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Concept
Monodromy
Summary
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 happens 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 be a covering with fiber . For a loop γ: [0, 1] → X based at x, denote a lift under the covering map, starting at a point , by . Finally, we denote by the endpoint , which is generally different from . There are theorems which state that this construction gives a well-defined group action of the fundamental group π1(X, x) on F, and that the stabilizer of is exactly , that is, an element [γ] fixes a point in F if and only if it is represented by the image of a loop in based at . This action is called the monodromy action and the corresponding homomorphism pi1(X, x) → Aut(H*(Fx)) into the automorphism group on F is the algebraic monodromy. The image of this homomorphism is the monodromy group. There is another map pi1(X, x) → Diff(Fx)/Is(Fx) whose image is called the topological monodromy group.
These ideas were first made explicit in complex analysis. In the process of analytic continuation, a function that is an analytic function F(z) in some open subset E of the punctured complex plane may be continued back into E, but with different values. For example, take
then analytic continuation anti-clockwise round the circle
will result in the return, not to F(z) but
In this case the monodromy group is infinite cyclic and the covering space is the universal cover of the punctured complex plane.
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