Geodetic datum

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

H_i(X, Z)

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 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 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 p-torsion in the homology.

Statement of the homology case

Consider the tensor product of modules H_i(X, Z) ⊗ A.The theorem states that there is an injective group homomorphism ι from this group to H_i(X, A), which has cokernel Tor(H_i-1(X, Z), A). In other words, there is a natural short exact sequence

$0\to H_i(X,\mathbf{Z})\otimes A\to H_i(X,A)\to \text{Tor}(H_{i-1}(X,\mathbf{Z}),A)\to 0.$

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

Let G be a module over a principal ideal domain R (e.g.,$\mathbf{Z}$ or a field.)

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

$0\to \mathrm{Ext}({\mathrm{H}}_{i-1}(X;R),G)\to {\mathrm{H}}^{}(X;G)\stackrel{h}{\to }{\mathrm{Hom}}_R(H_i(X;R),G)\to 0.$

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

In fact, suppose ${\mathrm{H}}_i(X;G)=\ker {\mathrm{\partial }}_i\otimes G/\mathrm{im}{\mathrm{\partial }}_{i+1}\otimes G$ and ${\mathrm{H}}^{*}(X;G)$ is defined as $\ker (\mathrm{Hom}(\partial ,G))/\mathrm{im}(\mathrm{Hom}(\partial ,G))$. Then h above is the canonical map: $h([f])([x])=f(x).$ 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 := Z₂.

Knowing that the integer homology is given by:

$H_i(X;\mathbf{Z})=\{\begin{array}{ll}\mathbf{Z}& i=0\text{ or }i=n\text{ odd,}\\ \mathbf{Z}/2\mathbf{Z}& 0<i<n,i\text{odd,}\\ 0& \text{else.}\end{array}$

We have Ext(R, R) = R, Ext(Z, R)= 0, so that the above exact sequences yield

$\mathrm{\forall }i=0\dots n,H^i(X;R)=R$.

In fact the total cohomology ring structure is

$H^{*}(X;R)=R[w]/⟨w^{n+1}⟩$.

Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex X, $H_i(X;\mathbf{Z})$ is finitely generated, and so we have the following decomposition.

$H_i(X;\mathbf{Z})\cong {\mathbf{Z}}^{{\beta }_i(X)}\oplus T_i.$

Where ${\beta }_i(X)$ are the betti numbers of X. One may check that

$\text{Hom}(H_i(X),\mathbf{Z})\cong \text{Hom}({\mathbf{Z}}^{{\beta }_i(X)},\mathbf{Z})\oplus \text{Hom}(T_i,\mathbf{Z})\cong {\mathbf{Z}}^{{\beta }_i(X)}$, and $\text{Ext}(H_i(X),\mathbf{Z})\cong \text{Ext}({\mathbf{Z}}^{{\beta }_i(X)},\mathbf{Z})\oplus \text{Ext}(T_i,\mathbf{Z})\cong T_i.$

This gives the following statement for integral cohomology:

$H^i(X;\mathbf{Z})\cong {\mathbf{Z}}^{{\beta }_i(X)}\oplus T_{i-1}.$

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that ${\beta }_i(X)={\beta }_{n-i}(X)$.

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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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