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

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Discussion Overview

The discussion revolves around the concept of the monodromy map associated with a covering space in topology, specifically in relation to a homomorphism from the fundamental group to the symmetric group. Participants are exploring the implications of a transitive group action and the relationship between the subgroup and the covering space.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant introduces a topological space X and a homomorphism ρ from the fundamental group to the symmetric group, questioning the nature of the monodromy map.
  • Another participant expresses confusion regarding the definition of the homomorphism and its transitive image.
  • A different participant explains the concept of a transitive group action and the role of the stabilizer in relation to the covering space and the monodromy action.
  • One participant mistakenly interprets Sn as the n-sphere rather than the symmetric group, indicating a potential misunderstanding of the notation.

Areas of Agreement / Disagreement

There appears to be some confusion and disagreement regarding the definitions and implications of the homomorphism and the nature of the monodromy map. Multiple interpretations are present, and the discussion remains unresolved.

Contextual Notes

Participants have not clarified certain assumptions about the definitions of the groups involved or the specific properties of the covering space. The discussion reflects varying levels of understanding of the concepts presented.

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

huh?
 
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.
 
i guess i thought Sn was the n sphere.
 

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