# cohomology vs homology

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