Iterated free infinite loop spaces

Question

Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted to do this again $QQX$, but examples such as spheres and eilenberg-maclane spaces get too complicated.

There is a natural maps $\iota :X \to QX$ which means we can have a tower:

$$X \to QX \to Q^2X \to \cdots$$

Taking the colimit I can imagine there being an object $Q^\infty X$. So my questions are:

What is known about repeated applications of the free infinite loop space functor?

Can it be iterated as I have described? What does this describe?

Answer 1

I suggest you look at Finkelsteins’s PhD thesis!

L. D. Finkelstein, On the stable Homotopy of infinite loop spaces. PhD thesis. Northwestern University. 1977

The objective of this thesis is to study $QA$ when $A$ is an infinite loop space which in particular you may take $A=QX$. Also, some papers of Kuhn consider iterated application of $Q$, in particular his papers on/related to Whitehead conjecture. Moreover, the Snaith splitting as mentioned in comments arises from a splitting of the space $QQX$. An application of Snaith splitting shows that understanding $QQQX$ would require one to describe the $r$-adic construction on $QX$ and I am not sure if this is studied in detail. I then conclude that $Q^\infty X$ would be too difficult to describe in terms which allows to do meaningful computations in terms of $X$.

Added One of the mail results of Finkelstein, also see section 2 of Kuhn’s paper on homology of James-Hopf maps, is that if $A$ is a suitable infinite loop space then there is a composition $$QA\to A\times QD_2A\to QA$$ which is a mod 2 equivalence. Note that the thesis is written using Barratt-Eccles $\Gamma^+$ functor. The more general case of this sequence, at an arbitrary prime, appears in Kuhn’s papers related to the Whitehead conjecture.

Answer 2

Let's assume $X$ is connected with friendly basepoint. The quick answer goes as follows. Imagine all subgroups $\Sigma_I < \Sigma_n$ of symmetric groups formed from smaller symmetric groups by iterating products and wreath products. For such an $I$, let $$D_I(X) = E\Sigma_{n+} \wedge_{\Sigma_I} X^{\wedge n}.$$ Then $Q^{\infty}(X)$ will be the product over all $I$ of the spaces $QD_I(X)$. As other folks have already hinted at, this follows from the stable splitting of $QX$.

By the way, the cosimplicial resolution $X \rightarrow Q(X) \cdots$ was used by Gunnar Carlsson to deduce the Sullivan conjecture from the Segal Conjecture, and by Greg Arone and Marja Kankaanrinta to give a second construction of the Goodwillie tower of the identity.

Dually, if $X$ is an infinite loopspace, one gets a simplicial infinite loopspace $X \leftarrow QX \cdots$. Back in the day, I thought about this quite a bit. When $X=S^1$, the Whitehead conjecture sequence formally splits off this. For general $X$, I proved a splitting result about the beginning of this in [Topology 23 (1985), 473-480]. This has a fun conceptual proof (no homology calculations!), works for all primes $p$, and includes Finkelstein's $p=2$ theorem and the Kahn-Priddy theorem at all primes. I would love to see someone generalize this in some interesting way.

Answer 3

If $X$ is connected then $QX$ is stably equivalent to the total extended power $DX$ (by the Snaith splitting, as others have mentioned), so $QQX=QDX$. If $X$ is not connected then $QX$ is still stably equivalent to a group completion of $DX$. So it is useful to think about iterating $D$, and then you can go back to thinking about $Q$ if you want.

A key example to consider is as follows: if $G$ is a groupoid, and $SG$ is the free symmetric monoidal category on $G$, then $\Sigma^\infty BG_+=K(SG)$, and $DBG=BSG$. If $G$ is just a group (considered as a one-object groupoid) then $SG$ is the groupoid of pairs $(P\xrightarrow{p}X)$, where $P$ and $X$ are finite sets, $G$ acts freely on $P$, and $p$ induces a bijection $P/G\to X$. In other words, $P$ is a $G$-torsor over $X$. More generally, we have $D^nBG=BS^nG$, where $S^nG$ is the groupoid of diagrams $$ (P\xrightarrow{p}X_{1}\xrightarrow{f_1}\dotsb\xrightarrow{}X_n), $$ where all the $X_i$ are finite sets, the maps $f_i$ are arbitrary, and $P$ is a $G$-torsor over $X_1$. The natural map $D^nBG\to D^{n+1}BG$ corresponds to the functor that just adds $X_{n+1}=\{0\}$ at the end. So $D^\infty BG=BS^\infty G$, where $S^\infty G$ is the analogous groupoid of diagrams $$ (P\xrightarrow{p}X_{1}\xrightarrow{f_1}X_2 \to X_3 \to \dotsb ) $$ where $|X_n|=1$ for $n\gg 0$.