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\begin{document}

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\section*{group cohomology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InHomotopyTypeTheory}{Fully general: in homotopy type theory}\dotfill \pageref*{InHomotopyTypeTheory} \linebreak
\noindent\hyperlink{InTermsOfHomologicalAlgebra}{For an ordinary group and abelian coefficients: In terms of homological algebra}\dotfill \pageref*{InTermsOfHomologicalAlgebra} \linebreak
\noindent\hyperlink{special_aspects_and_special_cases}{Special aspects and special cases}\dotfill \pageref*{special_aspects_and_special_cases} \linebreak
\noindent\hyperlink{InLowDegree}{Simplicial constructions and explicit formulas in low degree}\dotfill \pageref*{InLowDegree} \linebreak
\noindent\hyperlink{Degree1}{Degree-$1$ group cohomology}\dotfill \pageref*{Degree1} \linebreak
\noindent\hyperlink{Degree2}{Degree-$2$ group cohomology}\dotfill \pageref*{Degree2} \linebreak
\noindent\hyperlink{StructuredCohomology}{Structured group cohomology (topological groups and Lie groups)}\dotfill \pageref*{StructuredCohomology} \linebreak
\noindent\hyperlink{NonabelianGroupCohomology}{Nonabelian group cohomology}\dotfill \pageref*{NonabelianGroupCohomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{simple_examples}{Simple examples}\dotfill \pageref*{simple_examples} \linebreak
\noindent\hyperlink{cohomology_of_}{Cohomology of $\mathbb{Z}/2\mathbb{Z}$}\dotfill \pageref*{cohomology_of_} \linebreak
\noindent\hyperlink{cohomology_of___etc}{Cohomology of $U(n)$, $O(n)$, etc.}\dotfill \pageref*{cohomology_of___etc} \linebreak
\noindent\hyperlink{heisenberg_cocycle}{Heisenberg cocycle}\dotfill \pageref*{heisenberg_cocycle} \linebreak
\noindent\hyperlink{galileo_2cocycle}{Galileo 2-cocycle}\dotfill \pageref*{galileo_2cocycle} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak
\noindent\hyperlink{galois_cohomology}{Galois cohomology}\dotfill \pageref*{galois_cohomology} \linebreak
\noindent\hyperlink{lie_algebra_cohomology}{Lie algebra cohomology}\dotfill \pageref*{lie_algebra_cohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesOnStructuredGroupCohomology}{On structured group cohomology}\dotfill \pageref*{ReferencesOnStructuredGroupCohomology} \linebreak
\noindent\hyperlink{OnTopologicalGroups}{On various topological groups}\dotfill \pageref*{OnTopologicalGroups} \linebreak
\noindent\hyperlink{compact_lie_groups}{Compact Lie groups}\dotfill \pageref*{compact_lie_groups} \linebreak
\noindent\hyperlink{the_unitary_groups_}{The unitary groups $U(n)$}\dotfill \pageref*{the_unitary_groups_} \linebreak
\noindent\hyperlink{the_special_orthogonal_groups__}{The (special) orthogonal groups $O(n)$, $SO(n)$}\dotfill \pageref*{the_special_orthogonal_groups__} \linebreak
\noindent\hyperlink{for_spingroup_stringgroup_}{For spin-group, string-group, \ldots{}}\dotfill \pageref*{for_spingroup_stringgroup_} \linebreak
\noindent\hyperlink{the_exceptional_lie_groups}{The exceptional Lie groups}\dotfill \pageref*{the_exceptional_lie_groups} \linebreak
\noindent\hyperlink{loop_groups_of_compact_lie_groups}{Loop groups of compact Lie groups}\dotfill \pageref*{loop_groups_of_compact_lie_groups} \linebreak
\noindent\hyperlink{online_references}{Online references}\dotfill \pageref*{online_references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{group cohomology} of a [[group]] $G$ is the [[cohomology]] of its [[delooping]] $\mathbf{B}G$. This cohomology classifies [[group extensions]] of $G$.

More generally, the \emph{group cohomology} of an [[∞-group]] $G$ is the [[cohomology]] of its [[delooping]] $\mathbf{B}G$ and it classifies [[∞-group extensions]] of $G$ or equivalently [[principal ∞-bundles]] over $\mathbf{B}G$ (for [[coefficients]] with trivial [[∞-action]]) or [[associated ∞-bundles]] (for coefficients with nontrivial [[∞-action]]).

More in detail, if $A$ is any [[abelian group]] then a [[cocycle]] in $G$-group cohomology with [[coefficients]] in $A$ regarded as equipped with the trivial [[action]] is a morphism

\begin{displaymath}
c \colon \mathbf{B}G \to \mathbf{B}^n A
\end{displaymath}
and the [[cohomology group]] is the [[homotopy]] [[equivalence classes]] of this

\begin{displaymath}
H^n_{Grp}(G,A) \simeq \pi_0 \mathbf{H}(\mathbf{B}G,\mathbf{B}^n A)
  \,.
\end{displaymath}
More generally, $A$ here may be equipped with a $G$-[[action]] $\rho \colon A \times G \to A$. There is the the corresponding [[action groupoid]] or [[associated ∞-bundle]] $\mathbf{B}^n A\sslash G \to \mathbf{B}G$ and now a cocycle is a morphism $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^n A\sslash G$ fitting into a diagram

\begin{displaymath}
\itexarray{
    \mathbf{B}G && \stackrel{\mathbf{c}}{\longrightarrow}  && \mathbf{B}^n A \sslash G
    \\
    & \searrow &\swArrow& \swarrow 
    \\
    && \mathbf{B}G
  }
  \,.
\end{displaymath}
Equivalently this means that the group cohomology of $G$ with coefficients in an abelian group $A$ with $G$-[[action]] $\rho$ is the \emph{[[twisted cohomology]]} of the [[delooping]] $\mathbf{B}G$ with respect to the [[local coefficient ∞-bundle]] $\mathbf{B}^n A \sslash G$.

All this generalizes to $G$ itself any [[∞-group]] and $\mathbf{B}^n A$ replaced by any $G$-[[∞-action]] $\rho \colon V \times G \to G$ in which case a group cocycle is now a morphism

\begin{displaymath}
\itexarray{
    \mathbf{B}G && \stackrel{\mathbf{c}}{\longrightarrow}  && V \sslash G
    \\
    & \searrow &\swArrow& \swarrow 
    \\
    && \mathbf{B}G
  }
\end{displaymath}
hence a cocycle in the [[twisted cohomology]] of $\mathbf{B}G$ with coefficients in the [[local coefficient ∞-bundle]] given by the universal $\rho$-[[associated ∞-bundle|associated]] $V$-[[fiber ∞-bundle]].

In other words, the general notion of group cohomology of $G$ is just the most general notion of cohomology of $\mathbf{B}G$.

This general definition we discuss below in

\begin{itemize}%
\item \hyperlink{InHomotopyTypeTheory}{Fully general definition in homotopy type theory}.

\end{itemize}
The special case where $V = \mathbf{B}^n A$ is the $n$-fold [[delooping]] of an [[abelian group]] is important for applications and also because in this case powerful tools of [[homological algebra]] can be applied and group cohomology of ordinary groups may be computed in tersm of of [[Ext]]-functors. This we discuss in

\begin{itemize}%
\item \hyperlink{InTermsOfHomologicalAlgebra}{Of an ordinary group with abelian group coefficients - In terms of homological algebra}.

\end{itemize}
Finally one can break this further down into components In

\begin{itemize}%
\item \hyperlink{InLowDegree}{Special case: Explicit formulas in low degree}

\end{itemize}
we give some standard formulas for group cohomology in low degree.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{InHomotopyTypeTheory}{}\subsubsection*{{Fully general: in homotopy type theory}}\label{InHomotopyTypeTheory}

We give the general abstract definition in the language of [[(∞,1)-topos theory]] / [[homotopy type theory]].

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. Let $G \in Grp(\mathbf{H})$ be a [[group object in an (∞,1)-category|group object]], an [[∞-group]], in $\mathbf{H}$. Write $\mathbf{B}G \in \mathbf{H}$ for its [[delooping]].

An [[∞-action]] $\rho : V \times G \to V$ of $G$ on a $V \in \mathbf{H}$ is equivalently, as discussed there, exhibited by a [[fiber sequence]]

\begin{displaymath}
\itexarray{
    V &\to& V \sslash G
    \\
    && \downarrow^{\mathrlap{\bar \rho}}
    \\
    && \mathbf{B}G
  }
  \,.
\end{displaymath}
Regarded as an object in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ this is the [[categorical semantics]] of what in the [[syntax]] of [[homotopy type theory]] this is the [[dependent type]]

\begin{displaymath}
x \colon \mathbf{B}G \;\vdash \; V(x) \colon Type
  \,.
\end{displaymath}
Also, $\bar \rho$ is the \emph{[[local coefficient bundle]]} for $G$-group cohomology with [[coefficients]] in $V$ equipped with this $G$-[[∞-action]]. this means that the \emph{group cohomology} of $G$ with coefficients in $V$ is the hom in the [[slice (∞,1)-topos]] over $\mathbf{H}$ as [[base (∞,1)-topos]]

\begin{displaymath}
H^1_{Grp}(G,V) 
  \coloneqq
  \mathbf{H}_{/\mathbf{B}G}(\mathbf{B}G, V)
  \,,
\end{displaymath}
where we denote on the right by $\mathbf{B}G$ the [[terminal object]] in the slice $\mathbf{H}_{/\mathbf{B}G}$. Notice that in $\mathbf{H}$ this is the trivial [[fiber sequence]]

\begin{displaymath}
\itexarray{
     * &\to& \mathbf{B}G
     \\
     && \downarrow^{\mathrlap{id}}
     \\
     && \mathbf{B}G
  }
\end{displaymath}
This is the [[categorical semantics]] of what in the [[syntax]] of [[homotopy type theory]] is

\begin{displaymath}
\vdash \; \left(\prod_{x \colon \mathbf{B}G} \left(* \to V \right)\right) \colon Type
  \,.
\end{displaymath}
By the discussion at [[∞-action]], this expresses the [[invariant|∞-invariants]] of the [[conjugation action]] of $G$ on the morphisms $* \to V$ of the underlying objects. Since the action on the point is trivial, these are just the [[invariant|∞-invariants]] of $V$.

\begin{prop}
\label{GroupCohomologyInInfinityToposForTrivialAction}\hypertarget{GroupCohomologyInInfinityToposForTrivialAction}{}
In the special case that the $G$-[[∞-action]] on $V$ is trivial, the group cohomology is equivalently just the set of connected components of the [[derived hom space|hom space]]

\begin{displaymath}
H_{Grp}(G,V) \simeq \pi_0 \mathbf{H}(\mathbf{B}G, V)
  \,.
\end{displaymath}
In particular if $V = \mathbf{B}^n A$ for $A$ an abelian group, this is

\begin{displaymath}
H^n_{Grp}(G,A) \coloneqq H_{Grp}(G,\mathbf{B}^n A)
  \simeq
  \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)
  \,.
\end{displaymath}
\end{prop}
[[!include homotopy type representation theory -- table]]

\hypertarget{InTermsOfHomologicalAlgebra}{}\subsubsection*{{For an ordinary group and abelian coefficients: In terms of homological algebra}}\label{InTermsOfHomologicalAlgebra}

Let $G$ be an ordinary [[group]], specifically a [[group object]] in a [[topos]] $\mathcal{T}$ such that the [[abelian category]] $Ab(\mathcal{T})$ has [[projective object|enough projectives]]. If $G$ is an ordinary [[discrete group]] then this means that in the ambient [[set theory]] we assume the [[axiom of choice]] or ar least the [[presentation axiom]].

Write then

\begin{displaymath}
\mathbb{Z}[G] \in Ring
\end{displaymath}
for the [[group algebra]] of $G$ over the [[integers]]. Write

\begin{displaymath}
\mathcal{A} \coloneqq \mathbb{Z}[G] Mod
\end{displaymath}
for the category $\mathbb{Z}[G]$[[Mod]] of [[modules]] over $\mathbb{Z}[G]$.

