22
Jack Palmer Sanders
is a spectrum (M/N,YK where
4 4 4
Y.: S A (M./N. ) = (S A M.)/(S A N. ) - M„ ./N„ . is induced b1 y a.. If
1 1 1 l l 4+i 4+i -
I
M and N are convergent, then M/N is convergent by (1.15) and the
homotopy sequence of (M.,N.). The inclusion i: N -*• M and the projection
7T: M - M/N are maps of spectra. We need the following result from [3] .
1.17. Proposition. Given a convergent spectrum M and a convergent
subspectrum N, there is an exact sequence
i* TT*
-* 7T (N) IT (M) TT (M/N) -^— TT
n
(N) " ,
q q q q-1
where y is induced by the compositions
for sufficiently large k.**
Let N and N' be convergent subspectra of (M,a) and (M',a'),
respectively. Suppose there is a sequence {f Iq z 4z} of maps,
f : (M ,N ) - (M' ,N' ) , such that
q q q q q
S4 A (M ,N ) -£—* (M . ,N .)
q q q+4 q+4
1 A f J
I
f
q q+4
S4
A
(M'
,N' ) v (M' ^wN' .)
q q q+4 q+4
is homotopy commutative as pairs for all q. Then there are maps of spectra
f: M -*M', f I : N -* N', and an induced map F: M/N - M'/N'. We obtain the
following result by naturality.
1.18. Proposition. The diagram below is commutative.
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