I am trying to understand Stiefel-Whitney classes as obstructions for the existence of linearly independent sections of a vector bundle $E \to B$, that is a section of the Stiefel bundle $V_k (E) \to B$.
Ultimately I am interested in the problem of whether a vector bundle is trivialisable. This should be given by a section of the $n$-frames bundle $V_n (E) \to B$, where $n = \dim E$.
According to eg Hatcher, $w_i(E)$ is the mod 2 first obstruction to the existence of $n−i+1$ linearly independent sections of $E$. For $n=k$, the first Stiefel-Whitney class should measure the existence of $n$ linearly independent sections but what it measures is the orientability of $E$ (and there are many non-trivial orientable vector bundles out there).
Let me be more concrete about my questions:
- Why does a section of $V_n (E) \to B$ determines an orientation of $E$ instead of a collection of $n$ linearly independent sections?
- How to use the Stiefel-Whitney classes as obstructions for the existence of $n$ linearly independent sections (equivalently, to sufficient condition for the triviality of $E$)?