The 2-to-1 mapping ψ of SU(2) onto SO(3) found in Section 19.7 is more than merely a group homomorphism. It is also a mapping of the manifold of SU(2) onto the manifold of SO(3) of the kind known as a covering. It is locally a homeomorphism, in the sense that if P is any point on the manifold of SU(2), and Q is its image in the manifold of SO(3), then P has a neighborhood that is mapped homeomorphically by ψ onto a neighorhood of Q. Furthermore, if Q is any point of the second manifold, then there are always two points P in the first manifold, each of which has such a neighborhood. ψ is a two-sheeted covering of SO(3) by SU(2).
KeywordsLocal homeomorphism Projection p-sheeted covering good neighborhood principles of lifting universal covering manifold construction of mathematical models manifolds covered by a given manifold
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