In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups:

completely determine its homology groups with coefficients in A, for any abelian group A:

Here 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 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 of and the Betti numbers with coefficients in a field . These can differ, but only when the characteristic of is a prime number for which there is some -torsion in the homology.

Consider the tensor product of modules . The theorem states there is a short exact sequence involving the Tor functor

Furthermore, this sequence splits, though not naturally. Here is the map induced by the bilinear map .

If the coefficient ring is , this is a special case of the Bockstein spectral sequence.

Let be a module over a principal ideal domain (for example , or any field.)

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

As in the homology case, the sequence splits, though not naturally. In fact, suppose

and define

Then above is the canonical map:

An alternative point of view can be based on representing cohomology via Eilenberg–MacLane space, where the map takes a homotopy class of maps to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.^[1]

Let , the real projective space. We compute the singular cohomology of with coefficients in using integral homology, i.e., .

Knowing that the integer homology is given by:

We have and , so that the above exact sequences yield

for all . In fact the total cohomology ring structure is

A special case of the theorem is computing integral cohomology. For a finite CW complex , is finitely generated, and so we have the following decomposition.

where are the Betti numbers of and is the torsion part of . One may check that

and

This gives the following statement for integral cohomology:

For an orientable, closed, and connected -manifold, this corollary coupled with Poincaré duality gives that .

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.

For cohomology we have

where is a ring with unit, is a chain complex of free modules over , is any -bimodule for some ring with a unit , and is the Ext group. The differential has degree .

Similarly for homology,

for the Tor group and the differential having degree .

• 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.
• Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1–9. doi:10.1007/bf01113560. S2CID 122894881.
• Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

• Universal coefficient theorem with ring coefficients