On the Primitive Classes of K * KHALED S. FELALI Department of Mathematical Sciences, Umm Al-Qura University, Makkah Al-Mukarramah, Saudi Arabia

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1 JKAU: Sc., O vol. the Prmtve, pp Classes (49 of A.H. K (BU) / 999 A.D.) * 55 O the Prmtve Classes of K * (BU) KHALED S. FELALI Departmet of Mathematcal Sceces, Umm Al-Qura Uversty, Makkah Al-Mukarramah, Saud Araba ABSTRACT. We gve a characterzato of the prmtve classes of K * (BU()) terms of ts ratoal geerators ad use that to determe ew prmtve classes. Itroducto The BU-rg spectrum K determes a geeralzed homology theory K * wth coeffcet group K (pt) = [S, K] = π (K) It s well kow that K * (K) ca be regarded as a Hopf algebra over π * (K), ad for each X (space or spectrum) we ca defe a coacto map ψ : K * (X) K * (K) π* (K) K * (X) whch gves K * (K) the structure of a comodule over K * (K). The prmtve submodule PK * (X) s defed by PK * (X) = { α K * (X): ψ (α) = α } I [] we studed the case X = MU(2) ad gave a characterstc theorem whch determes the prmtve classes of K * (BU)(2)) terms of ts ratoal geerators. I [2] the same case s studed (amog other thgs) where K * (BU(2)) s detfed wth a certa submobule of Q[X, Y] whose homogeeous elemets determe the prmtve classes of K * (BU(2)). I [3] the authors geeralzed the case of [2] to the space BU[] where they detfy K * (BU()) wth a certa submodule of Q[x, x 2,... x ] whose homogeeous elemets aga determe the prmtve classes of K * (BU()). Although the above paper ca be regarded as the leadg ad most comprehesve study cocer BU(), but the prmtve classes whch were costructed there are all derved tally from BU(2). 55

2 56 Khaled S. Felal Here we geeralze the results of [] ad gve a characterzato of the prmtve classes of K * (BU()) terms of ts ratoal geerators ad use that to determe ew prmtve classes of K * (BU()) whch are ot derved from K * (BU(2)).. Notatos Let { β, β 2,..., β,... } be the usual π * (K)-bass of K * (CP ). For each postve teger we defe Γ = u (β ) b = u β (.) where u π 2 (K) s the usual geerator ad the product (β ) s duced by the tesor product : CP CP CP. Now usg the result of [4], [5] oe ca prove (see [] for detals) that were S r s the sterlg umber of the secod kd. Γ = Σ r! S r b r (.2) r = Let : CP BU be the caocal cluso ad deote the mages of β, Γ uder * also by β, Γ respectvely. The later classes of course ca be multpled K * (BU) by usg the Whtey sum maps: BU(m) BU() BU(m + ). The followg s a well kow (see[6; p.47] or [7; 6.3]). Theorem.3 () K * (BU()) s free over π * (K) wth a base cosstg of the moomals β β 2... β r such that > 0, 2 > 0,..., r 0, 0 r (The moomal wth r = 0 s terpreted as ) () K * (MU()) s free over π * (K) wth a base cosstg of the moomals such that > 0, 2 > 0,... > 0. β β 2... β r () K * (BU) s the polyomal algebra π * (K) [β, β 2,..., β,... ]. Remark.4 If we replace the β s by the b s the all the statemets gve the above theorem rema true. 2. Prmtvty K * (BU) I ths secto we shall see that the classes Γ, Γ 2,... play a mportat role the determato of the prmtve clases of K * (BU). Cosder the Hurewcz homomorphsm hk s : π * s (BU) K * (BU). It s easy to see that Γ s the mage of hk s. But ths s a atural homomorphsm of Potrjag rgs, hece

