Universal coefficient theorem

In algebraic topology, universal coefficient theorems (UCT) 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 ) {\displaystyle H_{i}(X,\mathbb {Z} )} completely determine its homology groups with coefficients in A, for any abelian group A: H i ( X , A ) {\displaystyle H_{i}(X,A)} Here H i {\displaystyle 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 {\displaystyle A} to be Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {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 {\displaystyle b_{i}} of X {\displaystyle X} and the Betti numbers b i , F {\displaystyle b_{i,F}} with coefficients in a field F {\displaystyle F} . These can differ, but only when the characteristic of F {\displaystyle F} is a prime number p {\displaystyle p} for which there is some p {\displaystyle p} -torsion in the homology.