I'm currently working through Nakahara's book and I've hit a snag on exercise 9.1.
Consider the real line bundle $ L $ of $ S^1 $ $ \left ( L,S^1,\pi,\mathbb{R}, G \right ) $
$ L $ is either the cylinder bundle or the Möbius bundle.
Consider the Whitney sum $ L \oplus L $
If $ L $ is the cylinder bundle $ S^1 \times \mathbb{R} $, it is straight forward from the triviality of this bundle to show that the Whitney sum is a trivial bundle as well.
However, in the exercise I am required to show that the Whitney sums of both of these line bundles are trivial.
I found that the structure group of the sum of the Möbius bundles was nontrivial and therefore I think I have to find that the associated principle bundle admits a global section. I think this is where I am getting confused. Someone has pointed out to me that the Whitney sum of Möbius bundles is a Möbius strip with an extra twist (which is apparently a trivial bundle, I haven't shown this yet either). This hasn't really clicked for me yet so an intuitive argument for that could help
The original question was:
"Let $ L $ be the real line bundle over $ S^1 $ (i.e. $ L $ is either the cylinder $ S^1 \times \mathbb{R} $ or the Möbius strip). Show that the Whitney sum $ L \oplus L $ is a trivial bundle. Sketch $ L \oplus L $ to confirm the result. "
