Simplicial currents
For a smooth manifold \(X\), the de Rham theorem provides a quasi-isomorphism from the complex \(\Omega^* (X)\) of differential forms into the complex of (smooth) singular cochains on \(M\). This theorem admits an extension to the category of simplicial manifolds, then the complex \(\Omega^* |X|\) of simplicial differential forms undertakes the role of differential forms. The dual version of the de Rham theorem gives a quasi-isomorphism from the complex of (smooth) singular chains on \(X\) to the complex \(\Omega_* (X)\) of compactly supported currents on \(X\). The authors introduce a complex \(\Omega_* |X|\) of the so-called singular currents on a simplicial manifold to obtain the isomorphism \(H(\Omega_* |X|)\to H_* (|X|)\) of the cohomologies of the fat realization \(|X|\) involving the identification of the coalgebra structures.
scientific article; zbMATH DE number 1043153