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is it true that when we compute homologies and cohomologies with coefficients in a field then homology and cohomology groups are isomorphic to each other?

That is valid when homology groups are free with integer coefficients.

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    Yes, these statements follow from the universal coefficient theorems.2011-06-01
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    _No._ What is true is that, as wckronholm says, the homology and cohomology are dual. This is not the same as saying they are isomorphic in infinite dimensions (e.g. consider $H_1$ and $H^1$ of a countable wedge of circles).2011-06-01

3 Answers 3