Extending the flux homomorphism to volume-preserving homeomorphisms

TL;DR

This paper extends the flux homomorphism to volume-preserving homeomorphisms, revealing a rigidity result where extended flux groups match the standard flux group, and explores implications for symplectic geometry and homeomorphism flexibility.

Contribution

The paper introduces an extension of the flux homomorphism to volume-preserving homeomorphisms and establishes a rigidity result linking extended and standard flux groups.

Findings

Extended flux homomorphism coincides with the standard flux group.

A new $(C^0, ext{delta})$-rigidity result is proved.

Implications for symplectic homeomorphisms with trivial flux.

Abstract

This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, \delta)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincar\'e duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic homeomorphism of $(T^2, \omega)$ with trivial flux, is a finite energy Hamiltonian homeomorphism of $(T^2, \omega)$. We discuss the cohomology groups $H^\ast(Homeo_0(M,\Omega), \mathcal{C}(M, \mathbb{R}) )$ of $ Homeo_0(M,\Omega)$ with coefficients in $ \mathcal{C}(M, \mathbb{R})$.