Localizing spaces at stable homotopy equivalences

Question

According to Bousfield's theory of localization, given any homology theory $E_*$ one can produce a reflective localization of the category of (pointed) CW complexes, in the sense that any such CW complex $X$ admits an essentially unique map $\eta_X:X \to L_E(X)$ such that

1. The map $\eta_X$ is an $E_*$-equivalence.
2. The space $L_E(X)$ is local with respect to the class of $E_*$-equivalences.

I am wondering about the case where we take $E_*$ to be stable homotopy theory, i.e. $E_*=\pi_*^{st}(-)$. My question is the following:

• Is there a concrete description of what it means for a space to be $E_*$-local in this case? My guess would be that such spaces are infinite loop spaces, but since a given space may admit several distinct infinite loop space structures, I get confused about what this actually means.
• How is this category of $E_*$-local spaces related to that of spectra? It would seem to ressemble the full subcategory spanned by the suspension spectra. But they can't be equivalent, since the functor $\Sigma^\infty:Spaces_* \to Spectra$ is not fully faithful (even homotopically).

Thank you in advance for your answers, which can hopefully correct some misunderstandings of mine.

Answer 1

Because suspension spectra, the sphere spectrum and the Eilenberg-MacLane spectrum are all $(-1)$-connected, a map $f\colon X\to Y$ of spaces induces an isomorphism of stable homotopy groups if and only if it induces an isomorphism of ordinary homology groups. Thus, localization with respect to $\pi^S_*$ is the same as localization with respect to $H_*$, which was studied in Bousfield's paper The localization of spaces with respect to homology. In particular, simply connected spaces are local, and more generally so are nilpotent spaces. For any space $X$, the map to Quillen's plus construction $X^+$ becomes an equivalence. By the Kan-Thurston theorem, for any connected space $X$ one can find a discrete group $G$ (which is usually a rather strange group) and a map $f\colon BG\to X$ which gives an isomorphism $H_*(BG)\to H_*(X)$ and thus also an isomorphism $\pi_*^S(BG)\to\pi_*^S(X)$.