THE K-THEORY OF PROJECTIVE STIEFEL MANIFOLDS 3 where, in the right side of the first equation 1ri = 0 if i 2n-k and 2 2n-k+l 2~-k 2j 1T2n-k = 15 +2 15 - j~l 2 1T2n-k-j PROOF. First, it is possible to give an explicit description for the res- triction of j to the maximal toruses. Then, considering that the representa- tion ring of a group is embedded in the one for its torus, one can use this description to obtain the lemma. Now, we can follow A. Roux [9] and apply some results on homological algebra due to Cartan and Eilenberg [3], to obtain the next proposition. PROPOSITION 5: The Hodgkin spectral sequence for x 4n,2k_1• collapses, and: E ,q = E • = 2 ... 0 if q is odd I E(B1•••••sk_ 2 .z 1 ,z 2 ,u)~Z[y,l5]/- if q is even * * q where the exterior generators live in E 1•* and the polynomial generators in 0 * E • • Furthermore: where -- a 2 · 2 y= 0 y = -2y 2k-115 = 0 152 = -22n-k+16+by 2ru = 0 z1y= z1u = 0 z215 = b/2a-r u z2u = 0 uy = -2u-2k-1-r15z1 ul5 = -22n-k+1u+2a-r yz2 a • a(4n ,2k-1) = min{2n-1,2i-1 +u 2(~n) i ~ 2n-k+1} r = min{a,k-1} and b = 2rn-2k+1 [1 +( -1 )k+1 (2n-1)] 2n-k THEOREM 6: As a z2-graded algebra K(X4n,2k-1) = E(Sl ••••• sk-2'zl,z2,u) x Z[y,l5]/- .where the exterior generators live in Kl and the polynomial generators in K 0• Furthermore, the relations - are described by the same formulas of proposition 5, except for the last two that now looks as follows: k-1-r uy = -2u-2 15z1+v1 ul5 = -22n-k+lu+2a-r yz2+v2 where v1 and v2 are integer linear combinations of y and 15.

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