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 $Z$ .

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

$0 \to H_{p} (G; Z) \otimes M \to H_{p} (G; M) \to T o r (H_{p - 1} (G; Z), M) \to 0$

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

$H_{p} (G; M) ≅ (H_{p} (G; Z) \otimes M) \oplus T o r (H_{p - 1} (G; Z), M)$

Typical case of finitely generated abelian groups

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

Suppose further that:

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

and

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

Then we have:

$H_{p} (G; M) ≅ M^{r_{p}} \oplus (T_{p} \otimes M) \oplus T o r (T_{p - 1}, M)$

where we have:

$T_{p} \otimes M ≅ ⨁_{1 \leq i \leq s} M / a_{i} M$

and

$T o r (T_{p - 1}, M) ≅ ⨁_{1 \leq i \leq t} A_{n n M} (b_{i})$

where $A_{n n M} (b_{i}) = {x \in M ∣ b_{i} x = 0}$

Thus, overall:

$H_{p} (G; M) ≅ M^{r_{p}} \oplus ⨁_{1 \leq i \leq s} M / a_{i} M \oplus ⨁_{1 \leq i \leq t} A_{n n M} (b_{i})$

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

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

Then the expressions simpliy further:

$T_{p} \otimes M ≅ T_{p}^{w} \oplus \sum_{1 \leq i \leq s, 1 \leq j \leq u} Z / g c d (a_{i}, c_{j}) Z$

and

$T o r (T_{p - 1}, M) ≅ ⨁_{1 \leq i \leq t, 1 \leq j \leq u} Z / g c d (b_{i}, c_{j}) Z$

Statement

For coefficients in an abelian group

Typical case of finitely generated abelian groups

Related facts