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 $p\geq 0$, 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:

$\operatorname {Hom} (T_{p},M)\cong \bigoplus _{1\leq i\leq s}\operatorname {Ann} _{M}(a_{i})$

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

Also:

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

Thus, we get overall that:

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

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:

$\operatorname {Hom} (T_{p},M)\cong \bigoplus _{1\leq i\leq s,1\leq j\leq u}\mathbb {Z} /\operatorname {gcd} (a_{i},c_{j})\mathbb {Z}$

and:

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

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