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Monodromy theory


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'.

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 π1(Xx) → 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 π1 (Xx) → Diff(Fx)/Is(Fx) whose image is called the geometric monodromy group.


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