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Let $M$ be a smooth manifold. How do I show that the tangent bundle $TM$ of $M$ is orientable?

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so sorry for my writing a question here, @ you for 2) in your answer, how do we know that "tau_M" and "xi" are orthogonal to each other, i.e. why each of the fibers of tau are orthogonal to those of xi? –  wqr May 12 '12 at 8:43
    
actually I read Characteristic Classes but I could not get a clue about the vectors in "fiber direction" and "manifold direction" –  wqr May 12 '12 at 8:51
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2 Answers

up vote 5 down vote accepted

Start with an atlas for $M$ and construct the corresponding atlas on $TM$, each of whose charts is constructed from one on $M$. Check that the transition functions in the latter have Jacobian with positive determinant.

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This is actually a nice computation to make! –  Mariano Suárez-Alvarez Apr 9 '12 at 3:40
    
that was my first approach to the question, I thought that there was a easier method... –  Jr. Apr 9 '12 at 3:52
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You should have told us that... –  Mariano Suárez-Alvarez Apr 9 '12 at 4:11
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(You can decompose $TTM$ as a vector bundle over $TM$ as the direct sum of $\pi^*(TM)$ (the pullback of $TM\to M$ along $\pi:TM\to M$) and the subbundle of $TTM\to TM$ of vertical vectors (those which are tangent to the fibers of $TM\to M$), and show that the determinant bundles of these two subbundles are isomorphic, so that the determinant bundle of $TTM$ is the tensor square of a line bundle. This implies orientability... This is just the same thing done complicatedly.) –  Mariano Suárez-Alvarez Apr 9 '12 at 4:13
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There is also a way to see it with fibre bundles and characteristic classes. It's possible the original poster is not familar, but other people might be and more importantly I need practice. It is based on two relatively basic facts (at least I'm mostly sure I've seen them before):

1) A smooth $n$-manifold $M$ is orientable iff the first Stiefel-Whitney class of its tangent bundle $\tau_M$ vanishes,

and

2) If $\xi$ is a smooth $k$-plane bundle with base space $M^n$, total space $E^{n+k}$ (both smooth manifolds) and projection $\pi:E\rightarrow M$, then $$\tau_E=\pi^*(\tau_M)\oplus\pi^*(\xi)$$

Then, if $TM$ is the total space of the $n$-plane bundle $\tau_M$ with projection map $\pi\colon TM\rightarrow M$, it is a smooth manifold with its own tangent bundle $\tau_{TM}$. Since $\pi$ is the projection map of $\tau_M$ we have $$\tau_{TM}=\pi^*(\tau_M)\oplus\pi^*(\tau_M)$$ so by the Whitney product formula $$\omega_1(\tau_{TM})=(\pi^*\omega_0)(\tau_M)\cup(\pi^*\omega_1)(\tau_M)+(\pi^*\omega_1)(\tau_M)\cup(\pi^*\omega_0)(\tau_M)=2\pi^*\omega_1(\tau_{M})=0\in H^1(TM;\mathbb{Z/2})$$

Hence the manifold $TM$ is orientable.

(But for all intents and purposes, writing down charts is the easiest way to go)

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This is the characteristic-class version of the comment to my answer :) Much preferable! –  Mariano Suárez-Alvarez Apr 9 '12 at 4:26
    
@you In which book should I find this content? thanks. –  Jr. Apr 9 '12 at 4:33
    
"Characteristic Classes" by John Milnor and James Stasheff. They treat characteristic classes axiomatically and develop some basic theory, and then later show that they exist. There's a lot of good content in there about characteristic classes and fibre bundles. If you want to learn more about the general theory of fibre bundles, check out Norman Steenrod's "Topology of Fibre Bundles." And it's probably also in Hatcher somewhere. –  you Apr 9 '12 at 4:35
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