Is there a DeRham _Homology_?

[WWGD]

Hi, I hope this is not too ignorant, but my Algebraic Topology is rusty:

Is there such a thing as DeRham _homology_? I always hear and read about

DeRham cohomology, but I have never heard of DeRham homology. Is there

such a thing?

[hunt_mat]

DeRahm cohomology is all to do with the cohomology of differential forms, you set up your Meyer-Veotoris sequence via the exterior derivative. There is alway a pairing between homology and cohomology, so I guess that there is that but there is no formal theory called DeRahm homology as far as I am aware.

[Bacle]

Why stop there with the (good) question? How about Cech homology?

[quasar987]

Cech homology exists, but it is not a homology theory in the sense of the Eilenberg-Steenrod axioms. The little book by Hocking and Young talks a bit about it.

[lavinia]

DeRahm cohomology is all to do with the cohomology of differential forms, you set up your Meyer-Veotoris sequence via the exterior derivative. There is alway a pairing between homology and cohomology, so I guess that there is that but there is no formal theory called DeRahm homology as far as I am aware.

The connection of the De Rham cochain complex to cohomology with real coefficiants is Stokes theorem. So the duality is through the isomorphism with real cohomology.

[Jamma]

Hmm, this is an interesting question and is applicable to something I have been thinking about. Apparently there is a notion of Cech homology- but it doesn't satisfy all of the properties you'd like (mainly, it doesn't do well with giving you sequences which should be exact). However, it can be altered, to a thing called (I think) strong homology, which does give a well behaved homology theory and for which cech cohomology is the dual theory.