Universal coefficient theorem for homology

Statement

For coefficients in an abelian group

Suppose $M$ is an abelian group and $X$ is a topological space. The universal coefficients theorem relates the homology groups for $X$ with integral coefficients (i.e., with coefficients in $\mathbb{Z}$) to the homology 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 this short exact sequence splits, so we obtain:

$H_n(X;M)\cong (H_n(X;\mathbb{Z})\otimes M)\oplus \mathrm{T}\mathrm{o}\mathrm{r}(H_{n-1}(X;\mathbb{Z}),M)$

For coefficients in a module over a principal ideal domain

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Related facts

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

Particular cases

Case of free abelian groups

If $H_{n-1}(X;\mathbb{Z})$ is a free abelian group, then we get:

$H_n(X;M)\cong H_n(X;\mathbb{Z})\otimes M$

As a corollary, if all the homology groups are free abelian, then the above holds for all $n$.