Question

[Question cross posted on stack-exchange]

I'm slowly working through Part III of the book, and I'm scratching my head a bit while reading the proof of Lemma 3.2 (here reproduced):

Let $X, A$ be a pair of CW-spectra, and $Y, B$ a pair of spectra such that $\pi_*(Y, B) = 0$. Suppose given a map $f: X \to Y$ and a homotopy $h: Cyl(A) \to Y$ from $f|A$ to a map $g: A \to B$. Then the homotopy can be extended over $Cyl(X)$ so as to deform $f$ to a map $X \to B$.

Adams begins by choosing representing functions for the maps. We choose $f' : X' \to Y$ and $h' : Cyl(A') \to Y$ for $X'$ cofinal in $X$ and $A'$ cofinal in $A$. However, later in the proof it seems that we must use the (assumed?) fact that $h'$ restricted to the bottom of the cylinder agrees with $f'$. In other words, since $f'$ and $h'$ were chosen at the beginning of the proof, I don't see how we can require $h' \circ i_0 = f'| A'$, where $i_0 : A' \to Cyl(A')$ is inclusion on the bottom of the cylinder.

Is there an unstated step such as "If $h' \circ i_0 \neq f' |A'$, then find cofinal sub-spectra and representing functions for which this is true"? Even that seems a bit fishy to me.

Answer 1

Well perhaps Adams did things a little out of order, but this should work:

We know that $h \circ i_0 = f \circ j$ where $j: A \rightarrow X$ is the inclusion map. Now choose representatives of the compositions, $k$ and $k'$, so that $k = k'$ on some cofinal spectrum in $A$. Next choose representatives for $h, f, i_0,$ and $j$ on some other cofinal spectra in $X, A$, and $Y$. It must be the case that $h \circ i_0$ agrees with $k$ on the intersection of the cofinal spectra you chose and similarly for $f \circ j$ and $k'$, so replace the cofinal spectra you chose with their intersection (on $A$), and you should be good.

Of course, probably the better way around all of this is to replace Adams' treatment of CW-spectra with something like Mays' (you don't need all the fancy machinery, but at least define a CW-spectrum using pushouts of stuff with cones on sphere spectra.) Then you can just copy the usual proof of this fact from the unstable case.