The monodromy representation $\pi_1(E)\to Mod(F)$ Given a fiber bundle $F\to X\to E$, with $F,E$ are Riemann surfaces. I know the monodromy gives a permutation of the fiber. But how to see we have the monodromy representatio: $\pi_1(E)\to Mod(F)$? Here Mod($F$) is the mapping class group of the fiber.

Update: Now I think by locally trivialization, we can get the monodromy gives a homeomorphism from the fiber to itself. But how do we know it preserves the orientation?
If you define the mapping class group as isotopy classes of orientation preserving homeomorphisms, then the bundle needs to be oriented.
If you define the mapping class group as isotopy classes of orientation preserving homeomorphisms, then the bundle needs to be oriented.