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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 24th 2021

added brief mentioning of the vertical tangent bundle and statement of the splitting formula

$T P \;\simeq\; \big( \pi^\ast T B \big) \oplus_P \big( T_\pi P \big)$
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 24th 2021

added more of an actual definition of the vertical tangent bundle (here, need to give a more canonical reference…)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 24th 2021

added statement of the example of the vertical tangent bundle of a vector bundle (here)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 24th 2021

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 27th 2021

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 27th 2021

• William Gollinger, Section 1.1.4 in: Madsen-Tillmann-Weiss Spectra and a Signature Problem for Manifolds, Münster 2016 (pdf)

as one place where the characterization of the vertical tangent bundle of a sphere-fiber bundle is made explicit

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 1st 2021

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 1st 2021

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 1st 2021

I have made more explicit how the decomposition $T P \simeq T_\pi P \oplus \pi^\ast T B$ comes from the fact that the defining short exact sequence of vector bundles $0 \to T_\pi P \longrightarrow T P \longrightarrow \pi^\ast T B \to 0$ splits.

Maybe we should have an entry short exact sequence of vector bundles.