$C^0$-Analogue of Ismagilov's Theorem

Abstract

We establish a $C^0$-analogue of Ismagilov's theorem on the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold, we prove that the first continuous cohomology of the identity component of the group of volume-preserving homeomorphisms, with coefficients in the Banach space of continuous zero-mean functions, is isomorphic to the first de Rham cohomology. The proof introduces a topological transport theory for volume-preserving isotopies, yielding a continuous volume flux homomorphism and an associated transport cocycle. As an application, we obtain a resolution of the volume-preserving $C^0$-Flux Conjecture.