6 Jack Palmer Sanders
convergent H-module, and there are an H-map Q(f): N - N(f) and a long
exact sequence
2 ( f ) *
3
. . . - TT
( N )
~ TT
( N )
TT
( N ( f )
) TT
n
(N)
-

.
z z+p z+p z- 1
(5.6) follows from this by induction.
The H-modules N(f , ,f.) of (5.6) depend on the representing maps
f. . To overcome this difficulty we define a second equivalence relation on
the collection of H-modules over M. We say A is weakly H-equivalent to
B if there is an H-map F: A - B such that F^: TT^(A) - TT^(B) is a
TT^(M)-isomorphism. This relation is neither symmetric nor transitive, but
it generates an equivalence relation which we also call weak H-equivalence.
We then prove that N(f , •••,f.) is unique up to weak H-equivalence.
If we assume in (5.6) that p. - +00 and v. - +00 as i - +°°, then
the sequence of convergent H-modules of (5.6) has the property that for
q e 4Z, there is a positive integer I(q) such that
N(fl'---'fI(q))q = N(fr---'fI(q)+l)q= •"' and
N(fl'"#/fI(
)
) ( R
'
q )
=N(f1,---rfI ) (R,q) = ••• . We obtain the
convergent H-module N(f°°) of (6.1) by defining N(f°°) = N (f.,,•••,f . . )
q 1 I (q) q
and N(f°°)(R,q) = N(f ,"«,f . )(R,q). We prove in this case also that
N(f°°) is unique up to weak H-equivalence.
In section 7 we define an H-pairing P: (A,B) - C of H-modules. There
is, for every (R,w,q), a space P(R,w,q) all of whose homotopy groups
vanish and a map e: P(R,w,q) - Map^(M A A A B , C ^ ) . There are maps
0 R w q w+q+ZR
c. a' for a e S(k), and W similar to those of an H-module. Given an
r ,s
H-pairing P: (A,B) - C and x e T T (M) represented by f: S - M , we
construct H-pairings P (f): (A(f),B) -C(f) and P2(f): (A,B(f)) -C(f).
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