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?