[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.

  • $\begingroup$ I assume Adam -> Adem $\endgroup$ – Igor Rivin Sep 22 '11 at 19:51
  • 3
    $\begingroup$ @Igor : No, the book is by Frank Adams. See en.wikipedia.org/wiki/Frank_Adams $\endgroup$ – Andy Putman Sep 22 '11 at 20:01
  • 1
    $\begingroup$ My typo... Should be Adams' text, not Adam's $\endgroup$ – Shaun Ault Sep 22 '11 at 20:43
  • $\begingroup$ @Andy I am aware of Frank Adams (though apparently not of his oevre...) $\endgroup$ – Igor Rivin Sep 23 '11 at 7:18

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.

  • $\begingroup$ Remember that the "maps" here are not in the homotopy category (otherwise why are we talking about their homotopies?), so "choose a representative" means something a bit nicer than choosing a rep. of a homotopy class of maps. $\endgroup$ – Dylan Wilson Sep 22 '11 at 23:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.