Extending the flux homomorphism to volume-preserving homeomorphisms

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 Poincaré 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 $H^{\ast }(Home{o}_0(M,\Omega ),\mathcal{C}(M,\mathbb{R}))$ of $Home{o}_0(M,\Omega )$ with coefficients in $\mathcal{C}(M,\mathbb{R})$.