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

David Marín (), Jean-François Mattei () and Eliane Salem ()
Additional contact information
David Marín: Universitat Autònoma de Barcelona, Departament de Matemàtiques
Jean-François Mattei: Université Paul Sabatier, Institut de Mathématiques de Toulouse
Eliane Salem: Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche

Chapter Chapter 4 in Handbook of Geometry and Topology of Singularities V: Foliations, 2024, pp 169-222 from Springer

Abstract: Abstract In this article we give an overview on the topology of singularities of holomorphic foliation germs in ℂ 2 $$\mathbb {C}^2$$ . 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.

Date: 2024
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-52481-3_4

Ordering information: This item can be ordered from
http://www.springer.com/9783031524813

DOI: 10.1007/978-3-031-52481-3_4

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().