Topological classification of singularities and a cone-like structure theorem
João Carlos Ferreira Costa
Universidade Estadual Paulista, IBILCE - São José do Rio Preto
Abstract:
In this talk we show some recent advances in the topological classification of
singularities problem. We want to compare results involving the classical topological
classification (via C0-𝒜-equivalence) and the topological 𝒦-classification (via C0-𝒦 equivalence or topological contact equivalence). For this study, we consider finitely
determined map germs. By Fukuda’s cone structure theorem, the topological type of a
finitely determined map germ f can be determined by the topological type of its link
associated, which is a stable map. Concerning in the topological 𝒦-classification
problem, we adapt the Fukuda’s cone construction in such way that now the link is not
required stable but, it is well defined up to homotopy and its homotopy type class
determines the C0-𝒦-class of f. If we consider the C0-𝒜-classification, we divided our
analysis in two parts: if f has or not isolated zeros. In both cases the link is a stable
map but its domain is different in each case. When f has non isolated zeros ( i.e., f-1(0)
≠ {0} ) the link associated to f is more complicated and we need a generalized version
of the cone-like structure theorem to describe the topology of f.