# Homotopy extension of $E_{\infty}$-spaces

Suppose that $$X$$ is a connected $$E_{\infty}$$-space, naturally $$\Omega X$$ is also an $$E_{\infty}$$-space. Can we classify all $$E_{\infty}$$-extensions of $$X$$ by $$\Omega X$$ (up to homotopy). I mean the following: we would like to classify of homotopy fiber sequences $$A\rightarrow B\rightarrow C$$ where $$A\sim \Omega X$$ and $$C\sim X$$ as $$E_{\infty}$$-spaces and $$B\rightarrow C$$ , $$A\rightarrow B$$ are maps of $$E_{\infty}$$-spaces in particular we assume that $$B$$ is an $$E_{\infty}$$-space. (space could mean a simplicial set or a topological space)

There are two obvious extensions of $$E_{\infty}$$-space the first one is when $$B\sim X\times \Omega X$$ and the second one is when $$B=PX$$ the path space. Is there others ? My first guess was that the set of extensions is a group and more precisely the set of homotopy classes of maps of spectra $$[X,X]$$ where $$X$$ is seen as a connective spectra. Maybe I'm wrong but the first extension corresponds to the homotopy class of the trivial map $$X\rightarrow \ast\rightarrow X$$ and the second extension corresponds to the identity map $$id: X\rightarrow X$$. But there should be other extension in bijective correspondence with $$[X,X]$$. Is it correct ?

• My guess is that it corresponds to homotopy classes of $E_{\infty}$-spaces from $X$ into the group completion of $X$. I have to sketches of an argument: First, if we also assume that $X$ is grouplike, then we can use that group-like $E_{\infty}$-spaces are the "same" as connective spectra and thus we are looking at (co)fiber sequences of spectra and can use that the homotopy category of spectra is triangulated. Jan 4 '20 at 7:20
• The second line of argument says that you are basically looking for $\Omega X$-principal bundles on $X$ and these should be classified by a map $X \to B \Omega X$. The latter is precisely the group completion of $X$ (for a modern reference see e.g. uni-muenster.de/IVV5WS/WebHop/user/nikolaus/papers/… ). For an "$E_{\infty}$-principal bundle" I expect $E_{\infty}$-maps. But this is just a guess. Jan 4 '20 at 7:23
• @LennartMeier Unless I'm mistaken $B\Omega X$ is the connected component of the identity of $X$ (the group completion is $\Omega BX$) Jan 4 '20 at 7:30
• @DenisNardin Of course, right. I was not paying attention. Jan 4 '20 at 8:30
• @LennartMeier $X$ is connected, in particular grouplike. It was also my first guess... Jan 4 '20 at 9:47

This is probably belaboring the obvious, but just take seriously the equivalence between grouplike $$E_{\infty}$$ spaces and connective spectra. See for example
Equivalence between $E_\infty$-spaces and connective spectra
The asumption that X is connected means that the associated spectrum is connected and not just connective. So we may as well just ignore $$E_{\infty}$$ spaces (much as I hate to do that!) and take $$X$$ to be a connected spectrum. One is asking for all fiber sequences $$\Sigma^{-1}X \to Y \to X$$ of spectra or equivalently all fiber sequences $$Y \to X \to X$$. That is, Y is equivalent to the fiber of a map $$X\to X$$. A quick triangulated category type argument shows that equivalence classes of such fibers correspond bijectively to maps $$X\to X$$, that is to elements of $$[X,X]$$.