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.