Notice that a [[module]]

\begin{displaymath}
A \in \mathbb{Z}[G] Mod
\end{displaymath}
is equivalently an [[abelian group]] equipped with a $G$-[[action]]. This or rather its $n$-fold [[suspension of a chain complex|suspension as a chain complex]]

\begin{displaymath}
A[n] \in Ch_\bullet(\mathbb{Z}[G]Mod)
\end{displaymath}
is the kind of [[coefficient]] for the group cohomology of $G$ to which the following statement applies.

\begin{remark}
\label{InvariantsByHomFunctor}\hypertarget{InvariantsByHomFunctor}{}
For $A$ a $G$-[[module]], the [[invariant|invariants]] of $A$ are equivalently the $\mathbb{Z}[G]$-module homomorphisms from $\mathbb{Z}$ equipped with the trivial module structure

\begin{displaymath}
Invariants(A) \simeq Hom_{\mathbb{Z}[G]}(\mathbb{Z}, A)
  \,.
\end{displaymath}
This equivalence is [[natural isomorphism|natural]] and hence the contravariant [[hom functor]] is equivalently the invariants-functor

\begin{displaymath}
Invariants(-)\simeq Hom_{\mathbb{Z}[G]}(\mathbb{Z}, -)
  \,.
\end{displaymath}
\end{remark}
By the fully general discussion \hyperlink{InHomotopyTypeTheory}{above}, group cohomology of $G$ with coefficients in some $A$ is the homotopy-version of the $G$-[[invariants]] of $A$. In the context of [[homological algebra]] and in view of remark \ref{InvariantsByHomFunctor}, this means that it is given by the [[derived functor]] of the [[hom functor]] out of the trivial $G$-module, hence by the [[Ext]]-functor:

\begin{defn}
\label{}\hypertarget{}{}
For $A$ an [[abelian group]] equipped with a $G$-[[action]], the degree-$n$ \emph{group cohomology} of $G$ with [[coefficients]] in $A$ is the $n$th-[[Ext]]-group

\begin{displaymath}
H^n_{Grp}(G,A) \coloneqq Ext^n_{\mathbb{Z}[G]}(\mathbb{Z}, A)
  \,,
\end{displaymath}
where on the right $\mathbb{Z} \in \mathbb{Z}[G] Mod$ is regarded as equipped with the trivial $G$-action.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[projective resolution]]} this means more explicitly the following: let $F_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z}$ be a [[projective resolution]] $Ch_\bullet(\mathbb{Z}[G]Mod)$ of $\mathbb{Z}$ equipped with the trivial $G$-[[action]], hence an [[exact sequence]]

\begin{displaymath}
\cdots \to F_3 \to F_2 \to F_1 \to F_0 \to \mathbb{Z} \to 0
\end{displaymath}
of $\mathbb{Z}[G]$-[[modules]]. Let

\begin{displaymath}
Hom_{\mathbb{Z}[G]Mod}(F_\bullet, A)
  = 
  \left[
    Hom_{\mathbb{Z}[G]Mod}(F_0, A) 
      \stackrel{d^0}{\longrightarrow} 
    Hom_{\mathbb{Z}[G]Mod}(F_1,A) 
      \stackrel{d^1}{\longrightarrow} 
    Hom_{\mathbb{Z}[G]Mod}(F_2,A) 
      \stackrel{d^2}{\longrightarrow} 
      \cdots
  \right]
\end{displaymath}
be the corresponding [[cochain complex]]. Then the degree-$n$ group cohomology of $G$ with coefficient in $A$ is the degree-$n$ [[cochain cohomology]] of this complex

\begin{displaymath}
H^n_{Grp}(G,A)
  \simeq
  H^n(Hom_{\mathbb{Z}[G]Mod}(F_0, A))
  \coloneqq
  ker(d^n)/im(d^{n-1})
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Give a [[normal subgroup]] $K \hookrightarrow G$ the [[invariant|invariants]]-functor may be decomposed as a [[composition]] of the functor that forms $K$-invariants with that which forms $(G/K)$-invariants for the [[quotient group]]. This decomposition gives rise to a [[Grothendieck spectral sequence]] for the group cohomology. This is called the \emph{[[Hochschild-Serre spectral sequence]]}.

\end{remark}
\hypertarget{special_aspects_and_special_cases}{}\subsection*{{Special aspects and special cases}}\label{special_aspects_and_special_cases}

The fully general definition \hyperlink{InHomotopyTypeTheory}{above} subsumes various cases that are not always discussed on the same footing in traditional literature. For emphasis we highlight these special cases separately.

\begin{itemize}%
\item \hyperlink{InLowDegree}{Simplicial constructions and explicit formulas in low degree}


\item \hyperlink{StructuredCohomology}{Structured group cohomology}


\item \hyperlink{NonabelianGroupCohomology}{Nonabelian group cohomology}



\end{itemize}
\hypertarget{InLowDegree}{}\subsubsection*{{Simplicial constructions and explicit formulas in low degree}}\label{InLowDegree}

We unwind the general abstract definition of group cohomology \hyperlink{InHomotopyTypeTheory}{above} in terms of constructions on [[simplicial sets]] (for cohomology of [[discrete groups]]) and [[simplicial presheaves]] (for cohomology of general [[group objects]]).

$\,$

Let $G$ be a [[discrete group]] and $A$ an [[abelian group|abelian]] [[discrete group]], regarded as equipped with the trivial $G$-[[action]]. Let $n \in \mathbb{N}$.

Write $\overline{W}G = G^{\times^\bullet}\in$ [[sSet]] for the [[nerve]] of the [[groupoid]] $*\sslash G$ and write $DK(A[n]) \in$ [[sSet]] for the [[image]] under the [[Dold-Kan correspondence]] of the [[chain complex]] which is the $n$-fold [[suspension of a chain complex]] of $A$.

\begin{prop}
\label{DiscreteGroupCohomologyBySimplicialHomotopy}\hypertarget{DiscreteGroupCohomologyBySimplicialHomotopy}{}
Then the degree-$n$ group cohomology of $G$ with [[coefficients]] in $A$ is the set

\begin{displaymath}
H^n_{Grp}(G,A) \simeq \pi_0 sSet(\overline{W}G, DK(A[n]))
\end{displaymath}
of [[homomorphisms]] of [[simplicial sets]] modulo [[simplicial homotopy]].

