group actions on spheres

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### Context
#### Spheres
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[[!include spheres -- contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The possible [[actions]] of well-behaved [[topological groups]] (such as [[compact Lie groups]]) on [[topological space|topological]] or [[smooth manifold|smooth]] [[n-spheres]] display various interesting patterns in their classification.

This entry is meant to eventually list and discuss some of these. For the moment it mainly just collects some references.

## Properties

### Free actions by finite groups and spherical space forms
 {#FreeActionsByFiniteGroupsAndSphericalSpaceForms}

We discuss some aspects of ([[G-space|continuous]]) [[free actions]] of [[finite groups]] on [[n-spheres]].

If the action is by [[isometries]] (with respect to the [[round sphere|round metric]]) then the [[topological quotient spaces]] of such free actions are traditionally known as *[[spherical space forms]]* (going back to [Killing 1891](spherical+space+form#Killing1891)). Later (maybe starting with [Madsen, Thomas & Wall 1976](#MadsenThomasWall76)) authors also speak of *topological spherical space forms* for quotients by actions which are continuous but not necessarily by isometries.

#### Basic examples
 {#BasicExamplesOfFreeActionsOnSpheres}

We make explicit some elementary but important examples of free continuous finite group actions on $n$-spheres.

\begin{example}
\label{ActionOfZModTwoByAntipodalInversion}
**(action of $\mathbb{Z}/2$ by antipodal involution)**
\linebreak
  For all $n \in \mathbb{N}$, the group [[cyclic group of order 2|$\mathbb{Z}/2$]] has a [[continuous function|continuous]] [[free action]]  on the [[n-sphere]], given by [[antipode|antipodal]] reflection (hence reversing the [[orientation]]).

The [[topological quotient space]] of this action is the [[real projective space]] $S^{n+1}/(\mathbb{Z}/2) \,\simeq\, \mathbb{R}P^n$.
\end{example}

\begin{example}
\label{FiniteCyclicGroupsActFreelyOnAddDimensionalSpheres}
**(Finite cyclic groups act freely on odd-dimensional spheres)**
\linebreak
 For every non-[[trivial group|trivial]] [[finite group|finite]] [[cyclic group]] $\mathbb{Z}/n \,\subset\, U(1)$, its left-multiplication action on the [[n-sphere|$2n+1$-sphere]], regarded as the [[unit sphere]] in the [[complex numbers|complex]] $n$-space
$$
  S^{2n+1}
    \,\simeq\,
  S\big(\mathbb{R}^{2n+2}\big)
    \,\simeq\,
  S\big(\mathbb{C}^{n+1}\big)
  \,,
$$
is (a [[G-space|continuous group action]] and) [[free action|free]].
\end{example}
This follows verbatim as the (slightly more interesting) Ex. \ref{FiniteADEGroupsActFreelyOnFourNPlusThreeSpheres} below, interchanging the [[complex numbers]] with the [[quaternions]].

\begin{example}
\label{FiniteADEGroupsActFreelyOnFourNPlusThreeSpheres}
**(Finite ADE-groups act freely on $S^{4n+3}$-spheres)**
\linebreak
For every non-[[trivial group|trivial]] [[finite subgroup of SU(2)]] $G \,\subset\, $ [[SU(2)]] $\simeq$ [[Sp(1)]] its [[quaternion|quaternionic]] left-multiplication action on the [[n-sphere|$4n+3$-sphere]], regarded as the [[unit sphere]] in the [[quaternion|quaternionic]] $n$-space
$$
  S^{4n+3}
    \,\simeq\,
  S\big(\mathbb{R}^{4n+4}\big)
    \,\simeq\,
  S\big(\mathbb{H}^{n+1}\big)
  \,,
$$
is (a [[G-space|continuous group action]] and) [[free action|free]].
\end{example}
\begin{proof}
Here are the elementary and straightforward details:

Under the given identification, points in $S^{4n+3}$ correspond to $(n+1)$-[[tuples]] of [[quaternions]] $\vec v \,=\, (v_1, \cdots, v_{n+1})$, $v_i \,\in\, \mathbb{H}$ (i.e. quaternionic [[vectors]]) such that

\[
  \label{UnitNormConditionOnQuaternionicVectors}
  \begin{aligned}
    \left\vert \vec v \right\vert^2
    &
    \;\coloneqq\;
    \vec v^\dagger \cdot \vec v
    \\
    &
    \;\coloneqq\;
    \bar v_1 \cdot v_1  +  \cdots  +  \vec v_{n+1} \cdot v_{n+1}
    \\
    &
    \;\overset{!}{=}\;
    1
  \end{aligned}
  \,,
\]

where $\overline{(-)}$ denotes [[quaternionic conjugation]].

Since [[Sp(1)]] is the [[subgroup]] of the quaternionic [[group of units]] on the unit-[[norm]] elements

$$
  Sp(1)
  \;\coloneqq\;
  \big\{
    q \,\in\, \mathbb{H}^\times
    \,\vert\,
    \bar q \cdot q \,=\, 1
  \big\}
  \;\subset\;
  \mathbb{H}^\times
$$

we have

$$
  \begin{aligned}
    \left\vert
      q \cdot \vec v
    \right\vert
    &
    \;=\;
    (q \cdot \vec v)^\dagger
    \cdot
    (q \cdot \vec v)
    \\
    &
    \;=\;
    \vec v^\dagger
      \cdot
    \underset{ = 1}{\underbrace{q^\dagger \cdot q}}
      \cdot
    \vec v
    \\
    &
    \;=\;
    \vec v^\dagger \cdot \vec v
    \\
    &
    \;=\;
    \left\vert \vec v\right\vert^2
    \,,
  \end{aligned}
$$

showing that the left multiplication action of $Sp(1)$ on $\mathbb{H}^{n+1}$ does restrict to an action on its unit sphere.
Moreover, since quaternion-multiplication is clearly [[continuous function|continuous]] with respect to the [[Euclidean topology]] on $\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4$, this yields a continuous action on $S^{4n+3}$.

Finally, since $\mathbb{H} \,\ni\, q \,\mapsto\, \bar q \cdot q \,\in\, \mathbb{R}$ is positive definite ($\bar q \cdot q = 0 \,\;\Leftrightarrow\;\, q = 0$ ), at least one of the components $v_i$ of $\vec v$ needs to be non-zero in order for  (eq:UnitNormConditionOnQuaternionicVectors) to hold. But on this component the left action $v_i \,\mapsto\, q \cdot v_i$ is left-multiplication in the [[group of units]] $\mathbb{H}^{\times} \,=\, \mathbb{H} \setminus \{0\}$ and hence is free, as the multiplication action of any group on itself is free.
\end{proof}

\begin{remark}
While an analogous argument as in Ex. \ref{FiniteADEGroupsActFreelyOnFourNPlusThreeSpheres} shows that with $G \subset Sp(1)$ also the [[direct product group]] $G^{n+1}$ canonically acts on $S^{4n+3}$ by componentwise left multiplication, for $n \geq 2$ this action is no longer free, as the $k$th factor subgroup now fixes the elements whose $k$th component vanishes. In fact, higher direct product powers of cyclic groups in general have *no* free action on spheres of any dimension, see Smith's $p^2$-condition (Prop. \ref{SmithPSquareCondition} below).
\end{remark}

#### Free involutions
 {#Involutions}

The [above examples](#BasicExamplesOfFreeActionsOnSpheres) of free actions on spheres are all on odd-dimensional spheres, except when the group is [[cyclic group of order 2|$\mathbb{Z}/2$]], hence when the action is by [[involutions]] (Rem. \ref{ZTwoActionsAreInvolutions} below), where the antipodal free action (Ex. \ref{ActionOfZModTwoByAntipodalInversion}) exists in all dimensions. Indeed, it holds in general that the only free actions of finite groups on even-dimensional spheres are [[involutions]] (Prop. \ref{OnlyZModTwoCanActFreelyOnEvenDimensionalSpheres} below), and, as in the canonical example, all free involutions on even-dimensional spheres are [[orientation]]-reversing, and those on odd-dimensional spheres are orientation-preserving (Prop. \ref{FreeZModTwoActionOnEvenDimSphereMustBeOrientationReversing} below).

