# Does an orientable subbundle of an orientable vector bundle always have a orientable complement?

If I have an orientable vector bundle $E$ and a subbundle $F$ on a manifold $M$, where both the bundles are orientable, does $F$ have a complement in $E$ which is also orientable? Does it have a complement bundle at all? That is, a subbundle of $E$ that is pointwise a complement of $F$.

What if $F$ is of codimension $1$? Is the complement always trivial in this case?

Does anything change if $E$ is specifically the tangent bundle $TM$?

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## 2 Answers

If $E=F\oplus G$, then $\Lambda^\det E=\Lambda^\det F\otimes\Lambda^\det G$. If both $\Lambda^\det E$ and $\Lambda^\det G$ are trivial, then so is $\Lambda^\det F$, because the three being line bundles, you can «divide by $\Lambda^\det G$» in that equality.

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By $\Lambda^\det$ I mean the maximal exterior power here. – Mariano Suárez-Alvarez♦ Aug 25 '12 at 3:36
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