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The Linearizability of Singular Foliations Is a Morita Invariant

Marco Zambon

Abstract

Hausdorff Morita equivalence is an equivalence relation on singular foliations, which induces a bijection between their leaves. Our main statement is that linearizability along a leaf is invariant under Hausdorff Morita equivalence. The proof relies on a characterization of tubular neighborhood embeddings using Euler-like vector fields.

The Linearizability of Singular Foliations Is a Morita Invariant

Abstract

Hausdorff Morita equivalence is an equivalence relation on singular foliations, which induces a bijection between their leaves. Our main statement is that linearizability along a leaf is invariant under Hausdorff Morita equivalence. The proof relies on a characterization of tubular neighborhood embeddings using Euler-like vector fields.

Paper Structure

This paper contains 12 sections, 15 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Singular foliations and their linearization
  3. Singular foliations
  4. The linearized foliation
  5. Interlude: Tubular neighborhood embeddings
  6. A linearization criterion
  7. Hausdorff Morita equivalence
  8. The main theorem
  9. Examples and an application
  10. Examples
  11. Restrictions of singular foliations to slices
  12. Hausdorff Morita equivalence and restrictions to slices

Key Result

Theorem 1

Let $(M_1,\mathcal{F}_1)$ and $(M_2,\mathcal{F}_2)$ be Hausdorff Morita equivalent singular foliations, and $L_1\subset M_1$, $L_2\subset M_2$ corresponding embedded leaves. Then $\mathcal{F}_1$ is linearizable around $L_1$ if and only if $\mathcal{F}_2$ is linearizable around $L_2$.

Theorems & Definitions (48)

  • Theorem : Theorem \ref{['thm:main']}
  • Proposition : Proposition \ref{['prop:pss']}
  • Corollary : Corollary \ref{['cor:slicelin']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • ...and 38 more