Monodromy

Monodromy referred to in the mathematics of how objects from the analysis, topology, or behave in algebraic and differential geometry, as soon as they move around a singularity.

Monodromy is closely connected with the theory of superposition and its Degenerierungen in branching points. Monodromietheorie is motivated by the phenomenon that certain features that you want to define, be multi-valued in the vicinity of singularities. This Monodromieeigenschaft can best be measured by the so-called monodromy group, a group of pictures, which operates on the values ​​of the function. This group operation encodes the behavior of the values ​​when running around the singularity.

Definition

Be a connected and locally coherent dotted topological space with base point x. Furthermore, let an overlay containing fiber. For a loop with the starting point is the foot lifter of having a starting point. Furthermore denote the end point, which may vary in general from.

It can be demonstrated that this construction leads to a well-defined group operation of the fundamental group on the fiber F. Here, the stabilizer of is exactly. This means that an element is a point in the fiber F is invariant if and only leaves when it is represented by the image of a loop with base point.

This group effect is described as Monodromiewirkung. The group homomorphism into the automorphism group of F is the monodromy. The image of the homomorphism is called the monodromy group.