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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017

    I wrote out some elementary details at basic complex line bundle on the 2-sphere.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2017

    Isn’t it just the Hopf fibration S1S3S2?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017
    • (edited May 28th 2017)

    Sure, it’s the complex line bundle canonically associated to the complex Hopf fibration.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2017

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2025
    • (edited Jun 6th 2025)

    Added a brief note (here) that the +1 eigenspaces of the family of Dirac operators DxijxjσjEnd(2) (for σ the Pauli matrices) as xS23, form the basic line bundle on S2. Still need to give a more canonical reference (or write out the proof).

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2025
    • (edited Jun 7th 2025)

    I have spelled out (here) a full proof, elementary as it may be

    (of the fact the basic line bundle on S2 is the +1 eigenspace bundle of the S2-family of operators xiixiσi, for σ the Pauli matrices).

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2025
    • (edited Jun 7th 2025)

    added the observation (here) that the induced S2-parameterized Fredholm operator S2Fred() (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.

    diff, v16, current