Universal coefficient theorem for cohomology

For an algebraic version of the theorem, see Groupprops:Universal coefficient theorem for group cohomology

Statement

For coefficients in an abelian group

Suppose $M$ is an abelian group and $X$ is a space with homology of finite type. The universal coefficients theorem relates the cohomology groups for $X$ with integral coefficients (i.e., with coefficients in $\mathbb{Z}$) to the cohomology groups with coefficients in $M$.

The theorem comes in two parts.

First, it states that there is a natural short exact sequence:

$0\to H^n(X;\mathbb{Z})\otimes M\to H^n(X;M)\to \mathrm{T}\mathrm{o}\mathrm{r}(H^{n+1}(X;\mathbb{Z}),M)\to 0$

Second, it states that the short exact sequence splits (non-canonically):

$H^n(X;M)\cong (H^n(X;\mathbb{Z})\otimes M)\oplus \mathrm{T}\mathrm{o}\mathrm{r}(H^{n+1}(X;\mathbb{Z}),M)$

Related facts

• Universal coefficient theorem for homology
• Dual universal coefficient theorem
• Kunneth formula for homology
• Kunneth formula for cohomology