Topology of Singular Foliation Germs in \(\mathbb {C}^2\)

In this article we give an overview on the topology of singularities of holomorphic foliation germs in \(\mathbb {C}^2\) . We describe several results of the authors on the topology of the leaves and the structure of the leaf space. We state criteria of topological conjugacy for any two foliation germs. These are based on the key notion of monodromy of a singular foliation, a topological invariant of geometric and dynamic nature. After a historical introduction, we focus on the simplest invariant sets (separatrices, separators and dynamical components) and we compare them to geometric blocks classical in the study of the topology of 3-dimensional manifolds. Subsequently, we introduce the notion of foliated connectedness, used in proving the incompressibility property of the leaves of the foliation, which plays a crucial role in the definition of the monodromy. We describe the ideas of the proofs of the main theorems leading to the topological classification of generic foliations that are generalized curves. Finally, we give an algebraic description of topological moduli spaces and we state the existence of complete families, with minimal redundancy given by an explicit action of a countable group on the finite dimensional parameter space.