Jack Palmer Sanders
and B(V), 3 £ S(j) , are analogous. We now sketch the essential features
of an H-module N = {N |q £ 4z}.
1. For every (R,q) , k _ 0, there is a space N(R,q) all of whose
homotopy groups vanish and a map e: N(R,q) -•Map (M A N / N ) .
2. For every (V,R,q), there is a map c: N(V,q+ER) x N(R,q) + N(V,R,q)
such that e(c(F,G)) is the composition
lAe(G) e(F)
M AMR ANq —— M AN —-— N
V V q+ZR q+ZR+ZV
3. For each q e 4Z, there is an element 1(N) e N(q) such that
e(l(N) ): N - N is the identity map.
q q q
4. For a e S (k) , there is a map a': N(avR),q) - N(R,q) such that
e(a'(F)) is the composition
T
a e (F)
M A N M . A N y N ^ ,
R q
a(Rx
) q q+ER
where T is the map which permutes the spaces.
5. If r + s = r , where 1 _ z _ k and r,s £ 4Z, there is a map
W : N(R,q) -* N(r. , ••,r . , r,s,r _ , «,r .q) such that e (W (F)) is
r,s 1 z-1 z+1 k r,s
the composition
M
A

A M A M A M A M A

A M A N
-»•
r i r z - l r S r z+l rk q
l A w A l
LLI
M A N
^!U
N
.
R q q+ZR
The maps e, c, a', and W are required to satisfy many properties. For
r,s
example, we require that c ° a' x B' = (a©3)' ° c:
N(a(V),q+lR) xN(6(R),q) -N(V,R,q).
An H-map F: A - B, where A and B are H-modules over M, is
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