Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they 'run round' a singularity. As the name implies, monodromy's fundamental meaning 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 is best measured by defining a monodromy group: a group of transformations acting on the data that codes what does happen as we 'run round'.

Related Topics:
Mathematics - Mathematical analysis - Algebraic topology - Differential geometry - Singularity - Covering map - Ramification - Function - Group

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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 disk D given

Related Topics:
Complex analysis - Analytic continuation - Analytic function - Punctured disk

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:0 < |z| < 1

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may be continued back into E, but with different values. For example if we take

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:F(z) = log z

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and E to be defined by

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: Re(z) > 0

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then analytic continuation anti-clockwise round the circle

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:|z| = 0.5

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will result in the return, not to F(z) but

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:F(z)+2πi.

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In this case the monodromy group is infinite cyclic. One important application is to differential equations, where a single solution may give further linearly independent solutions by analytic continuation. Linear differential equations defined in an open, connected set S in the complex plane have a monodromy group, which (more precisely) is a linear representation of the fundamental group of S, summarising all the analytic continuations round loops within S. The inverse problem, of constructing the equation (with regular singularities), given a representation, is called the Riemann-Hilbert problem.

Related Topics:
Infinite cyclic - Differential equation - Analytic continuation - Linear representation - Fundamental group - Regular singularities - Riemann-Hilbert problem

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In the case of a covering map, we look at it as a special case of a fibration, and use the homotopy lifting property to 'follow' paths on the base space X (we assume it path-connected for simplicity) as they are lifted up into the cover C. If we follow round a loop based at x in X, which we lift to start at c above x, we'll end at some c* again above x; it is quite possible that c ≠ c*, and to code this one considers the action of the fundamental group π1(X,x) as a permutation group on the set of all c, as monodromy group in this context.

Related Topics:
Fibration - Homotopy lifting property - Path-connected - Fundamental group - Permutation group

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In differential geometry, an analogous role is played by parallel transport. In a principal bundle B over a smooth manifold M, a connection allows 'horizontal' movement from fibers above m in M to adjacent ones. The effect when applied to loops based at m is to define a holonomy group of translations of the fiber at m; if the structure group of B is G, it is a subgroup of G that measures the deviation of B from the product bundle MxG.

Related Topics:
Parallel transport - Principal bundle - Smooth manifold - Connection - Holonomy

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