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Let $M$ be the Moebius vector bundle over $S^1$.

Is it possible to embedd the total space of $M\oplus (S^1\times \mathbb{R^1})$ over $S^1$ into $\mathbb{R}^3$?

I suppose this isn't possible but I don't know an argument.

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It's not possible, since the Whitney sum would be a nonorientable $3$-manifold, which cannot be embedded in $\mathbb R^3$.

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    Invariance of domain, that's what I meant!2012-02-13