Universal coefficient theorem

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 states that the integral homology groups

completely determine the groups

H_i(X,A).

Here H_i 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 , 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.

Statement

Consider the tensor product . The theorem states that there is an injective group homomorphism ι from this group to H_i(X,A), which has cokernel .

In other words, there is a natural short exact sequence

Furthermore, this is a split sequence (but the splitting is not natural).

The Tor group on the right can be thought of as the obstruction to ι being an isomorphism.

Universal coefficient theorem for cohomology

There is also a universal coefficient theorem for cohomology involving the Ext functor, stating that there is a natural short exact sequence

As in the homological case, the sequence splits, though not naturally.

References

• Allen Hatcher, Algebraic Topology , Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.