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). 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).