In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups:

H_i(X; Z)

completely determine its homology groups with coefficients in A, for any abelian group A:

H_i(X; A)

Here H_i might be the simplicial homology, or more generally the singular homology. The usual proof of this result 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 there is a short exact sequence involving the Tor functor

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

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

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

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:

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

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

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

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, H_i(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 β_i(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 β_i(X) = β_n−i(X).

Universal coefficient spectral sequence

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.

For cohomology we have

${\displaystyle E_{2}^{p,q}=Ext_{R}^{q}(H_{p}(C_{*}),G)\Rightarrow H^{p+q}(C_{*};G)}$

Where ${\displaystyle R}$ is a ring with unit, ${\displaystyle C_{*))$ is a chain complex of free modules over ${\displaystyle R}$, ${\displaystyle G}$ is any ${\displaystyle (R,S)}$-bimodule for some ring with a unit ${\displaystyle S}$, ${\displaystyle Ext}$ is the Ext group. The differential ${\displaystyle d^{r))$ has degree ${\displaystyle (1-r,r)}$.

Similarly for homology

${\displaystyle E_{p,q}^{2}=Tor_{q}^{R}(H_{p}(C_{*}),G)\Rightarrow H_{*}(C_{*};G)}$

for Tor the Tor group and the differential ${\displaystyle d_{r))$ having degree ${\displaystyle (r-1,-r)}$.