On a theorem of M. Fujii

Pub . Ma . UAB Vol . 29 Ns 2-3 No . 1985 INTRODUCTION : ON A THEOREMOF M . FUJII M .O . Aje unmobi In 1967 M . Fujii [21 compu ed he KO -i - ings o he complex p ojec i e spaces . We gi e a modi iedp oo he e using some esul s by S .G . Hogga [31 . Ou me hod seems di ec and easie o handle and i has been applied o compu e he KO -i - g oups o he complex lag mani olds o leng hs 2 and 3 [11 . The esul we ep o ed is Theo em 2 o Fujii [21 . Theo em [2, p . 1421 The KO -i - g oups o IP n-l (Q)  a e as ollow i  n - 2 (mod 4)  n - 0 (mod 4)  n odd 0  (2 +1)  a+ ZZ 2 (2 ) a  ( +1) zz n = 4 + 2  n = 4  n = 2 + 1 1 a 2 a 2 2  ( +1) a  + ZL 2 ( +1) zz + = [ n211 3  0  zz 2  0 4  ( +1) zz  ( +1) Z ; + a 2  ( +1) a zz 2 ( ) a+2 ;2 6  ( +1) a  ( +1) ZZ  ( ) 2Z Cohomology o Ip n-1 (0) Le XR be he eali ied bundle o he canonical bundle, X, o e IP n-1 (Q) . Then he second S ie el-Whi ney class w 2 (x R )  is hemod 2 educ ion o  C 1 (X) .  Pu  x = w 2 (x R ), hen an addi i e basis o H*(IP n-l (¢) ; T .L 2 )  is gi enby xl subjec o he condi ion xn = o . The Poinca é polynomial o IP n-l (0)  is gi en by (1)  P(IP n-1 (0)  . ) = 1 + 2 + 4 + . . . . . + 2(n-1) KO -i (IP n-1 (0» . F om (1), i is clea ha he 2k h Be i numbe , 0 2k =1  o 0 < k < n-1,  hus he anks o KO*(IPn-1 (0)) a e de e mined as ollowsusinglemma (2 .4) o [31 : o all alues o n . [ nn-11 2 ank KO o = ank KO-4 =  E R 4k = [ n1 1 + 1 k=o [ nn-11 2 Also, ank KO-2 = ank KO-6 =E 0 4k+2 = in211 + 1 k=o o n e en and [ nn-21 2 ank  KO -2 = ank  KO-6 =  E  R 4k+2 = [ n 2 2 1  + 1 = n21 k=o o n odd and his comple es he ee pa . Fo he o sion pa , conside he A iyah-Hi zeb uch spec alsequence which con e ges o KOp+q(IP,-1(¢)), see [2] . Conside he sequence o di e en ials (2) E p-2, q +1 2 Fo  q=0,4(mod 8),  E2 'q  gi es he ee pa c  KOp + q which is de e mined . Fo he o sion pa , we need only conside q= -l, -2(mod 8) . Fo  q = -1 - (mod 8) (2) becomes E p-2,8  d 2  Ep,8 -1 d2 Ep+2,8 -2 . 2  2  2 The map EZ'q - EP+2,q-1 is ze o o p = 0,4 (mod 8) (2a)  and is an isomo phism o  p = 2,6(mod 8)  i  E2 'q 70 . Thus (2b)2(n-2) = 0,4(mod 8) and o n odd, 2(n-2) - 2,6(mod 8), hus he di e en ial is an isomo phisms o n oddand ze o o n e en using (2a) and (2b) . Hence d 2 Eo,-1 - Eo,-1 3  2 and Ep ,-1 = 0 o he wise . Ep'q 2 d 2 (3)  E 2(n-1),-1 - { 0  n odd 3  2z 2 ;  n e en E p + 2,q -1 . 2 zz 2 o -all n . Fo n e en, 2(n-2),8 ) E 2(n-2),8  d 2  E 2(n-1),-1 2  2 -- ~ oz  g ~ -2 ~ - x~~ 8~)" (2"be : c : ¿ lé S' " (2 .4) in [3] w¿ - ' no o EE o(moa «)' we na e `-~^----- ' .' P' z~  ~u~'-' -  -  u~ _~ z, _  z  -'- ---  ---~- ' z ~ . . .~"^ =~~* `/1 o,-2 ~aud' . E ~ ` i 17 '12 1  `  un  . Also o o EE :' 2(mod 4), we ua e zz 2 1  U! ? X1  P, s z,1(mou  n),  p 0  o o ; o he w¡ae . mow,  o  i ~ ` ~(mnd 4) . ,  2 {o - l .> .~V  (m9d 8)' o  o5 :1  ](mod  «) '  u(u-1)  ~  4(mod  8)' o  u ~ - 2 (mod , 4J .,  .  .~ z(u-a ~ 2< md x>,  aud ' ~ o u ~ 0(muu 4), u(o-l) ~ 0(mod o) . Tbo9, ~ : iog (]) ni h (4) auu uaiog zenuna (z .1) in 131, . - .^ ^ we ba e` o uouu, . no-] = oou - Daz xo-4 ~ o -5  -§ ~o  ~  z z  - pac  mo  ~ o  auu no -7 ~ uoz - naz m" o ~ o oo -5 ~ oz o- naz mz-a ~ u aod nn -7 ~ oz z- pa  mx o ~ o KO -3 = 2Z 2 - pa KO -4 = 0 and KO-5 = zz2 - pa KO -6 = 0 . Now, we show ha .he  E 3 - e mssu i e o  E ,  o q = -1(mod 8) . . Le  E 0,-1 = a 2Z  ,  E2(n-1),-1 =  ~ zz ( n 3  2 3  2 Conside he di e en ial E 0,-1 E 0,-1 2 d 2 Thus E0' -1 = a a2 . e en) d  E '- , E '- = 0 excep - 0,2,4(mod 8) and d = 0 o = 0, 4(mod 8)  because i maps a ini e g oup o a ee g oup . Thus, we a e le wi h he case "= 2(mod 8) .  In his case, we claim ha d = 0 o - 2(mod 8) . P oo o claim : I su ices o show ha d 10 = 0 . F om he ze o di e en ial E2' -2 , we see ha E3' -1 = d (x s ) = sx s-1 d (x), inishing he claim . Also o n e Rn, we conside he di e en ial gene a ed by x o and since d 10 is a de i a ionwe ha e dio(x0) = 0 om he o mula is E 2(n-1)- , -2 d - ,  E 2(n-1), -1  E 2(n-1)- , -2 = 0   excep _ 0,2,6 (mod 8) using he p ope y o KO * (*) and d =  0 .  o  =  0,6  (mod  8),  see  [31 .  When  =  2 (mod  8 ) E2(n-1)- , -2 i s a ee g oup which su i es o E . . Thus d = 0 o all % 3 . We conside he il a ions whe e Re e ences KO 1 = Fo ' -1 D F 1 ' -2 D . . . . DF n-1 ' -n DF n,-n-1 =0 and  KO2n-3 = F 0, 2n-3 DF 1, 2n-4 D . . . D  n-1, n-2 Do a  0,-1  F O ' -1  0  2(n - 1), -1 _F 2(n-l) ' -1 ~G 2 = E m _  / F 1,-2,  ZL 2 = E~  / F2n-1,-2 Fp'q = Ke (KOP +q (X) X= IP n-1 (V) 1 Thus KO 1 = a 2 o all n . Also KO2n-3 = KO-3 o n = 0(mod 4) and KO2n-3 = KO-7 o n _ 2(mod 4) inishing he p oo o he heo em . KOP l q(Xp -l )) and E .  = 0 o ei he p o q odd . (1) AJETUNMOBI, M .O . : Ph .D . hesis Uni e si y o Ibadan, Nige ia . 1984 . (2) FUJII, - Michikazu : KO-g oups o p ojec i e spaces, OsakaJou n . Ma h . 4 (1967) pp . 141-149 . (3) HOGGAR, S .G . : On KO- heo y o G assmannian, Qua . Jou n . Ma h ., Ox o d Se . (2) 20(1969) pp . 447 - 463 . Rebu el 15 de 6ebnen del 1985 Ma hema ics Deoa men , Uni e si y o Ibadan Ibadan NIGERIA