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

Abstract

We establish a topological analogue of Ismagilov's theorem concerning the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold \(M\) with volume form \(Ω\), we consider the group \(\mathbb{G}^Ω(M)\) of homeomorphisms obtained as uniform limits of smooth volume-preserving isotopies. We show that its first continuous cohomology with values in the Banach space of zero-mean continuous functions is isomorphic to the first de Rham cohomology of \(M\). The proof develops a theory of transport for volume-preserving isotopies, producing a topological volume flux homomorphism and its associated transport cocycle. Under the additional hypothesis that \((M,Ω)\) satisfies the local \(C^0\)-generation property, and assuming the Müller-Sikorav approximation theorem (known for \(n\neq 4\)), the isomorphism extends to the full identity component \(\operatorname{Homeo}_0^Ω(M)\) of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.