Monodromies on links of normal complex surface singularities: tête-à-tête graphs and positive factorizations

In this talk we will give a survey on three of my works that deal with the understanding of the geometric monodromies on links of normal complex surface singularities induced by reduced holomorphic map germs. The first two works (which are co-authored with Javier Fernández de Bobadilla, María Pe Pereira and Norbert A'Campo; and Baldur Sigurdsson respectively) give combinatorial models that we call "tête-à-tête graphs" which capture the topological type of the Milnor fiber and the mapping class of the geometric monodromy. These graphs are 1-dimensional finite CW complexes equipped with a cyclic ordering of the edges adjacent to each vertex and a metric that satisfies a very special property. The third work generalizes a classical result from Picard-Lefschetz theory about monodromies of a isolated plane curve singularities, by proving that the monodromies on links of normal complex surface singularities admit positive factorizations, that is, they can be expressed as the product of right handed Dehn twists. An important result by Anne Pichon which characterizes topologically these monodromies, gives a connection between all these three works and the theory of mapping class groups making an impact outside the realm of singularity theory.