Monodromy group

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Definition

Suppose X is a path-connected locally path-connected topological space and \tilde{X} is a covering space of X with covering map p:\tilde{X} \to X. Suppose x_0\in X. Consider the set p^{-1}(x_0). There is a natural group action of the fundamental group \pi_1(X,x_0) on the set p^{-1}(x_0) defined as follows:

Any loop starting and ending at x_0 defines, for each point in p^{-1}(x_0), a unique path starting at the point. Consider the endpoint of this path. This gives a set map p^{-1}(x_0) \to p^{-1}(x_0). The inverse loop gives the inverse set map, so the set map is a permutation. Thus, for each loop, we get an element of \operatorname{Sym}(p^{-1}(x_0)). In other words, we have a map:

\Omega(X,x_0) \to \operatorname{Sym}(p^{-1}(x_0))

Two homotopic loops (in the based homotopy sense) induce the same permutation, so this descends to a map:

\pi_1(X,x_0) \to \operatorname{Sym}(p^{-1}(x_0))

Finally, we can verify that the map is a group homomorphism.

The image of this map, viewed as a subgroup of \operatorname{Sym}(p^{-1}(x_0)) (i.e., not just as an abstract group, but in the context of the action) is termed the monodromy group for the covering map at x_0. The group action itself is termed the monodromy action.