The $n$-th Stiefel–Whitney class $w_n \ne 0$ indicates that every section of the vector bundle (rank $n$) must vanish at some point

Question

What I read:

• If $w_n \ne 0$, where $n$ is the rank of the vector bundle, then there cannot exist one everywhere linearly independent section of the vector bundle.
• The $w_n \ne 0$ indicates that every section of the vector bundle (rank $n$) must vanish at some point.

My question：

• What does the every section means? Does it mean independent sections?

Answer 1

A collection of section $\{s_1, \dots, s_k\}$ of a vector bundle $E \to B$ are said to be linearly independent if $\{s_1(b), \dots, s_k(b)\}$ is a linearly independent subset of $E_b$ for every $b \in B$. When $k = 1$, we see that the collection $\{s\}$ is linearly independent if $\{s(b)\}$ is a linearly independent subset of $E_b$ for every $b$, i.e. $s(b) \neq 0$ for every $b \in B$. So a single linearly independent section is precisely a nowhere-zero section.

If a rank $n$ bundle $E \to B$ does not admit a linearly independent section, then by the above, it does not admit a nowhere-zero section. In particular, if $w_n(E) \neq 0$, then $E$ does not admit a nowhere-zero section, and hence every section has a zero.