# manifolds – bundles of trivial lines outside codimension 3

Let $$X$$ to be a complex CW (possibly a topological / smooth variety) of dimension $$n$$, $$L to X$$ a complex bundle of lines and $$Y subset X$$ a sub-complex (possibly a submultiple) contained in the codimension 3 skeleton of $$X$$, such as $$L$$ is trivial about $$X-Y$$. Is it true in this case that $$L$$ is trivial about the whole $$X$$?

This obviously applies to the linear holomorphic bundles of smooth manifolds, in which case this corresponds to the complex codimension 2, but I would like to be able to replace holomorphic by smooth or continuous.