\end{prop}
\begin{proof}
By prop. \ref{GroupCohomologyInInfinityToposForTrivialAction} the group cohomology is

\begin{displaymath}
H^n_{Grp}(G,A) \simeq \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)
  \,.
\end{displaymath}
By assumption the relevant [[(∞,1)-topos]] here is $\mathbf{H} =$ [[∞Grpd]], which for emphasis we might write ``[[Disc∞Grpd]]''. This is [[presentable (∞,1)-category|presented]] by the standard [[model structure on simplicial sets]], $Disc\infty Grpd \simeq L_{whe} sSet$.

By the discussion at \emph{[[delooping]]} and at \emph{[[∞-group]]}, a presentation in [[sSet]], necessarily [[cofibrant object|cofibrant]], of the [[delooping]] $\mathbf{B}G \in \mathbf{H}$ is the standard [[bar construction]]

\begin{displaymath}
\overline{W}G
  = 
  \left(
    \cdots \to G \times G \times G
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    G  \times G
    \stackrel{\longrightarrow}{\longrightarrow}
    G
    \to 
    {*}
  \right)
  \,,
\end{displaymath}
which is equivalently the [[nerve]] of the [[groupoid]] $*\sslash G$.

Moreover, by the discussion at \emph{[[Dold-Kan correspondence]]} a presentation of the [[Eilenberg-MacLane object]] $\mathbf{B}^n A$ is $DK(A[n]) \in sSet$, and this is a [[Kan complex]] and hence a [[fibrant object]] in the [[model category]] structure.

Therefore by the discussion at \emph{[[derived hom-space]]} we have that $sSet(\overline{W}G, DK(A[n]))$ is a [[Kan complex]] which presents the required hom-$\infty$-groupoid.

\end{proof}
For low values of $n$ it is useful and easily possible to describe these simplicial maps explicitly. This we turn to now.

\hypertarget{Degree1}{}\paragraph*{{Degree-$1$ group cohomology}}\label{Degree1}

A degree-one group cocycle $c$, $[c] \in H^1_{Grp}(G,A)$ is just [[group homomorphism]] $G \to A$ -- a [[character]] of $G$.

\hypertarget{Degree2}{}\paragraph*{{Degree-$2$ group cohomology}}\label{Degree2}

We discuss here in detail and in components the special case of degree-2 group cohomology of a [[discrete group]] $G$ with coefficients in $A$ an [[abelian group|abelian]] [[discrete group]] and regarded as being equipped with the trivial $G$-[[action]].

\begin{prop}
\label{2CocyclesAndCoboundaries}\hypertarget{2CocyclesAndCoboundaries}{}
Let $G$ be a [[discrete group]] and $A$ an [[abelian group|abelian]] [[discrete group]], regarded as being equipped with the trivial $G$-[[action]].

Then a \textbf{group 2-[[cocycle]]} on $G$ with coefficients in $A$ is a [[function]]

\begin{displaymath}
c \colon G \times G \to A
\end{displaymath}
such that for all $(g_1, g_2, g_3) \in G \times G \times G$ it satisfies the [[equation]]

\begin{equation}
c(g_1, g_2)
  -
  c(g_1, g_2 \cdot g_3)
  +
  c(g_1 \cdot g_2, g_3)
  - 
  c(g_2, g_3)
  = 
  0
  \;\;\;\; \in A
\label{2CocycleConditionForCentralExtension}\end{equation}
(called the \textbf{group 2-cocycle condition}).

For $c, \tilde c$ two such cocycles, a \textbf{[[coboundary]]} $h \colon c \to \tilde c$ between them is a [[function]]

\begin{displaymath}
h \colon G \to A
\end{displaymath}
such that for all $(g_1,g_2) \in G \times G$ the [[equation]]

\begin{equation}
\tilde c(g_1,g_2)
  = 
  c(g_1,g_2) + (d h)(g_1,g_2)
\label{Group2CoboundaryEquation}\end{equation}
holds in $A$, where

\begin{displaymath}
(d h)(g_1, g_2) \coloneqq h(g_1 g_2) - h(g_1) - h(g_2)
\end{displaymath}
is the \textbf{group 2-coboundary} encoded by $h$.

The degree-2 \textbf{group cohomology} is the set

\begin{displaymath}
H^2_{Grp}(G,A) = 2Cocycles(G,A) / Coboundaries(G,A)
\end{displaymath}
of [[equivalence classes]] of group 2-cocycles modulo group 2-coboundaries. This is itself naturally an [[abelian group]] under pointwise addition of cocycles in $A$

\begin{displaymath}
[c_1] + [c_2] = [c_1 + c_2]
\end{displaymath}
where

\begin{displaymath}
c_1 + c_2 \colon (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)
  \,.
\end{displaymath}
\end{prop}
This may be taken as the \emph{definition} of degree-2 group cohomology (with coefficients in abelian groups and with trivial action). The following proof shows how this \emph{follows} from the general simplicial presentation of prop. \ref{DiscreteGroupCohomologyBySimplicialHomotopy}.

\begin{proof}
By prop. \ref{DiscreteGroupCohomologyBySimplicialHomotopy} we have $H^2_{Grp}(G,A) \simeq \pi_0 sSet(\overline{W}G, DK(A[2]))$.

Notice that fully explicitly the 2-[[simplices]] in $\overline{W}G$ are

\begin{displaymath}
(\overline{W}G)_2
  =
  \left\{
  \left.
  \itexarray{
    && {*}
    \\
    & {}^{g_1}\nearrow && \searrow^{g_2}
    \\
    {*}
    &&\stackrel{g_1 g_2}{\longrightarrow}&& {*}
  }
  \right|
  g_1, g_2 \in G
  \right\}
  \,,
\end{displaymath}
and the 3-simplices are

\begin{displaymath}
(\overline{W}G)_3
  =
  \left\{
  \left.
  \itexarray{
    {*} &&\stackrel{g_2}{\longrightarrow}&& {*}
    \\
    \uparrow^{g_1} &&{}^{g_1 g_2}\nearrow&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_1 g_2 g_3}{\longrightarrow}&&
    {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;   
  \itexarray{
    {*} &&\stackrel{g_2}{\longrightarrow}&& {*}
    \\
    \uparrow^{g_1} &&\searrow^{g_2 g_3}&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_1 g_2 g_3}{\longrightarrow}&&
    {*}
  }
  \right|
  g_1, g_2, g_3 \in G
  \right\}
  \,.
\end{displaymath}
Therefore a homomorphism of simplical sets $c \colon \overline{W}G \to DK(A[2])$ is in degree 2 a function

