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?

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    $\begingroup$ What is deRham homology? $\endgroup$ Jan 17 '19 at 4:02
  • 3
    $\begingroup$ I think one uses currents instead of differential forms... $\endgroup$ Jan 17 '19 at 8:58
  • 5
    $\begingroup$ See Chapter IV of De Rham's book Differentiable manifolds. The result you want follows from Thm.16 in Sec. 21. $\endgroup$ Jan 17 '19 at 9:28
  • $\begingroup$ I think the book of Breadon, Geometry and topology contains a proof (for cohomology) $\endgroup$ Jan 17 '19 at 10:42
  • $\begingroup$ @FrancescoPolizzi: There is a description using currents; but the book I've alluded to above uses submanifolds. I appreciate currents are more general, and subsume submanifolds by way of Stokes theorem; however, I find the description of homology via submanifolds more intuitive than the simplicial approach in Hatcher. To my mind it makes a better beginning. Though of course one needs to know what a manifold is - but intuitively we know what this is. $\endgroup$ Jan 17 '19 at 17:20

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 $$H_{k, \, \mathrm{dR}}(X) \simeq H_{k, \,\mathrm{sing}}(X, \, \mathbb{R}).$$


[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.


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