Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$.

Equipping $Top_*$ with the Quillen model structure (weak homotopy equivalences, and Serre fibrations), and Bousfield localizing at $\mathbb{S}$ gives a model structure with the weak equivalences the isomorphisms on stable homotopy groups, i.e. $\pi_*^\mathrm{st}$-isomorphisms.

Is there a good characterization of the fibrant ($\mathbb{S}$-local) objects in this model structure?

In particular, are "nice" spaces fibrant in this model structure, e.g. nilpotent spaces, simply connected spaces, etc?

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    $\begingroup$ Isn't this the same as localizing at $H\mathbb{Z}$? Anyway: It looks like Eilenberg-MacLane spaces for abelian groups are local (since cohomology is a stable invariant), and local objects are closed under homotopy limits, so every nilpotent space is local. Since this is the same as HZ-localization, you can probably find lots more information in Bousfield's original paper inventing "Bousfield localization" for spaces ;) $\endgroup$ – Dylan Wilson Apr 3 at 14:37
  • $\begingroup$ @DylanWilson I believe this is precisely the same thing. Do you have a reference for this fact? I’ve tried showing it but got nowhere quick $\endgroup$ – Niall Taggart Apr 3 at 14:39
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    $\begingroup$ Elaborating on Dylan's comment: the $\mathbb{S}$-local model structure is actually the same as the $H\mathbb{Z}$-local one, since homology detects equivalences between bounded below spectra. $H\mathbb{Z}$-local spaces are called semi-$\mathbb{Z}$-complete in Bousfield and Kan (VII 2.2), but not much is said about them. Any $\mathbb{Z}$-complete space is $H\mathbb{Z}$-local, since by definition it is a homotopy limit of spaces of the form $\Omega^\infty(H\mathbb{Z}\wedge X)$. Any nilpotent space is $\mathbb{Z}$-complete, hence $H\mathbb{Z}$-local. $\endgroup$ – Marc Hoyois Apr 3 at 14:41
  • $\begingroup$ I suppose one way to show this is to show that an isomorphism on stable homotopy groups is an isomorphism on integral homology? Then both localized model structures are the same. $\endgroup$ – Niall Taggart Apr 3 at 14:57
  • $\begingroup$ @User1236262625 yes, this is true (for bounded below spectra), as Marc said- it's the Hurewicz theorem. $\endgroup$ – Dylan Wilson Apr 3 at 18:39

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