In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X , and its homology with coefficients in any abelian group A . It shows that the integral homology groups

: H i ( X , Z )

do in a certain, definite sense determine the groups

: H i ( X , A ).

Here H * might be the .

For example it is common to take A to be Z/2Z , so that coefficients are mod 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 (mathematics) 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.

The statement of the universal coefficient theorem runs as follows: consider

: H_i otimes A

where H i means H i ( X , Z ). Then there is an injective group homomorphism ι from it to H i ( X , A ). The theorem describes the cokernel of ι as

: Tor ( H i − 1, A ).

This Tor group can therefore be described as the obstruction to ι being an isomorphism, which could be thought of as the expected result.

There is also a universal coefficient theorem for cohomology, involving the Ext functor.