Monodromy theory

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'.

Let X be a connected and locally connected based topological space with base point x, and let ${\displaystyle p:{\tilde {X}}\to X}$ be a covering with fiber ${\displaystyle F=p^{-1}(x)}$. For a loop γ: [0, 1] → X based at x, denote a lift under the covering map (starting at a point ${\displaystyle {\tilde {x}}\in F}$) by ${\displaystyle {\tilde {\gamma }}}$. Finally, we denote by ${\displaystyle {\tilde {x}}\cdot \gamma }$ the endpoint ${\displaystyle {\tilde {\gamma }}(1)}$, which is generally different from ${\displaystyle {\tilde {x}}}$. 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 ${\displaystyle {\tilde {x}}}$ is exactly ${\displaystyle p_{*}(\pi _{1}({\tilde {X}},{\tilde {x}}))}$, that is, an element [γ] fixes a point in F if and only if it is represented by the image of a loop in ${\displaystyle {\tilde {X}}}$ based at ${\displaystyle {\tilde {x}}}$. This action is called the monodromy action and the corresponding homomorphism π₁(X, x) → Aut(H_*(F_x)) 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(F_x)/Is(F_x) whose image is called the geometric monodromy group.

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