s well known. For the Adams operations, the canonical map f : CP 2 1 1"^ -»- HP n induces a monomorphism: given by * n ?n+1 f ; KU(HP ) KU(CP ) f*(.c'?) = n + n . By [ l ] , th. 5.1, n = n 1 Hence f*(v) = f*(c'£ - 2) = n. + - - 2 , and: * k k * k >'c f u£(v) = U£ f (v) = ./£ f - 2) = i|£(n + r f 1 - 2) = ^ ( n ) + / ( r f 1 ) - 2 n k / Hence by n a t u r a l i t y , ^ ( v ) = -^k^) ' / We w i l l now compute L(HP n) and the ij^'s on L(HP n) with the help of the theorem §1.6. We proceed through a series of lemmas. 3.1.3 Lemma g g The cofibration HP n •> HP n -»• S n induces the exact sequences: (1) 0 + LCS""1) L(HP11) S LCHP""") •* 0 ~,„4n. g l ~, g2 ~, n-1. 29. # # (2) 0 -> LUCS ) + LU(HP ) V* LU(HP ) . + 0 . Proof: This results from a study of the long exact sequences of the cofibration i n KO- and KU-theory. For example, we show that g^ i s a monomorphism. One has the following part of exact sequence: + 1cocs 4 n + 1) 5 KO-.VP11) fir1 on? 1 1- 1) ^ KO(S 4 n) -F i r s t one shows inductively that KO (HP ) i s torsion or 0 . Indeed K0 _ 1(HP 1)- KO" 1(S 4) = 0 . Also KO(S 4 n + 1) =0 or Z 2 . Suppose that KO ''"(HP11 1) i s torsion or 0 . Then y : KO "''(HP11 1) ->• KO(S 4 n) - Z i s the 0 homomorphism. Hence 8 i s an epimorphism. But Ker 8 = Im a i s 0 or , so that KO (HP ) i s torsion or 0 , as wanted. From that, K0~ 1(HP n~ 1) K0(S 4 n) - Z i s the 0 homomorphism, hence g^ : K0(S 4 n) K0(HPn) i s a mono-morphism. The same considerations about ^ K 0 ( S 4 n + 5 ) +K0 - 5(HP n) -KO-^HP 1 1- 1) - K 0 ( S 4 n + 4 ) + show, that * ~ —4 4n ~* —4 n» g^ : KO (S ) -»- KO (HP ) i s also a monomorphism. Hence gj : L ( S 4 n ) - L(HP n) i s a monomorphism. (We use the natural isomorphism of §2.2). J 3.1 .4- Corollary L(HP n) i s torsion-free and U : L(HP n) -> LU(HPn) i s a monomorphism. 30. 1 4 Proof: L(HP ) - LCS ) i s torsion free (cf. §2.3 lemma). Then use induction on -n with the exact sequence (1) of lemma 3.1.3. The second assertion of the corollary i s due to §1.6. J Let us denote again the elements of LU(X) as pairs ( a , 3 ) a , g e KU(X) . Also define: 5. = \ l l even i odd 3.1.5. Lemma Im (U : 1!(HPn) +'LU(HPn)) i s the free abelian group 2 n 2 generated by: (2v,0) , (v ,0) (6 v ,0) , (0,v) , (0,2v ),, ( 0 , 6 n + 1 v n ) . Proof: Lemma 3.1.3 and the natural transformation U give the following commutative diagram with exact rows: 0 - L ( S 4 n ) I1 L(HP n) ! 2 LOIP 1 1- 1) - 0 K g # K g # i l J3 0 + LU(S 4 n) + 1 LU(HPn) + 2 LU(HPn 1) •+ 0 . Let a be a generator of KU(S ) - Z . Since: 8 2 ^ , 0 ) = (v\o) g^CO.v1) = (O^ 1) i = 1, 2,...n-1 g ^ C O ) = (0,0) g^CO.v11) = (0,0) one gets that: gj(a,0) = (v n,0) g*(0,a) = (0,v n) We also have that Im i s the free abelian group generated by (6^01,0) and (°» ( 5 n +i a) • ( c f* t 1 ] ) ' BY d e f i n i t i o n , (0,v) = (0, c'?-2) = l i a - e ) e Im U £ and also (2v,0) = (0,2)-(0,v) = U (e)'U (?-e) E Im U 2 . Hence (2v,0), (v 2,0),...,(6 nv n,0), (0,v), (0,2v 2),...,(0,6 n + 1,v n) e Im U 2 , 1 4 Now we can prove the lemma by induction on n . For HP ~ S , cf. §2.3, lemma. Assuming the result for HP n 1 , look at the s p l i t exact sequence: 0 -> Im U x -> Im U 2 -> Im U 3 ^ 0 We can define a s p l i t t i n g map H : Im Im U"2 by H(6 iv 1,0) = (6 v x,0) and H(0,6 i + 1v 1) = (0,6 j V 1 ) for i = l , . . . , n - l . Then we see that Im U 2 - g 2(Im V±) « H(Im Uj) i s as described i n the lemma Theorem: The ring L(HP n) i s generated by 1, e , and x = £ - e 2 n 11 with the relations e =4 and T = 0 . Moreover: Proof: By 3.1.5, Im ( U : L(HP n) •+ L U(HP n)) i s generated by (1,0), (0,2), (0,v) . Since U ( T ) = (0,v) and U(e) = (0,2) we have the structure of L(HP n) . k n Now use 3.1.2 to compute the ip 's on LU(HP ) . The obvious result i s : /(0,v) = \ (0, T f c(v)) k = 1, 3, 5... (T k(v), 0) k = 2, 4, 6... x i Noting that (v ,0) and (0,v ) can be written respectively as U(e) » V(T)X , T T, . U(e) • V(T)X , . , ^ and U(e) • — — - for l = 1, 2,...n , we use the fact that U commutes with the if ' SJ w e g e t the result stated i n the theorem. j~ Examples 2 2 ( T ) = T + 2ex 3 3 2 ^ (x) = T + 3ex + 9x 2 Proof: T 2(x) = x + 4x T 3(x) = x 3 + 6x 2 + 9x . 