- #1
olliemath
- 34
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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
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