While, therefore, all free involutions on a given $n$-sphere are [[homotopy|homotopic]] to the antipodal involution (Rem. \ref{FreeInvolutionsOnNSphereAllHaveTheSameHomotopyClass} below), at least in the [[categories]] of [[piecewise-linear structure|piecewise-linear]] and of [[smooth structure|smooth]] actions there are families of distinct exotic involutions on $n$-spheres (Ex. \ref{NonStandardFreeInvolutions} below).

\begin{proposition}
\label{OnlyZModTwoCanActFreelyOnEvenDimensionalSpheres}
**(only $\mathbb{Z}/2$ has free actions on even-dimensional spheres)**
\linebreak
  The only [[finite group]] with any [[free action|free]] [[G-space|continuous action]] on a [[sphere]] $S^{2n}$ of [[positive number|positive]] [[even number|even]] [[dimension of a manifold|dimension]] is [[cyclic group of order 2|$\mathbb{Z}_2$]].
\end{proposition}
\begin{proof}
  Given a fixed-point free-action, the [[quotient space]] [[coprojection]]
  $S^{2n} \xrightarrow{q} S^{2n}/G$ is a [[covering space]] with [[degree of a continuous function|degree]] $deg(q) = ord(G)$ equal to the [[order of a group|order]] of $G$. Passing to [[Euler characteristics]] $\chi(-)$ and using that

1. $\chi(-)$ is always an [[integer]] (by definition);

1. $\chi(S^{2n}) = 2$ (by [this Prop.](Euler+characteristic#EulerCharacteristicOfSpheres));

1. $\chi(-)$ of any finite covering coprojection is multiplication by the degree ([this Prop.](Euler+characteristic#EulerCharacteristicOfCoveringSpaces))

we obtain the [[equation]]

$$
  2
    \;=\;
  \chi(S^{2n})
    \;=\;
  ord(G) \cdot \chi(S^{2n}/G)
  \,.
$$

The only solutions for this algebraic equation (over the [[integers]]) have $ord(G) \,\in\, \{1, 2\}$. But $ord(G) = 1$ implies that $G$ is the [[trivial group]], whose action certainly has fixed points. Hence the only admissible solution is $ord(G) = 2$ and the only group of that order is [[cyclic group of order 2|$\mathbb{Z}_2$]]. That this does have at least one fixed-point free action is Ex. \ref{ActionOfZModTwoByAntipodalInversion}.
\end{proof}
\begin{remark}
  The nature of the [[fixed loci]] of finite group actions on even-dimensional spheres is discussed in [Craciun 2013](#Craciun13).
\end{remark}

\begin{remark}
\label{ZTwoActionsAreInvolutions}
**([[cyclic group of order 2|$\mathbb{Z}/2$]]-[[group action|actions]] are equivalently [[involutions]])**
\linebreak
  Any [[group action]] $\rho \,\colon\, \mathbb{Z}/2 \times S^n \to S^n$ of the [[cyclic group of order 2|$\mathbb{Z}/2$]] is determined by its value $\rho(\sigma) \,\colon\, S^n \to S^n$ on the single non-trivial element $\sigma \,\in\, \mathbb{Z}/2$, which is an [[involution]], and every involution corresponds to a unique $\mathbb{Z}/2$-action this way.
\end{remark}

\begin{proposition}
\label{FreeZModTwoActionOnEvenDimSphereMustBeOrientationReversing}
Every [[free action|free]] [[involution]]

* on an [[even number|even]]-dimensional sphere is [[orientation]]-reversing;

* on an [[even number|odd]]-dimensional sphere is [[orientation]]-preserving.

\end{proposition}
\begin{proof}
By the [[Lefschetz fixed point theorem]],
see [this example](Lefschetz+trace+formula#FixedPointTheoremForHomeomorphismsOfnSpheres) for details.
\end{proof}

\begin{remark}
\label{FreeInvolutionsOnNSphereAllHaveTheSameHomotopyClass}
  Prop. \ref{FreeZModTwoActionOnEvenDimSphereMustBeOrientationReversing} implies that the [[homotopy class]] of any fixed-point free [[involution]] $\sigma \,\colon\, S^{n} \to S^n$ is, as an element in the [[homotopy groups of spheres]],

* on an [[even number|even]]-dimensional $n$-sphere, $n = 2k$, equal to

  $$
    [\sigma] \,=\, - 1 \,\in\, \mathbb{Z} \,\simeq\, \pi_{2k}(S^{2k})
  $$

* on an [[odd number|odd]]-dimensional $n$-sphere, $n = 2k+1$, equal to

  $$
    [\sigma] \,=\, + 1 \,\in\, \mathbb{Z} \,\simeq\, \pi_{2k+1}(S^{2k+1})
    \,.
  $$

which means that every free involution on any $n$-sphere is [[homotopy|homotopic]] to the antipodal involution (Ex.\ref{ActionOfZModTwoByAntipodalInversion}).
\end{remark}
Nevertheless, there are in general several involutions on any given $n$-sphere which are *not* [[isomorphism|isomorphic]] to the antipodal involution as $\mathbb{Z}/2$-actions, certainly if regarded in the category of smooth (or just piecewise-linear) actions:

\begin{example}
\label{NonStandardFreeInvolutions}
**(non-standard free smooth involutions on $n$-spheres)**
\linebreak
There are, in the [[category]] [[smooth function|smooth]] $\mathbb{Z}/2$-[[group action|actions]], up to [[isomorphism]]:

* a single free involution on the [[3-sphere]] $S^3$ (hence the antipodal Ex. \ref{ActionOfZModTwoByAntipodalInversion})

  ([Livesay 1960](#Livesay60))

* at least 2 distinct free involutions on the [[4-sphere]] $S^4$

  ([Fintushel & Stern 1981](#FintushelStern81))

* 4 distinct free involutions on the [[5-sphere]] [[5-sphere|$S^5$]]

* 4 distinct free involutions on the [[6-sphere]] [[6-sphere|$S^6$]]

* $card(\mathbb{Z}/2 \oplus \mathbb{Z}/28 \oplus \mathbb{Z})$ distinct free involutions on the [[7-sphere]] [[7-sphere|$S^7$]]

  ([López de Medrano, Sec. V.6.1](#LopezdeMedrano71))

\end{example}
Here all spheres are equipped with their standard smooth structure (?), it is just the extra involutions which are non-standard.

