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, and its homology with coefficients in any abelian group are related as follows: the integral homology groups
completely determine the groups
Here 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 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.

Statement of the homology case

Consider the tensor product of modules. The theorem states there is a short exact sequence
Furthermore, this sequence splits, though not naturally. Here is a map induced by the bilinear map.
If the coefficient ring is, this is a special case of the Bockstein spectral sequence.

Universal coefficient theorem for cohomology

Let be a module over a principal ideal domain
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 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 from to to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.

Example: mod 2 cohomology of the real projective space

Let, the real projective space. We compute the singular cohomology of with coefficients in.
Knowing that the integer homology is given by:
We have, 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, 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.