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 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 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 EilenbergâÂÂMacLane space, where the map takes a homotopy class of maps to the corresponding homomorphism induced in homology. Thus, the EilenbergâÂÂMacLane space is a weak right adjoint to the homology functor.
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 .