Universal coefficient theorem

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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

$H_i(X, \mathbb{Z})$

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 $\mathbb{Z}/2\mathbb{Z}$, 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

Consider the tensor product $H_i(X, \mathbb{Z}) \otimes A$. The theorem states that there is an injective group homomorphism ι from this group to Hi(X,A), which has cokernel $\mbox{Tor}(H_{i-1}(X, \mathbb{Z}), A)$.

In other words, there is a natural short exact sequence

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

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

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

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

Example: mod 2 cohomology of the real projective space

Let $X = \mathbf {RP^n}$, the real projective space. We compute the singular cohomology of X with coefficients in

$R := \mathbf Z_2$.

knowing that the integer homology is given by:

$H_i(X; \mathbf{Z}) = \begin{cases} \mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\ \mathbf{Z/2Z} & 0

We have $\mathrm{Ext}(R, R)= R, \mathrm{Ext}(\mathbf Z, R)= 0$, so that the above exact sequences yield

$\forall i = 0 \ldots n , \ H^i (X; R) = R$.

In fact the total cohomology ring structure is

H * (X;R) = R[w] / < wn + 1 > .

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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