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Hi everyone,

Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it.

I was wondering if someone could recommend a cool reference that would, ideally:

  1. Cover several cohomology theories, including, say: singular, de Rham, Dolbeault, group cohomology, sheaf cohomology.
  2. Explain the relations between these theories.
  3. Not too hardcore to read (not too formal, containing examples, etc.)

Thanks for helping!

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This could be worthwhile to you: people.sissa.it/~bruzzo/notes/IATG/notes.pdf –  Callan McGill Nov 23 '12 at 0:43
seub -- all the theories you mention (singular, de Rham, Dolbeault, group cohomology, sheaf cohomology) are particular cases of sheaf cohomology, for which there are many references; if you're after a quick and informal introduction that skips some proofs but still conveys the main ideas I would suggest e.g. Kirwan and Woolf's Oxford lecture notes on intersection homology. Now, you also mention cohomology theories, which is a slightly different thing. It is very carefully explained e.g. in Switzer's Algebraic topology but I find the presentation a bit tedious. –  algori Nov 23 '12 at 1:10
What about Bott & Tu's Differential Forms in Algebraic Topology? It covers singular and de Rham in detail, and sheaf cohomology somewhat, and spends quite a bit of time relating these and giving examples of computations. –  skupers Nov 23 '12 at 3:57
You might be looking for Gelfand and Manin's "Homological Algebra". Here's a downloadable version: math.unam.mx/javier/AlgebraV.pdf –  Dan Petersen Nov 23 '12 at 7:42
Your point 2 is covered in pretty much any exposition which mentions two of the cohomology theories... If a text mentions two of these cohomologies is generally to connect them, because that is the whole point of having two constructions for the same thing! –  Mariano Suárez-Alvarez Nov 25 '12 at 20:17
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6 Answers

Let me make a few comments here starting with sheaf cohomology. It can be defined in several ways, but the definitions are less important the properties:

  1. $H^0$ of a sheaf coincides with its global sections.
  2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
  3. It can be computed by acyclic resolutions, such as fine resolutions on a paracompact Hausdorff space.

Using 3 plus the Poincaré lemma, we get a version of de Rham's theorem that sheaf cohomology of a manifold with coefficients in the constant sheaf $\mathbb{R}$ coincides with de Rham cohomology. A similar application of 3 shows that on a complex manifold, sheaf cohomology of the sheaf of holomorphic $p$-forms coincides with Dolbeault cohomology. Also, on a sufficiently nice space (e.g. a manifold) singular cohomology with coefficients in an abelian group $A$ coincides with sheaf cohomology with coefficients in $A$. This is also true if $A$ is replaced by a local system. A local system comes with a monodromy representation $\pi_1(X)\to Aut(A)$. There is map from group cohomology to singular cohomology $$H^i(\pi_1(X), A) \to H^i(X, A)$$ which is an isomorphism if $X$ is aspherical (Eilenberg-Maclane) but not in general (e.g. take $X=S^n, n>1$).

Postscript A more genuine answer, which would be even more unhelpful to the OP than the one I gave, is that I doubt there is a single coherent and easy treatment of all of these disparate topics. But there are plenty of good references (some of which have been mentioned) that cover algebraic topology, group cohomology and its applications, sheaf cohomology and its applications, homological algebra...

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Wow, this is the top answer right now. Interesting. I voted it down without hesitation, because the question was not "Hi guys, what do you know about cohomology". I am happy for you that you know all these things (so did I), but that doesn't help me much. –  seub Nov 24 '12 at 1:34
I understood "Explain the relations between these theories", as a request for an explanation. But if you understood all this, or think you do, then perhaps you should have said so... –  Donu Arapura Nov 24 '12 at 4:24
You forgot (why?) to quote the first part of the sentence : "I was wondering if someone could recommend a cool reference that would..." –  seub Nov 24 '12 at 10:16
I think your use of "acyclic resolutions" is circular. Acyclic objects are objects with trivial higher cohomology. Their use in resolutions comes from repeated applications of the short exact sequence property. I think you mean to say that injective objects are acyclic. But perhaps you are using a different definition? –  Will Sawin Nov 25 '12 at 18:20
@seub, I think that the tone of your comments to Donu's answer is quite not the correct one for someone who is asking for help. And asking professional mathematicians from all other the word who, most likely, you do not personally know, –  Mariano Suárez-Alvarez Nov 25 '12 at 20:16
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C. Weibel's Intro to hom alg is likely what one wants for a perspective on how these different examples fit into a larger context. In particular, thinking that the relevant algebra is dismissible is misguided: many of the seeming geometric issues are more shallow, in a good sense.

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thanks for your answer, please look at my comment up there. –  seub Nov 23 '12 at 8:58
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what about this: Une introduction aux (co)homologies, Cours & exercices by T. Masson: http://science.thilucmic.fr/spip.php?article16

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Actually, I came across that reference while searching the web yesterday, it does seem very attractive, maybe I should go for it. Have you read it? I'm not sure whether the fact that it was written by a "physicist" is good or bad news for me. If you're a French speaker (like me), I also came across this, which also looks attractive to me: www.yann-ollivier.org/maths/toutcohom.ps.gz –  seub Nov 23 '12 at 10:10
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I really like the book of Bott and Tu for the De Rham theory. Hatchers book - freely available on his site - contains nice treatments of singular and cellular (co) homologies.

About your comment. What relations between the theories are you looking for? The Eilenberg Steenrod axioms - http://www.encyclopediaofmath.org/index.php/Steenrod-Eilenberg_axioms - show that the singular, cellular, and de Rham theories are the same (you have to be a bit careful with the coefficients of course), on spaces where they are all defined. I believe the book of Bredon discusses this a bit, but I don't have it with me here (there is a very short passage in Hatcher). Group cohomology can be seen to be the cohomology of a certain space associated to the group. I does not really matter in which theory we compute these, because they will give the same results. I'm not familiar with the Daubault and sheaf cohomology, so I don't have anything relevant to add to this.

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thanks for your answer, please look at my comment up there. –  seub Nov 23 '12 at 8:58
There was a recent question about the possibility of characterizing sheaf cohomology axiomatically: mathoverflow.net/questions/113812/… –  Mark Grant Nov 23 '12 at 9:39
I would find it appreciable if the equivalences between theories were described explicitly (and with examples) for instance. Here's an example of the kind of things I'd like to see explained: if X is a K(G,1) space, then the cohomology of X (possibly with coefficients in something) is isomorphic to the group cohomology of G. –  seub Nov 23 '12 at 10:03
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I learned sheaf cohomology from Claire Voisin's Hodge Theory and Complex Algebraic Geometry I. This is a great book. As its name suggests, it also spends quite some time explaining Dolbeault cohomology, De Rham cohomology, singular cohomology, and how all these are defined/can be understood in terms of sheaf cohomology. (There's no group cohomology, as far as I recall.)

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Chapter 5 of Frank Warner's book, Foundations of Differentiable Manifolds, (http://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943) presents four different cohomology theories and why they are all isomorphic.

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