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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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up vote 5 down vote accepted

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

You can also see this by the fact that any sub bundle has a complement (choose a metric and take the orthogonal complement) and now use that a bundle is orientable iff its first stiefel-whitney class vanishes and that this class is additive under direct sums.

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