Physics Notes by Jakob Schwichtenberg

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group_theory:groups:so3

SO(3)

Topology of SO(3)

A great description can be found at page 409 in Lorentz Group, CPT and Neutrinos. Proceedings of the International Workshop Zacatecas, Mexico, 23 – 26 June 1999. Edited by: A E Chubykalo.

The rotation group SO(3) is not simply connected, as foreseen in Chap. 00. To see this fact, represent the rotation Rn(ψ) by the point x = tan ψ 4 n of an auxiliary space R 3 ; all these points are in the ball B3 of radius 1, with the identity rotation at the center and rotations of angle π on the sphere S 2 = ∂B3 , but because of Rn(π) = R−n(π) , (see Chap 00, sect. 1.1), diametrically opposed points must be identified. It follows that there exists in SO(3) closed loops that are non contractible: a path from x to x passing through two diametrically opposed points on the sphere S 2 must be considered as closed but is not contractible (Fig. 1.2.b). There exist two classes of non homotopic closed loops from x to x and the group SO(3) is doubly connected. Its homotopy group is π1(SO(3)) = Z2. In fact, we already know the universal covering group of SO(3): it is the group SU(2), which has been shown to be homeomorphic to the sphere S 3 , hence is simply connected, and for which there exists a homomorphism mapping it to SO(3), according to ±Un(ψ) = ±(cos ψ 2 − isin ψ 2 σ.n) 7→ Rn(ψ),

This is also discussed nicely at page 164 in the book Magnetic Monopoles by Shnir.

$SO(3)$ is still a sphere, but antipodal points are identified!