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Lefschetz fixed point theorem (for example) implies that a compact manifold with a nowhere vanishing vector field must have Euler characteristic zero. Is there a way to draw stronger algebraic conclusion from the existence of a basis of vector fields / a trivial tangent bundle?

I would like to apply something like that to show the tangent bundle of the Klein bundle is nontrivial (which I guessing to be the case from trying to draw some), but I don't know any relevant theorem.

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    $\begingroup$ For the second part, the determinant bundle $\det T_* K = \Lambda^2 T_* K \to K$ is nontrivial, since a trivialization of it (i.e., a nowhere-vanishing section) is exactly an orientation of $K$. Thus $T_* K$ itself is nontrivial. For the first part, what exactly do you have in mind? (For help in searching, a manifold with trivial tangent bundle is called 'parallelizable'. Any Lie group $X$, for example, is parallelizable; consider the maps $DL_g: T_1 X\to T_g X$ for $L_g(x) = gx$.) $\endgroup$ – anomaly Aug 20 '15 at 4:24
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A smooth manifold whose tangent bundle is trivializable is called parallelizable or frameable. On such a manifold, not only does the Euler characteristic vanish, but all characteristic classes vanish. Hence the nonvanishing of any characteristic class prevents a manifold from being parallelizable.

In particular, the Klein bottle is non-orientable, so its first Stiefel-Whitney class $w_1$, which measures non-orientability, doesn't vanish. More directly you can use the argument anomaly gives in the comments: a smooth manifold is orientable iff the top exterior power of its tangent bundle is trivializable.

There are more precise things you can say if you know that there are, say, $k$ linearly independent vector fields; this is a good exercise in characteristic classes.

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  • $\begingroup$ Where can I learn about characteristic classes? This has always been a scary word for me. $\endgroup$ – Lorenzo Najt Aug 20 '15 at 16:51
  • $\begingroup$ @AreaMan: the standard reference is Milnor and Stasheff (books.google.com/books/about/…). $\endgroup$ – Qiaochu Yuan Aug 20 '15 at 17:16

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