How can I calculate cohomology?

Question

I know what homology means and I know what (de Rham) cohomology means. But I can’t actually calculate cohomology of a given space (while I can do it for homology)

My question is that is there a relationship between homology and cohomology? (I mean can I derive something about cohomology is I know its homology?) If not, how can I calculate cohomology?

you know about the topological version of cohomology (which is the dual of homology?) one can use this quite well for computations. Also, you have stuff like poincare theorem, and so on. — Felix, Feb 19 '19 at 13:23
Actually, the only thing I know about cohomology is Poincare theorem. Can you explain more about the topological version of cohomology please? — Mike Park, Feb 28 '19 at 13:23

score 1 · Accepted Answer · answered Feb 19 '19 at 16:24

de Rham cohomology is only defined for smooth manifolds. It it is known that it agrees with singular cohomology on smooth manifolds.

Now you can apply the universal coefficient theorem for cohomology. See for example https://en.wikipedia.org/wiki/Universal_coefficient_theorem or any textbook on algebraic topology. This theorem establishes a short exact sequence $$0 \to \text{Ext}(H_{n−1}(X);\mathbb{Z})\to H^n(X) \to \text{Hom}(H_n(X);\mathbb{Z}) \to 0$$ In many cases you can compute $H^n(X)$ using this sequence. However, in general it is an extension problem with a non-unique solution.

How can I calculate cohomology?

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