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Monodromy
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{{Use American English|date = March 2019}} {{Short description|Mathematical behavior near singularities}} [[File:Imaginary log analytic continuation.png|thumb|The imaginary part of the [[complex logarithm]]. Trying to define the complex logarithm on <math>\C-\{0\}</math> gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of <math>\C-\{0\}</math> by a [[helicoid]] (an example of a [[Riemann surface]]).]] 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 [[Mathematical singularity|singularity]]. As the name implies, the fundamental meaning of ''monodromy'' comes from "running round singly". It is closely associated with [[covering map]]s and their degeneration into [[Ramification (mathematics)|ramification]]; the aspect giving rise to monodromy phenomena is that certain [[function (mathematics)|function]]s 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 (mathematics)|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''.<ref name="König2015">{{cite book|last1=König|first1=Wolfgang|last2=Sprekels|first2=Jürgen|title=Karl Weierstraß (1815–1897): Aspekte seines Lebens und Werkes – Aspects of his Life and Work|date=2015|publisher=Springer-Verlag|isbn=9783658106195|pages=200–201|url=https://books.google.com/books?id=7IHDCgAAQBAJ&q=Karl+Weierstra%C3%9F+(1815%E2%80%931897)|access-date=5 October 2017|language=de}}</ref> ==Definition== Let <math>X</math> be a connected and [[locally connected]] based [[topological space]] with base point <math>x</math>, and let <math>p: \tilde{X} \to X</math> be a [[covering map|covering]] with [[Fiber (mathematics)|fiber]] <math>F = p^{-1}(x)</math>. For a loop <math>\gamma: [0, 1] \to X</math> based at <math>x</math>, denote a [[homotopy lifting property|lift]] under the covering map, starting at a point <math>\tilde{x} \in F</math>, by <math>\tilde{\gamma}</math>. Finally, we denote by <math>\tilde{x} \cdot \gamma</math> the endpoint <math>\tilde{\gamma}(1)</math>, which is generally different from <math>\tilde{x}</math>. There are theorems which state that this construction gives a well-defined [[Group action (mathematics)|group action]] of the [[fundamental group]] <math>\pi_1(X, x)</math> on <math>F</math>, and that the [[stabilizer (group theory)|stabilizer]] of <math>\tilde{x}</math> is exactly <math>p_*\left(\pi_1\left(\tilde{X}, \tilde{x}\right)\right)</math>, that is, an element <math>[\gamma]</math> fixes a point in <math>F</math> [[if and only if]] it is represented by the image of a loop in <math>\tilde{X}</math> based at <math>\tilde{x}</math>. This action is called the '''monodromy action''' and the corresponding [[group homomorphism|homomorphism]] <math>\pi_1(X, x) \to \operatorname{Aut}(H_*(F_x))</math> into the [[automorphism group]] on <math>F</math> is the '''algebraic monodromy'''. The image of this homomorphism is the '''monodromy group'''. There is another map <math>\pi_1(X,x) \to \operatorname{Diff}(F_x)/\operatorname{Is}(F_x)</math> whose image is called the '''topological 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]] <math>F(z)</math> in some open subset <math>E</math> of the punctured complex plane <math>\mathbb{C} \backslash \{0\}</math> may be continued back into <math>E</math>, but with different values. For example, take : <math>\begin{align} F(z) &= \log(z) \\ E &= \{z\in \mathbb{C} \mid \operatorname{Re}(z)>0\}. \end{align}</math> Then analytic continuation anti-clockwise round the circle : <math>|z| = 1</math> will result in the return not to <math>F(z)</math> but to : <math>F(z) + 2\pi i.</math> 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 equation]]s <math>(x, y, z) = (\rho \cos \theta, \rho \sin \theta, \theta)</math> restricted to <math>\rho > 0</math>. 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]]s, where a single solution may give further linearly independent solutions by [[analytic continuation]]. Linear differential equations defined in an open, connected set <math>S</math> in the complex plane have a monodromy group, which (more precisely) is a [[linear representation]] of the [[fundamental group]] of <math>S</math>, summarising all the analytic continuations round loops within <math>S</math>. The [[inverse problem]], of constructing the equation (with [[regular singularity|regular singularities]]), given a representation, is a [[Riemann–Hilbert problem]]. For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators <math>M_j</math> corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices <math>j</math> are chosen in such a way that they increase from <math>1</math> to <math>p+1</math> when one circumvents the base point clockwise, then the only relation between the generators is the equality <math>M_1\cdots M_{p+1}=\operatorname{id}</math>. The [[Deligne–Simpson problem]] is the following realisation problem: For which tuples of conjugacy classes in <math>\operatorname{GL}(n,\mathbb{C})</math> do there exist irreducible tuples of matrices <math>M_j</math> 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]]. The problem has been considered by other authors for matrix groups other than <math>\operatorname{GL}(n,\mathbb{C})</math> as well.