TITLE: A list of machineries for computing cohomology
QUESTION [25 upvotes]: This is not a question, but I just hope to hear more from everyone here on it.
A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I have never seen this being made explicit.
What I have in mind at the moment:
"Basic" methods:
*) The definition: for example Simplicial cohomology makes the problem into one of pure linear algebra which can then be done by hand or by many computer program packages at the moment. For singular cohomology this is not really reasonable though.
*) The Axioms: Things Such as the Mayer–Vietoris sequence or the LES of a Pair. These two methods allow you to compute the cohomology of most cell complexes that you are likely to encounter early in your education. More detailed study of the maps in the sequences can get you even farther.
"Advanced" methods:
*) Spectral sequences. Leray-Serre seems to be the most commonly used, since many interesting spaces can be written in terms of fibrations.
*) Morse theory. Surprisingly effective for many difficult problems, especially if one can construct a good energy function, such that the critical sets and flows are simpler.
*) Weil conjecture. After Deligne's proof, one can go in the opposite direction and find Betti numbers by point-counting. Unfortunately it can not give the torsions as far as I know.
For the last two methods, I find Atiyah-Bott's celebrated paper on the moduli space of bundles an excellent demonstration.
Now I am looking forward to your inputs. How many important methods are missing here?

REPLY [6 votes]: 1) If you want to compute the rational or real cohomology of something, you can try to use rational homotopy theory. Rational homotopy theory says that the rational singular chain complex is (as a dga) chain equivalent to a very small chain complex, the minimal model, whose generators are in correspondence to the generators of the rational homotopy groups. So, if you have knowledge of the rational homotopy groups, you can try this. It works also quite well to study the cohomology of the (free) loop space of a space, because you can compute the minimal model of the (free) loop space of a space if you know the minimal model of the space. 
2) If you can show that your space is a $BG$, its cohomology equals the group cohomology of $G$ which is computable in some cases. But I suppose that for manifolds the direct usage of this method is not very efficient since the fundamental group of all acyclic manifolds is infinite and group cohomologies of infinite groups can be very difficult to compute by algebraic means. 
3) For a compact connected Lie group you can use the theorem that the De Rham cohomology equals the equivariant forms (see chapter V.12 in Bredon). 
4) Even if you have no strict Lie group structure, but only a multiplication which fulfills the axioms up to homotopy, i. e. an H-space, you can make use of this structure. With field coefficients, you have the structure of a Hopf algebra on the cohomology/homology and there are various structure theorems. E.g. an easy application of this method is that an H-space which is a finite CW-complex with non-trivial homology has zero Euler-characteristic.