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I was interested in the following question: if one has a fibration

$F\to E\to B$

there is associated a monodromy map, that is basically an action of the fundamental group $\pi_1(B)$ on the cohomology of the fibre $H^*(F)$. In case this is trivial, there are several simplifications in the cohomological relations between these manifolds. In particular, the associated Leray spectral sequence has second page isomorphic to $$E^{p,q}_2 = H^p(E) \otimes H^q(F) . $$ In the User's guide to spectral sequences, they talk of a condition thas has the same implication on the Leray spectral sequence: the system of local coefficients on $B$ determined by $F$ is simple (I suspect this means that $\cal{H^q}$$( F,\mathbb Q)$ is constant equal to $H^q(F,\mathbb Q)$, which indeed would imply the previous identity, since the bidegree on the Leray spectral sequence are multiplicative for $\otimes$).

My question is: what are the relations between these two concepts? I strongly suspect they are the same, or at least strongly related (the notion of simple system of local coefficients seems to me a more adapted notion to the language of spectral sequences), but can't find any reference.

Any help would be greatly appreciated!

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  • $\begingroup$ Look at the page 138 of the book where the simple system is defined. It says that the system is simple when the isomorphisms $h[\lambda]$ are identities. In particular, if $\lambda$ is a class of a loop you get the trivial system in your sense. $\endgroup$
    – Jarek Kędra
    Commented Jul 12, 2017 at 12:02
  • $\begingroup$ They are the same. If $\pi_1(B)$ acts trivially on $H^*(F)$ then the sheaf of groups $\mathcal{H}^*(F)$ on $B$ reduces to the constant sheaf $H^*(B)$ and the local coefficient system is said to be trivial. Thus the first condition implies the second statement which then gives the first statement. $\endgroup$
    – Tyrone
    Commented Jul 12, 2017 at 13:01
  • $\begingroup$ Thank you both! I had that impression, but wasn't being able of finding a reference. $\endgroup$
    – Jaime
    Commented Jul 19, 2017 at 21:04

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