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We all know that we can compute homology and cohomology with arbitrary coefficient if we already know the homology groups with coefficient in $\mathbb{Z}$. I wonder if it is possible if we know the cohomology groups with coefficient in some groups, then we are able to calculate the cohomology groups with coefficients in other groups. To be a little more specific, we can compute fairly easily the de Rham cohomology on manifolds. Does such groups give information about cohomology group with coefficients in other groups? Or inversely, if we some how know the cohomology with $\mathbb{Z}$ coefficients, can we calculate its cohomology groups with $\mathbb{R}$?

Qijun Tan
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    Sure there is such a theorem. Have you looked up all the various universal coefficient theorems in Spanier's textbook? When using real coefficients the primary thing you lose is torsion. – Ryan Budney Mar 04 '15 at 22:15

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