I am reading Lewis' paper "Is there a convenient category of spectra?". To prove the main result on the non-existence of such a nice category, he shows that otherwise the unit component of $QS^0= \varinjlim \Omega^n S^n$ would have to be weakly equivalent a product of Eilenberg-Mac Lane spaces. So far so good, but it isn't immediately clear to me:


**Q:** *Why can't the unit component of $QS^0$ be a product of Eilenberg-Mac Lane spaces?*


This should probably be very easy, given how the paper feels no need to include any justification for it.



$QS^0$ isn't a product of Eilenberg-Mac Lane
If $X$ is a product of Eilenberg-MacLane spaces then the map 
$$ \eta^\*\colon \pi\_2(X) = [S^2,X]\to[S^3,X]=\pi\_3(X) $$ is easily seen to be zero (where $\eta\colon S^3\to S^2$ is the Hopf map). However, standard calculations give: 
\begin{align\*}
 \pi\_2(QS^0) & =\pi\_2^S(S^0)=\mathbb{Z}/2.\eta^2\\
 \pi\_3(QS^0) &= \pi\_3^S(S^0)=\mathbb{Z}/24.\nu
\end{align\*}
with $\eta^3=12\nu$. Thus, $\eta^\*$ is nonzero in this context.


