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](https://mathoverflow.net/questions/3740/cogroup-objects) 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").



When can you desuspend a homotopy cogroup?
Whilst the n-lab page on [co-H-spaces](http://ncatlab.org/nlab/show/co-H-space) could be described as a little meagre, it does nonetheless contain a reference to a paper in the Handbook of Algebraic Topology. Various parts of this book are in the "preview" in google books, in particular page 1153 which contains the magical sentence:



> 
> We can now construct cell complexes $S^n \cup\_\alpha e^m$ which are co-H-spaces but not suspensions.
> 
> 
> 


This follows a theorem which classifies when such spaces are co-H-spaces and suspensions.


The specific article is the aptly named "Co-H-spaces" by Martin Arkowitz, and is chapter 23 of Handbook of Algebraic Topology, edited by Ioan James. I would recommend this as a first place to look to find out more about these spaces.


Incidentally, I don't believe your remark about delooping. I'm almost certain that there are H-spaces that can't be delooped so either there are H-spaces that aren't $A\_\infty$-groups or your claim is incorrect. Unfortunately, I'm not currently surrounded by my usual stock of algebraic topology textbooks so can't look this up (and you can do an internet search as well as I can). Can you supply a reference for the delooping claim?


