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?
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