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Suppose $\Sigma^2\subset M^3$ is an one-sided, non-orientable embedding. Let $\pi:\tilde{\Sigma}\rightarrow\Sigma$ be the orientable double cover. Is it necessary that the pull back of normal bundle $\pi^*N\Sigma$ is a trivial bundle on $\tilde{\Sigma}$?

I know an answer here: Pullback of normal bundle by a covering map.. However, it does not work. It just shows that manifolds that is not simply connected admit a non-trivial line bundle. But our question here is a special line bundle, the above counterexample does not show that it is non-trivial.

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    Let $K$ be the Klein bottle. The covering map $S^1 \times S^1 \rightarrow K$ is nontrivial on first homology, correct? If so, I believe you can take the embedding of $K$ into the total space of the line bundle over $K$ with $w_1$ in the image of the covering. – Connor Malin Aug 20 '22 at 18:15

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