Or, more generally, if $E$ is a spectrum, when is $E$ equivalent to $\bigvee_r \Sigma^r H(\pi_r E)$?

@lenticcatachresis one easy obstruction is the multiplication on the homotopy of a (homotopy) ring spectrum

@AaronMazel-Gee I don't understand. The result is true for spectra with rational homotopy groups, right? e.g. the rationalized sphere, which has a product

@lenticcatachresis $H^1(S;Z/2) = 0,H^1(\bigvee_r \Sigma^r H\pi_r^s; Z/2) = Z/2$ coming from $\pi_1 = Z/2$

@BertramArnold great, thanks. is there a "moral" reason?

also, I have another question. In a paper by Burdick, Conner and Floyd from 67 they show that any generalized homology theory which comes as: first functorially assign a chain complex and then take homology, where we assume that the first functor maps a cofibration to a short exact sequence of complexes, is actually the homology defined by the generalized Eilenberg-Mac Lane spectrum associated to the graded abelian group given by the coefficients

however, Schwede shows in exercise E.II.13 of his symmetric spectra v3 notes, that we can actually associate a functor landing in chain complexes to a spectrum such that spectrum-homology coincides with the homology of this chain complex

this seems like a contradiction at first glance

@lenticcatachresis In the exercise it says "Let A be a symmetric spectrum of abelian groups", that should imply that the corresponding spectrum splits as a sum of Eilenberg-MacLane spectra

thanks

@BertramArnold I have a follow-up question. Is $S$-homology, i.e. stable homotopy, equal to $H\pi_*^s$-homology? Here $HG_*=\bigvee_r \Sigma^r H(G_r)$ for $G_*$ a graded abelian group. Your proof show that the spectra are not equivalent since they have different $\mathbb Z/2$-cohomology, but a priori I don't see how this implies that define different homology theories

The Z/2-cohomology of $S$ is just the $S$-homology of HZ/2, so evaluating both theories on HZ/2 should do the trick

Ah, yes, yes. I'm finally converging to my real question: I meant the homology theories they define on spaces (i.e. on suspension spectra)

their values agree on spheres, but they don't agree on maps between spheres

for instance, the Hopf map S^3 -> S^2 of course induces multiplication by eta in stable homotopy, but it gives the zero map on any ordinary homology theory

great, thanks!