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I am reading chapter on Characteristic classes from the book Vector bundles and K-theory by Allen Hatcher.

When giving motivation for what does characteristic classes measure author says that

The first obstacle to triviality is orientability.

I do not understand what it means. I know what is it to say vector bundle is trivial and I know what is it to say avector bundle is orientable.

But, I do not understand what it means to say first obstacle to triviality is orientability.

Any thoughts on this is welcome.

  • This answer explains very well. (And it tells you about the "second" obstruction! And so on.) (You may know that the word "parallelizable manifold" means that the tangent bundle of the manifold is trivial.) https://math.stackexchange.com/questions/46297/which-manifolds-are-parallelizable/46306 – Eric Auld Jul 16 '19 at 16:44

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If you are familiar with trivial and orientable bundles, then I assume you have seen a result that a trivial bundle is orientable. This implies that in order for a bundle to be trivial, it has to be orientable; if orientability fails, so does triviality. This is a simple observation, so it is a "first obstacle". It is just a way of saying that orientability is a necessary but insufficient condition for triviality.

Joonas Ilmavirta
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  • Thank you. I did not see yet the result that trivial bundle is orientableyet but I feel it has to be true. Thanks. –  Feb 21 '18 at 16:24
  • @PraphullaKowshik I'm glad to be able to help. That explains the confusion. I would have expected to have that result proven before stating what you quoted. The proof might also be an exercise; it is good exercise for getting used to bundles, and I warmly recommend trying to prove it. Please ask a follow-up question if you have trouble finding a proof. – Joonas Ilmavirta Feb 21 '18 at 16:28
  • Yes yes. Sure. :) –  Feb 21 '18 at 16:41