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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, monodromy is the study of how objects from
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
,
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
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 functions 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

Let be a connected and locally connected based
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
with base point , and let p: \tilde \to X be a covering with
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
F = p^(x). For a loop based at , denote a lift under the covering map, starting at a point \tilde \in F, by \tilde. Finally, we denote by \tilde \cdot \tilde the endpoint \tilde(1), which is generally different from \tilde. There are theorems which state that this construction gives a well-defined group action of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
on , and that the stabilizer of \tilde is exactly p_*\left(\pi_1\left(\tilde, \tilde\right)\right), that is, an element fixes a point in if and only if it is represented by the image of a loop in \tilde based at \tilde. This action is called the monodromy action and the corresponding
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
into the automorphism group on is the algebraic monodromy. The image of this homomorphism is the monodromy group. There is another map whose image is called the topological monodromy group.


Example

These ideas were first made explicit in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. In the process of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
, a function that is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
in some open subset of the punctured complex plane may be continued back into , but with different values. For example, take : \begin F(z) &= \log(z) \\ E &= \ \end then analytic continuation anti-clockwise round the circle : , z, = 1 will result in the return, not to but : F(z) + 2\pi i In this case the monodromy group is infinite cyclic and the covering space is the universal cover of the punctured complex plane. This cover can be visualized as the helicoid (as defined in the helicoid article) restricted to . 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, where a single solution may give further linearly independent solutions by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
. 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
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 mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems h ...
. For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators ''Mj'' corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices ''j'' are chosen in such a way that they increase from 1 to ''p'' + 1 when one circumvents the base point clockwise, then the only relation between the generators is the equality M_1\cdots M_=\operatorname. The Deligne–Simpson problem is the following realisation problem: For which tuples of conjugacy classes in GL(''n'', C) do there exist irreducible tuples of matrices ''Mj'' 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 Vladimir may refer to: Names * Vladimir (name) for the Bulgarian, Croatian, Czech, Macedonian, Romanian, Russian, Serbian, Slovak and Slovenian spellings of a Slavic name * Uladzimir for the Belarusian version of the name * Volodymyr for the Ukra ...
. The problem has been considered by other authors for matrix groups other than GL(''n'', C) as well. and the references therein.


Topological and geometric aspects

In the case of a covering map, we look at it as a special case of a
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...
, 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
1(''X'', ''x'') as a
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
on the set of all ''c'', as a 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 ''M'' × ''G''.


Monodromy groupoid and foliations

Analogous to the
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...
it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy classes of) lifts of paths in the base space ''X'' of a fibration p:\tilde X\to X. The result has the structure of a groupoid over the base space ''X''. The advantage is that we can drop the condition of connectedness of ''X''. Moreover the construction can also be generalized to foliations: Consider (M,\mathcal) a (possibly singular) foliation of ''M''. Then for every path in a leaf of \mathcal 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

Let F(''x'') denote the field of the
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in the variable ''x'' over the field F, which is the field of fractions of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
F 'x'' An element ''y'' = ''f''(''x'') of F(''x'') determines a finite
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''F(''x'') : F(''y'') This extension is generally not Galois but has
Galois closure In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
''L''(''f''). The associated Galois group of the extension 'L''(''f'') : F(''y'')is called the monodromy group of ''f''. In the case of F = C
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
theory enters and allows for the geometric interpretation given above. In the case that the extension ''C(''x'') : C(''y'')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 A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
* Monodromy theorem * Mapping class group (of a punctured disk)


Notes


References

*{{springer, author=V. I. Danilov, title=Monodromy, id=M/m064700 * "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. Brow
Topology and Groupoids
(2006). * P.J. Higgins, "Categories and groupoids", van Nostrand (1971

* H. Żołądek, "The Monodromy Group", Birkhäuser Basel 2006; doi: 10.1007/3-7643-7536-1 Mathematical analysis Complex analysis Differential geometry Algebraic topology Homotopy theory