First obstacle to triviality is orientability

Question

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.

Answer 1

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.