33. §3.2. Computation for CP n We s t i l l denote by p a generator of 1cU(CPn) . Let g be a generator of 'KUCS ) - Z . Let p Q = r(y) e £b(CPn) and y 2 = r(yg 2) e £o~4(CPn) , where r : KU(X) -> KO(X) i s as i n §1.6. Let v 2 = $~ 1(y 2) e KSp(CPn) . 3.2.1. Theorem Let n be even. ^(CP 1 1) i s the free abelian group generated H -1 n -1 by p Q, P Q , . . . , P 2 , v 2 , v 2p Q,...,v 2p2 . The m u l t i p l i c a t i v e 2 2 structure i s completed by the r e l a t i o n = P Q • Moreover: / ( y Q ) = T k(y Q) k = 1, 2, * kCv 2) = -I v2 — T. (p n) k = 1, 3,, y Q k v r 0 y Proof; 16] gives the structure of KO(CPn) « KO~ 4(CP n) . Hence we have the structure of L(CP n) by §2.2. In order to compute the ip k ,s , we remark that TCCP 1 1 . ) i s torsion-free (for n even). Therefore, U : L(CP n) -> LU(CP n) i s a monomorphism. We f i r s t determine t h i s map. Let $ : KU(X) •> KU (X) be the complex Bott isomorphism as described i n £3J §2.2 for instance. The following diagram i s 34. commutative (compare {5J) to" 4(X) $ KU~4(X) KSp(X) KU(X) where c, c' as i n §1.6. Moreover c » r = 1 + (~) : KU(X) KU(X) (cf [6] for instance). With the help of these remarks, we find out that: U(u Q) = (c o r (u), 0) = (y+y, 0) U(v 2) = (0, $ 2 c' $ x ( v 2 ) ) = (0, $ 2 c'(y 2)) = (0, $" 2(g 2y + g 2y)) = (0, *~ 2(g 2(y+y))) 2 2 since g = g (cf [6]) = (0, y+y)" By theorem 3.1.1 we can compute i f k : LU(CP n) -> LU(CP n) /(y,0) = ( ( l + y ) k - l , 0) /(0,y) = 4 (0, (l+y) K-l) { ((1+y) -1, 0) lc 1 j 2 y • lc X ^  3 ^  • • lc 2 ^  4,. 35. From t h i s , we get: /(y+y, 0) = CT k(y+y), 0) k = 1, 2, / ( 0 , y+y) = \ (0, T k(y+y)) k = 1, 3, (T k( M+ii), 0) k = 2, 4,, (Details as i n proof of 3.1.2). F i n a l l y , since ((y+y) 1, 0) = U ( y 0 ) 1 (0, (y+7) 1) = U ( y 0 ) i _ 1 U(v 2) for i = 1, 2,...— , we get the result as stated i n the theorem, j 3.2.2. Theorem (i) For n = 4t + 1 , "L(C7U) i s the direct sum of the free 2 2t 2 t - l abelian group generated by y Q, y Q,...,y 0 , v 2 > v 2y Q,...,y 2y Q 2t+l and the c y c l i c group of order two generated by y^ . The 2 2 mu 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 V2 = ^0 " ( i i ) For n = 4t + 3 , L(CP ) i s the direct sum of the free 2 2t+l 2t abelian group generated by y Q, y Q,...,y 0 , v^, ^2VQ''''>v2y0 a n d 2t+l the c y c l i c group of order two generated by ' ^ e m u l t i p l i c a t i v e 2 2 structure i s completed by = y^ . Moreover 36. * CP0) = T k(y Q) / ( v 2 ) -~0 W k = 1, 2, k = 2, 4, k = 1, 3, where the suitable coefficient has to be taken modulo 2 i n each case. Proof; [6] and our §2.2 give again the structure of lT(CPn) . , k < . ,n+l . For the i|> 's , the natural inclusion CP CP induces an epimorphism: iT(cp n + 1) - iT(cp n) n + 1 i s even. Hence, using the result just found for <L(CPn+"'") , i t i s clear out to get the result for L(CP ) , by na t u r a l i t y of the \p s I 3.2.3. Remark The result for the truncated complex projective spaces are £ n+£ 77 n+£ now also obvious. Looking at the cofibration CP -> CP -> CP /' £ , one observes that 77 induces an embedding; * ~-„„n+£. „. ~, ii+£. 77 : I A C P / C P £ ) •> L C C P ) M -5 £ /v/ —1 £ since KO (CP ) = KO (CP ) = 0 . It i s then clear how to get the n+£ results for CP / £ . Bibliography J. F. Adams, Vector f i e l d s on Spheres, Annals of Math., Vol. 75 (1962), pp. 603-632. J. F. Adams, Lectures on Lie Groups, W. A. Benjamin, i n c . , New York (1969). M. Atiyah, K-Theory, W. A. Benjamin, i n c . , New York (1967). M. A. Atiyah and D. 0. T a l l , Group Representations, X-Rings and  the J-Homomorphism, Topology, Vol. 8 (1969) pp. 253-297. R. Bott, Quelques remar ques sur l e s theoremes de p£riodicite, B u l l . Soc. Math. France Vol 87 (1959), pp. 293-310. M. F u j i i , K^-Groups of Projective Spaces, Osaka J . Math., Vol. 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 , Vol. 47 (1972), pp. 36-52. 

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