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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!

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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. –  Jason DeVito Nov 13 '12 at 20:37
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1 Answer 1

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.

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My English isn't well... Hope you can understand what I mean –  lee Nov 24 '12 at 3:02
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