Is there a stable Hurewicz Theorem?

Question

Given a $(n-1)$-connected spectrum $E$ , is the natural morphism ${\pi _k}\left( E \right) \to {\pi _k}\left( {H\mathbb{Z} \wedge E} \right)$ an isomorphism for $k \leq n$?

I think yes, but I can't find a reference.

Answer 1

Yes, this is true. For instance, it follows from the fact that any $(n-1)$-connected spectrum is equivalent to a CW-spectrum with no cells of dimension less than $n$ (see for instance Proposition 2.3 of these notes of Hatcher). Or in less model-dependent language, any connective spectrum can be built up as a filtered (homotopy) colimit of connective finite spectra. The result is true for finite spectra by the Hurewicz theorem for spaces, and so taking the colimit it also holds for spectra.