A Remark on Distributions and the De Rham Theorem

We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this isomorphism extends to other cases, such as relative and absolute cohomology spaces of manifolds with corners. The de Rham theorem ([2, 3]) is generally interpreted as the isomorphism, for a compact oriented manifold X, between the cohomology of the de Rham complex of smooth forms (1) 0 −→ C∞(X) −→ C∞(X; Λ) −→ · · · −→ C∞(X; Λ) −→ 0, where dimX = n, and the simplicial, or more usually the Čech, cohomology of X. This isomorphism is constructed using a double complex; for proofs of various stripes see [5], [4] or [6]. The distributional de Rham cohomology, the cohomology of the complex (1) with distributional coefficients (currents in the terminology of de Rham), (2) 0 −→ C−∞(X) −→ C−∞(X; Λ) −→ · · · −→ C−∞(X; Λ) −→ 0, is naturally Poincaré dual to the smooth de Rham cohomology using the integration map (α, β) 7−→ ∫ X α ∧ β. Here we show that there is a relatively simple retraction argument which shows that the homology of (2) is isomorphic to the simplicial homology, for any simplicial decomposition, in the dual dimension. The map from simplicial to distributional de Rham cohomology takes a simplex to its Poincaré dual (see for example [1]). There are many possible variants of the proof below and in particular it is likely to apply to intersection type homology theories on compact manifolds with corners. I would like to thank Yi Lin for pointing out an error in the proof of Lemma 4 in an earlier version. 1. Distributions and currents We use some results from distribution theory which are well known. These are mainly to the effect that a simplex is ‘regular’ as a support of distributions. Lemma 1. Any extendible distribution on the interior of an n-simplex S ⊂ R, i.e. an element of the dual of Ċ∞(S; Ω) = {u ∈ C∞(Rn); supp(u) ⊂ S}, is the restriction of a distribution on R with support in S.