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I'm reading Switzer's "Algebraic Topology", which talks about the (homology) ASS in chapter 19. His ASS (where he puts $S^0$ in the first slot) either converges to $\pi_n(Y)$, or to $\pi_n(Y)/\cap_{s\geq 0}F^{s,n+s}$ (which happens e.g. when our theory $E$ is the spectrum $H\mathbb{F}_p$ representing $H^*(-;\mathbb{F}_p)$).

Switzer doesn't talk about this at all, but I've seen/heard before that one can also present the ASS as converging to something denoted $[L_EX, L_EY]$. I don't know much about the functor $L_E:Top\rightarrow Top$, but I believe it is characterized as follows: If $E_*X=0$ then we say that $X$ is $E$-acyclic. Now, a space $Y$ is called $E$-local if whenever $X$ is $E$-acyclic, $[X,Y]=0$. Then, the $E$-localization $L_EY$ of $Y$ is an initial object in the category of $E$-local spaces under $Y$.

So here is what I'm wondering. The next result after the ASS is the following Proposition 19.11: If $\iota_*:\pi_q(S^0)\rightarrow \pi_q(E)$ is an isomorphism for $q\leq 0$ and an epimorphism for $q=1$ and if $\pi_q(Y)=0$ for $q\lt N$ for some $N\in \mathbb{Z}$, then $A^{s,t}=D^s=0$ for all $s,t$, and hence the Adams spectral sequence converges to $\pi_*(Y)$. (Here $\iota:S^0\rightarrow E$ is the unit map.) This seems very much like a fact about localization in disguise -- presumably it'd be saying something about $L_EY$, perhaps that $Y=L_EY$. Except that I don't know what $L_ES^0$ is, which I'd imagine should be the first thing anyone would try to figure out.

So I guess my questions are:

  1. Do I have the right characterization of localization?
  2. Assuming the answer to (a) is yes (or I guess even if it isn't), what is this proposition saying about localization?
  3. Replacing $S^0$ with an arbitrary $X$, in Switzer's setup we can hope for the ASS to give us $[X,Y]/\cap_{s\geq 0}F^{s,*+s}$ -- how can we pass between this quotient and $[L_EX,L_EY]$? Is there anything more geometric to say than that the denominator is the subgroup of maps inducing 0 in homology?
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I am pretty sure that Switzer is talking about spectra and not spaces. – Sean Tilson Jan 20 '11 at 1:36
Right sorry, it's easy to confuse them. I mean, anywhere you have a space $Y$ you really mean $\Sigma^\infty Y$, right? – Aaron Mazel-Gee Jan 20 '11 at 5:52
up vote 7 down vote accepted

If you're talking about strong convergence, your statement that the ASS converges to a quotient of $\pi_*(Y)$ is not correct, even for $Y=S^0$ and $E=H\mathbb{F}_p$, where it converges to the p-adic homotopy groups $\pi_*(S^0) \otimes \mathbb{Z}_p$. Which agrees with the localization in this case. Weak convergence you do have, but that's really less than useful.

1) For spectra, this is a correct characterization of localizations; for spaces, not quite. A space Y is E-local if for every E-equivalence $X \to X'$, the induced map $[X',Y] \to [X,Y]$ is a bijection. An $E$-equivalence, of course, is a map that induces an isomorphism in $E$-homology.

2) I don't have a copy of Switzer handy, but the kind of homology theory $E$ in the proposition (connective with $\pi_0 = \mathbb{Z}$) will produce exactly the same kind of localizations as singular homology with coefficients in $\mathbb{Z}$ if your spectrum $Y$ is connective, as you assume; this is proved, I think, in Bousfield's original papers on localization. In other words, the localization is the identity.

It is not in general true that the Adams spectral sequence converges in any sense to the localization, though. It converges to the completion (nilpotent completion, to be exact) in the sense of Ravenel 1984. And if you're into convergence criteria, the kind of convergence is conditional convergence, in particular if there's no derived $E^\infty$ term then it converges strongly. Unfortunately, the nilpotent completion is only an approximation to the localization. The nilpotent completion of a connective spectrum at $H\mathbb{Z}$, however, is the spectrum itself; this is a Postnikov tower argument. So no problem in that case, and no contradiction to Switzer's proposition.

3) $[L_E X,L_E Y] = [X,L_E Y]$ by the universal property. For finite spectra $X$, the ASS behaves pretty much the same as for $X=S^0$. For infinite $X$, I believe the situation becomes more complicated, but I'm not sure about the details.

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In your last sentence, what is meant by infinity $X$? – Sean Tilson Jan 19 '11 at 13:43
I meant infinite $X$ (fixed). Infinitely many cells. – Tilman Jan 19 '11 at 15:05
Thanks for the nice answer by the way. – Sean Tilson Jan 19 '11 at 21:40
Thanks a lot, this is very helpful. Two questions. A space is $E$-local iff its suspension spectrum is, right? Also, what do these terms strong/weak/conditional convergence mean? – Aaron Mazel-Gee Jan 21 '11 at 7:41
No - a nonlocal space may well have a local suspension spectrum. I can't think of a concrete example right now, but an $H\mathbb{Z}$-local space has to have a nilpotent fundamental group, which is something you can't see in the suspension spectrum. The other direction is more obviously false: $S^0$ (two points) is local for any theory, while the sphere spectrum isn't. For the different kinds of convergence questions, I'd like to point you to the excellent paper "conditionally convergent spectral sequences" by Mike Boardman. – Tilman Jan 21 '11 at 18:08

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