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.
Best Answer
It's a somewhat broad question, but yes there are connections between various things on your list under quite general conditions. Since it's a big topic, I'll mostly be content to list references since you asked for them. If your manifold is simply connected then closed $1$-forms are exact as you surmised. So the first de Rham cohomology which is the quotient of closed $1$-forms by exact forms can be thought of an obstruction to simple connectivity. In fact it coincides with $Hom(\pi_1(X), \mathbb{R})$. For higher forms, de Rham cohomology, which coincides with real simplicial or singular cohomology, measures something else. There is quite lot of material on this topic. For example, the book by Bott and Tu was my favourite source for this in grad school. But the book by Guilleman and Pollack might be better to start with.
The second topic, relating this to harmonic or analytic functions is Hodge theory as Paul Siegel points out. The wikapedia page that he linked to seems a little terse however. If you want to to learn more about this story, I would suggest picking up a book on compact Riemann surfaces (e.g by Forster, Narasimhan, and let's not forget Weyl).
I've made this "community wiki" so people should feel free to add more references as appropriate.