What do higher cohomologies mean concretely (in various cohomology theories)?

Superficially I think I understand the definitions of several cohomologies: (1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't read anything about it) (2) Cech cohomology on Riemann surfaces, or schemes (3) Group cohomology in number theory and I have some rough understanding of interpreting cohomology functors as derived functors.

So my question is: what do higher cohomology groups mean concretely?

Some specifics:

For (1): closed forms modulo exact forms, but is there anything more concrete? It does solve some differential equations, but...is there more to it?

For (2): Serre duality implies the 1st cohomology group is dual to linearly independent meromorphic functions satisfying certain conditions wrt a divisor. How about higher cohomologies? My primary source is Forster's book, so the Serre duality treated there might not be the most general possible.

For (3): $H^0(G,A)$ is $G$-invariant elements of $A$, 1st cohomology is the $A$-torsors, 2nd cohomology is extensions of $A$ by $G$. How about higher cohomologies? My primary source is Artin's "Algebraic numbers and algebraic functions", and "Cohomology of number fields". The latter book (p.20, 2nd Edition) states very roughly that (my interpretation, sincere apologies to the authors if I misunderstand anything), higher cohomologies may not have concrete interpretations, but they play significant roles in understanding lower cohomologies and proving results about them.

PS: My background is (in case it's needed), very very rudimentary knowledge in analysis, algebra, algebraic geometry and number theory, but have not seriously learned any algebraic topology (though have seen the proof of Brouwer fixed point theorem via singular homology). I might have a tendency for the analytical and algebraic understanding of things (e.g. my primary impression of cohomology is that it's the obstruction of exactness, the need to extend exact sequences).

A side question: is it advisable to actually seriously learn algebraic topology to get a better idea of cohomology theories?

Thank you very much.

Solution 1:

Your last question has an easy answer : it is very desirable to learn algebraic topology! It's the most concrete setting for a cohomology theory, and it would be hard to appreciate much (for instance) sheaf cohomology without at least some understanding of the algebraic topology background.

With that said, the following theorem of Thom gives what I consider the most concrete description of higher singular homology groups, at least over $\mathbb{Q}$. This is probably the best thing to try to understand first.

THEOREM : Let $X$ be a smooth manifold. Then for any $v \in H_k(X;\mathbb{Q})$, there exists a nonzero rational number $q$, a smooth compact, oriented $k$-manifold $M^k$, and a continuous map $\phi : M^k \rightarrow X$ such that $v = q \cdot \phi_{\ast}([M^k])$. Here $[M^k]$ is the orientation class of the manifold.

In other words, you should think of $k$-dimensional homology classes as being some kind of "weighted singular $k$-dimensional submanifold" of your space. If $X$ is a manifold of dimension at least $2k+1$, in fact, you can assume that the maps $\phi$ described about are embeddings, so the homology classes are actually weighted submanifolds.

Of course, you asked about cohomology, not homology. However, for compact $n$-manifolds we have Poincare duality which gives a natural identification between $H^{n-k}(M;\mathbb{Q})$ and $H_{k}(M;\mathbb{Q})$. You can thus transfer your intuition for homology over to cohomology.

Solution 2:

Learning algebraic topology, at least to the level of singular homology and cohomology, and Poincare duality, is essential if you plan to use homological techniques in your work with any sort of serious understanding.

It is not just that the simplicial techniques involved in singular homology and cohomology are important in themselves (although they certainly are!), but they provide a model for much else (e.g. the definition of Cech cochains). Also, group cohomology is an instance of singular cohomology (of a classifying space for the group), and so this provides one interpretation of the higher cohomology groups, and a method for proving things about them (and also a potential reason to care about them).

As another example, de Rham cohomology is perhaps best understood in terms of the de Rham theorem, which relates it to the singular cohomology of the manifold (and shows that there is "more to it" --- namely, the de Rham theorem is one of the first examples of the phenomenon of topological obstructions to solving certain differential equations, something which is a pretty general theme in mathematics). It can also be related to certain coherent cohomology spaces (in the context of smooth projective algebraic varities, e.g. Riemann surfaces), and then one also obtains a relationship between Poincare duality and Serre duality, which sheds light on both.