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I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for the homology the indexes are decreasing and in cohomology are increasing).

Why, in some geometric applications it is done a radical selection and it is important to distinguish homology from cohomology?

For example if one constructs a Cech homology instead af Cech cohomology, what are the differences?

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See also mathoverflow.net/questions/20877/… . –  Qiaochu Yuan Jun 20 '12 at 10:20
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One short answer is that cohomology (of spaces, not of general chain complexes) carries a ring structure, the cup product. There are spaces with the same homology and cohomology as groups, but where the ring structure on the cohomologies is different; in this case one can use the cup product to distinguish them.

There is probably a lot one can say as far as a long answer. For example, for nice spaces cohomology is representable by Eilenberg-MacLane spaces, so one can say a lot about cohomology by studying these spaces (e.g. classifying cohomology operations using the Yoneda lemma), and analogous things don't seem to be true of homology.

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so we have this graded ring: en.wikipedia.org/wiki/Cohomology_ring which doesn't exist for homology. –  fair-coin tossing Jun 20 '12 at 10:32
    
In nice situations (see mathoverflow.net/questions/415/does-homology-have-a-coproduct) homology admits a dual structure called a comultiplication, but comultiplications are a little less intuitive to work with. –  Qiaochu Yuan Jun 20 '12 at 10:36
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