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 functionss 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 happens as we "run round" in one dimension. Lack of monodromy is sometimes called polydromy.

Definition

be a connected and locally simply connected based topological space with base point be a covering. Define its base fiber.

, can be lifted.

.

This construction gives a well-defined left group action. This is the monodromy action. The stabilizer if and only if, is still a loop.

is discontinuously deformed, in such a way that it cannot.

is the topological monodromy group.

As usual in algebraic topology, there is an algebraic version. The homomorphism into the automorphism group on the homology is the algebraic monodromy. The image of this homomorphism is the (algebraic) monodromy group.

Example

These ideas were first made explicit in complex analysis. In the process of analytic continuation, a function that is an analytic function, but with different values. For example, take

Then analytic continuation anti-clockwise round the circle

but to

In this case the monodromy group is the infinite cyclic group, and the covering space is the universal cover of the punctured complex plane. This cover can be visualized as the helicoid with parametric equations. The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane.

Differential equations in the complex domain

One important application is to differential equations, where a single solution may give further linearly independent solutions by analytic continuation in the complex plane have a monodromy group, which (more precisely) is a linear representation of the fundamental group. The inverse problem, of constructing the equation (with regular singularities), given a representation, is a Riemann–Hilbert problem.

. The Deligne–Simpson problem from these classes satisfying the above relation? The problem has been formulated by Pierre Deligne and Carlos Simpson was the first to obtain results towards its resolution. An additive version of the problem about residua of Fuchsian systems has been formulated and explored by Vladimir Kostov as well.

Topological and geometric aspects

In the case of a covering map, we look at it as a special case of a fibration, and use the homotopy lifting property (we assume it path-connected, and to code this one considers the action of the fundamental group as a permutation group, as a monodromy group in this context.

In differential geometry, an analogous role is played by parallel transport. In a principal bundle over a smooth manifold, a connection is to define a holonomy.

Monodromy groupoid and foliations

Analogous to the fundamental groupoid. The result has the structure of a groupoid.

Moreover the construction can also be generalized to foliations we can consider its induced diffeomorphism on local transversal sections through the endpoints. Within a simply connected chart this diffeomorphism becomes unique and especially canonical between different transversal sections if we go over to the germ of the diffeomorphism around the endpoints. In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy.

Definition via Galois theory

denote the field of the rational functions over the field, which is the field of fractions of the polynomial ring determines a finite field extension.

This extension is generally not Galois but has Galois closure. The associated Galois group.

Riemann surface 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.

See also

• Braid group

• Monodromy matrix

• Monodromy theorem

• Mapping class group (of a punctured disk)

Notes

References

• "Group-groupoids and monodromy groupoids", O. Mucuk, B. Kılıçarslan, T. ¸Sahan, N. Alemdar, Topology and its Applications 158 (2011) 2034–2042 doi:10.1016/j.topol.2011.06.048

• R. Brown Topology and Groupoids (2006).

• P.J. Higgins, "Categories and groupoids", van Nostrand (1971) TAC Reprint

• H. Żołądek, "The Monodromy Group", Birkhäuser Basel 2006; doi: 10.1007/3-7643-7536-1

• Allen Hatcherhttps://pi.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology