Universal coefficient theorem for group homology

Statement

For coefficients in an abelian group

Suppose $G$ is a group and $M$ is an abelian group. The universal coefficients theorem for group homology describes the homology groups for trivial group action of $G$ on $M$ in terms of the homology groups for trivial group action of $G$ on $\mathbb {Z}$.

Explicitly, it states that there is a natural short exact sequence of abelian groups:

$0\to H_{p}(G;\mathbb {Z} )\otimes M\to H_{p}(G;M)\to \operatorname {Tor} (H_{p-1}(G;\mathbb {Z} ),M)\to 0$

The sequence splits (though not naturally) to give that:

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

Typical case of finitely generated abelian groups

Suppose $H_{p}(G;\mathbb {Z} )\cong \mathbb {Z} ^{r_{p}}\oplus T_{p}$ for some finite group $T_{p}$ and $H_{p-1}(G;\mathbb {Z} )\cong \mathbb {Z} ^{r_{p-1}}\oplus T_{p-1}$ for some finite group $T_{p-1}$.

Suppose further that:

$T_{p}\cong \mathbb {Z} /a_{1}\mathbb {Z} \oplus \mathbb {Z} /a_{2}\mathbb {Z} \oplus \dots \oplus \mathbb {Z} /a_{s}\mathbb {Z}$

and

$T_{p-1}\cong \mathbb {Z} /b_{1}\mathbb {Z} \oplus \mathbb {Z} /b_{2}\mathbb {Z} \oplus \dots \oplus \mathbb {Z} /b_{t}\mathbb {Z}$

Then we have:

$H_{p}(G;M)\cong M^{r_{p}}\oplus (T_{p}\otimes M)\oplus \operatorname {Tor} (T_{p-1},M)$

where we have:

$T_{p}\otimes M\cong \bigoplus _{1\leq i\leq s}M/a_{i}M$

and

$\operatorname {Tor} (T_{p-1},M)\cong \bigoplus _{1\leq i\leq t}\operatorname {Ann} _{M}(b_{i})$

where $\operatorname {Ann} _{M}(b_{i})=\{x\in M\mid b_{i}x=0\}$

Thus, overall:

$H_{p}(G;M)\cong M^{r_{p}}\oplus \bigoplus _{1\leq i\leq s}M/a_{i}M\oplus \bigoplus _{1\leq i\leq t}\operatorname {Ann} _{M}(b_{i})$

If, further, $M$ is a finitely generated abelian group, of the form:

$M\cong \mathbb {Z} ^{w}\oplus \mathbb {Z} /c_{1}\mathbb {Z} \oplus \mathbb {Z} /c_{2}\mathbb {Z} \oplus \dots \oplus \mathbb {Z} /c_{u}\mathbb {Z}$

Then the expressions simpliy further:

$T_{p}\otimes M\cong T_{p}^{w}\oplus \sum _{1\leq i\leq s,1\leq j\leq u}\mathbb {Z} /\operatorname {gcd} (a_{i},c_{j})\mathbb {Z}$

and

$\operatorname {Tor} (T_{p-1},M)\cong \bigoplus _{1\leq i\leq t,1\leq j\leq u}\mathbb {Z} /\operatorname {gcd} (b_{i},c_{j})\mathbb {Z}$

Related facts

• Universal coefficients theorem for group cohomology
• Dual universal coefficients theorem for group cohomology