My question is connected with the answer to this question:
https://mathoverflow.net/questions/344491/zero-differential-in-serre-spectral-sequence-for-configuration-spaces
I cannot see why from the fact that $\pi_1(C_n)\to\pi_1(C_{n-1})$ is a split epimorphism we have that the action of $\pi_1(C_{n})$ on the cohomology of a fiber is trivial.
This seems like a sort of a general statement:
If in a fibration $F\to E\to B$ there exists a section $B\to E$, then the action of the fundamental group of $B$ on the cohomology of $F$ is trivial. Is it true? And if so, why?
Best Answer
It is not true in general that if a bundle has a section then the monodromy on the cohomology of the fiber is trivial. For example, a projectivized vector bundle always has two sections ($0$ and $\infty$) but the monodromy on the top cohomology (for odd rank) is trivial iff the bundle is orientable.
In the case of the bundle $C_n \to C_{n-1}$, it is not hard to explicitly describe the monodromy. The action on $\pi_1(F)$ is by conjugation (by what?) and hence trivial on $H^1$. Since the fiber is a wedge of circles this is all that needs to be checked.