Understanding Monodromy Map of a Covering Space: Can Anyone Help?

[angy]

Hi!

Suppose we have a topological space X, a point x\in X and a homomorphism \rho:\pi(X,x) \rightarrow S_n with transitive image. Consider the subgroup H of \pi(X,x) consisting of those homotopy classes [\gamma] such that \rho([\gamma]) fixes the index 1\in \{1,\ldots,n\}. I know that H induces a covering space p:Y\rightarrow X. However, I can't understand why the monodromy map of p is exactly \rho.

Can anyone help me?

 

[mathwonk]

"a homomorphism ρ:π(X,x)→Sn with transitive image."

huh?

 

[lavinia]

Suppose there is a transitive group action on a set of points. And let H be the stabilizer of a point. Then the action of G on the coset space, G/H, is isomorphic to the action of G on the set of points.

G acts transitively - via the monodromy action -on the fiber of the covering corresponding to the subgroup,H. H is the stabilizer of the fiber under this action.

 

[mathwonk]

i guess i thought Sn was the n sphere.