[Math] A list of machineries for computing cohomology

ag.algebraic-geometryat.algebraic-topologybig-listcohomologydg.differential-geometry

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?

Best Answer

Some suggestions:

  • Sheaf cohomology via derived functors (this generalizes both Cech cohomology and de Rham cohomology).

  • Hodge decomposition for smooth projective varieties/compact Kähler manifolds can be very useful; see for example this question.

  • Homotopy classes of maps to the corresponding Eilenberg-Mac Lane spaces (cf. Brown representation theorem).

  • The Hochschild-Kostant-Rosenberg theorem says, roughly, that Hochschild homology of the algebra of functions on your manifold/variety/whatever = differential forms, and so gives an alternate viewpoint on de Rham cohomology.

  • There is a QFT-inspired point of view (on de Rham cohomology, K-theory, and conjecturally tmf) due to Stolz-Teichner, see this survey for example.

  • The Lefschetz hyperplane theorem relates the cohomology of varieties with that of their hyperplane sections.

  • Duality theorems such as Poincare duality and Serre duality can be helpful, as well as index theorems such as Riemann-Roch.

Also check out:

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