I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please?
The first Stiefel-Whitney class of a vector bundle is an element in the first cohomology group of the base space $H^1(B,Z_2)$. This element can be seen as a map $w_1 : \pi_1(B) \rightarrow Z_2$ (because of the universal coefficient theorem and the fact that $H_1$ is the abelianization of $\pi_1$), so it assigns to each loop based at a fixed point an element of $Z_2$. Now this number is $0$ if and only if that loop is orientation preserving (locally we have an orientation and when we go around a loop and come back to our first place, we can ask if orientation has changed or not)