I was inspired by the following algebraic topology orals question:

"Is $S^1$ the loop space of another space?"

This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop space of any $K(G,n)$ is a $K(G,n-1)$.

I then also remembered that the loop space functor is a functor from pointed topological spaces and continuous maps to the category of H-spaces and continuous homomorphisms. H-spaces being topological spaces that satisfy the axioms of a group up to homotopy (see Spanier, Chapter 1, Section 5).

I have three questions:

- Is there a useful criterion for when an H-space is actually a topological group?
- Seeing that $S^1$,$S^3$, and $S^7$ are the only spheres that support group structures, it doesn't seem coincidental that $S^1$ is a loop space, because it is in fact an H-space. Since $CP^{\infty}$ is the loop space of $K(Z,3)$ it too is an H-space, but is it known if it is a topological group?
- Even if not, is there a way (other than concatenation of loops) to "see" this structure on $CP^{\infty}$?

Thanks!

notsupport a homotopy-associative H-space structure. I.M. James proves (Trans. AMS, March 1957) in "Multiplication on Spheres (II)", Theorem 1.4: "There exists no homotopy-associative multiplication on S<sup>n</sup> unless n = 1 or 3." $\endgroup$