It is well known that $S^n$ admits an H-space structure if and only if $n=0,1,3,7$. I'm interested in whether there are other suspensions $\Sigma X$ that admit H-space structures:


**Question 1** For which $X$ (not a sphere) is $\Sigma X$ an H-space? And what about $\Sigma X$ that are *associative* H-spaces?


My motivation is that we have a construction (in the framework of homotopy type theory, and presumably portable to a wide range of model categories) that gives an H-space structure on the join $\Sigma X \* \Sigma X$ whenever $\Sigma X$ has a homotopy-associative H-space structure (that is compatible with an involution on $X$ – for details, see [these slides](http://www.andrew.cmu.edu/user/ulrikb/mpim-2016.pdf)). Thus, it would be interesting to know some more examples where this construction applies. 


This also leads to a follow-up question (mostly in case the answer to Q1 is “none”):


**Question 2** If we go to a localization, do we get more answers to Q1? What about in other (non-stable) model categories?


Any references would be appreciated.


Finally, let me share a little scratch-work in trying to answer Q1 (feel free to ignore!): Any H-space structure $\mu : \Sigma X \times \Sigma X \to \Sigma X$ gives rise a Hopf map $H(\mu) : \Sigma X \* \Sigma X \to \Sigma^2 X$. Assuming $X$ is pointed, we get a map $\Sigma^3(X \wedge X) \to \Sigma^2 X$. Applying $K$-cohomology to the cofiber sequence should yield information restricting $X$, but I failed to get much milage out of it without a notion of “bidegree” for $\mu$ (generalizing the case of $X$ a sphere).



H-space structures on non-sphere suspensions?
If $Y$ is a connected CW-complex of finite type which is both an H-space and a co-H-space, then $Y$ has the homotopy type of $S^1$, $S^3$, $S^7$ or a point. This is a result of Robert West:


*Robert W. West*, [**$H$-spaces which are co-$H$-spaces**](http://www.ams.org/mathscinet-getitem?mr=285006), *Proc. Amer. Math. Soc.* **31** (1972), 580--582.


It follows that if $X$ is a finite type CW-complex such that $\Sigma X$ is an H-space, then $\Sigma X$ is homotopy equivalent to one of these spaces.


On the other hand, Adams and Walker give an example of a $4$-dimensional infinite CW-complex $Y$ which is both an Eilenberg--Mac Lane space and a Moore space of type $(\mathbb{Q},3)$:


*J. F. Adams and G. Walker*, [**An example in homotopy theory**](http://www.ams.org/mathscinet-getitem?mr=166786), *Proc. Cambridge Philos. Soc.* **60** (1964), 699--700. 


This $Y$ is a suspension by construction, and an H-space by virtue of being an Eilenberg--Mac Lane space of an abelian group.


