Universal coefficient theorem for homology

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:

$$\! 0 \to H_n(X; \mathbb{Z}) \otimes M \to H_n(X;M) \to \operatorname{Tor}(H_{n-1}(X;\mathbb{Z}),M) \to 0$$

Second, it states that this short exact sequence splits, so we obtain:

$$H_n(X;M) \cong (H_n(X;\mathbb{Z}) \otimes M) \oplus \operatorname{Tor}(H_{n-1}(X;\mathbb{Z}),M)$$

Related facts

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

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 $$n$$.