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 AM AN —— M AN —-— N V R q 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 r k 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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