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I wrote out some elementary details at basic complex line bundle on the 2-sphere.
Isn’t it just the Hopf fibration S1→S3→S2?
Sure, it’s the complex line bundle canonically associated to the complex Hopf fibration.
I added in the description that this is the tautological line bundle of the complex projective line, which I think is probably the simplest description. Whether that belongs in the Idea section or not, I’ll let someone else decide.
added the observation (here) that the induced S2-parameterized Fredholm operator S2⟶Fred(ℋ) (which represents the basic line bundle as an index bundle) is naturally Spin(3)≃SU(2)-equivariant and as such should represent even the equivariant tautological line bundle on the 2-sphere in equivariant K-theory.
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