Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Representations of the Fundamental Group

  1. Jun 17, 2009 #1
    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 [tex]\gamma[/tex] in X with base point x and a point y in [tex]F_x[/tex] we can lift to a uniqe curve [tex]\gamma'[/tex] in |F| with initial point y. Setting [tex]\gamma\cdot y=\gamma'(1)[/tex] we obtain an action of [tex]\pi(X,x)[/tex] on [tex]F_x[/tex] which is called the monodromy action. [tex]F_x[/tex] is a vector space, but I don't see how we know that the map [tex]y\mapsto\gamma\cdot y[/tex] is linear.
    Or possibly this is not the right approach?

    Any help is greatly appreciated - O
     
  2. jcsd
  3. Jun 17, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Representations of the Fundamental Group
  1. Fundamental Group (Replies: 1)

  2. Fundamental Group (Replies: 1)

  3. The Fundamental Group (Replies: 19)

  4. Fundamental Group (Replies: 3)

Loading...