Universal coefficient theorem

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:



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



Here $H_{i}(X; Z)$ 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 $H_{i}(X; A)$, 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 $H_{i}$ of $X$ and the Betti numbers $Z/2Z$ 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 $b_{i}$. The theorem states there is a short exact sequence involving the Tor functor


 * $$ 0 \to H_i(X; \mathbf{Z})\otimes A \, \overset{\mu}\to \, H_i(X;A) \to \operatorname{Tor}_1(H_{i-1}(X; \mathbf{Z}),A)\to 0.$$

Furthermore, this sequence splits, though not naturally. Here $μ$ is the map induced by the bilinear map $b_{i,F}$.

If the coefficient ring $A$ is $H_{i}(X; Z) ⊗ A$, 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., $H_{i}(X; Z) × A → H_{i}(X; A)$ 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


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

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

In fact, suppose


 * $$H_i(X;G) = \ker \partial_i \otimes G / \operatorname{im}\partial_{i+1} \otimes G$$

and define:


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

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 $Z/pZ$ 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 $Z$, the real projective space. We compute the singular cohomology of $X$ with coefficients in $K(G, i)$.

Knowing that the integer homology is given by:


 * $$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 $X = P^{n}(R)$, so that the above exact sequences yield


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

In fact the total cohomology ring structure is


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

Corollaries
A special case of the theorem is computing integral cohomology. For a finite CW complex $X$, $R = Z/2Z$ 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 $Ext(R, R) = R, Ext(Z, R) = 0$ are the Betti numbers of $X$ and $$T_i$$ is the torsion part of $$H_i$$. One may check that


 * $$ \operatorname{Hom}(H_i(X),\mathbf{Z}) \cong \operatorname{Hom}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \operatorname{Hom}(T_i, \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)},$$

and


 * $$\operatorname{Ext}(H_i(X),\mathbf{Z}) \cong \operatorname{Ext}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \operatorname{Ext}(T_i, \mathbf{Z}) \cong T_i.$$

This gives the following statement for integral cohomology:


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

For $X$ an orientable, closed, and connected $n$-manifold, this corollary coupled with Poincaré duality gives that $H_{i}(X; Z)$.

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

For cohomology we have


 * $$E^{p,q}_2=Ext_{R}^q(H_p(C_*),G)\Rightarrow H^{p+q}(C_*;G)$$

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

Similarly for homology


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

for Tor the Tor group and the differential $$d_r$$ having degree $$(r-1,-r)$$.