# Representations of the Fundamental Group

1. Jun 17, 2009

### olliemath

This is not important, but it's been bugging me for a while.
I'm struggling to see how the locally constant sheaves of vector spaces on X give rise to representations of the fundamental group of X.

The approach I've been thinking of is the following. Given a locally constant sheaf F on X, the associated topological space |F| is a covering space for X. Thus, given a loop $$\gamma$$ in X with base point x and a point y in $$F_x$$ we can lift to a uniqe curve $$\gamma'$$ in |F| with initial point y. Setting $$\gamma\cdot y=\gamma'(1)$$ we obtain an action of $$\pi(X,x)$$ on $$F_x$$ which is called the monodromy action. $$F_x$$ is a vector space, but I don't see how we know that the map $$y\mapsto\gamma\cdot y$$ is linear.
Or possibly this is not the right approach?

Any help is greatly appreciated - O

2. Jun 17, 2009

### Hurkyl

Staff Emeritus
You have to use the fact that you're working in a sheaf of vector spaces, not just a sheaf of sets where you've put a vector space structure on one of the fibers.