Similarly, there are non-standard involutions on $n$-spheres in the category of [[piecewise-linear structures]] ([López de Medrano, p. 2 (11 of 114)](#LopezdeMedrano71)) following a similar pattern.

\begin{example}
\label{NonStandardFreeTopologicalInvolutions}
**(non-standard free continuous involutions on $n$-spheres)**
There are also plenty of non-standard involutions on $n$-spheres in the topological category (both for even and for odd $n$):

By the analog of the [[Poincaré conjecture]]-theorem in higher dimensions (e.g. [Newman 1966, Thm. 7](#Newman66)), every [[topological manifold]] of the [[homotopy type]] of a [[real projective space]] has a double [[covering space]] which is a manifold of the homotopy type of an $n$-sphere, and hence is actually [[homeomorphism|homeomorphic]] to an $n$-sphere.

Therefore, topological involutions on $n$-spheres should have the same classification up to homeomorphism as homeomorphism types of topological manifolds which are homotopy equivalent to $\mathbb{R}P^n$s, and large classes of non-trivial examples  of these are known ([Belegradek, MO:407390](https://mathoverflow.net/a/407390/381)).
\end{example}

#### General obstructions and existence
 {#GeneralObstructionToAndExistenceOfFreeActions}

Not all [[finite groups]] have any free continuous action on any $n$-sphere: One [[obstruction]] is Smith's "$p^2$-condition" (Prop. \ref{SmithPSquareCondition} below), another is Milnor's "$2 p$-condition" (Prop. \ref{Milnor2PCopndition} below). But these are the only two obstructions, and a finite group that evades these is guaranteed to act not just continuously on a sphere of a single dimension, but smoothly on spheres of any dimension which is a multiple of its [[Artin-Lam induction exponent]] (Prop. \ref{MadsenThomasWallTheorem} below).

\begin{proposition}
\label{SmithPSquareCondition}
**(Smith's $p^2$-condition)**
\linebreak
  For $p$ a [[prime number]],
  the [[direct product group]] $\mathbb{Z}/p \times \mathbb{Z}/p$ and more generally the higher powers $(\mathbb{Z}/p)^{\geq 2}$ of the prime [[cyclic group]] do *not* have any [[G-space|continuous]] [[free action]] on any [[n-sphere]].
\end{proposition}
([Smith 1944, p. 107 (4 of 5)](#Smith44))

\begin{remark}
By the [[fundamental theorem of finitely generated abelian groups]],
Prop. \ref{SmithPSquareCondition} immediately implies that if a [[finite group]] has any [[G-space|continuous]] [[free action]] on any [[n-sphere]], then it must satisfy the condition that all its [[subgroups]] of [[order of a group|order]] $p^2$ are [[cyclic group|cyclic]], i.e. [[isomorphism|isomorphic]] to $\mathbb{Z}/p^2$.

This condition has come to be called the "$p^2$-condition" (Def. \ref{pqCondition} below).
\end{remark}

\begin{definition}\label{pqCondition}
**($p q$ condition)**
For $p, q$ a [[pair]] of [[prime numbers]], not necessarily distinct,
a [[finite group]] $G$ is said to satisfy the *$p q$-condition* if all [[subgroups]] of [[order of a group|order]] $p \cdot q$ are [[cyclic groups]]:
$$
  H \,\subset\, G
  \;\;\;
  \text{with}
  \;\;\;
  ord(H) \,=\, p q
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  H \,\simeq\, \mathbb{Z}/(p q)
  \,.
$$
\end{definition}

\begin{proposition}\label{EquivalentVersionsOfPSquareCondition}
**(equivalent statements of the $p^2$-condition)**
\linebreak
  For a [[finite group]] $G$, the following are equivalent:

  1. For all [[prime numbers]] $p$, $G$ satisfies the $p^2$-condition (Def. \ref{pqCondition}).

  1. For all [[prime number|primes]] $p$,  the [[Sylow p-subgroup]] of $G$ is either cyclic or (if $p = 2$) [[generalized quaternion group|generalized quaternion]].

  1. $G$ has "periodic cohomology" in that there exists $k \in \mathbb{N}$ such that its [[integral cohomology|integral]] [[group cohomology]] [[cohomology groups|group]] in degree $k$ is [[cyclic group|cyclic]]:

     $$
       \underset{k \in \mathbb{N}}{\exists}
       \;
       \underset{N_k \in \mathbb{N}}{\exists}
       \;\;
       H^k_{grp}(G;\, \mathbb{Z}) \,\simeq\, \mathbb{Z}/{n_k}
     $$

     In this case the possible "periods" $k$ form a [[subgroup]] of $\mathbb{Z}$ ([Cartan & Eilenberg 1956, p. 261 (283 of 421)](homological+algebra#CartanEilenberg), with $H^{-k}(G;\, \mathbb{Z}) \coloneqq H_k(G;\, \mathbb{Z})$ by [p. 232 (254 of 421)](homological+algebra#CartanEilenberg))

1. Every [[abelian group|abelian]] [[subgroup]] of $G$ is [[cyclic group|cyclic]].

\end{proposition}
([Cartan & Eilenberg 1956, Thm. IV 11.6, p. 262 (284 of 421)](homological+algebra#CartanEilenberg))

\begin{proposition}\label{Milnor2PCopndition}
**(Milnor's $2 p$-condition)**
\linebreak
  If a [[finite group]] has any [[topological G-space|continuous]] [[free action]] on any [[n-sphere]], then

1. every element of [[order of an element|order]] 2 is in its [[center of a group|center]]

1. $G$ satisfies the $2p$-condition (Def. \ref{pqCondition}) for all $p$.

\end{proposition}
([Milnor 1957, Cor 1 & p. 627](#Milnor57))

\begin{proposition}
**(Zassenhaus's $p q$-condition)**
\label{ZassenHauspqCondition}
  If a [[finite group]] has an *orthogonal* [[free action]] on some [[n-sphere]], namely a free action through a [[group homomorphism]] $G \xrightarrow O(n+1)$ to the [[orthogonal group]] regarded with its canonical action on the [[unit sphere]] $S^n \,\simeq\, S(\mathbb{R}^{n+1})$, then $G$ satisfies all $p q$-conditions (Def. \ref{pqCondition}) for all pairs of primes $p,q$.

Moreover, if $G$ is [[solvable group|solvable]], then the converse holds: Every solvable group satisfying all $p q$-conditions has a free orthogonal action on some $n$-sphere.
\end{proposition}
([Zassenhaus 1935](#Zassenhaus35), [Vincent 1947](#Vincent47))

\begin{proposition}\label{MadsenThomasWallTheorem}
**(Madsen-Thomas-Wall theorem)**
\linebreak
  A [[finite group]] $G$ has a [[G-space|continuous]] [[free action]] on some [[n-sphere]] if and only if it satisfies

1. the $p^2$-condition (see Prop. \ref{EquivalentVersionsOfPSquareCondition})

1. the $2 p$-condition (see Prop. \ref{Milnor2PCopndition})

for all [[prime numbers]] $p$ (Def. \ref{pqCondition}).

In this case there exists also a [[smooth structure]] on some [[n-sphere]] (possibly an [[exotic smooth structure]]) such that $G$ has a [[smooth function|smooth]] [[free action]] on it.

