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").

3more comments