PL approximations of symplectic manifolds

TL;DR

This paper explores the relationship between smooth and piecewise linear (PL) symplectic manifolds, demonstrating that smooth symplectic structures can be approximated by PL structures through triangulations.

Contribution

It introduces the concept of PL symplectic manifolds and proves that smooth symplectic manifolds can be approximated by PL symplectic manifolds via triangulations.

Findings

Smooth symplectic manifolds admit arbitrarily fine triangulations in general position.

Smooth symplectic structures can be $C^0$-approximated by PL symplectic manifolds.

Volume forms of smooth symplectic manifolds can be triangulated by those of PL approximations.

Abstract

This paper is a contribution to piecewise linear (PL) symplectic topology. We define the notion of PL symplectic manifold as being a combinatorial manifold endowed with a piecewise constant Whitney symplectic form and investigate possible relations between the two categories of symplectic spaces. We prove that smooth symplectic manifolds admit arbitrarily fine smooth triangulations in general position with respect to the symplectic form and can be $C^0$-approximated by PL symplectic manifolds. We cannot prove that smooth symplectic structures can be triangulated, except in trivial cases, but we can prove that their associated volume form can be triangulated by the volume form of some of these approximating PL manifolds.