Monodromy action

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what does happen as we "run round" in one dimension. Lack of monodromy is sometimes called polydromy.

Let $X$ be a connected and locally connected based topological space with base point $x$ , and let $p:{\tilde {X}}\to X$ be a covering with fiber $F=p^{-1}(x)$ . For a loop $γ: [0, 1] \to X$ based at $x$ , denote a lift under the covering map, starting at a point ${\tilde {x}}\in F$ , by ${\tilde {\gamma }}$ . Finally, we denote by ${\tilde {x}}\cdot {\tilde {\gamma }}$ the endpoint ${\tilde {\gamma }}(1)$ , which is generally different from ${\tilde {x}}$ . There are theorems which state that this construction gives a well-defined group action of the fundamental group $π 1 (X, x)$ on $F$ , and that the stabilizer of ${\tilde {x}}$ is exactly $p_{*}(\pi _{1}({\tilde {X}},{\tilde {x}}))$ , that is, an element $[γ]$ fixes a point in $F$ if and only if it is represented by the image of a loop in ${\tilde {X}}$ based at ${\tilde {x}}$ . This action is called the monodromy action and the corresponding homomorphism $π 1 (X, x) \to Aut(H * (F x))$ into the automorphism group on $F$ is the algebraic monodromy. The image of this homomorphism is the monodromy group. There is another map $π 1 (X, x) \to Diff(F x)/Is(F x)$ whose image is called the geometric monodromy group.

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