Dual universal coefficient theorem for group cohomology

Statement

For coefficients in an abelian group

Suppose $G$ is a group and $A$ 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 $A$ as follows:

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

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

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

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

For coefficients in the integers

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

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

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