2 Jack Palmer Sanders
could define a module pairing w(f): (MU,MU(f)) - MU(f), the mapping cone
construction could be repeated for x e T T (MU) to obtain a spectrum
MU(f,f) and an exact triangle relating H^(X;MU(f)) and H^(X;MU(f,f') ) .
In trying to obtain the pairing w(f), however, one sees that the maps w
r
rQ
must satisfy several additional homotopy commutativity properties. In
general, the more times one wants to repeat the mapping cone construction,
the more homotopy commutativity properties must be satisfied. A main purpose
of this paper is to show that the maps associated with any of the Thorn
spectra MO, MSO, MU, MSU, and MSp satisfy sufficiently many homotopy
commutativity properties that this mapping cone construction can be repeated
any finite number of times and, with some additional dimensional require-
ments, a countably infinite number of times.
More generally, suppose that M is a ring spectrum. In (2.1) we
define an H-module over M. There are two useful features of an H-module N
over M. First, N is a module over M, and so for x e T T (M) represented
P
p+v
by f: S - M , a mapping cone spectrum N(f) can be constructed.
Secondly, the spectrum N(f) is also an H-module over M, and hence one
can by induction construct other H-modules over M. We now state our main
results.
5.6. Proposition. Let N be a convergent H-module over M. Given
p.+v.
xn , X-,««« , x . e T f (M) , p . 0, r e p r e s e n t x . by f . : S -*• M ,
1 2
I D .
* i l * l v .
l l
where v. _ 0. Then there are a sequence N, N(f ), N(f ,f ),••• of
convergent H-modules over M, a sequence of H-maps
Q(f.) : N(f , ,f. ) - N(f , , f . ) , and for each i _ 1, a long exact
sequence
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