Questions tagged [monodromy]

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 $\displaystyle {\tilde {x}}\cdot {\tilde {\gamma }}$ 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 $π_1(X, x)$ on $F$, and that the stabilizer of $\displaystyle {\tilde {x}}$ is exactly $\displaystyle p_{*}\left(\pi _{1}\left({\tilde {X}},{\tilde {x}}\right)\right)$, 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 $π_1(X, x) → \text{Aut}(H*(Fx))$ into the automorphism group on $F$ is the algebraic monodromy. The image of this homomorphism is the monodromy group. There is another map $π_1(X, x) → \text{Diff}(Fx)/\text{Is}(Fx)$ whose image is called the topological monodromy group.