TITLE: Homology of spectra vs homology of infinite loop spaces
QUESTION [16 upvotes]: Let $X$ be a CW complex and let $\Sigma^\infty X$ denote its suspension spectrum. By definition, the $n$th singular homology group of $\Sigma^\infty X$ with coefficients in $\mathbb{Z}$ is $\pi_n(\Sigma^\infty X \wedge H\mathbb{Z})$.
Now, connective spectra are in bijection with infinite loop spaces. The infinite loop space corresponding to $\Sigma^\infty X$ is $QX := \Omega^\infty \Sigma^ \infty X = \lim_{n \to \infty} \Omega^n \Sigma^n X$. Is the homology of $\Sigma^\infty X$ (as a spectrum) isomorphic to the homology of $QX$ (as a CW complex)? I would expect this.
But what confuses me is a theorem is Rudyak's On Thom spectra, orientability, and cobordism (Theorem 7.11.(i)). Rudyak states that for every rational spectrum $E_\mathbb{Q}$ the Hurewicz map $\pi_\ast(E_\mathbb{Q}) \rightarrow H_\ast(E_\mathbb{Q})$ is an isomorphism. This seems to prove that the homotopy/homology groups of spectra do not agree with homotopy/homology groups of their corresponding infinite loop spaces.
First, $\pi_\ast(Q X) \otimes \mathbb{Q}$ is isomorphic to the stable rational homotopy $\pi_\ast^S(X) \otimes \mathbb{Q}$ of $X$. Then, by the Milnor-Moore theorem, $H_\ast(QX,\mathbb{Q}) \cong \mathbb{Q}[\pi_\ast^S(X) \otimes \mathbb{Q}]$. So we'd have that $H_\ast(X,\mathbb{Q})$ is the free commutative-graded algebra on the rational stable homotopy groups of $X$ for any CW complex $X$. But it is easy to write down examples of CW complexes whose rational homology is not free e.g. projective spaces. So what, then, is the relationship between $\pi_\ast(\Sigma^\infty X \wedge H\mathbb{Z})$ and $H_\ast(QX, \mathbb{Z})$?

REPLY [2 votes]: First of all, I think the concern of the OP is the fact that the homology of the infinite loop space doesn't agree with that of the spectra.  So let's start with this.  Let $Y$ be a spectrum, then it is homology is related to its infinite loop spaces' homology by
$$H_*(Y)=\lim _i H_{*+i}(\underline{Y}_{i+j})$$ where $Y_k$ is the $k$-th infinite loop space associated to $Y$.  Notably we have $Y_{k-1}=\Omega Y_k$, so it should be more or less clear that the homology of a spectrum can almost never agree with that of the associated infinite loop spaces.
Now, let's get to $H_*(QX)$.  As has been noted, its rational homology is easy.  So basically we need to know $p$-complete homology for all prime numbers $p$ to recover integral homology.  In other words, we need to know the Bockstein spectral sequence for all prime $p$.  The input of the Bockstein spectral sequence $H_*(QX;Z/p)$ was computed by Kudo-Araki for $p=2$, Dyer-Lashof for odd prime.  The rest was worked out by May in Chapter 3, Theorem 3.12 of Frederick R. Cohen, Thomas J. Lada, J. Peter May "The homology of iterated loop space" Springer Lecture Notes in Mathematics 533.