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