I apologize if this question is obvious as I am just beginning to learn this field. I am also abusing the dependance on $p$ for notational ease. The Chromatic Convergence theorem establishes that $S=\mathrm{holim}_{n}L_{E(n)}S$. This gives that the maps $f_n^m:\pi_m(S)\to \pi_m(L_{E(n)}S)$ contain all the homotopy groups, atleast for $n>>0$. The groups on the left are finite (for all the ones that matter), and so they will be seen in $\pi_m(L_{E(n)}S)$ after some finite $n$, so that our localization is $m$connected after some finite $n$. My question is, what are the lower bounds on this number? That is, do we have information on how quickly the chromatic tower will give us all the homotopy information above a certain index?
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5$\begingroup$ As a word of warning: even once $\pi_m S \to \pi_m L_{E(n)} S$ becomes injective, it can fail to be surjective indefinitely without violating the convergence theorem. So, even if you had some lower bound on n and a computation of $\pi_m L_{E(n)} S$, it will be hard to sort out what information is "real" and what is chromatic crosstalk. $\endgroup$ – Eric Peterson Oct 19 '15 at 18:51