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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 23rd 2021

added the statement that

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

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

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 24th 2021

added pointer to p. 403 in

where this statement appears somewhat between the lines.

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

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

• 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 $n$Lab.

• 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.

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