# Fibrant objects in $\mathbb{S}$-local model structure on $Top_*$

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?

• 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 ;) – Dylan Wilson Apr 3 at 14:37
• @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 – Niall Taggart Apr 3 at 14:39
• 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. – Marc Hoyois Apr 3 at 14:41
• 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. – Niall Taggart Apr 3 at 14:57
• @User1236262625 yes, this is true (for bounded below spectra), as Marc said- it's the Hurewicz theorem. – Dylan Wilson Apr 3 at 18:39