For what kind of problems was de Rham cohomology introduced?

Question

I'm new to the world of (co)homology theories, and I have some difficulty understanding the intuitive motivation for introducing the de Rham cohomology.

More explicitly, I studied singular (co)homology essentially as a tool to 'study the problem' of 'holes' of a topological space, sheaf cohomology as a machinery to study the existence of global sections with some local property and Cech cohomology as a computational tool for sheaf cohomology. In particular for paracompact and locally contractible spaces, all of these cohomologies are equivalent. The de Rham cohomology is defined for smooth manifolds, and is not difficult to prove that de Rham cohomology is equal to the sheaf, Cech and singular (also Alexander-Spanier) cohomologies. Hence my question:

Since for all kinds of spaces we can define the de Rham cohomology and it is always equivalent to the singular, etc. cohomologies, why do we introduce it?

Two immediate answers could be: 1.) Poincaré duality; 2.) It can be useful to have another machinery for calculating cohomology groups. But I'm interested to know if there is some more intuitive motivation beyond this construction like 'holes' for singular and 'extending local section' for sheaves.

Answer 1

The short answer to your question is that the differential forms that come up in the definition of de Rham cohomology are interesting and useful objects, and are fun to play with too.

Here's a longer answer. Besides just the (co)homology groups that come out of a (co)homology theory, the (co)chain complexes out of which that theory is built are interesting and useful objects. In the singular theory, that means singular (co)chains. In de Rham cohomology, that means differential forms. The study of differential forms is rich and interesting and it is packed with information and applications to the study of differential manifolds. There is a lot more to this theory than just the application of differential forms to the computation of cohomology groups.

One nice example of this is the Gauss-Bonnet theorem. Consider $S$ a closed oriented 2-dimensional Riemannian manifold of genus $g$. Let $K$ be the curvature function. One can form a (closed) 2-form $K \, dA$ representing a certain scalar multiple of the fundamental class in the de Rham cohomology group $H^2(S) \approx \mathbb R$; that scalar is given by the integral $\int_S K \, dA$. The Gauss-Bonnet theorem gives a beautiful and useful form to that scalar multiple: $$\frac{1}{2\pi} \int_S K \, dA = \chi(S) = 2-2g $$ You can apply this theorem to study geometric structures on surfaces. For example, one corollary is that $S$ has a Euclidean structure (a Riemannian metric of curvature $0$) only if $\chi(S)=0$, equivalently $g=1$ and $S$ is a torus. Another is that $S$ has a hyperbolic structure (a Riemannian metric of curvature $-1$) only if $\chi(S)<0$, equivalently $g \ge 2$.

There are a lot more examples of practical connections between differential forms and various kinds of special cohomology classes. I'll mention very briefly a generalization of the Gauss-Bonnet theory known as the Chern-Weil theory. This allows one to use differential forms to compute certain topological invariants of a complex vector bundle over a manifold $M$. These invariants are known as the Chern classes, and they come in the form of a sequence of cohomology classes of $M$. The Chern-Weil theory again gives formulas for these classes in de Rham cohomology by integration of certain differential forms over $M$.

Answer 2

I believe it is anachronistic to consider sheaf and singular theories and then "introduce" de Rham cohomology; I don't know the history well, but I am confident that the de Rham -- singular comparison and de Rham -- soft sheaf-theoretic resolution comparison theorems came after de Rham's cohomology was considered for the first time.

But, armed with more modern perspectives, we can still ask why it is relevant. For me, de Rham cohomology as a way of computing 'ordinary' cohomology gives geometric interpretations to cohomology class, geometric interpretations to things like the cup products, geometric interpretations to characteristic classes, ... and since you are "often" working with forms and their derivatives, you really do care about the question of whether or not closed forms are exact - knowing that this is purely a topological question reducing to the real cohomology is firstly beautiful, and secondly useful.

Cohomology could be critiqued as abstract, with lots of derived nonsense hiding on the fringes, and de Rham's theory is certainly the most concrete one I am aware of. There is a value in that.

I need to read the book: "Differential forms in algebraic topology" by Bott and Tu; perhaps you should read this too!

Answer 3

Homology of a manifold (or a topological space, more generally) $X$ is a commutative group, whereas cohomology (with differential forms) is a real vector space. As a result, the structure of $H^n(X)$ is simpler than the structure of $H_n(X)$.

For instance, when using the Mayer-Vietoris sequence, from a cohomology perspective, you are dealing with linear maps, and so it is possible to calculate cohomology in many cases by taking advantage of the rank-nullity theorem from linear algebra. With homology you do not have access to such tools.

It is true that you are losing information when you use (de Rahm) cohomology. But it is precisely this loss of information which makes it convenient! Sometimes the loss of information is tolerable that you can still do the calculations that you want.

Actually, if you ever seen (mod 2) homology the idea of losing information is quite similar. The (mod 2) homology might just be simple enough to compute, and yet still can give useful computations that you need.

I like to think of homology vs cohomology as equality vs inequality. In analysis we often rarely have a nice pretty formula for a sum/integral, so we use an inequality instead. An inequality will lose information but often it is just enough information to be useful.

Answer 4

In my opinion, de-Rham stuff is just the same thing as singular stuff, the goal is to find holes in the space. Singular homology does the job by throwing singular simplices at it, while de-Rham searches for holes as an obstruction to fundamental theorem of calculus, or more generally as an obstruction to finding primitive functions and doing it via integration.

Answer 5

In general, since diffeomorphic differentiable manifolds enjoy the same (de Rham) cohomology, the contrapositive is that those with different cohomology aren't diffeomorphic.

So this is somewhat a categorization or uniformization problem.

Similarly homology and homotopy functors btw (in algebraic topology etc.)