Specifically, such free smooth actions exist in particular on all $S^{2 k \cdot e(G) - 1}$, for all $k \in \mathbb{N}$, where $e(G) \,\in\, \mathbb{N}_+$ is the  [[Artin-Lam induction exponent]] of $G$, hence exist on spheres of arbitrarily large dimension.
\end{proposition}

([Madsen, Thomas and Wall 1976, Thm. 0.5-0.6](#MadsenThomasWall76), [1983, Thm. 5](#MadsenThomasWall83), reviewed in [Hambleton 2014, Thm. 6.1](#Hambleton14))

Classification results on groups satisfying these conditions are discussed in [Wall 2013](#Wall13).

\begin{example}
\label{FiniteSubgroupsOfSUTwoSatisfyTheSmithAndTheMilnorCondition}
**(Finite subgroups of $SU(2)$ satisfy the $p^2$ and $2p$ conditions)**
\linebreak
  Every non-[[trivial group|trivial]] [[finite subgroup of SU(2)]] $G \,\subset\, SU(2) \,\simeq\, Sp(1)$
has a free smooth action on $S^{4k+3}$, for all $k \in \mathbb{N}$, by Ex. \ref{FiniteADEGroupsActFreelyOnFourNPlusThreeSpheres}. By Prop. \ref{MadsenThomasWallTheorem} this implies that all subgroups of $G$ of order $p^2$ or $2 p$ must be cyclic.

Indeed, by the [[ADE-classification]] of [[finite subgroups of SU(2)]], the only possible non-cyclic subgroups are:

1. the [[binary dihedral groups]] $2 D_{2(r+3)}$ of order $2^2 \cdot (r + 3)$,

1. the [[binary tetrahedral group]] of order $24 \,=\, 2^3 \cdot 3$,

1. the [[binary octahedral group]] of order $48 \,=\, 2^4 \cdot 3$,

1. the [[binary icosahedral group]] of order $120 \,=\, 2^3 \cdot 3 \cdot 5$.

None of these orders is of the form $p^2$ or $2 p$, in accord with Prop. \ref{MadsenThomasWallTheorem}.

On the other hand, the plain [[dihedral groups]] $D_{2(r+3)} \,\subset\, SO(3)$ have order $2 \cdot (r + 3)$ and hence [[finite subgroups of SO(3)]] may violate Milnor's $2 p$-condition (Prop. \ref{Milnor2PCopndition}). Certainly their canonical actions on [[2-sphere|$S^2$]] are not free (as each element must act by a rotation which necessarily fixes an [[antipode|antipodal]] pair of points).
\end{example}

\linebreak

### Fixed loci of the circle group acting on spheres

+-- {: .num_prop}
###### Proposition

Given an continuous [[action]] of the [[circle group]] on the [[topological space|topological]] [[4-sphere]], its [[fixed point]] space is of one of two types:

1. either it is the [[0-sphere]] $S^0 \hookrightarrow S^4$

1. or it has the [[rational homotopy theory|rational homotopy type]] of an even-dimensional sphere.

=--

([Félix-Oprea-Tanré 08, Example 7.39](#FelixOpreaTanre08))

(...)

## Literature

See also the references at *[[spherical space form]]*.

### Characterization of finite free actions by homeomorphisms

Discussion of [[free actions]] by [[finite groups]]:

The original article identifying the $p^2$-condition for continuous actions:

* {#Smith44} P. A. Smith, *Permutable Periodic Transformations*, Proceedings of the National Academy of Sciences of the United States of America Vol. 30, No. 5 (May 15, 1944), pp. 105-108 ([jstor:87918](https://www.jstor.org/stable/87918))

The original article identifying the $2 p$-condition for continuous actions:

* {#Milnor57} [[John Milnor]], *Groups Which Act on $S^n$ Without Fixed Point*, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 ([jstor:2372566](https://www.jstor.org/stable/2372566))

See also:

* [[Richard G. Swan]], *Periodic Resolutions for Finite Groups*, Annals of Mathematics Second Series **72** 2 (1960) 267-291 [[doi:10.2307/1970135](https://doi.org/10.2307/1970135)]

Existence results:

* {#Wall78} [[C. T. C. Wall]], _Free actions of finite groups on spheres_, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/wall7.pdf))

* {#MadsenThomasWall76} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *The topological spherical space form problem II: existence of free actions*, Topology Volume 15, Issue 4, 1976, Pages 375-382 (<a href="https://doi.org/10.1016/0040-9383(76)90031-8">doi:10.1016/0040-9383(76)90031-8</a>)

* {#MadsenThomasWall83} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *Topological spherical space form problem III: Dimensional bounds and smoothing*,  Pacific J. Math. 106(1): 135-143 (1983) ([pjm:1102721110](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-106/issue-1/Topological-spherical-space-form-problem-III-Dimensional-bounds-and-smoothing/pjm/1102721110.full))

Classification results:

* {#Wall13} [[C. T. C. Wall]], *On the Structure of Finite Groups with Periodic Cohomology*, 381-413 in: *Lie Groups: Structure, Actions, and Representations*, Progress in Mathematics **306** Birkhäuser 2013 ([doi:10.1007/978-1-4614-7193-6_16](https://doi.org/10.1007/978-1-4614-7193-6_16))

following

* {#Wolf74} [[Joseph Wolf]], Part III of: _Spaces of constant curvature_, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 ([doi:10.1090/chel/372](https://doi.org/10.1090/chel/372))

Discussion of free actions on [[product topological space|products]] of spheres:

* [[Alejandro Adem]], _Constructing and deconstructing group actions_, in [[Paul Goerss]], [[Stewart Priddy]], *Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory*, Contemporary Mathematics **346** AMS 2004 ([arXiv:0212280](http://arxiv.org/abs/math/0212280), [doi:10.1090/conm/346](http://dx.doi.org/10.1090/conm/346))

Discussion of the fixed point-sets of finite group actions on even-dimensional spheres:

* {#Craciun13} Gheorghe Craciun, *Most homeomorphisms with a fixed point have a Cantor set of fixed points*, Archiv der Mathematik volume 100, pages 95–99 (2013) ([doi:10.1007/s00013-012-0466-z](https://doi.org/10.1007/s00013-012-0466-z))

### Classification of free involutions:

* {#Livesay60} [[George R. Livesay]]:  *Fixed Point Free Involutions on the 3-Sphere*, Annals of Mathematics Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 603-611 ([jstor:1970232](https://www.jstor.org/stable/1970232))

* {#LopezdeMedrano71} [[Santiago López de Medrano]], *Involutions on Manifolds*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **59**, Springer 1971 ([doi:10.1007/978-3-642-65012-3](https://link.springer.com/book/10.1007/978-3-642-65012-3))

* {#FintushelStern81} [[Ronald Fintushel]], [[Ronald J. Stern]], *An Exotic Free Involution on $S^4$*, Annals of Mathematics Second Series, Vol. 113, No. 2 (Mar., 1981), pp. 357-365 ([jstor:2006987](https://www.jstor.org/stable/2006987))

### Classification of finite free actions by isometries

The original article identifying the $p q$-conditions for orthogonal actions:

