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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.

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

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 $BU_1\times BU_{n-1}\times BU_n$ 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.

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.

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 $BU_1\times BU_{n-1}\times BU_n$ 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.

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 $BU_1\times BU_{n-1}\times BU_n$ 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.