Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its "integral homology groups":
completely determine its "homology groups with coefficients in" , for any abelian group :
Here might be the simplicial homology, or more generally the singular homology: 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 involving the Tor functor 
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 (e.g., 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 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:
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 formula_9 is the torsion part of formula_10. 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 .