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'.

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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

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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.

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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.

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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.

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Mathematical analysis: Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers...

Algebraic topology: Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces....

Differential geometry: Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for...