TITLE: When can you desuspend a homotopy cogroup?
QUESTION [17 upvotes]: Any topological group $G$ has a classifying space, whose loopspace is a (homotopy) group which is homotopy equivalent to $G$ in a way that preserves the group structure.  More generally, if $G$ is an $A_\infty$-group (a space with a binary operation which satisfies the group axioms up to coherent homotopy), it similarly can be delooped to a classifying space.
Now suppose you have a cogroup in the category of pointed spaces.  If it is actually literally a cogroup, it's not hard to show it must be a point.  However, up to (coherent) homotopy, the suspension of any space is a cogroup.  Is the converse true?  Can you desuspend any $A_\infty$-cogroup?  Are there any examples of homotopy cogroups (possible not $A_\infty$) which are not suspensions?  More generally, are there any criteria that you can use to prove that a space does not have the homotopy type of a suspension?  The only one I know is that all cup products must vanish, but this also holds automatically for a homotopy cogroup (indeed, for any "co-H-space").

REPLY [9 votes]: Hopkins' result alluded to above gives a coordinate-free approach a lá Segal. The paper I wrote with Schwänzl and Vogt:
Comultiplication and suspension. Topology Appl. 77 (1997), no. 1, 1–18,
gives an actual co-operadic approach: we show that a 2-connected space which has a homotopy everything co-action by the Stasheff operad (i.e., a co-$A_\infty$-structure) is always a suspension.
This paper answers the first of the questions posed above.
There is an open question about this matter which I'd like to bring up. Hopkins claims his result holds for 1-connected spaces, but I and my co-authors could only get our result to work for 2-connected ones. Hopkins' proof, which was based on the Bousfield spectral sequence, never appeared (our proof involved the higher Blakers-Massey theorems). 
Hopkin's result is stated, but not proved, in: Formulations of cocategory and the iterated suspension. Algebraic homotopy and local algebra (Luminy, 1982), 212-226, Astérisque, 113-114, Soc. Math. France, Paris, 1984. Thus we don't know if the 1-connected case has really been settled or not.
Additional Note: Schwänzl, Vogt and I also show that if the space is highly connected with respect to its CW dimension (i.e., $(n-1)$-metastable), then the existence of a co-$A_n$-structure on it is tantamount to the existence a co-$A_\infty$-structure. It follows that it too desuspends.