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T _ l , a t + 1 ..... al, st corresponds that a to ~: M ~ X to the g e n e r a t o r can be c h o s e n sequence, if n e c e s s a r y ) . so that alter a of so that by means of We h a v e a c o m m u t a t i v e dia- gram E J M s E w where p Then sists of s p h e r i c a l classes pose ~ ~(~) non-trivial Denote by if o' suspension = ~ ' ( ~ t ) , r*~(~) consists sets. l ~ ~(~i ) . T h e n we h a v e of s p h e r i c a l {vi} is a s p h e r i c a l of in terms of the ~l,'',~a T's con- classes.

We seek a w a y to compute cohomology denote operations the o p e r a t i o n of Adams. Thus #is (See Chapter 3 of [ ~ ]). Let a s s o c i a t e d with the A d e m r el at io n d e f i n e d on those classes Sq 2(u) #(u) = Sq 2 Sq l(u) u~H*(A) such that = O. is then a coset of the s ub gr ou p Sq 2 H n+1(A) in Our m e t h o d uses the s e c o n d a r y Sq 2 Sq 2 + Sql(Sq 2 Sq I) = O. (4) And k(~). + Sq I H n+2(A) H n+3 (A), a s s u m i n g dim u = n. We prove T h e o r e m 6. Let A and ~ Sq I H 4s+2(A) c Sq 2 H 4s+l(A), ~(~) = be as in then #(W4s(~)) , (3), wi th s>O.

7. P. Serre, Seminaire Cartan: Algebres et homotopie, 1954/55. ~ (Vn I-IV, Quart. r ,m )' pp. 249-268. cohomology 86 (1957), operations, 197-211. (2) Trans. Amer. , 54 (1951), pp. 425-505. - MacLane fibr6s, Groupes d'homotopie et classes de groupes Ann. of Math. 58 (1953), pp. 258-244. 9. I0. E. J. Math. Steenrod, Cohomology invariants 50 (1949), 954-988. of mappings, Ann. Ann. of ab61iens, of Math. ii. E. Thomas, On cross-sections to fiber spaces, Proc. Acad. , 54 (1965), 40-41. 12. Postnikov invariants and higher order cohomology operations, to appear.