Is $\mathbb{H}P^\infty$ an H-space or not?

Question

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?

Answer 1

First, let me make the weaker claim that $\mathbb{HP}^{\infty}$ is not naturally an H-space. The natural H-space structures on $\mathbb{RP}^{\infty}$ resp. $\mathbb{CP}^{\infty}$ come from the fact that they classify isomorphism classes of real resp. complex line bundles, which naturally have group structures given by taking the tensor product. This no longer holds for quaternionic line bundles since $\mathbb{H}$ is no longer commutative, so there's no longer an obvious natural candidate for an H-space structure on $\mathbb{HP}^{\infty}$.

But in fact there is no H-space structure at all. The reason is that, as observed by Alex Youcis in the comments, we have $\mathbb{HP}^{\infty} \cong BSU(2)$ and hence $\Omega \mathbb{HP}^{\infty} \cong SU(2)$, so if $\mathbb{HP}^{\infty}$ had an H-space structure then the H-space structure on $SU(2)$ coming from its Lie group structure would be homotopy commutative by the Eckmann-Hilton argument. But Araki, James, and Thomas showed that no compact connected nonabelian Lie group is homotopy commutative as an H-space.