\begin{displaymath}
c_2
  \;\;
  :
  \;\; 
  \left(
  \itexarray{
    && {*}
    \\
    & {}^{g_1}\nearrow && \searrow^{g_2}
    \\
    {*}
    &&\stackrel{g_2 g_1}{\to}&& {*}
  }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
  \itexarray{
    && {*}
    \\
    & {}^{{*}}\nearrow &\Downarrow^{c(g_1,g_2)}& \searrow^{{*}}
    \\
    {*}
    &&\stackrel{{*}}{\to}&& {*}
  }
  \right)
\end{displaymath}
i.e. a map $c : G \times G \to K$. To be a simplicial homomorphism this has to extend to 3-[[simplices]] as:

\begin{displaymath}
\begin{aligned}
  c_3
  \;\;\;
  &:
  \;\;\;
  \left(
  \itexarray{
    {*} && \stackrel{g_2}{\longrightarrow} && {*}
    \\
    \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3}
    \\
    {*}
    && \stackrel{g_3 g_2 g_1}{\longrightarrow} && {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;   
  \itexarray{
    {*} &&\stackrel{g_2}{\to}&& {*}
    \\
    \uparrow^{g_1} &&\searrow^{g_3 g_2}&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_3 g_2 g_1}{\to}&&{*}
  }
  \right)
  \\
  & \mapsto
  \left(
  \itexarray{
    {*} &\stackrel{}{\longrightarrow} & &\stackrel{}{\longrightarrow}& {*}
    \\
    \uparrow &\Downarrow^{c(g_1,g_2)}
    &\nearrow&\Downarrow^{c(g_2,g_3)}& 
    \downarrow
    \\
    {*}
    &\longrightarrow&&\longrightarrow&{*}
  }
  \;\;\;\;
  \stackrel{}{\Rightarrow}
  \;\;\;\;   
  \itexarray{
    {*} &\longrightarrow&&\longrightarrow& {*}
    \\
    \uparrow &\Downarrow^{c(g_1,g_2 g_3)}
    &\searrow &\Downarrow^{c(g_2, g_3)}& \downarrow
    \\
    {*}
    &\longrightarrow&&\longrightarrow&{*}
  }
  \right)
  \end{aligned}  
  \,.
\end{displaymath}
Since there is a unique 3-cell in $DK(A[2])$ whenever the oriented sum of the $A$-labels of the boundary of the corresponding tetrahedron vanishes, the existence of the 3-cell on the right here is precisely the claimed cocycle condition.

A similar argument gives the coboundaries

\end{proof}
We discuss now how in the computation of $H^2_{Grp}(G,A)$ one may concentrate on the \emph{normalized} cocycles.

\begin{defn}
\label{Normalized2Cocycle}\hypertarget{Normalized2Cocycle}{}
A group 2-cocycle $c \colon G \times G \to A$, def. \ref{2CocyclesAndCoboundaries} is called \textbf{normalized} if

\begin{displaymath}
\forall_{g_0,g_1 \in G}
  \;\;
  \left(g_0 = e \;or\; g_1 = e \right)
  \Rightarrow
  \left( c(g_0,g_1) = e \right)
  \,.
\end{displaymath}
\end{defn}
\begin{lemma}
\label{2CocycleOngeisSameasOnee}\hypertarget{2CocycleOngeisSameasOnee}{}
For $c \colon G \times G \to A$ a group 2-cocycle, we have for all $g \in G$ that

\begin{displaymath}
c(e,g) = c(e,e) = c(g,e)
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
The cocycle condition \eqref{2CocycleConditionForCentralExtension} evaluated on

\begin{displaymath}
(g^{-1},  g,  e) \in G^3
\end{displaymath}
says that

\begin{displaymath}
c(g^{-1}, g) + c(e, e) = c(g, e) + c(g^{-1}, g )
\end{displaymath}
hence that

\begin{displaymath}
c(e,e) = c(g, e)
  \,.
\end{displaymath}
Similarly the 2-cocycle condition applied to

\begin{displaymath}
(e,  g,  g^{-1}) \in G^3
\end{displaymath}
says that

\begin{displaymath}
c(e,g) + c(g,g^{-1}) = c(g,g^{-1}) + c(e,e)
\end{displaymath}
hence that

\begin{displaymath}
c(e,g) = c(e,e)
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Every group 2-cocycle $c \colon G \times G \to A$ is cohomologous to a normalized one, def. \ref{Normalized2Cocycle}.

\end{prop}
\begin{proof}
By lemma \ref{2CocycleOngeisSameasOnee} it is sufficient to show that $c$ is cohomologous to a cocycle $\tilde c$ satisfying $\tilde c(e,e) = e$. Now given $c$, Let $h \colon G \to A$ be given by

\begin{displaymath}
h(g) \coloneqq c(g,g)
  \,.
\end{displaymath}
Then $\tilde c \coloneqq c + d c$ has the desired property, with \eqref{Group2CoboundaryEquation}:

\begin{displaymath}
\begin{aligned}
    \tilde c(e,e)
    & \coloneqq
    (c + d h)(e,e) 
    \\
    & = 
    c(e,e)  + c(e \cdot e, e \cdot e) - c(e,e) - c(e,e)
    \\  
    & = 
    0
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{StructuredCohomology}{}\subsubsection*{{Structured group cohomology (topological groups and Lie groups)}}\label{StructuredCohomology}

If the groups in question are not plain groups ([[group object]]s internal to [[Set]]) but groups with extra structure, such as [[topological group]]s or [[Lie group]]s, then their cohomology has to be understood in the corresponding natural context.

In parts of the literature cohomology of structured groups $G$ is defined in direct generalization of the formulas above as homotopy classes of morphisms from the simplicial object

\begin{displaymath}
\left(
    \cdots G \times G\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}G \stackrel{\longrightarrow}{\longrightarrow} *
  \right)
\end{displaymath}
to a simplicial object $N (\mathbf{B}^n A)$.