3 O the Prmtve Classes of K * (BU) 57 t follows that all the classes Γ, Γ 2,... are the mage of hk s ad hece Im h s _ K Z [Γ, Γ 2,..., Γ,... ] 2. Now sce K * (BU) s torso-free, the map: K * (BU) K * (BU) Q s a moomorphsm. By deotg the mage of Γ uder ths map also by Γ we have Im h s KQ _ Q [Γ, Γ 2,..., Γ,... ] where h s KQ : π * s (BU) Q K * (BU) Q = (KQ) * (BU). I fact by a smple applcato of the Atyah-Hrzebruch spectral sequece E 2 u, v = ~ H u (BU; π s v ) πs u+v (BU) oe ca prove by comparg the raks that (see [] for detals) Im h s KQ = Q [Γ, Γ 2,..., Γ,... ] (2.2) Now sce π * (K) s torso-free h s KQ maps πs * (BU) Q somorphcally oto P(KQ) * (BU). Hece we have proved the followg: Proposto 2.3 P(KQ) * (BU) Q [Γ, Γ 2,..., Γ,...] The followg result ca be proved exactly the same as (2.3) or alteratvely oe ca use the above result ad the stable equvalece BU = MU() of [8; Th..4.2] to = prove t. Proposto 2.4 P(KQ) * (MU()) s free over Q wth a base cosstg of all moomals Γ Γ 2... Γ such that > 0, 2 > 0,..., > 0. Note that such a moomal s ot dvsble K * (MU()), but more complcated expressos may be well be. I fact we have the followg (see [ ; 6.32]). Theorem 2.5 A elemet A K * (MU() s prmtve f ad oly f t ca be wrtte the form A = Σ λ Γ m, Γ m,2... Γ m, λ Q such that whe we rewrte t terms of the π * (K)-base { b b 2... b }, the duced formula has tegral coeffcets. Proof Sce K * (MU()) s torso-free we have a moomorphsm α: K * (MU()) K * (MU ()) Q (KQ) * (MU()). Now let

4 58 Khaled S. Felal A = Σ λ Γ m, Γ m,2... Γ m, λ Q The by the above theorem A s P(KQ) * (MU()). Now f A also satsfes the codto of the theorem the t s the mage of α ad so t represets a elemet of PK * (MU()) as requred. Coversely f A s PK * (MU()) the t s also P(KQ) * (MU()) where we detfy A wth ts mage uder the moomorphsm α. Hece by (2.4) we ca wrte A the form A = Σ λ Γ m, Γ m,2... Γ m, λ Q Note that the codto of the theorem s satsfed sce A essetally s K * (MU()). Theorem 2.6 A elemet A K * (MU()) s prmtve f ad oly f t ca be wrtte the form A = Σ λ Γ m, Γ m,2... Γ m, λ Q such that λ, λ 2,..., λ k are ratoal umbers satsfy the followg codto! 2!... r! m, m2, m, λ kϕ () kϕ ( 2)... k ϕ( ) ϕ s a teger for all -tuples (k, k 2,... k ) of postve tegers cotas r- dstct elemets repeated, 2,... r tmes, respectvely, where ϕ rus over all the permutatos of (, 2,... ). Proof Suppose that A s a prmtve class of K * (MU()). The by the above theorem we ca wrte t the form A = Σ λ Γ m, Γ m,2... Γ m, λ Q such that whe we wrte t terms of the π * (K)-base {b b 2... b }, the duced formula has tegral coeffcets. Now by (.2) we have rj, A = λ ( r, j)! Sm b j r., j, j= r, j= Let a (k, k 2,..., k ) be the coeffcet of b k b k2... b k. The we have

5 O the Prmtve Classes of K * (BU) 59 ak (, k2,..., k) = k ( j) ( k j! Sm ) ( ).!!... j r! λ ϕ ϕ, 2 ϕ j= To complete the log ad tedous proof of the theorem oe use the ducto o k = k + k k together wth the formula (see [9; p. 226]). r rs r t! = ( ) ( r t) t t= 0 See [; 6.33] for the proof of the two dmesoal case. Remark 2.7. As we metoed before the prmtve classes of K * (BU() are represeted [3] by ratoal polyomals satsfy certa codtos. The prmtve class A the above theorem s correspodg to the ratoal polyomal f (x,..., x ) defed by m, m, 2 m, f( x,..., x ) = λ x ( 2) x ( 2)... x ( )! ϕ ϕ ϕ ϕ 3. Some Prmtve Elemets K * (BU) Here we shall use theorem (2.6) to determe the prmtve classes of K * (MU()) of the form λ(γ m Γm+s G m+s ), where λ Q. By the above theorem such a class s prmtve f ad oly f the followg expresso λ = ( )! ( kk... k ) m 2 k s kk ( 2... k ) s! 2!... r! = s a teger for each -tuple (k, k 2,... k ) of postve tegers cotag r dstct elemets repeated, 2,..., r tmes, respectvely. Notatos 3.. Let ( ) X( k, k,..., k) = k s ( k k... k) s, 2 2 = ( ) Xk = X( k,,..., ) It s easy to show that Proposto 3.2

