# Homotopy of Spectra Maps Induced by Homotopy of Functions

So, I'm still working through Adams' lecture notes, and here's something that I haven't been able to immediately suss out:

It is clear that for $F$ a spectrum, $K$ a finite CW-complex and $E$ its associated suspension spectrum, a function $f:\Sigma^nK\to F_n$ induces a map of spectra (since we can just take everything above $\Sigma^nK$ to be the cofinal subspectrum and the suspensions induce all the higher maps).

Now, we are ultimately going to be considering the object $A=\mathrm{colim}_n[\Sigma^nK,F_n]$. So, suppose two maps $f:\Sigma^nK\to F_n$ and $g:\Sigma^mK\to F_m$ induce the same object in $A$. Then they must (by the nature of the directed system we index our colimit over) coincide at some finite point, i.e. for some $p$ the maps $\Sigma^pK\overset{\Sigma^{p-n}f}\to\Sigma^{p-n}F\to F_p$ and $\Sigma^pK\overset{\Sigma^{p-n}g}\to\Sigma^{p-n}F\to F_p$ are homotopic or define the same object in $[\Sigma^pK,F_p]$.

Adams' next claim is that this homotopy can be extended to the entire spectra (on suitable cofinal subspectra of course) and perhaps it's just my lack of expertise with suspensions and stuff, but I can't quite get it to work out. Something like, if we have a homotopy at a given level then we can just suspend that homotopy to get it on every higher level (i.e. a cofinal subspectrum).

Thanks! -Jon

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Hi Jon, it's two and a half years in the future, and I've just slightly edited your post to fix some typos. I just wanted to let you know :) –  lentic catachresis Mar 28 '14 at 15:44

Welp, figured it out...

The idea is pretty obvious. I think all the time I spent writing up the question in exact detail jarred it loose.

If we have a homotopy at one level, that means that there is a map $h:\Sigma^pK\times I\to F_p$. Since this homotopy is pointed, it is constant on the base point, so we might as well consider it as a map $h:\Sigma^pK\wedge I^+\to F_p$. Thus, we have the next map $\Sigma h:\Sigma(\Sigma^pK\wedge I^+)\cong \Sigma^{p+1}K\wedge I^+\to F^{p+1}$ and so on all the way up, which is exactly what we need!

Anyway... sorry this is a really obvious question.

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This often happens for me too, that writing things down helps me think through the issue. Half the time I'm not even sure what's holding me back. Anyways, enjoy the notes but don't read about products! (I assume you've heard at least a little bit about the various categories of spectra...?) –  Aaron Mazel-Gee Oct 7 '11 at 7:48
Indeed, thanks for the warning though. I will be skipping his smash products section (though I still haven't decided what exactly to replace it with, though I've heard maybe symmetric spectra are the best choice for now, since EKMM is perhaps significantly more grueling). –  Jon Beardsley Oct 10 '11 at 16:45
Well, it all depends on what you're trying to do. I spent a fair bit of time just taking it on faith that sufficiently nice constructions exist; you can get pretty far without actually using explicit constructions. By the way, my friend Cary is most of the way through writing a very nice set of notes that (among other things) seeks to organize ideas surrounding the various constructions of the Stable Homotopy Category: math.stanford.edu/~carym/stable.pdf –  Aaron Mazel-Gee Oct 11 '11 at 7:25
Also, there are a bunch of other cool notes from our Berkeley-Stanford student topology seminar here: math.berkeley.edu/~aaron/xkcd . In particular, I'd definitely recommend Eric's 2-parter "The Geometry of Formal Varieties", which sets the tone for the use of algebraic geometry in algebraic topology. (I think it's in the archives.) –  Aaron Mazel-Gee Oct 11 '11 at 7:27
Oh man cool seminar. I wish I was in California! –  Jon Beardsley Oct 11 '11 at 16:04