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Tyrone
Tyrone
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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.

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.

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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Tyrone
Tyrone
  • 5.6k
  • 1
  • 29
  • 50

Reference request: Basic H-Space properties of $SO(3)$

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.