Is the 3-sphere isomorphic to Spin(3)?

Question

I am trying to learn some spin geometry stuff and getting a bit confused. The unit quaternions can be thought of as a group structure on $S^3$ which gives the group $\text{Spin}(3)$.

Is there some sense in which $S^3$ is isomorphic to or equal to $\text{Spin}(3)$?

Answer 1

Yes. This group appears to possess a bunch of alternative definitions, like

• The unit quaternions
• The special unitary group $\text{SU}(2)$
• The spin group $\text{Spin}(3)$

See https://en.wikipedia.org/wiki/Spin_group#Accidental_isomorphisms