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
- Hi(X; Z)
completely determine the groups
- Hi(X; A)
Here Hi 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/2Z, 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 bi of X and the Betti numbers bi,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
Furthermore, this sequence splits, though not naturally. Here μ is a map induced by the bilinear map Hi(X; Z) A → Hi(X; A).
If the coefficient ring A is Z/pZ, 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.)
As in the homology case, the sequence splits, though not naturally.
In fact, suppose
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.
Example: mod 2 cohomology of the real projective space
Let X = Pn(R), the real projective space. We compute the singular cohomology of X with coefficients in R = Z/2Z.
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
A special case of the theorem is computing integral cohomology. For a finite CW complex X, Hi(X; Z) is finitely generated, and so we have the following decomposition.
where βi(X) are the Betti numbers of X and is the torsion part of . One may check that
This gives the following statement for integral cohomology:
- 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.