Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological symplectic manifolds and bi-Lipschitz structures

Dan Cristofaro-Gardiner, Boyu Zhang

Abstract

We show that a topological symplectic manifold has a canonically associated bi-Lipschitz structure. As a corollary, we obtain the first examples of non-existence and non-uniqueness for topological symplectic structures. Our arguments hold for any topological manifold admitting an atlas with transition maps that are $C^0$--limits of bi-Lipschitz homeomorphisms.

Topological symplectic manifolds and bi-Lipschitz structures

Abstract

We show that a topological symplectic manifold has a canonically associated bi-Lipschitz structure. As a corollary, we obtain the first examples of non-existence and non-uniqueness for topological symplectic structures. Our arguments hold for any topological manifold admitting an atlas with transition maps that are --limits of bi-Lipschitz homeomorphisms.
Paper Structure (20 sections, 33 theorems, 45 equations)

This paper contains 20 sections, 33 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Future directions
  3. Preliminaries
  4. First considerations
  5. $\mathcal{C}$-smoothings of $\overline{\mathcal{C}}$-embeddings
  6. Local contractibility
  7. $\overline{\mathcal{C}}$ implies globally $\overline{\mathcal{C}}$
  8. Properties of universal $\overline{\mathcal{C}}$ isotopies
  9. Local contractibility revisited
  10. Proofs of the main results
  11. The adaptation of local contractibility
  12. Structure of Sullivan's argument
  13. Our refinement: extension after restriction revisited
  14. Handle straightening revisited
  15. Refined local contractibility
  16. ...and 5 more sections

Key Result

Theorem 1.1

Every topological bi-Lipschitz four-manifold has a canonically associated bi-Lipschitz structure.

Theorems & Definitions (70)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 60 more