Monodromy

In mathematics, monodromy describes how objects from analysis , topology or in algebraic and differential geometry behave as soon as they move around a singularity.

Monodromy is closely related to the theory of overlays and their degenerations in branching points . Monodrome theory is motivated by the phenomenon that certain functions that one would like to define become multivalued in the vicinity of singularities. This monodromic property can best be measured by the so-called monodromic group , a group of mappings that operate on the values ​​of the function. This group operation codes the behavior of the values ​​when circulating around the singularity.

definition

Let be a connected and locally connected dotted topological space with a base point . Continue to be an overlay with fiber . For a loop with a starting point, let the lift from with a starting point . Also denote the end point , which can generally be different from . ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle p:{\tilde {X}}\to X}$ ${\displaystyle F=p^{-1}(x)}$ ${\displaystyle \gamma :[0,1]\longrightarrow X}$ ${\displaystyle x}$ ${\displaystyle {\tilde {\gamma }}}$ ${\displaystyle \gamma }$ ${\displaystyle {\tilde {x}}\in F}$ ${\displaystyle {\tilde {x}}\cdot \gamma }$ ${\displaystyle {\tilde {\gamma }}(1)}$ ${\displaystyle {\tilde {x}}}$

It can be shown that this construction leads to a well-defined group operation of the fundamental group on the fiber . Here the stabilizer is accurate . This means that an element leaves a point in the fiber invariant if and only if it is represented by the image of a loop in with a base point . ${\displaystyle \pi _{1}(X,x)}$ ${\displaystyle F}$ ${\displaystyle {\tilde {x}}}$ ${\displaystyle p_{*}(\pi _{1}({\tilde {X}},{\tilde {x}}))}$ ${\displaystyle [\gamma ]}$ ${\displaystyle F}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle {\tilde {x}}}$

This group effect is described as a monodrome effect . The group homomorphism in the automorphism group of is monodromy . The image of homomorphism is called the monodromy group. ${\displaystyle \pi _{1}(X,x)\longrightarrow {\text{Aut}}(F)}$ ${\displaystyle F}$

literature

• VI Danilov: Monodromy . Springer-Verlag, ISBN 978-1-55608-010-4