Dual universal coefficient theorem for group cohomology

Statement

For coefficients in an abelian group

Suppose $G$ is a group and $M$ is an abelian group. The dual universal coefficients theorem relates the homology groups for trivial group action of $G$ on $\mathbb{Z}$ and the cohomology groups for trivial group action of $G$ on $M$ as follows:

First, for any $p\ge 0$, there is a natural short exact sequence of abelian groups:

$0 \to \operatorname{Ext}(H_{p-1}(G;\mathbb{Z}),M) \to H^p(G;M) \to \operatorname{Hom}(H_p(G;\mathbb{Z}),M) \to 0$

Second, the sequence splits (not necessarily naturally), and we get:

$H^p(G;M) \cong \operatorname{Hom}(H_p(G;\mathbb{Z}),M) \oplus \operatorname{Ext}(H_{p-1}(G;\mathbb{Z}),M)$

For coefficients in the integers

This is the special case where $M = \mathbb{Z}$. In this case, we case:

$H^p(G;\mathbb{Z}) \cong \operatorname{Hom}(H_p(G;\mathbb{Z}),\mathbb{Z}) \oplus \operatorname{Ext}(H_{p-1}(G;\mathbb{Z}),\mathbb{Z})$

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:

$T_p \cong \mathbb{Z}/a_1\mathbb{Z} \oplus \dots \mathbb{Z}/a_s\mathbb{Z}$

and

$T_{p-1} \cong \mathbb{Z}/b_1\mathbb{Z} \oplus \dots \mathbb{Z}/b_t\mathbb{Z}$

Then we have:

$H^p(G;M) \cong M^{r_p} \oplus \operatorname{Hom}(T_p,M) \oplus \operatorname{Ext}(T_{p-1},M)$

where we further have:

$\operatorname{Hom}(T_p,M) \cong \bigoplus_{1 \le i \le s} \operatorname{Ann}_M(a_i)$

where $\operatorname{Ann}_M(a_i) = \{ x \in M \mid a_ix = 0 \}$, i.e., the $a_i$-torsion of $M$.

Also:

$\operatorname{Ext}(T_{p-1},M) \cong \bigoplus_{1 \le i \le t} M/b_iM$

Thus, we get overall that:

$H^p(G;M) \cong M^{r_p} \oplus \bigoplus_{1 \le i \le s} \operatorname{Ann}_M(a_i) \oplus \bigoplus_{1 \le i \le t} M/b_iM$

Finally, suppose:

$M \cong \mathbb{Z}^w \oplus \mathbb{Z}/c_1\mathbb{Z} \oplus \dots \mathbb{Z}/c_u\mathbb{Z}$

In this case, the expressions simplify further:

$\operatorname{Hom}(T_p,M) \cong \bigoplus_{1 \le i \le s, 1 \le j \le u} \mathbb{Z}/\operatorname{gcd}(a_i,c_j)\mathbb{Z}$

and:

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

Typical case of finitely generated abelian groups and coefficients in the integers

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 $T_{p-1}$. Then:

$H^p(G;\mathbb{Z}) \cong \mathbb{Z}^{r_p} \oplus T_{p-1}$

In other words, we pick the torsion-free part of $H_p$ and the torsion part of $H_{p-1}$ (roughly speaking).

Related facts

Similar facts for group cohomology

• Universal coefficients theorem for group homology
• Universal coefficients theorem for group cohomology
• Kunneth formula for group homology
• Kunneth formula for group cohomology

Similar facts for other homology and cohomology theories

• Dual universal coefficients theorem for cohomology (for topological spaces)
• Universal coefficients theorem for homology (for topological spaces)
• Universal coefficients theorem for cohomology (for topological spaces)

Applications

• This can be used to show the equivalence of the facts: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms and first homology group for trivial group action equals tensor product with abelianization
• Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization