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One knows from the Universal Coefficient Theorem that Integral Homology can be used to derive homology with coefficients in any other groups like e.g. Reals, Rationals, Z/2Z etc.

Suppose you have access to Homology group computed over real coefficients (& hence Betti Numbers computed over reals). To begin with I am not interested in the torsion information.

My question is the following: Then what can you tell about the Integral Betti Numbers, given the information about Real Betti numbers? Is there some kind of a partial converse to the Universal Coefficient Theorem?

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The Betti numbers over a field will only differ from the Betti numbers when the homology contains some $p$-torsion and the field is of characteristic $p$ (this is straight from the Universal Coefficient Theorem).

Then since $\mathbb{R}$ has characteristic $0$...

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Thanks Juan! That makes sense... – Brian D'Souza Apr 8 '12 at 16:57

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