Why can't the 3-type of $S^2$ be delooped? In 2.2 of this paper, the argument runs through the fact that the 3-type of $S^2$ cannot be delooped. Why not?
I understand that this is probably a fairly basic fact of homotopy theory (hence neither that paper nor this paper, which it cites, providing an explanation or source for the fact). All I know of homotopy stems from studying higher categories & groupoids, so I'm weak on the basics. Unfortunately, my google-fu has failed me, and I haven't been able to find a source that discusses why the 3-type of the 2-sphere cannot be delooped. I'd appreciate a pointer to an appropriate text, and I'd appreciate a quick explanation of why this should be so even more.

This is an answer to a different question which I can't find: when is X an H-space?
Let G = \pi_1(X) and H = \pi_2(X)
Interpret the Postnikov invariant beta  as a (homotopy class of maps)
K(G,1) to K(H,3)
Then X is an H-space if and only if beta is primitive
i.e. for m the multiplication on K(G,1)
if and only if m^* beta = beta \otimes 1 + 1 \otimes beta
In this special case of K(G,1)  K(H,3)
This is also sufficient for beta to be the looping of B beta \in H^4(BG,H)
More generally there is a result due to Art Copeland that if X is induced
by f :K(G,p) to K(H,q) with q  \geq p then X has an H-space structure if and only if
f represents a primitive class
Outside that range, things are more subtle
Not sure what you mean by H-group
