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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2021

    added the statement that

    The stable tangent bundle of a unit sphere bundle S(𝒱)S(\mathcal{V}) in a real vector bundle 𝒱pM\mathcal{V} \overset{p}{\longrightarrow} M (Example \ref{UnitSphereBundles}) over a smooth manifold MM is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

    T stabS(𝒱)S(p) *(T stabM M𝒱). T^{stab} S(\mathcal{V}) \; \simeq \; S(p)^\ast \big( T^{stab} M \oplus_M \mathcal{V} \big) \,.

    Still need to add a more canonical reference and/or a proof.

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    added pointer to p. 403 in

    where this statement appears somewhat between the lines.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021
    • (edited Mar 24th 2021)

    I have written out (here) a proof of this Milnor-trivial statement

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2021
    • (edited Mar 25th 2021)

    I have spelled out (here) a purely homotopy-type theoretic proof that the once-stabilized vertical tangent bundle to a sphere bundle associated to a vector bundle is the pullback of that vector bundle.

    (This is, somewhat implicitly, from Sec. 3 of our Twisted Cohomotopy implies M5-brane anomaly cancellation. Making it more explicit now in v2.)

    Incidentally, the tikzd diagrams don’t all come out scaled quite as intended: it seems that scaling just the columns with, say, [colum sep=tiny], scales also the rows, here on the nnLab.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2021

    added a concluding remark, to highlight:

    Prop. \ref{StableTangentBundleOfUnitSphereBundle} implies that every stable characteristic class of the tangent bundle of an orthogonal sphere-fiber bundle – i.e all polynomials in its Pontryagin classes – are basic, i.e. pulled back from the base space.

    diff, v10, current

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