TITLE: How to compute the cohomology of a local system?
QUESTION [11 upvotes]: Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well.
Suppose that we are given a monodromy representation $\rho: \pi_1(X,x) \longrightarrow \text{GL}(V)$.
How does one compute the cohomology $H^i(X,L)$ of the local system $L$ associated to $\rho$?

REPLY [2 votes]: If you're looking for something very concrete in terms of things like explicit boundary map computations, you might be interested in looking at Section 31 of Steenrod's Topology of Fibre Bundles. As in the other answers, though, it sort of depends on what sort of information you already possess or can get at for your space.