* {#Zassenhaus35} [[Hans Zassenhaus]], *Über endliche Fastkörper*, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg. Vol. 11. No. 1. Springer-Verlag, 1935 ([pdf](https://link.springer.com/content/pdf/10.1007/BF02940723.pdf))

* {#Vincent47} [[Georges Vincent]], *Les groupes linéaires finis sans points fixes*, Commentarii Mathematici Helvetici 20.1 (1947): 117-171 ([pdf](https://link.springer.com/content/pdf/10.1007/BF02568125.pdf))

Classification of free finite group actions by [[isometries]], hence with [[quotient spaces]] being [[spherical space forms]]:

* {#Wolf74} [[Joseph Wolf]], _Spaces of constant curvature_, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 ([doi:10.1090/chel/372](https://doi.org/10.1090/chel/372))

review:

* {#Hambleton14} [[Ian Hambleton]], _Topological spherical space forms_, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston ([arXiv:1412.8187](https://arxiv.org/abs/1412.8187), [ISBN:9781571463012](https://www.intlpress.com/site/pub/pages/books/items/00000439/index.php))

streamlined re-proof:

* {#Allock15} [[Daniel Allcock]], _Spherical space forms revisited_, Trans. Amer. Math. Soc. 370 (2018), 5561-5582  ([arXiv:1509.00906](https://arxiv.org/abs/1509.00906), [doi:10.1090/tran/7167](https://doi.org/10.1090/tran/7167))

The subgroups of [[special orthogonal group|SO(8)]] which act freely on $S^7$ have been classified in [Wolf 1974](#Wolf74) and lifted to actions of [[Spin group|Spin(8)]] in

* {#Gadhia07} [[Sunil Gadhia]], _Supersymmetric quotients of M-theory and supergravity backgrounds_, PhD thesis, School of Mathematics, University of Edinburgh, 2007 ([spire:1393845](http://inspirehep.net/record/1393845/))

Further discussion of these actions of $Spin(8)$ on the [[7-sphere]] is in

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

where they are related to the [[black brane|black]] [[M2-brane]] [[BPS state|BPS]]-solutions of [[11-dimensional supergravity]] at [[ADE-singularities]].

### Discussion of circle-actions

Discussion of [[circle group]]-actions on spheres:

* {#FelixOpreaTanre08} [[Yves Félix]], [[John Oprea]], [[Daniel Tanré]], _Algebraic Models in Geometry_, Oxford University Press 2008

[[!redirects group action on spheres]]

[[!redirects free group actions on spheres]]
[[!redirects free group action on spheres]]

[[!redirects finite group actions on spheres]]
[[!redirects finite group action on spheres]]

[[!redirects group action on an n-sphere]]
[[!redirects group actions on an n-sphere]]

[[!redirects group action on n-spheres]]
[[!redirects group actions on n-sphere]]

[[!redirects group action on n-spheres]]
[[!redirects group actions on n-spheres]]

[[!redirects free group actions on n-spheres]]
[[!redirects free group action on n-spheres]]


group algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _group algebra_ of a [[group]] $G$ over a [[ring]] $R$ is the [[associative algebra]] whose elements are [[formal linear combinations]] over $R$ of the elements of $G$ and whose multiplication is given on these [[basis]] elements by the group operation in $G$.

## Definition

### For discrete groups
 {#ForDiscreteGroups}

Let $G$ be a [[discrete group]]. Let $R$ be a [[commutative ring]].

+-- {: .num_defn }
###### Definition

The **group $R$-algebra** $R[G]$ is the [[associative algebra]] over $R$

1. whose [[underlying]] $R$-[[module]] is the the [[free module]] over $R$ on the [[underlying set]] of $G$;

1. whose multiplication is given on [[basis]] elements by the group operation.

=--

+-- {: .num_remark }
###### Remark

By the discussion at [[free module]], an element $r$ in $R[G]$ is a [[formal linear combination]] of [[basis]] elements in $G$ with [[coefficients]] in $R$, hence a formal sum

$$
  r = \sum_{g \in G} r_g \cdot g
$$

with $\forall_{g \in G} (r_g \in R)$ and only finitely many of the coefficients different from $0 \in R$.

The addition of algebra elements is given by the componentwise addition of coefficients

$$
  r + \tilde r =
  \sum_{g \in G} (r_g + \tilde r_g) g
$$

and the multiplication is given by the [[convolution product]]

$$
  \begin{aligned}
    r \tilde r
    & =
    \sum_{g \in G}
      \sum_{\tilde g \in G}
      (r_g \tilde r_{\tilde g}) g \cdot \tilde g
    \\
    & =
    \sum_{q \in G}
    \left(
      \sum_{k \in G}  (r_{q\cdot k^{-1}} r_k)
    \right) q
    \,.
  \end{aligned}
$$

=--

+-- {: .num_remark }
###### Remark

The [[formal linear combinations]] over $R$ of element in $G$ may equivalently be thought of as [[functions]]

$$
  r_{(-)} \colon U(G) \to U(R)
$$

from the underlying set of $G$ to the underlying set of $R$ which have _finite support_. Accordingly,  often the underlying set of the group $R$-algebra is written as

$$
  U(R[G]) = Hom_{Set}^{fin\;supp}(U(G), U(R))
$$

and for the [[basis elements]] one writes

$$
  \chi_g \colon U(G) \to U(R)
  \,,
$$

the **characteristic function** of an element $g \in G$, defined by

$$
  \chi_g \colon \tilde g \mapsto
  \left\{
    \array{
      1 & | g = \tilde g
      \\
      0 & | otherwise
    }
  \right.
  \,.
$$

In terms of this the product in the group algebra is called the **[[convolution product]]** on functions.

=--

+-- {: .num_remark }
###### Remark

The notion of group algebra is a special case of that of a [[groupoid algebra]], hence of [[category algebra]].

=--

###For profinite groups

The completed group ring of a [[profinite group]] is a pseudocompact ring.
Let $\hat{\mathbb{Z}}$ be the [[profinite completion]] of the ring of integers, $\mathbb{Z}$, then $\hat{\mathbb{Z}}$  is itself a [[pseudocompact ring]] as it is the inverse limit of its _finite_ quotients.  Now let $G$ be a profinite group.

The _completed group algebra_, $\hat{\mathbb{Z}}[\![G]\!]$, of $G$ over $\hat{\mathbb{Z}}$ is the inverse limit of the ordinary group algebras, $\hat{\mathbb{Z}}[G/U] $, of the finite quotients, $G/U$ (for $U$ in the directed set, $\Omega(G)$, of open normal subgroups of $G$), over $\hat{\mathbb{Z}}$;

$$\hat{\mathbb{Z}}[\![G]\!] = lim_{U\in \Omega(G)} \hat{\mathbb{Z}}[G/U].$$

For $R$ a pseudocompact ring, it is then easy to construct the corresponding pseudo-compact group algebra of $G$ over $R$; see the paper by Brumer.

### For topological groups
 {#ForTopologicalGroups}

Discussion of group algebras in the generality of [[locally compact topological groups]]:

e.g. [Dixmier (1977) §13.2](#Dixmier77)

## Properties

### General

+-- {: .num_prop }
###### Proposition

A group algebra is in particular a [[Hopf algebra]] and a $G$-[[graded algebra]].

=--

The following states a [[universal property]] of the construction of the group algebra.

