If $X$ is a space, we can form $QX=\varinjlim \Omega^n\Sigma^nX$ which is an infinite loop space with homotopy groups $\pi\_i(QX)=\pi^{s}\_i(X)$ the stable homotopy groups of $X.$ But these are the *unstable* homotopy groups of $QX.$


**Q:** Is there any similar expression for the *stable* homotopy groups of $QX$?


Just to get some feeling, **I would already be very happy to know just what $\pi\_0^s(QS^0)$ is**. If I understand things correctly, it should be the same as $\pi\_0^s(B\Sigma\_\infty^+),$ but I still don't know how to compute it.


I guess $\pi\_i^s(QX)$ seems like some sort of "secondary stable homotopy group" of $X$. But there is no reason to stop there. How about groups $\pi\_i(Q^nX)$ which should be "$n$-ary stable homotopy groups" of $X$ - what sort of information do they carry?


Of course then we could start forming other very silly objects such as about some sort of stabilizations $\pi\_{i+n}(Q^nX)$ as $n\to\infty$ (does that make sense and/or produce anything sensible?) and so on, but at that point I am burried so deep in things that I don't understand that it's a bit pointless.



Stable homotopy groups of $QX$
The "Snaith splitting" gives the following spectrum level statement: for a pointed connected space $X$, there is a weak equivalence:
$$
\Sigma^\infty\_+ (\Omega^\infty \Sigma^\infty X) \simeq \bigvee\_{n \geq 0} \Sigma^\infty (X^{\wedge n})\_{h\Sigma\_n}
$$
There is also an unbased version:
$$
\Sigma^\infty (\Omega^\infty \Sigma^\infty X) \simeq \bigvee\_{n \geq 1} \Sigma^\infty (X^{\wedge n})\_{h\Sigma\_n}
$$
This gives an isomorphism on the level of stable homotopy groups:
$$
\pi\_\*^{s}((QX)\_+) = \pi\_\*(\Sigma^\infty\_+ \Omega^\infty \Sigma^\infty X) \cong \bigoplus\_{n \geq 0} \pi\_\*^s((X^{\wedge n})\_{h\Sigma\_n})
$$


When the space $X$ is not connected, one needs to add a group-completion to get the correct statement. The net effect is that on $\pi\_0$ we get an isomorphism between $\pi\_0^s((QX)\_+)$ and the group algebra on the free abelian group on $\pi\_0(X)$: it is freely generated as a commutative ring by generators coming from $\pi\_0(X)$ and their inverses. In the case of $QS^0$ this becomes the Laurent polynomial ring $\Bbb Z[t^{\pm 1}]$.


