Universal coefficient theorem - Definition and Overview

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 simplicial homology or more general singular homology theory: the result itself 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/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 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

<math> H_i \otimes A<math>

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.