# Topological moduli space for germs of holomorphic foliations

## TL;DR

This paper classifies germs of singular holomorphic foliations on complex surfaces using topological invariants and introduces a new algebraic object called group-graph to describe the moduli space, which can be finite-dimensional under generic conditions.

## Contribution

It defines a new algebraic structure, the group-graph, to compute the moduli space of topological classes of foliations based on fixed invariants.

## Key findings

- The moduli space can be infinite dimensional but is finite under generic conditions.
- The paper describes the algebraic and topological structure of the moduli space.
- Introduces the group-graph as a tool for classification.

## Abstract

This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structures.