+-- {: .num_remark}
###### Remark

There is an [[adjunction]]

$$
  (R[-]\dashv (-)^\times)
  \;\colon\;
  Alg_R
    \underoverset
      {
        \underset{
          \;\;\;
          (-)^\times
          \;\;\;
        }{
          \longrightarrow
        }
      }
      {
        \overset{
          \;\;\;
          R[-]
          \;\;\;
        }{
          \leftarrow
        }
      }
      {}
  Grp
$$

between the [[category]] of [[Algebras]] ([[associative algebras]] over $R$) and that of [[Groups]], where $R[-]$ forms [[group rings]] and $(-)^\times$ assigns to an $R$-algebra its [[group of units]].

=--

+-- {: .num_remark}
###### Remark

Let $V$ be an [[abelian group]]. A [[homomorphism]] of rings $R[G] \to End(V)$ of the group ring to the [[endomorphism ring]] of $V$ is equivalently a $R[G]$-[[module]] structure on $V$.

Any [[homomorphism]] of groups $p \colon G \to Aut(V)$ to the [[automorphism group]] of $V$ extends to to a morphism of rings.

This observation is used extensively in the theory of [[group representation|group representations]]. See also at _[module -- Abelian groups with G-action as modules over a ring ](http://ncatlab.org/nlab/show/module#AbelianGroupsWithGAction)_.

=--

\begin{prop}\label{ComplexGroupAlgebraOfFiniteGroupAsDirectSumOfEndomorphismAlgebras}
  For $G$ a [[finite group]] with [[isomorphism classes]] of [[irreducible representations]] $[V] \in Irreps_{\mathbb{C}}(G)_{/\sim}$ over the [[complex numbers]], the complex [[group algebra]] of $G$ is [[isomorphism|isomorphic]] to the [[direct sum]] of the linear [[endomorphism algebras]] of the [[complex vector spaces]] underlying the [[irreps]]:
$$
  \mathbb{C}[G]
  \;\simeq\;
  \underset{
    [V] \in Irreps_{\mathbb{C}}(G)_{/\sim}
  }{ \oplus }
  End_{\mathbb{C}}(V)
$$
\end{prop}
(e.g. [Fulton-Harris 91, Prop. 3.29](#FultonHarris91))
\begin{proof}
  For every representation $V$, the defining [[group action]]
  $$
    G
      \overset{}{\longrightarrow}
    Aut_{\mathbb{C}}(V)
      \hookrightarrow
    End_{\mathbb{C}}(V)
  $$
  [[extension|extends]] uniquely to an [[algebra homomorphism]]
  $$
    \mathbb{C}[G]
      \overset{ \phi_V }{\longrightarrow}
    End_{\mathbb{C}}(V)
    \,.
  $$
  Observe that this is a [[surjection]], since if it were not then we could split off a  non-trivial [[cokernel]], contradicting the assumption that $V$ is irreducible.

  We claim that the resulting homomorphism to the [[direct sum]]
  $$
    \mathbb{C}[G]
      \overset{ (\phi_V)_{[V]} }{\longrightarrow}
    \underset{
      [V] \in Irreps_{\mathbb{C}}(G)_{/\sim}
    }{\oplus}
    End_{\mathbb{C}}(V)
  $$
  is an [[isomorphism]]: By the previous comment it is [[surjective]], hence it is sufficient to observe that the [[dimension of a vector space|dimension]] of the group algebra equals that of the right hand side, hence that
$$
  dim\big(
    \mathbb{C}[Sym(G)]
  \big)
  \;=\;
  \underset{[V]}{\sum} \big( dim(V)\big)^2
  \,.
$$
This is indeed the case, by [this property](regular+representation#RegularRepDecomposedIntoIrreps) of the [[regular representation]].
\end{proof}

+-- {: .num_theorem}
###### Theorem
([[Maschke's theorem]])

Let $G$ be a [[finite group]], let $R = k$ be a field.

Then $k[G]$ is a [[semi-simple algebra]] precisely if the [[order]] of $G$ is not divisible by the [[characteristic]] of $k$.

=--

### Of central extensions & Twisted group algebras
 {#OfCentralExtensionsAndTwistedGroupAlgebras}

A [[quotient algebra]] of the group algebra of a [[central extension]] $G^\omega$ of a group $G$ corresponding to a [[group cocycle|group 2-cocyle]] $\omega \,\colon\, G \times G \to k$ is the *$\omega$-twisted* group algebra of $G$ (eg. [Nachbin 1993, Ch 2, Thm. 4.1](#Nachbin93)).

\begin{example}
\label{ComplexGroupAlgebraOfCircleCentralExtension}
**(complex group algebra of $\mathrm{U}(1)$-central extension)**
\linebreak
For $\big(G, (-)\cdot(-), \mathrm{e}\big)$ a [[discrete group]], consider a [[central extension]]

$$
  1
    \to
  \mathbb{R}/\mathbb{Z}
    \hookrightarrow
  G^\omega
    \twoheadrightarrow
  G
    \to
  1
$$

classified by a [[circle group]]-valued [[group cocycle|group 2-cocyle]] with [[underlying]] [[function]]

$$
  \omega
    \,\colon\,
  G \times G \to \mathbb{R}/\mathbb{Z}
  \,,
$$

which we may and do assume to be normalized:

$$
  \omega(g,\mathrm{e}) \,=\, 0,\,
  \;\;\;
  \omega(\mathrm{e},g) \,=\, 0
  \,.
$$

In terms of this cocycle, the group operation on the [[underlying]] [[set]]

$$
  G^\omega
    \;\simeq\;
  G \,\times\, \mathbb{R}/\mathbb{Z}
$$

is given by

$$
  (g, r) \cdot (g',r')
  \;\equiv\;
  \big(
    g \cdot g'
    ,\,
    r + r' + \omega(r,r')
  \big)
 \,.
$$

> We regard $\mathbb{R}/\mathbb{Z}$ as a [[discrete group]]. Since the cocycle will typically (certainly if $G$ is [[finite group|finite]]) take values only in a [[finite group|finite]] [[cyclic group|cyclic]] [[subgroup]] $\mathbb{Z}/n \hookrightarrow \mathbb{R}/\mathbb{Z}$ one may want to take $G^\omega$ to be just the finite group given by the resulting $\mathbb{Z}/n$-[[central extension]]. The discussion here applies verbatim in either case.

If we denote the generators of the [[complex numbers|$\mathbb{C}$-valued]] [[group algebra]] of $G^\omega$ by

$$
  \array{
    G^\omega
      &\longrightarrow&
    \mathbb{C}\big(G^\omega\big)
    \\
    (g,r) &\mapsto& U(g,r)
  }
$$

then we have in $\mathbb{C}\big(G^\omega \big)$ the relations

$$
  U(g,r)
  \;=\;
  U(g,0) \cdot U(\mathrm{e},r)
  \;=\;
  U(\mathrm{e},r)
    \cdot
  U(g,0)
$$
(by normality of $\omega$) and hence
$$
  \begin{array}{rcl}
  U(g, 0) \cdot U(g', 0)
  &=&
  U\big(g g', \omega(g,g')\big)
  \\
  &=&
  U\big(\mathrm{e}, \omega(g,g')\big)
  \cdot
  U(g g', 0)
  \,.
  \end{array}
$$

This looks, up to the [[center|central]] correction factor $U\big(\mathrm{e}, \omega(g,g')\big)$, like the group algebra of $G$.

