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Rigidity of Nilpotent Lie Foliations: Cohomological Obstructions and Classification

Ameth Ndiaye

Abstract

In this article, we develop a systematic cohomological framework for the study of the rigidity of nilpotent Lie foliations with respect to solvable deformations. We introduce the deformation complex associated to a pair of Lie algebras $(\mathfrak{g}, \mathfrak{h})$ and show that the main obstruction to deforming a nilpotent Lie foliation into a non-nilpotent solvable foliation lies in the cohomology group $H^2(\mathfrak{g},\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])$. We establish a necessary and sufficient algebraic criterion for rigidity within the family of foliations modelled on the generalized Heisenberg groups $H_{2k+1}$. This result unifies and generalizes the construction of Dathe--Ndiaye (2012) as well as its subsequent extensions. We complete the article with a full classification of nilpotent Lie foliations of codimension at most six according to their deformation behaviour.

Rigidity of Nilpotent Lie Foliations: Cohomological Obstructions and Classification

Abstract

In this article, we develop a systematic cohomological framework for the study of the rigidity of nilpotent Lie foliations with respect to solvable deformations. We introduce the deformation complex associated to a pair of Lie algebras and show that the main obstruction to deforming a nilpotent Lie foliation into a non-nilpotent solvable foliation lies in the cohomology group . We establish a necessary and sufficient algebraic criterion for rigidity within the family of foliations modelled on the generalized Heisenberg groups . This result unifies and generalizes the construction of Dathe--Ndiaye (2012) as well as its subsequent extensions. We complete the article with a full classification of nilpotent Lie foliations of codimension at most six according to their deformation behaviour.
Paper Structure (13 sections, 6 theorems, 19 equations, 1 table)

This paper contains 13 sections, 6 theorems, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie Foliations
  4. Deformations of Lie Foliations
  5. Nilpotent and Solvable Lie Algebras
  6. Deformation Complex and Cohomological Obstructions
  7. Chevalley--Eilenberg Cohomology
  8. The Solvable Deformation Complex
  9. Computation of $H^2(\mathfrak{h}_{2k+1},\,\mathfrak{h}_{2k+1}/\mathfrak{z})$
  10. Heisenberg Foliations and Proof of Theorem \ref{['thm:A']}
  11. Construction of Heisenberg Foliations
  12. Proof of Theorem \ref{['thm:A']}
  13. Classification in Codimension $\leq 6$

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a Lie foliation of codimension $q \geq 4$ on a compact manifold $M$, modelled on a generalized Heisenberg group $H_{2k+1}$ with $2k+1 \leq q$. Then $\mathcal{F}$ admits no non-nilpotent solvable deformation if and only if where $\mathfrak{z}$ denotes the center of the Heisenberg algebra $\mathfrak{h}_{2k+1}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Generalized Heisenberg Algebra
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 12 more