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?



Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?
As Serre seems to be fond of saying, life begins at $H^3$.


The first appearance of a $3$-cocycle seems to be in Teichmüller's article *Über die sogenannte nichkommutative Galoissche Theorie...* in **Deutsche Mathematik 5**. He was trying to extend Galois Theory to extensions of a commutative field which are not themselves commutative. Teichmüller's formula for the $3$-cocycle helped Eilenberg and MacLane come up with the right definition of group cohomology in 


<http://www.jstor.org/stable/1968966> (Group Extensions and Homology, Samuel Eilenberg and Saunders MacLane, Annals of Mathematics Second Series, Vol. 43, No. 4 (Oct., 1942), pp. 757--831).


Let me just give one example where $H^3$ is useful. Let $p$ be a prime number, and consider a finite extension $K$ of the field $\mathbf{F}\_p((t))$ (or of $\mathbf{Q}\_p$, which you seem to know). Let $\bar K$ be a separable algebraic closure of $K$, and $G\_K=\mathrm{Gal}(\bar K|K)$. It is an absolutely crucial result that $H^3(G\_K,\mathbf{Z})=0$. That's what allows you to lift projective representations $G\_K\to\mathrm{PGL}\_n(\mathbf{C})$ to linear representations $G\_K\to\mathrm{GL}\_n(\mathbf{C})$. See for example 


<https://eudml.org/doc/142305> (André Weil, Exercises dyadiques, Inventiones mathematicae (1974) Volume: 27, pp. 1--22).