<ref>{{Citation|author=V. P. Kostov|title=The Deligne–Simpson problem — a survey|journal=J. Algebra|volume=281|year=2004|issue=1|pages=83–108|mr=2091962|doi=10.1016/j.jalgebra.2004.07.013|arxiv=math/0206298|s2cid=119634752}} and the references therein.</ref> ==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]] to "follow" paths on the base space <math>X</math> (we assume it [[path-connected]] for simplicity) as they are lifted up into the cover <math>C</math>. If we follow round a loop based at <math>x</math> in <math>X</math>, which we lift to start at <math>c</math> above <math>x</math>, we'll end at some <math>c^*</math> again above <math>x</math>; it is quite possible that <math>c \neq c^*</math>, and to code this one considers the action of the [[fundamental group]] <math>\pi_1(X, x)</math> as a [[permutation group]] on the set of all <math>c</math>, as a '''monodromy group''' in this context. In differential geometry, an analogous role is played by [[parallel transport]]. In a [[principal bundle]] <math>B</math> over a [[smooth manifold]] <math>M</math>, a [[connection (mathematics)|connection]] allows "horizontal" movement from fibers above <math>m</math> in <math>M</math> to adjacent ones. The effect when applied to loops based at <math>m</math> is to define a '''[[holonomy]]''' group of translations of the fiber at <math>m</math>; if the structure group of <math>B</math> is <math>G</math>, it is a subgroup of <math>G</math> that measures the deviation of <math>B</math> from the product bundle <math>M \times G</math>. ===Monodromy groupoid and foliations=== [[File:Monodromy action.svg|thumb|upright=0.3|A path in the base has paths in the total space lifting it. Pushing along these paths gives the monodromy action from the fundamental groupoid.]] Analogous to the [[Fundamental group#Fundamental groupoid|fundamental groupoid]] 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 <math>X</math> of a fibration <math>p:\tilde X\to X</math>. The result has the structure of a [[groupoid]] over the base space <math>X</math>. The advantage is that we can drop the condition of connectedness of <math>X</math>. Moreover the construction can also be generalized to [[foliation]]s: Consider <math>(M,\mathcal{F})</math> a (possibly singular) foliation of <math>M</math>. Then for every path in a leaf of <math>\mathcal{F}</math> 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 (mathematics)|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 <math>\mathbb{F}(x)</math> denote the field of the [[rational function]]s in the variable <math>x</math> over the [[field (mathematics)|field]] <math>\mathbb{F}</math>, which is the [[field of fractions]] of the [[polynomial ring]] <math>\mathbb{F}[x]</math>. An element <math>y = f(x)</math> of <math>\mathbb{F}(x)</math> determines a finite [[field extension]] <math>[\mathbb{F}(x) : \mathbb{F}(y)]</math>. This extension is generally not Galois but has [[Galois closure]] <math>L(f)</math>. The associated [[Galois group]] of the extension <math>[L(f) : \mathbb{F}(y)]</math> is called the monodromy group of <math>f</math>. 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}(x):\mathbb{C}(y)]</math> is already Galois, the associated monodromy group is sometimes called a [[Covering map|group of deck transformations]]. This has connections with the [[Grothendieck's Galois theory|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== {{Reflist}} ==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. Brown [http://groupoids.org.uk/topgpds.html Topology and Groupoids] (2006). * P.J. Higgins, "Categories and groupoids", van Nostrand (1971) [http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html TAC Reprint] * H. Żołądek, "The Monodromy Group", Birkhäuser Basel 2006; doi: 10.1007/3-7643-7536-1 [[Category:Mathematical analysis]] [[Category:Complex analysis]] [[Category:Differential geometry]] [[Category:Algebraic topology]] [[Category:Homotopy theory]]
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