# 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:

$T_p \otimes M \cong \bigoplus_{1 \le i \le s} M/a_iM$

and

$\operatorname{Tor}(T_{p-1},M) \cong \bigoplus_{1 \le i \le t} \operatorname{Ann}_M(b_i)$

where $\operatorname{Ann}_M(b_i) = \{ x \in M \mid b_ix = 0 \}$

Thus, overall:

$H_p(G;M) \cong M^{r_p} \oplus \bigoplus_{1 \le i \le s} M/a_iM \oplus \bigoplus_{1 \le i \le t} \operatorname{Ann}_M(b_i)$

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:

$T_p \otimes M \cong T_p^w \oplus \sum_{1 \le i \le s, 1 \le j \le u} \mathbb{Z}/\operatorname{gcd}(a_i,c_j)\mathbb{Z}$

and

$\operatorname{Tor}(T_{p-1},M) \cong \bigoplus_{1 \le i \le t, 1 \le j \le u} \mathbb{Z}/\operatorname{gcd}(b_i,c_j)\mathbb{Z}$