Monodromy

In 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 )+2π i .

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 singularity), 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 (mathematics) 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 M x G .

=Definition via Galois theory=

Let mathbb{F}(x) denote the field of fractions of the ring mathbb{F}[x] where mathbb{F} is also a field. An element f(y) in mathbb{F}(y) determines a finite field extension

:mathbb{F}(x) hookrightarrow mathbb{F}(y)

by setting

:f(y) = x

which is generally not Galois but which has Galois closure

:L_{f} , !.

The associated Galois group of the extension L_f/mathbb{F}(x) is called the monodromy group of the extension.

In the case of mathbb{F} = mathbb{C} Riemann surface theory enters and allows for the geometric interpretation given above. In the case that the extension mathbb{C}(y) is already Galois, the associated monodromy group is sometimes called a Covering map.

This has connections with the Grothendieck s Galois theory leading to the Riemann existence theorem.