Combinatorial models in the topological classification of singularities of mappings

Abstract: The topological classification of finitely determined map germs f from $(R^n,0)$ to $(R^p,0)$ is discrete (by a theorem due to R. Thom), hence we want to obtain combinatorial models which codify all the topological information of the map germ. According to Fukuda's work, the topology of such germs is determined by the link, which is obtained by taking the intersection of the image of f with a small enough sphere centered at the origin. If $f^{-1}(0)=\left\{0\right\}$, then the link is a topologically stable map from $S^{n-1}$ to $S^{p-1}$ (or stable if $(n,p)$ are nice dimensions) and f is topologically equivalent to the cone of the link. When $f^{-1}(0)$ is not equal to $\left\{0\right\}$, the situation is more complicated. We analyze the particular case of mappings from $R^2$ to $R^3$, where the link is a doodle (a closed curve on the sphere with only transverse double points) and the combinatorial model is provided by the Gauss word. We will review some recent results about topological classification, singularities of ruled surfaces, topological triviality of families and topological finite determinacy.

Spanishcommutative algebraalgebraic geometryanalysis of PDEsalgebraic topologydifferential geometryfunctional analysisgeneral topologygeometric topologyprobabilityrings and algebras

Audience: general audience

Seminários de Matemática da UFPB