Group Cohomology - Main Theorem

The cohomological dimension and transfer maps are fundamental structural results in group cohomology.

Definition6.10Cohomological Dimension

The cohomological dimension of a group $G$ is: $\text{cd}(G) = \sup\{n : H^n(G, M) \neq 0 \text{ for some } G\text{-module } M\}$

For free groups, $\text{cd}(G) \leq 1$ . For surface groups, $\text{cd}(G) = 2$ .

Theorem6.11Stallings-Swan Theorem

A group $G$ has cohomological dimension $\leq 1$ if and only if $G$ is a free group.

Remark

This theorem shows that cohomological dimension detects geometric properties of groups, connecting group cohomology to geometric group theory.

Theorem6.12Transfer (Corestriction) Theorem

For a finite-index subgroup $H \leq G$ with $[G:H] = n$ , there exists a transfer map: $\text{Cor}: H^k(H, M) \to H^k(G, M)$

satisfying $\text{Cor} \circ \text{Res} = [G:H] \cdot \text{id}$ .

In particular, if $[G:H]$ is coprime to the order of $H^k(G, M)$ , restriction is injective.

Theorem6.13Eckmann-Shapiro Theorem

For a subgroup $H \leq G$ and $H$ -module $M$ : $H^n(G, \text{Ind}_H^G M) \cong H^n(H, M)$

where $\text{Ind}_H^G M = \mathbb{Z}[G] \otimes_{\mathbb{Z}[H]} M$ . This isomorphism is natural and compatible with cup products.

ExampleApplication to Finite Groups

For finite group $G$ and trivial module $\mathbb{Z}$ : $|G| \cdot H^n(G, \mathbb{Z}) = 0 \quad \text{for all } n \geq 1$

This follows from $\text{Cor} \circ \text{Res} = |G| \cdot \text{id}$ with $H = \{e\}$ .

Theorem6.14Lyndon-Hochschild-Serre Spectral Sequence

For a normal subgroup $N \triangleleft G$ and $G$ -module $M$ : $E_2^{p,q} = H^p(G/N, H^q(N, M)) \Rightarrow H^{p+q}(G, M)$

This spectral sequence is fundamental for computing group cohomology via normal subgroups.