6 60 Khaled S. Felal s ( ) X( k, k ) X ( k k... k ) ( k ),..., k k s 2 = 2 = 2 [ ] s 2 s( 2) s( 3) s ( ( ) Xk = ( k ) ) ( k ) + 2( k ) ( 2) ( k ) + 2 Next we recall the defto of a umercal fucto m(t) defed o the postve tegers. Let v p (k) be the expoet of the prme p k, so that k = Π p p v p (k). Defto 3.3. [0] If t s a postve teger, we defe m(t) by v2( m( t)) = 2 + t odd v2( t) t eve for p odd vp( m( t)) = p ot dvde + vp( t) p dvdes t t Let M (t) be the hghest commo factor of the expressos k (k t ) where k rus over the postve tegers. Oe ca prove the followg Proposto 3.4. [0; p. 43] For each prme p we have, v p (M (t) = M {, v p (m(t)) } I partcular whe s large eough we have M(t) = m(t). Returg to our case we wat to fd the dvsblty the expresso From 3.2 () t s easy to show that whe s a odd prme we ca choose s (may take s = ) such k m X k s a multplcato of M m (, s) where we defe v p (M m (, s)) = M { m, v p (.m 2 (s)) } (3.5) Note that whe m s bg eough we have M m (, s) =.m 2 (s). Now s a odd prme. Hece ( )! / 2!... r! s a teger. Therefore t follows from the above remark together wth fomula () of (3.2) that the expresso (*) s dvsble by M m (,s). We have proved the followg: Proposto 3.6. ( )!! 2!... r! ( kk 2... k) m X( k, k,..., k ) (*) 2 Let p be a odd prme. For each postve teger m there s a prmtve class K * (MU(p)) of the form

7 O the Prmtve Classes of K * (BU) 6 Γm p p Γm+ p( p ) Γm+ p Mm( p, p ) Fally we wat to meto that the prmtve classes costructed here are oly examples of the use of theorem (2.6) ad we ca form more of them usg the same method. Also all of these classes ca be costructed the laguage of [3] usg the same method (see remark (2.7)), but here the prmtve classes are more recogzable. Refereces [] Felal, K.S., Itersecto pots of mmersed mafolds, Thess, Uv. of Machester, 982. [2] Ray, N. ad Schwartz, L., Costructo d élémets das π s *(BU(2)), Bull. Soc. Math. Frace (983), [3] Baker, A., Clarke, F., Ray, N. ad Schwartz, L., O the Kummer cogrueces ad the stable homotopy of BU, Tras. Amerca Math. Sco., 36 (989), o. 2, [4] Clarke, F., Self-map of BU, Math. Proc. Camb. Phl. Soc., 89 (98), [5] Schwartz, I., Opératos d Adams e K-homologe et applcatos, Bull. Soc. Math. Frace, 09 (98) [6] Adams, J.F., Stable Homotopy ad Geeralzed Homology, Chcago Lectures Notes Math. (974). [7] Swtzer, R.M., Algebrc Topology-Homotopy ad Homology, Grudlehre Math. Vol. 22, Sprger- Verlag (975). [8] Sath, V., Algebrac Cobordsm ad K-theory, Memors A.M.S. vol. 2 (979). [9] Rorda, J., Combatoral Idettes, Wley, New York (968). [0] Adams, J.F., O the group J(X) II, Topology 3 (965), 37-7.

8 62 Khaled S. Felal K * (BU) W d'«remk W c '«u BH«wöO bu ÈdI«Â WFU,WO{Ud«ÂuKF«r WœuF«WOdF«WJKL*«W dj*«wj BU() U«d?HK K * (BU()) Wd?'«rEMK W c?'«ub?h«æhk?*«vk e?d W?IU??«U?«b«lO?L? Ê ô ±π ÂU? cm U?N??«œ b Æ = 2 UbM W7U)«WU(«Æ 3 UNO ÊuJ rem W c'«ubh«f œb Y««c w