Editing Monodromy (section)

==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'''.