Universal coefficient theorem
In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A. It states that 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.
Consider the tensor product of modules Hi(X, Z) ⊗ A.The theorem states that there is an injective group homomorphism ι from this group to Hi(X, A), which has cokernel Tor(Hi-1(X, Z), A). In other words, there is a natural short exact sequence
Furthermore, this is a split sequence (but the splitting is not natural).
The Tor group on the right can be thought of as the obstruction to ι being an isomorphism.
Universal coefficient theorem for cohomology
Let G be a module over a principal ideal domain R (e.g., or a field.)
There is also a universal coefficient theorem for cohomology involving the Ext functor, stating that there is a natural short exact sequence
As in the homological case, the sequence splits, though not naturally.
In fact, suppose and is defined as . Then h above is the canonical map:
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 := Z2.
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, is finitely generated, and so we have the following decomposition.
Where are the betti numbers of X. One may check that
- , and
This gives the following statement for integral cohomology: