# Universal coefficient theorem

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

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.

## Statement of the homology case

Consider the tensor product of modules Hi(X; Z) ⊗ A. The theorem states there is a short exact sequence

${\displaystyle 0\to H_{i}(X;\mathbf {Z} )\otimes A\,{\overset {\mu }{\to }}\,H_{i}(X;A)\to \operatorname {Tor} (H_{i-1}(X;\mathbf {Z} ),A)\to 0.}$

Furthermore, this sequence splits, though not naturally. Here μ is a map induced by the bilinear map Hi(X; Z) × AHi(X; A).

If the coefficient ring A is Z/pZ, this is a special case of the Bockstein spectral sequence.

## Universal coefficient theorem for cohomology

Let G be a module over a principal ideal domain R (e.g., 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

${\displaystyle 0\to \operatorname {Ext} _{R}^{1}(\operatorname {H} _{i-1}(X;R),G)\to H^{i}(X;G)\,{\overset {h}{\to }}\,\operatorname {Hom} _{R}(H_{i}(X;R),G)\to 0.}$

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

In fact, suppose

${\displaystyle H_{i}(X;G)=\ker \partial _{i}\otimes G/\operatorname {im} \partial _{i+1}\otimes G}$

and define:

${\displaystyle H^{*}(X;G)=\ker(\operatorname {Hom} (\partial ,G))/\operatorname {im} (\operatorname {Hom} (\partial ,G)).}$

Then h above is the canonical map:

${\displaystyle h([f])([x])=f(x).}$

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.[1]

## 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 = Z/2Z.

Knowing that the integer homology is given by:

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

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

${\displaystyle \forall i=0,\cdots ,n:\qquad \ H^{i}(X;R)=R.}$

In fact the total cohomology ring structure is

${\displaystyle H^{*}(X;R)=R[w]/\left\langle w^{n+1}\right\rangle .}$

## Corollaries

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

${\displaystyle H_{i}(X;\mathbf {Z} )\cong \mathbf {Z} ^{\beta _{i}(X)}\oplus T_{i},}$

where βi(X) are the Betti numbers of X and ${\displaystyle T_{i}}$ is the torsion part of ${\displaystyle H_{i}}$. One may check that

${\displaystyle \operatorname {Hom} (H_{i}(X),\mathbf {Z} )\cong \operatorname {Hom} (\mathbf {Z} ^{\beta _{i}(X)},\mathbf {Z} )\oplus \operatorname {Hom} (T_{i},\mathbf {Z} )\cong \mathbf {Z} ^{\beta _{i}(X)},}$

and

${\displaystyle \operatorname {Ext} (H_{i}(X),\mathbf {Z} )\cong \operatorname {Ext} (\mathbf {Z} ^{\beta _{i}(X)},\mathbf {Z} )\oplus \operatorname {Ext} (T_{i},\mathbf {Z} )\cong T_{i}.}$

This gives the following statement for integral cohomology:

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

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that βi(X) = βni(X).

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