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
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
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.
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