4 ENRIQUE ANTONIANO PROOF. As we can see from proposition 5, the algebra *'0 E ' . is a quotient of a universal algebra, i.e. a Z-graded algebra which is the tensor product of an exterior algebra with generators in dimension 1 by a polynomial algebra with generators in dimension 0. Also, the K-theory of a finite complex is a quotient of such a universal algebra when one considers only the natural z2-graduation. Since the algebra K is an extension of the algebra E, it follows that both·are quotients of the same universal algebra say by ideals IK and IE. Now we observe that in our case, the ideal IE, is generated by elements in dimensions 0, 1 and 2, so this is also true for IK. By inspection, we find the relations in theorem 6. Let ~ denote the Hopf bundle over X k' which is the line bundle asso- n, ciated with the double covering V k+ X k' and let p e K(X k) be the n, n, n, complexification of (E -1) 6 KO(X k). n, COROLLARY 7. The order of p e K( x 4n,s) is 2a( 4n,s), where a(4n,s) =min{2n-1, 2i-1+u2 2 ") i~[¥]} PROOF. For s=2k-1, we have the injection [9]: E~ ,O + K(X4n ,2k-1) y I+ p so p has the same order as y. 4n-2k On the other hand, the fibration S + x 4n,2k+ x 4n,2k_1 is tota 11 y non cohomologous to zero in K-theory, [5], so p has the same order in K(X4n,2k) as in K(X4n,2k_1), as the corollary claims. Let 213(4n,s) be the order of (~-1) e KO(X4n,sh then we have that (3(4n,2)=a(4n,s)+e:, with e:=O or 1. Now, if n2, s4n-2 and (n,s) (3 8) or (4,8), it is possible to show that 4ns is not divisible by 213{4n,s) and since. 4ns(~-l) is the stable tangent bundle for the manifold x 4n,s' [7], we conclude that it is not stably parallelizable. Joining these facts to the explicit constructions of vector fields given by P. Zvengrowski in [10] and some others given later by himself [2], we get the following corollary. COROLLARY 8. About the parallelizability of projective Stiefel manifolds, we have the next table:
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