# Universal coefficient theorem

In algebraic topology, **universal coefficient theorems** establish relationships between homology and cohomology theories. For instance, the *integral homology theory* of a topological space X, and its *homology with coefficients* in any abelian group A are related as follows: the integral homology groups

*H*_{i}(*X*;**Z**)

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 **Z**/2**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}* 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 of the homology case

Consider the tensor product of modules *H*_{i}(*X*; **Z**) ⊗ *A*. The theorem states there is a short exact sequence

Furthermore, this sequence splits, though not naturally. Here μ is a map induced by the bilinear map *H _{i}*(

*X*;

**Z**) ×

*A*→

*H*(

_{i}*X*;

*A*).

If the coefficient ring A is **Z**/*p***Z**, this is a special case of the Bockstein spectral sequence.

## Universal coefficient theorem for cohomology

Let G be a module over a principal ideal domain R (e.g., **Z** or a field.)

There is also 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 h above is the canonical map:

An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to *K*(*G*, *i*) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a *weak right adjoint* to the homology functor.^{[1]}

## Example: mod 2 cohomology of the real projective space

Let *X* = **P**^{n}(**R**), the real projective space. We compute the singular cohomology of X with coefficients in *R* = **Z**/2**Z**.

Knowing that the integer homology is given by:

We have Ext(*R*, *R*) = *R*, Ext(**Z**, *R*) = 0, so that the above exact sequences yield

In fact the total cohomology ring structure is

## Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex X, *H _{i}*(

*X*;

**Z**) is finitely generated, and so we have the following decomposition.

where *β _{i}*(

*X*) are the Betti numbers of X and is the torsion part of . One may check that

and

This gives the following statement for integral cohomology:

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that *β*_{i}(*X*) = *β*_{n−i}(*X*).

## Notes

- ↑ (Kainen 1971)

## 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. - Kainen, P. C. (1971). "Weak Adjoint Functors".
*Mathematische Zeitschrift*.**122**: 1–9. doi:10.1007/bf01113560.