A list of machineries for computing cohomology

Question

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?

Answer 1

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:

• http://ncatlab.org/nlab/show/cohomology

• equivalence of Grothendieck-style versus Cech-style sheaf cohomology

• What is a cohomology theory (seriously)?

Answer 2

cohomological descent is an important technique, especially when computing the cohomology of some singular spaces/stacks. Of course it can be viewed as a general form of the Cech-to-derived spectral sequence.

$\bf{Edit}$: two examples.

1. Let $X$ be a space, and let $\{U_0,U_1,...,U_n\}$ be an open cover of $X.$ Define a simplicial space $X_{\bullet}$ as follows. Let $X_0$ be the disjoint union of the $U_i$'s, and let $X_1$ be the disjoint union of the 2-intersections $U_{ij}=U_i\cap U_j,$ and so on; the transition maps among the $X_i$'s are induced by various inclusions, like $U_{ij}\to U_i.$ Let $F$ be a sheaf on $X.$ Then there is a spectral sequence $$ E_1^{pq}=H^q(X_p,F_p)\Rightarrow H^{p+q}(X,F), $$ where $F_q$ is the pullback of $F$ to $X_q.$ One sees that when $F$ has no higher cohomology in finite intersections of this open cover, the Cech cohomology computes the true (i.e. derived) cohomology, since $q$ has to be 0 and we have a horizontal complex, which is the Cech complex, whose cohomology is the Cech cohomology, and the spectral seq degenerates. This recovers a well-known fact.

2. Let $X$ be the classifying space $BG$ of a finite group $G,$ and let $EG\to BG$ be the universal $G$-bundle. Let $X_0=EG,$ and let $X_1$ be the fiber product of $EG$ with $EG$ over $BG,$ and let $X_i$ be the $i$-fold fiber product. Then $X_1\simeq EG\times G,$ and in general $X_i\simeq EG\times G^i.$ Let $V$ be a local system on $X;$ it corresponds to a representation of $G.$ This time $EG\to BG$ is not an open cover in the sense of example 1, but is a covering for some other "Grothendieck topology" (in the world of alg geom, it's called etale topology, where "inclusions of open sets" are local isomorphisms), and the spectral sequence still work. Since $EG$ is contractible, $X_i$ is just a finite set of points, and there is no higher cohomology (i.e. $q$ must be zero). The horizontal complex in the $E_1$-page is the cochain complex of cycles of $G^i$ into $V,$ the cohomology of which computing the group cohomology, and one recovers the fact that the cohomology of $BG$ agrees with the group cohomology of $G$ in the coefficient sheaf.

Answer 3

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.