This is what is described \hyperlink{InLowDegree}{above} for \emph{[[discrete groups]]}. But this does \textbf{not} in general give the right answer for structured groups: while the [[simplicial set]] $\overline{W}G = G^{\times^\bullet}$ is [[cofibrant object|cofibrant]] in the relevant model category presenting the ambient [[(∞,1)-topos]] [[Disc∞Grpd]], for $G$ a structured group the [[simplicial object]] given by the same formula is not in general already cofibrant. It needs to be further resolved, instead.

Specifically, for a [[Lie group]] $G$, the object

\begin{displaymath}
\left(
    \cdots G \times G\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}G \stackrel{\longrightarrow}{\longrightarrow} *
  \right)
\end{displaymath}
has to be considered as an [[Lie ∞-groupoid]]: an object in the [[model structure on simplicial presheaves]] over a [[site]] such as [[Diff]] or [[CartSp]]. As such it is in general \textbf{not} both cofibrant and fibrant. To that extent plain morphisms out of this object do \textbf{not} compute the correct [[derived hom-space]]s. Instead, the right definition of structured group cohomology uses the correct fibrant and cofibrant replacements.

Doing requires more work. This is discussed at

\begin{itemize}%
\item \emph{[[Lie group cohomology]]}

\end{itemize}
See below at \emph{\hyperlink{ReferencesOnStructuredGroupCohomology}{References - For structured groups}} for pointers to the literature.

\hypertarget{NonabelianGroupCohomology}{}\subsubsection*{{Nonabelian group cohomology}}\label{NonabelianGroupCohomology}

If the coefficient group $K$ is nonabelian, its higher [[delooping]]s $\mathbf{B}^n K$ to not exist. But [[n-groupoid]]s approximating this non-existant delooping do exists. Cohomology of $\mathbf{B}G$ with coefficients in these is called [[nonabelian group cohomology]] or [[Schreier theory]]. See there for more details.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\hypertarget{simple_examples}{}\paragraph*{{Simple examples}}\label{simple_examples}

\begin{itemize}%
\item [[carrying]]

\end{itemize}
\hypertarget{cohomology_of_}{}\paragraph*{{Cohomology of $\mathbb{Z}/2\mathbb{Z}$}}\label{cohomology_of_}

For group cohomology of the [[group of order 2]] $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ see at \emph{Groupprops}, \emph{\href{http://groupprops.subwiki.org/wiki/Group_cohomology_of_cyclic_group:Z2}{Group cohomology of cyclic group Z2}}

\hypertarget{cohomology_of___etc}{}\paragraph*{{Cohomology of $U(n)$, $O(n)$, etc.}}\label{cohomology_of___etc}

We consider for $G$ a [[topological group]] such as

\begin{itemize}%
\item the [[unitary group]] $U(n)$;


\item the [[special unitary group]] $SU(n)$;


\item the [[symplectic group]] $Sp(n)$



\end{itemize}
the corresponding group cohomology in terms of the cohomology of the [[classifying space]]/[[delooping]] $B G$.

For all $n \in \mathbb{N}$ we have

\begin{displaymath}
\begin{aligned}
 H^\bullet(BU(n);\mathbb{Z}) &= \mathbb{Z}[c_1,\cdots,c_n] 
 \\
 H^\bullet(BSU(n);\mathbb{Z}) &= \mathbb{Z}[c_2,\cdots,c_n] 
 \\
 H^\bullet(BSp(n);\mathbb{Z}) &= \mathbb{Z}[p_1,\cdots,p_n]
\end{aligned}
\end{displaymath}
where $c_i\in H^{2i}$ and $p_i\in H^{4i}$.

\hypertarget{heisenberg_cocycle}{}\paragraph*{{Heisenberg cocycle}}\label{heisenberg_cocycle}

The additive group on the [[Cartesian space]] $\mathbb{R}^2$ with group operation

\begin{displaymath}
(a,b) + (a',b') = (a + a' , b + b')
\end{displaymath}
carries a degree-2 group cocycle $\omega$ with values in $\mathbb{R}$ given by

\begin{displaymath}
\omega : ((a_1,b_1), (a_2,b_2)) \mapsto a_1 \cdot b_2
  \,.
\end{displaymath}
The cocycle condition for this is the identity

\begin{displaymath}
a_1 \cdot (b_2 + b_3) + a_2 \cdot b_3 
  =
  a_1 \cdot b_2 + (a_1 + a_2) \cdot b_3
\end{displaymath}
The [[group extension]] classified by this cocycle is the [[Heisenberg group]].

\hypertarget{galileo_2cocycle}{}\paragraph*{{Galileo 2-cocycle}}\label{galileo_2cocycle}

\begin{itemize}%
\item [[Galileo 2-cocycle]]

\end{itemize}
\hypertarget{classes_of_examples}{}\subsubsection*{{Classes of examples}}\label{classes_of_examples}

\hypertarget{galois_cohomology}{}\paragraph*{{Galois cohomology}}\label{galois_cohomology}

The group cohomology of [[Galois groups]] is called \emph{[[Galois cohomology]]}. See there for more details.

\hypertarget{lie_algebra_cohomology}{}\paragraph*{{Lie algebra cohomology}}\label{lie_algebra_cohomology}

We may regard a [[Lie algebra]] as an [[infinitesimal object|infinitesimal]] group. Under this perspective [[Lie algebra cohomology]] and [[infinity-Lie algebra cohomology]] is a special case of (higher) group cohomology. See there for details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{group cohomology}

\begin{itemize}%
\item [[nonabelian group cohomology]], [[groupoid cohomology]]


\item [[Galois cohomology]]



\end{itemize}

\item [[group homology]]


\item [[group extension]]

\begin{itemize}%
\item [[Baer sum]]

\end{itemize}

\item [[Lie group cohomology]]

\begin{itemize}%
\item [[∞-group cohomology]]


\item [[∞-Lie groupoid cohomology]]



\end{itemize}

\item [[equivariant cohomology]]


