Tangent bundles of bundles (Ex)

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Let F \to E \stackrel{\pi}{\to} B be a smooth fiber bundle so that all spaces F, E and B are manifolds.

Exercise 0.1. Show that TE, the tangent bundle of E, splits as the sum of two bundles

\displaystyle  TE \cong \pi^*TB \oplus T_{\pi}E

where T_{\pi}E consists of those tangent vectors tangent to the fibres of \pi.

Question 0.2. Is the bundle T_{\pi}E the pullback of some bundle over B?

Exercise 0.3. Suppose that \pi \colon E \to B is itself a smooth vector bundle. Determine TE in terms of TB and \pi regarded as a vector bundle.

Exercise 0.4. Suppose that \pi \colon E \to B is the sphere bundle of a vector bundle. Determine the stable tangent bundle of E in terms of \pi and TB.

As an explicit example, recall that there are principal bundles S^1 \to S^{4k+3} \to \CP^{2k+1} and S^3 \to S^{4k+3} \to \Hh P^k. It follows that there is an S^2 fibre bundle S^2 \to \CP^{2k+1} \to \Hh P^k with structure group S^3 acting on S^2 via S^3 \to S^3/S^1 \cong S^2.

Exercise 0.5. Compute the total Pontrjagin class of \Hh P^n, quaternionic projective space. (This was first achieved in [Hirzebruch1953]).
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