Monodromy
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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'.
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
- 0 < |z| < 1
may be continued back into E, but with different values. For example if we take
- F(z) = log z
and E to be defined by
- Re(z) > 0
then analytic continuation anti-clockwise round the circle
- |z| = 0.5
will result in the return, not to F(z) but
- F(z)+2πi.
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.
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.
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.
Formal definition
Let <math>\mathbb{F}(x)<math> denote the field of fractions of the ring <math>\mathbb{F}[x]<math> where <math>\mathbb{F}<math> is also a field. An element <math>f(y) \in \mathbb{F}(y)<math> determines a finite field extension <math>\mathbb{F}(x) \hookrightarrow \mathbb{F}(y)<math> by setting <math>f(y) = x<math> which is generally not Galois but which has Galois closure <math>L_{f} \, \!<math>. The associated Galois group of the extension <math>L_f/\mathbb{F}(x)<math> is called the monodromy group of the extension. In the case of <math>\mathbb{F} = \mathbb{C}<math> Riemann surface theory enters and allows for the geometric interpretation given above. In the case that the extension <math>\mathbb{C}(y)<math> is already Galois, the associated monodromy group is sometimes called a group of deck transformations. This has connections with the Galois theory of covering spaces leading to the Riemann Existence Theorem.