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A classification of neighborhoods around leaves of a singular foliation

Simon-Raphael Fischer, Camille Laurent-Gengoux

TL;DR

The paper develops a formal, jet-level classification of singular foliations near a fixed leaf by introducing leaf data comprising a Galois cover, a symmetry extension, and a principal bundle. It proves a one-to-one correspondence between formal foliations along a leaf with a given transverse model and leaf data, with a simplified version reducing to an outer holonomy plus a finite-dimensional extension. A reconstruction procedure glueing formal neighborhoods via $H$-cocycles solidifies the correspondence, and several concrete cases are worked out: simply-connected leaves, transversally quadratic transverse models, flat $\mathcal{F}$-connections, and torus leaves. The framework leverages diffeological groups, Yang-Mills groupoids, and formal Ehresmann connections to capture holonomy and symmetry data, providing a robust, modular approach to understanding singular foliations near a leaf. This formal-jet perspective clarifies how transverse structure, inner/outer symmetries, and Galois covers govern the local classification and suggests avenues for real-analytic or centerless extensions.

Abstract

We classify singular foliations admitting a given leaf and a given transverse singular foliation.

A classification of neighborhoods around leaves of a singular foliation

TL;DR

The paper develops a formal, jet-level classification of singular foliations near a fixed leaf by introducing leaf data comprising a Galois cover, a symmetry extension, and a principal bundle. It proves a one-to-one correspondence between formal foliations along a leaf with a given transverse model and leaf data, with a simplified version reducing to an outer holonomy plus a finite-dimensional extension. A reconstruction procedure glueing formal neighborhoods via -cocycles solidifies the correspondence, and several concrete cases are worked out: simply-connected leaves, transversally quadratic transverse models, flat -connections, and torus leaves. The framework leverages diffeological groups, Yang-Mills groupoids, and formal Ehresmann connections to capture holonomy and symmetry data, providing a robust, modular approach to understanding singular foliations near a leaf. This formal-jet perspective clarifies how transverse structure, inner/outer symmetries, and Galois covers govern the local classification and suggests avenues for real-analytic or centerless extensions.

Abstract

We classify singular foliations admitting a given leaf and a given transverse singular foliation.
Paper Structure (18 sections, 31 theorems, 111 equations, 1 figure)

This paper contains 18 sections, 31 theorems, 111 equations, 1 figure.

Table of Contents

  1. Formal singular foliations along a leaf
  2. Formal neighborhoods and formal singular foliations
  3. Formal diffeomorphisms
  4. Formal singular foliations near a point
  5. Inner and outer symmetries
  6. Principal bundles over symmetries of a singular foliation
  7. The transverse singular foliation
  8. The Yang-Mills groupoid
  9. Classification I: The Theorem
  10. From formal singular foliation to leaf data
  11. Reconstruction
  12. Conclusion
  13. Classification II: Particular Cases
  14. Simply-connected leaves
  15. Transversally quadratic singular foliations
  16. ...and 3 more sections

Key Result

Lemma 1.4

Iglesias-Zemmour Any formal neighborhood of a connected manifold $L$ is isomorphic (as a sheaf of filtered algebra) to infinite jets along $L$ of smooth functions on the normal bundle $\pi \colon T \rightarrow L$.

Figures (1)

  • Figure 1: Parallel transport along loops following a foliation ${\mathcal{F}}$ with concentric circle as transverse model. Whenever the loop is contractible, each horizontal lifts ends in the circle it started from. Whenever the loop is not contractible, its horizontal lifts might end up in a different circle.

Theorems & Definitions (84)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Lemma 1.4
  • Definition 1.5
  • Remark 1.6
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • Definition 1.10
  • ...and 74 more