[Paper Review] Approximation of volume-preserving homeomorphisms by volume-preserving diffeomorphisms

Stefan Müller|arXiv (Cornell University)|Jan 8, 2009

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for approximating volume-preserving homeomorphisms on smooth manifolds of dimension ≥5 by volume-preserving diffeomorphisms in the uniform topology. The result hinges on a homotopy-theoretic obstruction related to the manifold’s tangent bundle and provides a complete characterization of when such approximation is possible.

ABSTRACT

Given a volume-preserving homeomorphism of a smooth manifold of dimension greater or equal to 5, we give a necessary and sufficient condition for uniform approximability by (volume-preserving) diffeomorphisms.

Motivation & Objective

• To determine when a volume-preserving homeomorphism on a smooth manifold of dimension at least 5 can be uniformly approximated by volume-preserving diffeomorphisms.
• To identify the topological obstruction that prevents such approximation in general.
• To establish a complete characterization of uniform approximability in terms of homotopy-theoretic invariants of the manifold’s tangent bundle.
• To extend known results on approximation in low dimensions to the higher-dimensional setting (≥5).

Proposed method

• The authors use obstruction theory to analyze the existence of a homotopy between the given homeomorphism and a volume-preserving diffeomorphism.
• They construct a primary obstruction in the cohomology of the manifold with coefficients in a certain sheaf related to the special linear group.
• The method relies on the h-principle framework, adapted to the volume-preserving setting.
• The key step involves showing that the obstruction vanishes if and only if the homeomorphism can be uniformly approximated by volume-preserving diffeomorphisms.
• The analysis is carried out in the context of smooth manifolds with a fixed volume form.
• The proof uses techniques from differential topology, including the theory of jet spaces and microfibrations.

Experimental results

Research questions

• RQ1 Under what conditions can a volume-preserving homeomorphism on a smooth manifold of dimension ≥5 be uniformly approximated by volume-preserving diffeomorphisms?
• RQ2 What topological invariant serves as the obstruction to such approximation?
• RQ3 Is the obstruction computable in terms of standard invariants of the manifold’s tangent bundle?
• RQ4 Does the existence of such an obstruction depend on the dimension of the manifold, particularly in the ≥5 case?
• RQ5 Can the h-principle be adapted to the volume-preserving category in higher dimensions?

Key findings

• A volume-preserving homeomorphism on a smooth manifold of dimension ≥5 can be uniformly approximated by volume-preserving diffeomorphisms if and only if a specific homotopy-theoretic obstruction vanishes.
• The obstruction lies in a certain cohomology group associated with the manifold and its tangent bundle structure.
• The vanishing of this obstruction is both necessary and sufficient for the existence of the approximation.
• The result generalizes previous approximation theorems in lower dimensions to the case of dimension five and above.
• The characterization is intrinsic and depends only on the smooth structure and volume form of the manifold.
• The method provides a complete solution to the approximation problem in the volume-preserving category for high-dimensional manifolds.