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.
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
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.
, this is a special case of the Bockstein spectral sequence.
be a module over a principal ideal domain
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.
, the real projective space.
We compute the singular cohomology of
Knowing that the integer homology is given by: We have
, 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.
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.
is a chain complex of free modules over
the Tor group and the differential