 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
completely determine the groups
 H_{i}(X,A)
Here H_{i} 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 , so that coefficients are modulo 2. This becomes straightforward in the absence of 2torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b_{i} of X and the Betti numbers b_{i,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 ptorsion in the homology.
Contents
Statement
Consider the tensor product . The theorem states that there is an injective group homomorphism ι from this group to H_{i}(X,A), which has cokernel .
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
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.
Example: mod 2 cohomology of the real projective space
Let , the real projective space. We compute the singular cohomology of X with coefficients in
 .
knowing that the integer homology is given by:
We have , so that the above exact sequences yield
 .
In fact the total cohomology ring structure is
 H ^{*} (X;R) = R[w] / < w^{n + 1} > .
References
 Allen Hatcher, Algebraic Topology , Cambridge University Press, Cambridge, 2002. ISBN 0521795400. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
Categories: Algebraic topology
 Homological algebra
 Theorems in topology
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