76

Y Choi

P ( Q ? Q^xo : a , b>0 ) = P(Ofo|[l] * [-2"^*] : а,Ь > 0)

P ( QfQ^zo : a , b>0 ) = PiQfQ^W^l-l"^"]: a,b >0)

P ( Qr'xo : a>0 ) = P(ß2"(Ö2[l] * [-2]) : fl > 0)

In fact, if we use the Eilenberg-Moore spectral sequence with £2 = Cotor H*(ß25p,„(„),z/(2))(Z/(2), Z/(2)), the above results say that spectral sequence collapses from the E2-term. So we can choose the generator x,, j,, Zi such that a(Xi) = щ+и сгСуЛ = Vi+ъ ^fo) = Wi+i,

Proof We will prove this theorem by the induction on k, i.e., from H^{QlSpin {Ы + к)\Ъ/{2)) to H^{QlSpin{%n + A: + l);Z/(2)). Like the double loop case we will prove four cases when к = 0,1,2 and 3. The proofs of the remain 4 cases, when Ä: = 4,5,6 and 8, are almost same as above к = 0,1,2 and 3 cases. Consider the morphism of fibrations

n'^Spin / SpiniSn - ^k ) -^ O^SpiniSn-{-k) —> Q^Spin

i i II

O'^Spin / SpiniSn + k + l) —> f2lSpin(Sn + к •¥ I) —> Ü^Spin

By the connectivity of H^{f2^Spin/Spin{%n +/: + !)) we have the non-trivial differential from ьы-ъ-^к to a (8w -4+/:)-dimensional element, we call it c%n-A^k^ in H^(Ü"^Spin/Spin(Sn +^);Z/(2)) for the Serre spectral sequence of the first column fibration. Here we exclude the case from Spin3 to SpinA. In that case the result comes from the fact Spin4 ^ Spin3 x Spin3. Since there is no (8n — 3 + k) dimensional generator in H^(f2^Spin) for /: = 0,1,2 f*icsn-4+k) фО , к =0,1,2. So by the naturality of the differential there is nonzero differential from tsn+k-з to a (Sn+k — 4) dimensional primitive element in H^{QlSpin{%n-¥k)\ Z/(2)) for ^ = 0,1,2 for the following fibration

QlSpiniU +k) ^ Ü^SpiniSn + l-¥k) ^^ Q^S^'^^K (Case 1) к =0. We have the nonzero differential from isn-з to a (8/t — 4) sional primitive element in H^{QlSpin{%n)\Z/{2)). But we have two possible elements X8„_4, z%n-A in H^{QlSpin{M)\Z/{2)). By the same method as Case 2 in the proof of Theorem 3.1, we should choose z^n-A- Since Я*(i?^5^") = P(öf t8„-3 : a > 0) 0P(ßf Ö2^i^8«-3 -а^ЬУ 0),

т ( боЧ^8« - з ) ) = ßf(Z8«-4),a>0

r ( Ôfa8n - 3 ) ) = ßf(Z8«-4),a>0. ^ • ^

For next we will prove that оз(^8п-4) = О- Assume that it is not zero. Since Ô3Z8n-4 is primitive, by the dimension reason the only possible case is that Ô3(z8n-4) = öij8n-3. By the Nishida relation,