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

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 vo