Mapping Class Group d'un germe de courbe plane singuli\`ere

TL;DR

This paper demonstrates that topological conjugations of singular holomorphic curve germs can be homotopically extended to their desingularizations and provides an explicit description of a subgroup of the mapping class group associated with such singularities.

Contribution

It establishes a homotopic extension property for conjugations of singular curve germs and explicitly describes a finite index subgroup of their mapping class group.

Findings

Topological conjugations extend to desingularizations.

Explicit presentation of a finite index subgroup of the mapping class group.

Homotopic equivalence of conjugations to extendable maps.

Abstract

We prove that every topological conjugation between two germs of singular holomorphic curves in the complex plane is homotopic to another conjugation which extends homeomorphically to the exceptional divisors of their minimal desingularizations. As an application we give an explicit presentation of a finite index subgroup of the mapping class group of the germ of such a singularity.