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So, I'm a little confused about one such statement made in class today : if M is a smooth manifold without boundary such that the tangent bundle of M is trivial, then M is orientable. Is this always true ?

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If you think of an orientation of a manifold as an orientation of the tangent space at each point which varies continuously, then if your tangent bundle is of the form $M\times \mathbb R^n$, you can use a fixed orientation on $\mathbb R^n$ to orient each $T_pM$. So yes, this is true in general. :)

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Always ?, is it possible for me to find a counterexample to this statement ? –  user8169 Mar 12 '11 at 23:14
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@Danny: The proof implies it is always true, so there will not be a counterexample. Maybe if you explain why you think this statement is dubious, I (or someone else) could better address your concerns. Also, you probably realize that this is not an "if and only if." There are lots of orientable manifolds with twisted tangent bundles. –  Grumpy Parsnip Mar 13 '11 at 0:39
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Like $S^2$ .... –  Ryan Budney Mar 13 '11 at 3:35
    
@Derek: Did I call you Danny by accident? I'm so sorry about that! –  Grumpy Parsnip Mar 22 '11 at 17:50
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