Group Cohomology - Examples and Constructions

Explicit computations reveal patterns in group cohomology and connect to other areas of mathematics.

ExampleCohomology of Symmetric Groups

For the symmetric group $S_n$ with trivial coefficients: $H^1(S_n, \mathbb{Z}) = 0 \quad \text{for } n \geq 3$

Higher cohomology is related to stable homotopy groups of spheres via the Barratt-Priddy-Quillen theorem.

ExampleGalois Cohomology

For a Galois extension $L/K$ with Galois group $G = \text{Gal}(L/K)$ , Galois cohomology studies $H^n(G, L^*)$ where $L^*$ is the multiplicative group.

Hilbert's Theorem 90: $H^1(G, L^*) = 0$

This is fundamental in algebraic number theory and class field theory.

Definition6.8Shapiro's Lemma

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

where $\text{Ind}_H^G M = \mathbb{Z}[G] \otimes_{\mathbb{Z}[H]} M$ is the induced module.

Remark

Shapiro's Lemma shows that cohomology commutes with induction, making computations more tractable by reducing to subgroups.

ExampleCohomology and Central Extensions

$H^2(G, M)$ classifies central extensions: $1 \to M \to E \to G \to 1$

where $M$ is in the center of $E$ . The Schur multiplier $H^2(G, \mathbb{C}^*)$ governs projective representations.

Definition6.9Hochschild-Serre Spectral Sequence

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

This spectral sequence computes group cohomology via a normal subgroup.

ExampleApplications to Topology

Group cohomology computes singular cohomology of classifying spaces: $H^n(BG; M) \cong H^n(G, M)$

where $BG$ is the classifying space of $G$ . This connects algebra to topology.