For a smooth manifold X, the deRham theorem provides a quasi-isomorphism from the complex Ω~*(X) of differential forms to the complex of (smooth) singular cochains on X. Furthermore (under this isomorphism) the wedge-product in Ω~*(X) induces the cup-product in cohomology; but Ω~*(X) has the advantage of being an associative, graded commutative algebra already on the chain level. In the dual case the deRham theorem gives a quasi-isomorphism from the complex of (smooth) singular chains on X to the complex Ω_*(X) of compactly supported currents on X. (We use this non-standard notation rather than D′(X) or D′_*(X).).
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