TITLE: Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?
QUESTION [7 upvotes]: I am reading Class field theory - Bonn Lectures by Neukirch.
Given a $G$ module $A$ he defines Cohomology groups $H^i(G,A) : i\in \mathbb{Z}$ by considering some complete resolution of $G$ modules and applying hom functor.
He then calculated $H^{-1}(G,A),H^0(G,A),H^1(G,A),H^2(G,A)$ 
It says $H^2(G,A)$ is related to group extensions. Given an abelian group $A$ written multiplicatively and an arbitrary subgroup $G$, we want to find all group extensions $\hat{G}$ of $A$ such that $A$ is normal subgroup pf $\hat{G}$ and $\hat{G}/A\cong G$. He says this is related to $H^2(G,A)$.
It says if the group $G$ acts trivially on $A$ then $H^1(G,A)=\rm{Hom}(G,A)$, in particular for $A=\mathbb{Q}/\mathbb{Z}$ we have character group $H^1(G,A)=\chi(G)$.
It says $H^0(G,A)=A^G/N_GA$ the norm residue group of $G$ module $A$ with notation $A^G=\{a\in A : \sigma a=a~\forall \sigma\in G\}$ and $N_GA=\{\sum_{g\in G}ga:a\in A\}$. It says this is useful in class field theory.
It says $H^{-1}(G,A)$ has some other concrete form.
It says $H^{-2}(G,\mathbb{Z})=G^{ab}$ abelianization of $G$.
My question is are there any higher cohomology groups that are of interest at least in some special cases?

REPLY [6 votes]: Here's a number field example, analogous to Dalawat's local field example. Let $K$ be a number field. One can prove that $H^3(G_{\overline K/K},\overline K^*)=0$. (Although interestingly, there may be finite Galois extensions $L/K$ with $H^3(G_{L/K},L^*)\ne0$.) Anyway, Tate's cohomological construction of the pairing on the Tate-Shafarevich group $III(E/K)$ of an elliptic curve $E/K$ proceeds by first constructing a 2-cochain and then using the fact that $H^3(G_{\overline K/K},\overline K^*)=0$ to find a "correction term" that turns it into a 2-cocycle, thereby giving the desired element of $\text{Br}(K)$. So the whole construction depends on the calculation of an $H^3$.