\item [[Friedlander-Milnor isomorphism conjecture]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Standard textbook references on group cohomology include

\begin{itemize}%
\item [[Kenneth Brown]], \emph{Cohomology of Groups} , Graduate Texts in Mathematics, 87 (1972)

\end{itemize}
and specifically with an eye towards cohomology of [[finite groups]]

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Characters and cohomology of finite groups}, Publications Math\'e{}matiques de l'IH\'E{}S, 9 (1961), p. 23-64 (\href{http://www.numdam.org/item?id=PMIHES_1961__9__23_0}{Numdam})


\item [[Alejandro Adem]], [[R. James Milgram]], \emph{Cohomology of Finite Groups}, Springer 2004



\end{itemize}
Exposition includes

\begin{itemize}%
\item [[Alejandro Adem]], \emph{Lectures on the cohomology of finite groups} (\href{http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf}{pdf})


\item [[Narthana Epa]], \emph{Platonic 2-groups}, 2010 (\href{http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf}{pdf})



\end{itemize}
An introduction to group cohomology of a group $G$ as the cohomology of the [[classifying space]] $B G$ is for instance in

\begin{itemize}%
\item Joshua Roberts, \emph{Group cohomology: a classifying space perspective} (\href{http://www.piedmont.edu/math/jroberts/notes/qualifyingtalk.pdf}{pdf})


\item \emph{Advanced course on classifying spaces and cohomology of groups} (\href{www-irma.u-strasbg.fr/~henn/notes.ps}{ps})



\end{itemize}
Discussion of the cohomology of [[discrete groups]] with abelian coefficients in terms of [[crossed modules]] instead of [[chain complexes]] (an intermediate step in the [[Dold-Kan correspondence]]) is in chapter 12 of

\begin{itemize}%
\item R. Brown, P. Higgins, R. Sivera, \emph{Nonabelian algebraic topology} (\href{http://www.bangor.ac.uk/%7Emas010/pdffiles/rbrsbookb-e040609.pdf}{pdf}, \href{http://www.bangor.ac.uk/~mas010/nonab-a-t.html}{web})

\end{itemize}
Cohomology of [[simplicial groups]] is discussed for instance in

\begin{itemize}%
\item Sebastian Thomas, \emph{On the second cohomology group of a simplicial group} (\href{http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/Thomas_On_the_second_cohomology_group_of_a_simplicial_group.pdf}{pdf})

\end{itemize}
Much of what is called ``[[nonabelian cohomology]]'' in the existing literature concerns the case of nonabelian group cohomology with coefficients in the [[automorphism 2-group]] $AUT(H)$ of some possibly nonabelian group $H$.

This is the topic of [[Schreier theory]].

A random example for this use of terminology would be

\begin{itemize}%
\item Roggenkamp, Scott, \emph{Automorphisms and nonabelian cohomology} (\href{http://www.math.virginia.edu/~lls2l/automorphisms_and_nonabelian.pdf}{pdf})

\end{itemize}
For a conceptual discussion of nonabelian group cohomology see

\begin{itemize}%
\item [[John Baez]], [[Mike Shulman]], \emph{[[Lectures on n-Categories and Cohomology]]} (\href{http://arxiv.org/abs/math/0608420}{arXiv})

\end{itemize}
Group cocycles classify [[group extensions]]. This is often discussed only for 2-cocycles and extensions by ordinary groups. Higher cocycles classify extensions by [[2-group]]s and further by [[infinity-groups]]. In the context of [[crossed complexes]], which are models for \emph{strict} $\infty$-groups, this is discussed for instance in

\begin{itemize}%
\item [[Johannes Huebschmann]], \emph{Crossed $n$-fold extensions and group cohomology} (\href{http://dx.doi.org/10.1007/BF02566688}{web})

\end{itemize}
\hypertarget{ReferencesOnStructuredGroupCohomology}{}\subsubsection*{{On structured group cohomology}}\label{ReferencesOnStructuredGroupCohomology}

In

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Continuous cohomology of groups and classifying spaces} Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 (\href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540920}{web})

\end{itemize}
$n$-cocycles on a topological group $G$ with values in a topological abelian group $A$ are considered as continuous maps $G^{\times n}\to A$ (p. 3 ).

A definition in terms of [[derived functor|Ext-functors]] and comparison with the naive definition is in

\begin{itemize}%
\item David Wigner, \emph{Algebraic cohomology of topological groups} Transactions of the American Mathematical Society, volume 178 (1973)(\href{http://egg.epfl.ch/~nmonod/bonn/Wigner_1973.pdf}{pdf})

\end{itemize}
A classical reference that considers the cohomology of Lie groups as topological spaces is

\begin{itemize}%
\item [[Armand Borel]], \emph{Homology and cohomology of compact connected Lie groups} (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063923/pdf/pnas01596-0040.pdf}{pdf})

\end{itemize}
A corrected definition of topological group cohomology has been given by Segal

\begin{itemize}%
\item [[Graeme Segal]], \emph{Cohomology of topological groups} In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377\{387. Academic Press, London, (1970).


\item [[Graeme Segal]], \emph{A classifying space of a topological group in the sense of Gel'fand-Fuks. Funkcional. Anal. i Prilozen., 9(2):48\{50, (1975).}



\end{itemize}
For [[reductive algebraic groups]]:

\begin{itemize}%
\item [[Hélène Esnault]], [[Bruno Kahn]], [[Marc Levine]], [[Eckart Viehweg]], \emph{The Arason invariant and mod 2 algebraic cycles}, J. Amer. Math. Soc. 11 (1998), 73-118 (\href{https://www.uni-due.de/~bm0032/publ/Arason.pdf}{pdf},\href{http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00248-3/}{publisher page})

\end{itemize}
\hypertarget{OnTopologicalGroups}{}\subsubsection*{{On various topological groups}}\label{OnTopologicalGroups}

Some references pertaining to the [[cohomology]] of the [[classifying space]]/[[delooping]] $B G$ for $G$ a [[topological group]] ([[characteristic class]]es).

\hypertarget{compact_lie_groups}{}\paragraph*{{Compact Lie groups}}\label{compact_lie_groups}

Cohomology of the [[classifying space]] $B G$ for $G$ the [[topological group]] underlying a [[compact topological space|compact]] [[Lie group]].

\begin{itemize}%
\item [[John Milnor]], [[Jim Stasheff]], \emph{Characteristic Classes} , Princeton University Press and University of Tokyo Press, Princeton, New Jersey, (1974).

