In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures.
Closed forms/exact forms
real parts of analytic functions/harmonic functions
Analytic functions/analytic functions that have holomorphic antiderivatives.
One can see that for open connected subsets U of R^2, to have any of these being trivial is equivalent to U being simply connected, and any of these conditions imply U is homeomorphic to either C or the unit disk.
In higher dimensions, or in general in real manifolds, it only makes reasonable sense (to me as a graduate student) to still talk about closed/exact and simple connectivity. Are there any connections here? I know simple connectivity, even of an open subset of R^3, no longer implies trivial De Rham cohomology. But what about the converse?
In R^2, what about the nontrivial case? that is, are any of the above groups isomorphic to any other for a general open connected subset of C? If so, do any of the isomorphisms carry more structure than just the structure of the abelian group? Do the isomorphisms become homomorphisms for higher dimensional spaces?
I am interested not so much in factual answers, but more in proofs or references to proofs.