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There is one point I don't understand about G-torsor.

  1. Jul 14, 2007 #1
    There is one point I don't understand about G-torsor.

    A Lie group G acts freely and transitively on a manifold F.
    F x G -> F
    (f, g) -> fg, f(g1g2) = (fg1)g2
    is a smooth map.

    fix an element f of F
    then the map h
    F -> G
    fg -> g
    is a homeomorphism.

    I know h is open from the continuity of the map
    {f} x G -> F
    g -> fg

    How to see h is continuous?
     
    Last edited: Jul 14, 2007
  2. jcsd
  3. Jul 20, 2007 #2
    please help!!!!! I'm desperate. I will help out in the Homework and coursework questions section if anybody can help me.
     
  4. Aug 2, 2007 #3
    Your description of the map h: F -> G seems to have a typo. I guess you meant it to be f -> fg.

    h is factored through Stab(g), right? Now Stab(g) is trivial and h is surjective since F is a G-torsor. Thus h is one-to-one.

    By definition h is continuous, so h is bijective and continuous. Since in addition F and G are locally compact and Hausdorff, h is a homeomorphism.
     
  5. Aug 6, 2007 #4
    No, there is no typo, I have typed a little too fast. Let me use Latex and state my question clearer.

    There is one point I don't understand about G-torsor.

    A Lie group G acts freely and transitively on a manifold F.
    [tex]\rho: F \times G \longrightarrow F[/tex]
    [tex]\rho(f, g) = fg[/tex]
    [tex]f(g_1g_2) = (fg_1)g_2[/tex]

    fix an element f of F
    then the map
    [tex]h_f: \{fg | \forall g\in G\} \longrightarrow G[/tex]
    [tex]h_f(fg) = g[/tex]
    is a homeomorphism.

    I know [tex]h_f[/tex] is open from the continuity of the map
    [tex]\rho_f = h_f^{-1}[/tex]
    [tex]\rho_f: \{f\} \times G \longrightarrow F[/tex]
    [tex]\rho_f(g) = fg[/tex]

    How to see h is continuous?

    Your h is my [tex]\rho_f[/tex]. Were you saying [tex]\rho_f[/tex] is open because F, G are (required to be) locally compact and Hausdorff?
     
  6. Aug 31, 2007 #5
    I couldn't find any theorem which guarantees that.

    closests two are:

    (1) if G is compact and F is Hausdorff, [tex]\rho_f[/tex] is open
    (2) if G is locally compact, F is locally compact and Hausdorff, F is a topological group under the induced group operations, [tex]\rho_f[/tex] is open
     
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