Universal coefficient theorem

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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 shows that the integral homology groups

Hi(X,Z)

do in a certain, definite sense determine the groups

Hi(X,A).

Here H∗ 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.

The statement of the universal coefficient theorem runs as follows. Consider

H_i \otimes A

where Hi means Hi(X,Z). Then there is an injective group homomorphism ι from it to Hi(X,A). The theorem describes the cokernel of ι as

Tor(Hi − 1,A).

This Tor group can therefore be described as the obstruction to ι being an isomorphism, which could be thought of as the 'expected' result.

This can be summarized saying there is a natural short exact sequence

0 \rightarrow H_i\otimes A\rightarrow H_i(X,A)\rightarrow\mbox{Tor}(H_{i-1},A)\rightarrow 0

Furthermore, this is a split sequence (but the splitting is not natural).


There is also a universal coefficient theorem for cohomology involving the Ext functor, stating that there is a natural short exact sequence

0 \rightarrow \mbox{Ext}(H_{i-1},A)\rightarrow H^i(X,A)\rightarrow\mbox{Hom}(H_i,A)\rightarrow 0

As in the homological case, the sequence splits, though not naturally.

[edit] References

  • 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.
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