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Weinstein neighbourhood theorems for stratified subspaces

Yael Karshon, Sara B. Tukachinsky, Yoav Zimhony

Abstract

By analogy with Weinstein's neighbourhood theorem, we prove a uniqueness result for symplectic neighbourhoods of a large family of stratified subspaces. This result generalizes existing constructions, e.g., in the search for exotic Lagrangians. Along the way, we prove a strong version of Moser's trick and a (non-symplectic) tubular neighbourhood theorem for these stratified subspaces.

Weinstein neighbourhood theorems for stratified subspaces

Abstract

By analogy with Weinstein's neighbourhood theorem, we prove a uniqueness result for symplectic neighbourhoods of a large family of stratified subspaces. This result generalizes existing constructions, e.g., in the search for exotic Lagrangians. Along the way, we prove a strong version of Moser's trick and a (non-symplectic) tubular neighbourhood theorem for these stratified subspaces.

Paper Structure

This paper contains 32 sections, 46 theorems, 203 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Normal form theorems
  3. Results
  4. Applications
  5. Existing normal form theorems
  6. New normal form theorems
  7. Outline
  8. Acknowledgements
  9. Stratified differential spaces
  10. Differential spaces
  11. Stratified differential spaces
  12. Isotropic and coisotropic stratified subspaces
  13. Extending tangent bundle isomorphisms to diffeomorphisms
  14. Extendable and locally extendable pairs
  15. Local extendability implies extendability
  16. ...and 17 more sections

Key Result

Theorem A

For $i=0,1$, let $\left(M_i, \omega_i\right)$ be symplectic manifolds and $\left(A_i, \mathcal{S}_i\right)$ stratified subspaces of them, smoothly locally trivial with conical fibers. Assume that $A_i$ are strongly coisotropic and that there exists a stratified diffeomorphism $g:\left(A_0, \mathcal{ such that $\mathcal{G}$ restricts on $A_0$ to a stratified diffeomorphism Moreover, this restricti

Figures (2)

  • Figure 5.1: A schematic drawing of neighbourhoods: the stratified subspace $A^{\leq d}$ is a union of the three lines, which represent strata of dimension $d$, and the vertex in the middle, which represents $A^{\leq (d-1)}$.
  • Figure A.1: A schematic drawing of \ref{['lemma:closed_open_sets_euler']}: the dashed arrows represent the orbits of $m^t_{\mathcal{E}}:\Psi(O) \to \Psi(O)$ for $t\in[0,1]$. If $K \cap \mathop{\mathrm{supp}}\nolimits{\varphi}$ meets a fiber ${\left(m^0_{\mathcal{E}}\right)}^{-1}(n)$ then ${\left(m^0_{\mathcal{E}}\right)}^{-1}(n) \cap \mathop{\mathrm{supp}}\nolimits{\varphi} \subset U$.

Theorems & Definitions (118)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 108 more