Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space $X$ , its integral homology groups:

H i (X; Z)

completely determine its homology groups with coefficients in $A$ , for any abelian group $A$ :

H i (X; A)

Here $H i$ might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients $A$ may be used, at the cost of using a Tor functor.

For example it is common to take $A$ to be $Z /2 Z$ , so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers $b i$ of $X$ and the Betti numbers $b i, F$ with coefficients in a field $F$ . These can differ, but only when the characteristic of $F$ is a prime number $p$ for which there is some $p$ -torsion in the homology.