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I dug into the literature but could not find references for some of the basic H-space properties of $SO(3)$. Basic properties that I am looking for include

  • What H-maps are there $SO(3)\rightarrow S^3$?
  • For what integer $N$ is the $N^{th}$ power map $N:SO(3)\rightarrow SO(3)$ an H-map?
  • What is the group $[SO(3)\times SO(3),SO(3)]$?
  • What is the known about the commutator $c:SO(3)\wedge SO(3)\rightarrow SO(3)$?

If anyone knows where calculations of these things - or any other relevant interpreting tidbits - may be found it would be very welcome. I was surprised not to find any of this basic information.

Something I did find was Naylor's paper "Multiplications on $SO(3)$" (there are 768), which calculates order the homotopy set $[SO(3)\wedge SO(3),SO(3)]$ to obtain its conclusion. Also James's numerous papers on homotopy commutativity are of some interest and feel free to include with this Hamanaka's "Homotopy-Commutativity in Rotation Groups" as relevant.

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    $\begingroup$ Regarding the first point: If we consider the composition $S^3 \to SO(3) \to S^3$, this has even degree. At least if the composite $f\colon S^3 \to S^3$ is a group homomorphism, it induces a map $Bf\colon \mathbb{HP}^\infty \to \mathbb{HP}^\infty$ of classifying spaces. If I understand it correctly, it must have odd degree when it is non-constant by Feder, S.; Gitler, S. Mappings of quaternionic projective spaces. This suggests that every $H$-map $SO(3) \to S^3$ might be constant. $\endgroup$
    – Lennart Meier
    Commented Sep 27, 2016 at 9:29
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    $\begingroup$ I'm not sure that an H-map need be a loop map. $\endgroup$
    – Tyrone
    Commented Sep 27, 2016 at 16:18
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    $\begingroup$ Here are two papers with relevant information. Arkowitz,Ewing,Schiffman, Quart. J. Math. 26 (1975) 295-307; McGibbon,same journal 31 (1980) 341-350. One consequence is that the degree of an H-map from SO(3) to S^3 must be congruent to 0 or 8 mod 12. $\endgroup$
    – John R. Harper
    Commented Feb 16, 2017 at 14:26

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