Dual universal coefficient theorem for group cohomology

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).

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