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

[kakarukeys]

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?

 

[kakarukeys]

please help! I'm desperate. I will help out in the Homework and coursework questions section if anybody can help me.

 

[LorenzoMath]

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.

 

[kakarukeys]

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?

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.
$$\rho: F \times G \longrightarrow F$$
$$\rho(f, g) = fg$$
$$f(g_1g_2) = (fg_1)g_2$$

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

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

How to see h is continuous?

Your h is my $$\rho_f$$. Were you saying $$\rho_f$$ is open because F, G are (required to be) locally compact and Hausdorff?

 

[kakarukeys]

I couldn't find any theorem which guarantees that.

closests two are:

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