Monodromy

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 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 does happen as we 'run round' in one dimension. Lack of monodromy is sometimes called polydromy. Let X be a connected and locally connected based topological space with base point x, and let p : X ~ → X {displaystyle p:{ ilde {X}} o X} be a covering with fiber F = p − 1 ( x ) {displaystyle F=p^{-1}(x)} . For a loop γ: → X based at x, denote a lift under the covering map, starting at a point x ~ ∈ F {displaystyle { ilde {x}}in F} , by γ ~ {displaystyle { ilde {gamma }}} . Finally, we denote by x ~ ⋅ γ ~ {displaystyle { ilde {x}}cdot { ilde {gamma }}} the endpoint γ ~ ( 1 ) {displaystyle { ilde {gamma }}(1)} , which is generally different from x ~ {displaystyle { ilde {x}}} . There are theorems which state that this construction gives a well-defined group action of the fundamental group π1(X, x) on F, and that the stabilizer of x ~ {displaystyle { ilde {x}}} is exactly p ∗ ( π 1 ( X ~ , x ~ ) ) {displaystyle p_{*}(pi _{1}({ ilde {X}},{ ilde {x}}))} , that is, an element fixes a point in F if and only if it is represented by the image of a loop in X ~ {displaystyle { ilde {X}}} based at x ~ {displaystyle { ilde {x}}} . This action is called the monodromy action and the corresponding homomorphism π1(X, x) → Aut(H*(Fx)) into the automorphism group on F is the algebraic monodromy. The image of this homomorphism is the monodromy group. There is another map π1 (X, x) → Diff(Fx)/Is(Fx) whose image is called the geometric monodromy group. 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 complex plane ℂ  {0} may be continued back into E, but with different values. For example, take