Talk:Tangent bundles of bundles (Ex)
Solution 0.1.
There is a short exact sequence of vector bundles over
![\displaystyle 0 \to T_{\pi}E \stackrel{i}{\rightarrow} TE \stackrel{j}{\rightarrow} \pi^*TB \to 0](/images/math/1/6/4/1646e3eb272f432219c1953bc75dcebe.png)
where is the inclusion and
is defined by
. We choose a Riemannian metric on
, then the orthogonal projection to
gives a splitting of the exact sequence. Which implies that
![\displaystyle TE \cong \pi^*TB \oplus T_{\pi}E](/images/math/9/9/f/99f8ef8a484325d0ef22c79dbe461aed.png)
Solution 0.2.
A necessary condition for being the pullback of some bundle over
is that
is trivial. This is seen by restricting the bundle to a point in
. On the other hand, obviously when
is a vector bundle or the bundle
is trivial then
is the pullback of a vector bundle over
. We don't know if this is true in general.
Solution 0.3.
If is itself a smooth vector bundle, then
, therefore
![\displaystyle TE \cong \pi^*TB \oplus T_{\pi}E\cong\pi^*(TB\oplus E).](/images/math/6/5/2/6529c214004400a08b57646d665bf9f9.png)
Solution 0.4.
Denote the associated vector bundle by with projection
, then
![\displaystyle TE \oplus \underline{\mathbb R} = i^*\pi_{\xi}^*(TB \oplus \xi),](/images/math/8/0/d/80df8a51befda50ad5f9a66d005522a5.png)
where is the inclusion of the sphere bundle into the vector bundle.
Solution 0.5.
Apply the description of the tangent bundle of sphere bundles in the previous solution to the sphere bundle we get
, where
is the associated vector bundle. Now taking Pontrjagin classes on both sides, we get an identity (since there is no torsion in cohomology)
![\displaystyle p(\CP^{2k+1})=\pi^*(p(\Hh P^k) \cdot p(\xi)).](/images/math/8/7/e/87e7d25470c060eb6eb1d90fd8ad827f.png)
We know that and
, where
is determined by restricting to the case
(where
whose Pontrjagin class is known to be trivial):
with
a generator. Therefore in
we have the equation
![\displaystyle (1+x^2)^{2k+2}(1+4x^2)^{-1}=\pi^*p(\Hh P^k).](/images/math/2/4/2/24261985f969aaca862d87fead33a01c.png)
It's seen from the Gysin sequence that is injective. This is sufficient to determine
.