To bring out this relation, consider now the [[quotient algebra]] of $\mathbb{C}\big(G^\omega\big)$ by the canonical [[augmentation ideal]] of the group algebra of the extension group, ie. by the two-sided [[ideal]] in $\mathbb{C}\big(G^\omega\big)$ generated by the [[kernel]] of the $\mathbb{C}$- [[algebra homomorphism]]

$$
  \begin{array}{ccc}
    \mathbb{C}\big(\mathbb{R}/\mathbb{Z}\big)
    &\xrightarrow{\;\; \epsilon \;\;}&
    \mathbb{C}
    \\
    U\big(
     \mathrm{e},
     r
    \big)
    &\mapsto&
    e^{2 \pi \mathrm{i} r }
    \mathrlap{\,.}
  \end{array}
$$

In other words, in this quotient of $\mathbb{C}\big(G^\omega\big)$ we enforce the [[relations]]
$$
  \begin{array}{rcl}
    U(\mathrm{e}, r)
    &\sim&
    exp(2 \pi \mathrm{i} r )
    \,
    U(\mathrm{e}, 0)
    \\
    &=&
    exp(2 \pi \mathrm{i} r )
  \end{array}
$$
(using in the last step that $(\mathrm{e}, 0)$ is the [[neutral element]] of $G^\omega$, so that $U(\mathrm{e}, 0)$ is the [[unit element]] in its group algebra).

Therefore the [[quotient algebra]] $\mathbb{C}\big(G^\omega\big)\big/ ker(\epsilon)$ is that generated by $U(G) \,=\, U\big(G \times \{0\}\big)$ subject to the relations

$$
  U(g) \cdot U(g')
  \;=\;
  e^{ 2 \pi \mathrm{i} \omega(g,g') }
  \,
  U(g\cdot g')
  \,.
$$

This is the $\omega$-twisted group algebra $\mathbb{C}^\omega(G)$ of $G$:

$$
  \mathbb{C}\big(
    G^\omega
  \big)
  \big/
  \mathrm{ker}(\epsilon)
  \;\;
  \simeq
  \;\;
  \mathbb{C}^\omega(G)
  \,.
$$
\end{example}

With due care, this situation generalizes from [[discrete groups]] to suitable (eg. [[locally compact topological groups|locally compact]]) [[topological groups]] ([Edwards & Lewis 1969a](#EdwardsLewis69a), [1969b](#EdwardsLewis69b)).

\begin{example}
**([Binz, Honegger & Rieckers 2007](#BinzHoneggerRieckers07))**
\linebreak
It is in this way that the group algebra of a ([[underlying]] [[discrete group|discrete]]) [[Heisenberg group]] (which is a [[central extension]] of an [[abelian group]]) is related to the corresponding [[Weyl group]] (whose [[Weyl relations]] are those of a twisted additive group algebra).
\end{example}

### Relation to universal enveloping algebras
 {#RelationToUniversalEnvelopingAlgebra}

Concerning group algebras of [[algebraic groups]]:

> The [[universal enveloping algebra]] of a [[Lie algebra]] is the analogue of the usual [[group algebra]] of a group. It has the analogous function of exhibiting the [[category]] of [[Lie algebra modules]] as a category of [[modules]] for an [[associative algebra]]. This becomes more than an analogy when the universal enveloping algebra is viewed with its full [[Hopf algebra]] structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible [[algebraic group]] over a [[field]] of [[characteristic 0]], contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion.

(quoted from [Hochschild 1981, p. 221](universal+enveloping+algebra#Hochschild81), see Thm. 3.1 on p. 230 there)

## Related concepts

* [[Cohn localization]]

* [[groupoid algebra]], [[category algebra]]

* [[augmentation ideal]]

* [[pseudocompact ring]]

* [[Day convolution]]

* [[∞-group ∞-ring]]

## References

### General

Original discussion:

* [[Hermann Weyl]], §III.13 in: *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics*, Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]

Monographs

for the case of [[locally compact topological groups]]:

* {#Dixmier77} [[Jacques Dixmier]], §13.2 in: *$C^\ast$-algebras*, North Holland (1977)

* {#Landsman17} [[Klaas Landsman]], §C.18 in: *Foundations of quantum theory -- From classical concepts to Operator algebras*, Springer Open (2017) [[doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf)]

for the case of [[finite groups]]:

* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], Section 3.4 of: *Representation Theory: a First Course*, Springer (1991) [[doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9)]

Lecture notes:

* [[Kiyoshi Igusa]], _Algebra II, part D: representations of groups_, ([pdf](http://people.brandeis.edu/~igusa/Math101bS07/Math101b_notesD1a.pdf))

* Andrei Yafaev, _Group algebras_ ([pdf](http://www.ucl.ac.uk/~ucahaya/GroupAlgebras.pdf))

On twisted group algebras and their relation to plain group algebras of [[group extensions]]:

* C. M. Edwards, *$C^\ast$-algebras of central group extensions I*, Annales de l'institut Henri Poincaré. Section A, Physique Théorique **10** 3 (1969) 229-246 [[numdam:AIHPA_1969__10_3_229_0](http://www.numdam.org/item/?id=AIHPA_1969__10_3_229_0)]

* {#EdwardsLewis69a} C. M. Edwards, [[John T. Lewis]], *Twisted group algebras I*, Communications in Mathematical Physics **13** (1969) 119–130 [[doi:10.1007/BF01649871](https://doi.org/10.1007/BF01649871), [euclid:cmp/1103841535](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-13/issue-2/Twisted-group-algebras-I/cmp/1103841535.full)]

* {#EdwardsLewis69b} C. M. Edwards, [[John T. Lewis]], *Twisted group algebras II*, Communications in Mathematical Physics **13** (1969) 131–141 [[doi:10.1007/BF01649872](https://doi.org/10.1007/BF01649872)]

* {#Nachbin93} Leopoldo Nachbin (ed.), *Twisted Group Algebras*, Chapter 2 in: *Group Representations Volume 2*, North-Holland Mathematics Studies **177** (1993) 65-103 [doi:10.1016/S0304-0208(09)70147-1]

The [[universal localization]] of group rings (see also at _[[Snaith's theorem]]_) is discussed in

* [[M. Farber]], [[Pierre Vogel]], _The Cohn localization of the free group ring_, Math. Proc.  Camb. Phil. Soc. (1992) 111, 433  ([pdf](http://www.maths.ed.ac.uk/~aar/papers/fv.pdf))

* Davidson, Nicholas, _Modules Over Localized Group Rings for Groups Mapping Onto Free Groups_ (2011). Boise State University Theses and Dissertations. Paper 170. ([web](http://scholarworks.boisestate.edu/td/170/))

On the case of profinite groups:

* A. Brumer, _Pseudocompact algebras, profinite groups and class formations_,  J. Algebra __4__ (1966) 442-470, [MR202790](http://www.ams.org/mathscinet-getitem?mr=202790), doi [pdf](http://deepblue.lib.umich.edu/bitstream/handle/2027.42/33410/0000811.pdf?sequence=1)

