TITLE: Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it?
QUESTION [13 upvotes]: When I read the GTM 082, "The Splitting Principle of the complex vector bundle", I see that in the proof we split off one subbundle at a time by pulling back to the projectivization of a quotient bundle. I don't know why we need to pull this back to the projectivization of the quotient bundle. I mean if there is always a line bundle embeded in the given complex vector bundle, we can just split it on the original vector bundle, every time we take quotient of these two bundle, we get a quotient bundle with dimensional reduced by 1. So in the similar way we can conclude that every bundle can be decomposed into the direct sum of several line bundle. So is it true? Can we always find a line bundle embedded into a given complex vector bundle?

REPLY [8 votes]: I think thinking in terms of classifying spaces will help clarify the situation. We know that a rank $n$ complex vector bundle $V$ on $X$ is the same thing as a homotopy class of maps $f:X\to BU_n$. A choice of decomposition $V\cong L\oplus W$ where $L$ is a line bundle corresponds to a lift of $f$ to $B(U_1\times U_{n-1})\cong BU_1\times BU_{n-1}$. The homotopy fiber of the map $BU_1\times BU_{n-1}\to BU_n$ is $U_n/(U_1\times U_{n-1})\cong \mathbb{CP}^{n-1}$. This is a non-trivial principal fibration and so it has no section.
In particular you can take the universal vector bundle on $BU_n$ (or if you want, its restriction to some finite skeleton given by a suitable complex Grassmannian) and this will give you a vector bundle that admits no line bundle.
Note that $BU_1\times BU_{n-1}$ is precisely the projectivization of the universal vector bundle on $BU_n$, so this is just another way of phrasing the splitting principle.
More precisely you can find a sequence of obstructions in $H^{i+1}(X;\pi_i\mathbb{CP}^{n-1})$ for $i\ge2$ and you can find a sub-line bundle if and only if these obstruction vanish. For example you can always find a such a line bundle if the (cohomological) dimension $X$ is less or equal than 2