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After some time spent on it I now have some understanding of de Rham cohomology and can actually calculate some cohomology groups. However I have now and then encountered many other cohomology theories: compact, harmonic, rational, Cech ...

I understand that they all share a similar structure, however it seems to me that the calculation techniques can be quite different: for example homotopy invariance is a key tool to use in de Rham cohomology, but it is not valid in compact cohomology.

So, is it that at a higher level of knowledge the similarities between the various theories are such that some experiences in computing cohomology groups in one of them can be easily transferred to the others or is just life hard?

I should point out that my aim is certainly not to perform research in this area, but rather to be able to calculate the cohomology groups of some non-trivial space (but nothing terribly complicated) when the need arises.

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    $\begingroup$ en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms $\endgroup$
    – Ben McKay
    Commented Nov 27, 2013 at 19:58
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    $\begingroup$ Thanks Ben, I actually knew that cohomology can be axiomatised, part of my question was indeed to understand wether the common structure is large enough that learning to calculate in one makes it very easy learning how to calculate in the other or not, sorry if that was not clear. I am more interested in using cohomology as a tool rather then in its more formal aspects. $\endgroup$
    – GFR
    Commented Nov 27, 2013 at 20:03
  • $\begingroup$ It depends on the specific problem you are interested in.For example, in algebraic geometry, when the ground field is not necessarily C (complex numbers), you need étale cohomology (constructed by GROTHENDIECK via étale topology) (instead of the ordidary complex one you must first learn about), to solve , let us say, Weil's conjectures.It is a theorem that over C the two cohomlogies are isomorphic ! $\endgroup$
    – Al-Amrani
    Commented Nov 27, 2013 at 20:20
  • $\begingroup$ I thought that what I am asking -given that cohomology is not my field but a tool and I need to employ it as a relatively basic level, is it possible to transfer basic results from one cohomology theory to the other or does one need heavy machinery to do that or are only higher level results which do transfer- is a question that often arises in research: you need a tool which is not in your main area, is it worth learning it? In my case the tool is basic use of different cohomology theories. However I am fine with this being moved if you think it is not appropriate. $\endgroup$
    – GFR
    Commented Nov 27, 2013 at 20:54
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    $\begingroup$ I think it's unfair to point the poster to the Eilenberg-Steenrod axioms without pointing out that compactly supported cohomology (which s/he asks about specifically) does not satisfy those axioms and so is not a "cohomology theory" in that sense. $\endgroup$
    – Greg Friedman
    Commented Nov 28, 2013 at 6:03

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