2

John R. Harper

Remark. If M = {x,,Pxx0 ,3PxxQ} then U(M) ~ T , the algebra over

° ^ 5 P

over G considered in the introduction.

Spaces with cohomology U(N). When we write H*(X) • = U(N), we have

understood a specific embedding j: N - H*(X), since in general

there is no canonical embedding. To avoid cluttering notation, j

is suppressed when clear by context. The elements

j(N) c

H*(X)

form a p-simple system of generators for H*(X).

1.1.3. X-ACTION [3],[6],[20]. For M e objUfo/G, an action A: M + M is

given gradewise by A|M2^ = P' ^ and A|M2J+1 = 3P^ .

l.l.U. A routine use of the Adem relations gives the following relation of

the A-action to the action of the Steenrod algebra,

(a) 3Ax = 0

AP'^/P x, j = 0 mod p

(b) P'^Ax = \ A3P ^ P x, j

E

1 mod p and dim x odd

0 otherwise

Hence im A is an G-module.

Remark,For p odd, ker A need not be an G-module. For example

M = (x2,3x2, px3x2, 3P13x2 .

has x e kerA but 3x i ker A.

1.1.5. If Me obj um/G satisfies 3M =0, then ker A is an d-module,

Proof. It follows from 1.1.k. that ker A is closed under the

action of P . The Adem relation

pl&p5 l = pJ*$ + (j-l)3P^

and the hypotheses imply ker A is closed under the action of 3.

1.1.6. ALGEBRAIC LOOP AND SUSPENSION FUNCTORS [6],[20]. The category Ufo/G

has an adjoint pair of functors Z and ft where £ is the usual

suspension and 0, is defined gradewise for connected M by

(ftM)k = (M/AM)k+1 .

If f: M - N, then Ef and ftf are the obvious maps. We write

a : M - ftM for the projection.