Topological symplectic manifolds and bi-Lipschitz structures

TL;DR

This paper demonstrates that topological symplectic manifolds inherently possess bi-Lipschitz structures, leading to new examples showing the non-existence and non-uniqueness of topological symplectic structures.

Contribution

It establishes a canonical bi-Lipschitz structure on topological symplectic manifolds and explores implications for their existence and uniqueness.

Findings

Existence of a canonical bi-Lipschitz structure on topological symplectic manifolds

First examples of non-existence of topological symplectic structures

First examples of non-uniqueness of topological symplectic 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 $C^0$--limits of bi-Lipschitz homeomorphisms.