2
$\begingroup$

I have an exercises as follows: Let $E$ be a trivial bundle on $S^n$. Prove that the Whitney sum $TS^n\oplus E$ is also trivial. The hint is using the normal bundle of $TS^n$, but I don't know how to use it. Some one can help me? Thanks a lot!

$\endgroup$
1
  • 1
    $\begingroup$ Hint: If $i:S^n\rightarrow \mathbb{R}^{n+1}$ is the usual embedding, then $i^*T\mathbb{R}^{n+1}\cong TS^n \oplus \nu$ where $\nu$ is the normal bundle. $\endgroup$ – Jason DeVito Nov 13 '12 at 20:37
4
$\begingroup$

You just notice that the Whitney sum of normal bundle and tangent bundle is trivial...As they add up to be the underlying Euclidean space of your sphere. And give a diffeomorphism between the line bundle and normal bundle. Actually you can use this to prove that the product of two spheres, one is odd dimensional, must have a trivial bundle.

$\endgroup$
1
  • $\begingroup$ My English isn't well... Hope you can understand what I mean $\endgroup$ – lee Nov 24 '12 at 3:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.