Extending the flux homomorphism to volume-preserving homeomorphisms

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 $\mathcal{C}(M, \mathbb{R})$.