I would like to know more about relationships between the mapping class group of an orientable surface with negative Euler's characteristic and moduli spaces. In particular, I would like to have a rigorous explanation of the following three facts:
having a family of surfaces given by a fibration $F\to E\to B$ (e.g. take B as $S^1$ and F as a compact surface of genus $g>1$), how to construct a map from $\pi_1(B)$ to $MCG(F)$.
how point 1 should give a relation between the $MCG$ and the topological type of $E$
how the cohomology of the $MCG$ is related with the cohomology of the moduli space of curves
In other words, I know just some chatting about that, but I would like to learn more deeply how to formulate and prove those facts.
Thanks a lot in advance, bye!