The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology.
Now de Rhams theorem asserts that there is an isomorphism between de Rham cohomology of smooth manifolds and that of singular cohomology; and so what appears to be an invariant of smooth structure, is actually an invariant of topological structure.
Is there a similar theorem showing an isomorphism between de Rham homology and singular homology?
Best Answer
I guess that by de Rham homology you mean the homology groups $H_{k, \, \mathrm{dR}}(X)$ constructed on a closed manifold $X$ by using the complex of currents.
In that case, [1, Theorem 2 page 582] shows that there is an isomorphism between $H^{n-k}_{\mathrm{dR}}(X)$ and $H_{k, \, \mathrm{dR}}(X)$, where the cohomology is the usual one (constructed by using the complex of differential forms) and $n = \dim X$.
Now, using the standard De Rham isomorphism between $H^{n-k}_{\mathrm{dR}}(X)$ and the singular cohomology group $H^{n-k}_{\mathrm{sing}}(X, \, \mathbb{R})$, together with the Poincaré duality $H^{n-k}_{\mathrm{sing}}(X, \, \mathbb{R}) \simeq H_{k, \,\mathrm{sing}}(X, \, \mathbb{R})$, we deduce the desired isomorphism $$H_{k, \, \mathrm{dR}}(X) \simeq H_{k, \,\mathrm{sing}}(X, \, \mathbb{R}).$$
References
[1] Giaquinta, Mariano; Modica, Giuseppe; Souček, Jiří, Cartesian currents in the calculus of variations I. Cartesian currents, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 37. Berlin: Springer. xxiv, 711 p. (1998). ZBL0914.49001.