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:

$0\to \mathrm{E}\mathrm{x}\mathrm{t}(H_{p-1}(G;\mathbb{Z}),M)\to H^p(G;M)\to \mathrm{H}\mathrm{o}\mathrm{m}(H_p(G;\mathbb{Z}),M)\to 0$

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

$H^p(G;M)\cong \mathrm{H}\mathrm{o}\mathrm{m}(H_p(G;\mathbb{Z}),M)\oplus \mathrm{E}\mathrm{x}\mathrm{t}(H_{p-1}(G;\mathbb{Z}),M)$

For coefficients in the integers

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

$H^p(G;\mathbb{Z})\cong \mathrm{H}\mathrm{o}\mathrm{m}(H_p(G;\mathbb{Z}),\mathbb{Z})\oplus \mathrm{E}\mathrm{x}\mathrm{t}(H_{p-1}(G;\mathbb{Z}),\mathbb{Z})$

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:

$H^p(G;M)\cong M^{r_p}\oplus \mathrm{H}\mathrm{o}\mathrm{m}(T_p,M)\oplus \mathrm{E}\mathrm{x}\mathrm{t}(T_{p-1},M)$

where we further have:

$\mathrm{H}\mathrm{o}\mathrm{m}(T_p,M)\cong {⨁}_{1\le i\le s}{\mathrm{A}}_{\mathrm{n}}(a_i)$

where ${\mathrm{A}}_{\mathrm{n}}(a_i)=\{x\in M\mid a_{i}x=0\}$, i.e., the $a_i$-torsion of $M$.

Also:

$\mathrm{E}\mathrm{x}\mathrm{t}(T_{p-1},M)\cong {⨁}_{1\le i\le t}M/b_{i}M$

Thus, we get overall that:

$H^p(G;M)\cong M^{r_p}\oplus {⨁}_{1\le i\le s}{\mathrm{A}}_{\mathrm{n}}(a_i)\oplus {⨁}_{1\le i\le t}M/b_{i}M$

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:

$\mathrm{H}\mathrm{o}\mathrm{m}(T_p,M)\cong {⨁}_{1\le i\le s,1\le j\le u}\mathbb{Z}/\mathrm{g}\mathrm{c}\mathrm{d}(a_i,c_j)\mathbb{Z}$

and:

$\mathrm{E}\mathrm{x}\mathrm{t}(T_{p-1},M)\cong T_{p-1}^w\oplus {⨁}_{1\le i\le t,1\le j\le u}\mathbb{Z}/\mathrm{g}\mathrm{c}\mathrm{d}(b_i,c_j)\mathbb{Z}$

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