H-space structures on non-sphere suspensions? 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).
 A: Here are some comments about the case where $X$ is not assumed to have finite type.  Put $Y=\Sigma X$.  For any field $K$, the groups $H_*(Y;K)$ form a Hopf algebra in which all elements of the augmentation ideal are primitive.  If $u$ and $v$ lie in the augmentation ideal, then $u$, $v$ and $uv$ are all primitive, which gives $u\otimes v+(-1)^{|u||v|}v\otimes u=0$.  If $u$ and $v$ are nonzero, it follows that $|u|=|v|$ and $Ku=Kv$, and $|u|$ is odd unless $K$ has characteristic $2$.  Thus, $H_*(Y;K)$ is either $K$ or $K\oplus Ku$ for some element $u$, usually of odd degree.  In particular, we see that $Y$ is rationally trivial or an odd-dimensional sphere.
Now consider the groups 
\begin{align*}
 A(p)_k &= \widetilde{H}_k(Y;\mathbb{Z}_{(p)}) \\
 B(p)_k &= A(p)_k/p \\
 C(p)_k &= \widetilde{H}_k(Y;\mathbb{Z}/p) \\
 D(p)_k &= \text{ann}(p,A(p)_{k-1})
\end{align*}
so that 


*

*$A(p)\otimes\mathbb{Q}$ has dimension $0$ or $1$ over $\mathbb{Q}$

*$C(p)$ has dimension $0$ or $1$ over $\mathbb{Z}/p$

*There is a short exact sequence $B(p)\to C(p)\to D(p)$, so $B(p)$ and $D(p)$ have dimension $0$ or $1$, and at least one of them is zero.


There are a number of different possibilities here.


*

*If $D(p)=0$ then $A(p)$ is torsion-free and so injects in $A(p)\otimes\mathbb{Q}$.  This means that $A(p)=0$ or $A(p)\simeq\mathbb{Z}_{(p)}$ or $A(p)\simeq\mathbb{Q}$.

*If $B(p)=0$ then $A(p)$ is divisible, and so is injective as a $\mathbb{Z}_{(p)}$-module.  If we let $T(p)$ denote the torsion part of $A(p)$ then we find that $T(p)$ is also divisible and therefore injective and therefore a summand in $A(p)$.  We therefore have $A(p)=T(p)\oplus Q(p)$, where $Q(p)$ is a $\mathbb{Q}$-module.  It follows that $A(p)\otimes\mathbb{Q}\simeq Q(p)$, so $Q(p)$ is $0$ or $\mathbb{Q}$.  If $D(p)=0$ then $T(p)=0$.  If $D(p)=\mathbb{Z}/p$ then I think it follows that $T(p)=\mathbb{Z}/p^\infty$.


The most interesting question arising from this analysis is as follows.  Let $Y$ be the Moore space with $H_2(Y)=\mathbb{Z}/p^\infty$ for some prime $p$.  To avoid trouble from the low-dimensional homotopy groups of spheres, we may want to take $p\geq 5$.  Note that $\widetilde{H}_*(Y;K)=0$ unless $K$ has characteristic $p$, and that $\widetilde{H}_*(Y;\mathbb{Z}/p)$ is a copy of $\mathbb{Z}/p$ in dimension $3$.  Does $Y$ have an $H$-space structure?  I do not see an easy way to answer that.  Note that $Y\wedge Y$ is a Moore space with $H_5(Y\wedge Y)=\mathbb{Z}/p^\infty$, but that does not immediately give a good hold on $[Y\times Y,Y]$. 
A: 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, 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, 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.
