By Armando Rojo

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204 @ k operad m subject t o c e r t a i n axioms which s t a t e how'the A r e l a t e to the i n t e r n a l . . . . . . . 154 and BTop a t p > 2 . . 162 The homology of BCoker J , BSF, and BJ This notion is b a s e d on Such. a n action c o n s i s t s of m a p s Orthogonal groups of finite fields Orientation sequences a t p ring spaces. ll0 The orthogonal, unitary, and syrnplectic groups m sequences I at p = 2 j = j l . . j k. jk) denote the s e t of a l l = { i l , .

A. Epstein. Cohomology Operations. P r i n c e t o n University P r e s s . 1962. , originally obtained f r o m t h e g e o m e t r y of F. The f i r s t t h r e e sections a r e devoted t o t h e s e formulas. e s t a b l i s h e s notations and gives f o r m u l a s f o r t h e evaluation of the "multiplicative Pontryagin product N. E. Steenrod and E . Thomas. f r o m cyclic groups. Cohomology operations d e r i v e d Comment. Math. Helv. 3 2 (1957), 3 o n elements decomposable i n t e r m s I of the "additive" Pontryagin product 24.

Since a i s a l i n e a r combination of length two elements of R j>O Iterating, we r e a c h t e r m s w h e r e the e r r o r summation i s z e r o a f t e r finitely many steps. i J Since P*Q [I] i s a l i n e a r combination of monomials K Q [I] Thus 2 * i s just t h e t r a n s l a t e of the product the one component. * - 2 1 = Q Q [I]. Since deg as )/ 4, t h e f i r s t alternative m u s t h o l d (QS[1] i n H*X1 1 1 = 0 , a s . = Q Q [I] , o r F o r the second formula, such that P(K) = P(J), the r e s u l t follows b y induction o n the length of I.