# When localisation preserves isomorphy of homotopy groups

Let $$E$$ be a generalized cohomology theory. Let's agree that $$E$$ satisfies property $$(*)$$ if for any two finite CW complexes with isomorphic homotopy groups their $$E$$-localizations also have isomorphic homotopy groups. Is there some non-tautological condition implying property $$(*)$$?

• The only localization I know with this property is rationalization, and of course there is for trivial reasons. – Denis Nardin Mar 4 at 6:12
• @AknazarKazhymurat There is a variant of p-localization/completion for non simply connected spaces, but I doubt it has the property in question (I'm not 100% sure what it does to the fundamental group, but it is quite destructive) – Denis Nardin Mar 4 at 10:42

The question is very unlikely to have a satisfactory answer in the level of generality asked. Let $$X$$ and $$Y$$ be the two spaces with isomorphic homotopy groups, and let $$LX$$ and $$LY$$ denote their localizations with respect to $$E$$. For certain generalized cohomology theories $$E$$, we have formulas for $$\pi_n(LX)$$ and $$\pi_n(LY)$$, if $$X$$ and $$Y$$ are nilpotent, going back to Bousfield's paper The localization of spaces with respect to homology. An excellent summary is provided by Dwyer, and teaches us, for example, that for $$E = \mathbb{Z}/p$$, one has $$\pi_n(LX) \cong \pi_n(X) \otimes \mathbb{Z}_p$$, and for any subring $$R$$ of $$\mathbb{Q}$$, taking $$E = HR$$ gives $$\pi_n(LX) \cong \pi_n(X) \otimes R$$. However, this is only with an assumption about $$\pi_1(X)$$ (i.e. that $$X$$ is simply connected, or at least nilpotent). Bousfield's 1997 paper Homotopical Localizations of Spaces gets a similar result for nullification with respect to a Moore space (see Theorem 7.5).
So, basically, if the OP restricts attention to nilpotent spaces and nicely behaved $$E$$, then everything should be fine, but it is extremely unlikely to get an answer for general $$E$$. For counterexamples of what can go wrong for specific $$E$$, you can look to work of Carles Casacuberta (e.g. Example 4.5 here, or the many examples and counterexamples here).