\end{itemize}
\begin{itemize}%
\item Mark Feshbach, \emph{Some General Theorems on the Cohomology of Classifying Spaces of Compact Lie Groups} Transactions of the American Mathematical Society Vol. 264, No. 1 (Mar., 1981), pp. 49-58 (\href{http://www.jstor.org/stable/1998409}{JSTOR})


\item [[Donald Yau]], \emph{Cohomology of unitary and symplectic groups} (\href{http://www.mathnet.ru/links/c8897bc54ec0a1a49d03ca681d53db7b/ljm147.pdf}{pdf})


\item D. Benson, [[John Greenlees]], \emph{Commutative algebra for cohomology rings of classifying spaces of compact Lie groups} (\href{http://hopf.math.purdue.edu/Benson-Greenlees/Liegroupca.pdf}{pdf})


\item [[Eric Friedlander]], Guido Mislin, \emph{Cohomology of classifying spaces of complex Lie groups and related discrete groups} Commentarii Mathematici Helvetici Volume 59, Number 1, 347-361,



\end{itemize}
\hypertarget{the_unitary_groups_}{}\paragraph*{{The unitary groups $U(n)$}}\label{the_unitary_groups_}

For $U$ the [[unitary group]], the [[integral cohomology]] of the [[classifying space]] $B U(n)$ consists of the [[Chern class]]es, one in every even degree.

\hypertarget{the_special_orthogonal_groups__}{}\paragraph*{{The (special) orthogonal groups $O(n)$, $SO(n)$}}\label{the_special_orthogonal_groups__}

The cohomology of $B O(n)$ ([[orthogonal group]]) and $B SO(n)$ ([[special orthogonal group]]) with coefficients in $\mathbb{Z}_2$ is discussed in (\hyperlink{MilnorStasheff}{MilnorStasheff, 1974}).

The cohomology of $B O(n)$ with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_{2 m}$ was found in

\begin{itemize}%
\item Emery Thomas , \emph{On the cohomology of the real Grassman complexes and the characteristic classes of the $n$-plane bundle} , Trans. Amer. Math. Soc. 96 (1960), 67--89.

\end{itemize}
The [[ring]]-structure on the cohomology with integer coefficients was given in

\begin{itemize}%
\item E. Brown (Jr.), \emph{The cohomology of $B SO(n)$ and $B O(n)$ with integer coefficients} Proc. AMS Soc. 85 (1982)


\item Mark Feshbach, \emph{The integral cohomology rings of the classifying spaces of $O(n)$ and $SO(n)$, Indiana Univ. Math. J. 32 (1983), 511--516.}



\end{itemize}
For [[twisted cohomology|local coefficients]] see

\begin{itemize}%
\item [[Martin ?adek]], \emph{The cohomology of $B O(n)$ with twisted integer coefficients}, J. Math. Kyoto Univ. 39 (1999), no. 2, 277--286 (\href{http://projecteuclid.org/euclid.kjm/1250517912}{Euclid})


\item Richard Lastovecki, \emph{Cohomology of $B O(n_1) \times \cdots \times B O(n_m)$ with local integer coefficients} Comment.Math.Univ.Carolin. 46,1 (2005)21--32 (\href{http://www.emis.de/journals/CMUC/pdf/cmuc0501/lastovec.pdf}{pdf})



\end{itemize}
\hypertarget{for_spingroup_stringgroup_}{}\paragraph*{{For spin-group, string-group, \ldots{}}}\label{for_spingroup_stringgroup_}

The [[Whitehead tower]] of the [[orthogonal group]] starts out with

[[fivebrane group]] $\to$ [[string group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]]

Group cohomology of the [[spin group]] (cohomology of the [[classifying space]] $B Spin$) is discussed in

\begin{itemize}%
\item Emery Thomas, \emph{On the cohomology groups of the classifying space for the stable spinor group} , Bol. Soc. Mex. (1962), 57 - 69


\item Masana Harada, Akira Kono, \emph{Cohomology mod 2 of the classifying space of $Spin^c(n)$} Publications of the Research Institute for Mathematical Sciences archive Volume 22 Issue 3, Sept. (1986) (\href{http://dl.acm.org/citation.cfm?id=23393}{web})



\end{itemize}
Group cohomology of the [[string group]] (cohomology of the [[classifying space]] $B String$) is discussed in

\begin{itemize}%
\item \emph{On the integral cohomology of the seven-connective cover of B O} Bull. Austral. Math. Soc. Vol 38 (1988) (\href{http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ38_01%2FS0004972700027179a.pdf&code=8374f17b7aa6ffb83f42f347c677729f}{pdf})

\end{itemize}
\hypertarget{the_exceptional_lie_groups}{}\paragraph*{{The exceptional Lie groups}}\label{the_exceptional_lie_groups}

Cohomology of [[classifying space]]s of [[exceptional Lie group]]s.

\begin{itemize}%
\item Akira Kono, Mamoru Mimura, \emph{Cohomology mod 3 of the classifying space of the Lie group $E_6$} , Math. Scand. 46 (1980) (\href{http://www.mscand.dk/article.php?id=2533}{pdf})

\end{itemize}
\hypertarget{loop_groups_of_compact_lie_groups}{}\paragraph*{{Loop groups of compact Lie groups}}\label{loop_groups_of_compact_lie_groups}

\begin{itemize}%
\item Daisuke Kishimoto, Akira Kono, \emph{Cohomology of the classifying spaces of loop groups} (\href{http://www.math.kyoto-u.ac.jp/preprint/2004/13kishimoto.pdf}{pdf})

\end{itemize}
\hypertarget{online_references}{}\subsubsection*{{Online references}}\label{online_references}

Some of the above materiel is taken from discussion at

\begin{itemize}%
\item MO, \emph{\href{http://mathoverflow.net/questions/75389/group-cohomology-of-compact-lie-group-with-integer-coeffient}{Group cohomology of compact Lie group with integer coeffient}}

\end{itemize}
[[!redirects group cocycle]] [[!redirects group cocycles]]



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