Universal coefficient theorem

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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.