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:

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

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

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

For coefficients in a module over a principal ideal domain

Fill this in later

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 $n$.