### In physics

On group algebras as [[strict deformation quantization|strict]] [[deformation quantizations]] of [[Lie-Poisson manifolds]]:

* [[Marc A. Rieffel]], *Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures*, American Journal of Mathematics **112** 4 (1990) 657-685 [[doi:10.2307/2374874](https://doi.org/10.2307/2374874), [jstor:2374874](https://www.jstor.org/stable/2374874)]

* {#Rieffel90} [[Marc Rieffel]], Ex. 7, Ex. 8 in: _Deformation quantization and operator algebras_, in: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411-423, Proc. Sympos. Pure Math. __51__, Part 1, Amer. Math. Soc. (1990) [[pdf](http://math.berkeley.edu/~rieffel/papers/deformation.pdf), [[Rieffel-DefQuantization.pdf:file]] [MR91h:46120](http://www.ams.org/mathscinet-getitem?mr=1077400)]

Strengthening of the original result, including generalization to [[groupoid algebras]] of [[Lie groupoids]] [[Lie integration|integrating]] given [[Lie algebroids]]:

* [[Nicolaas P. Landsman]], §3.4 in: *Classical and quantum representation theory* [[arXiv:hep-th/9411172](https://arxiv.org/abs/hep-th/9411172)]

* [[Nicolaas P. Landsman]], Ex. 2 in: *Lie Groupoid $C^\ast$-Algebras and Weyl Quantization*, Communications in Mathematical Physics **206** (1999) 367–381 [[doi:10.1007/s002200050709](https://doi.org/10.1007/s002200050709)]

* [[Nicolaas P. Landsman]], B. Ramazan, Ex. 11.1 in: *Quantization of Poisson algebras associated to Lie algebroids*, in: *Groupoids in Analysis, Geometry, and Physics*, Contemporary Mathematics **282** (2001) [[arXiv:math-ph/0001005](https://arxiv.org/abs/math-ph/0001005), [ams:conm/282](http://www.ams.org/books/conm/282)]

See also:

* [[Klaas Landsman]], *Strict deformation quantization of a particle in external gravitational and Yang-Mills fields*, Journal of Geometry and Physics __12__ 2 (1993) 93-132 [doi:10.1016/0393-0440(93)90010-C]

* [[Nicolaas P. Landsman]], p. 27 in: *Classical and quantum representation theory* [[arXiv:hep-th/9411172](https://arxiv.org/abs/hep-th/9411172)]

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* {#BinzHoneggerRieckers07} [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]

On the [[supersymmetry|supersymmetric]] [[WZW model]] using group algebra:

* [[Matthias Blau]], [[George Thompson]], *Equivariant Kähler Geometry and Localization in the $G/G$ Model*, Nucl. Phys. B **439** (1995) 367-394 [doi:10.1016/0550-3213(95)00058-Z, [arXiv:hep-th/9407042](https://arxiv.org/abs/hep-th/9407042)]

[[!redirects group algebras]]

[[!redirects group ring]]
[[!redirects group rings]]

[[!redirects twisted group algebra]]
[[!redirects twisted group algebras]]

[[!redirects group convolution algebra]]
[[!redirects group convolution algebras]]


group averaging

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[gauge theory]], _group averaging_ is the idea of turning an arbitrary [[function]] of the [[field (physics)|fields]] into an [[invariant]] ([[gauge invariance|gauge invariant]]) function by suitably averaging it over the [[action]] of the group of [[gauge transformations]], hence by taking the [[integral]] against a [[measure]] on the group of [[gauge transformations]] of the function pulled back by the gauge action.

For groups of gauge transformations which are finite dimensional [[compact Lie groups]] (typical in [[quantum mechanics]]) this just works, but most groups of gauge transformations appearing in [[quantum field theory]] are infinite-dimensional (at best) and non-compact, hence much of the literature on group averaging in gauge theory is concerned with making sense of regularizing the naive prescription.

(Beware the terminology: the _[[gauge group]]_ is frequently finite-dimensional and compact (typically in [[Yang-Mills theory]]) but the group of gauge transformations is typically something like the group of suitable functions from spacetime into the gauge group. Some mathematical texts say "gauge group" for "group of gauge transformations", but this is not what physics texts do and should probably better be avoided.)

## Related concepts

* [[norm map]]

## References

Textbook accounts include

* {#Olver99} [[Peter Olver]], around (4.8) in _Classical invariant theory_, 1999

Reviews include

* Domenico Giulini, _Group Averaging and Refined Algebraic Quantization_, Nucl.Phys.Proc.Suppl. 88 (2000) 385-388 ([arXiv:gr-qc/0003040](http://arxiv.org/abs/gr-qc/0003040))


group character

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _multiplicative [[character]]_ of a [[group]] $G$ is a [[group homomorphism]] into the [[circle group]] $U(1)$, or more generally into the [[group of units]] $k^\times$ of a given [[ground field]] (for instance $\mathbb{C}^\times = U(1)$):

$$
  \chi
  \;\colon\;
  G \longrightarrow k^\times
  \,.
$$

Since $k^\times$ is an [[abelian group]], this means that group characters are in particular [[class functions]].

Dually a co-character is a homomorphism out of $k^\times$ into $G$.

The collection of characters is itself an [[abelian group]] under the pointwise multiplication, this is called the **character lattice**
$ Hom(G,k^\times) $
of the group. Similarly the **cocharacter lattice** is $Hom(k^\times, G)$.

For [[topological groups]] one considers [[continuous map|continuous]] characters. Specifically, for a [[locally compact Hausdorff]] group $G$ (often further assumed to be an [[abelian group]]), a __character__ of $G$ is continuous homomorphism to the [[circle]] group $\mathbb{R}/\mathbb{Z}$. If $G$ is [[profinite group|profinite]], then this is the same as an continuous homomorphism to the [[discrete space|discrete]] group $\mathbb{Q}/\mathbb{Z}$.  (See [MO](http://mathoverflow.net/questions/86089/two-definitions-of-character-of-topological-groups).)

## Properties

### Restriction of group characters to maximal tori -- weights
 {#Weights}

Let $G$ be a [[connected topological space|connected]] [[compact Lie group]].

By the general [properties of maximal tori](maximal+torus#Properties) in this case, it follows that every group character $G \to U(1)$ is already fixed by its restriction along a [[maximal torus]] inclusion

$$
  T \hookrightarrow G \to U(1)
  \,.
$$

Now the group characters of the [[abelian group|abelian]] maximal torus $T\simeq U(1)^n$ are [[n-tuples]] of group characters of the [[circle group]] $U(1)$, which are [[integers]] -- the _[[weight (in representation theory)|weights]]_.

Explicitly, under a given identification of the [[circle group]] as a [[quotient]] of the additive group of [[real numbers]]

$$
  U(1) \simeq \mathbb{R}/h \mathbb{Z}
$$

for $h \in (0,\infty)$, then the character $\lambda$ on $U(1)^n\simeq T$ labeled by $(x_1, \cdots, x_n) \in \mathbb{Z}^n$ is

$$
  \lambda(t_1,\cdots t_n)
  =
  \exp(\tfrac{i}{\hbar} \sum_{j = 1}^n t_j n_j )
$$

(where $\hbar \coloneqq h/2\pi$ is "[[Planck's constant]]").

(e.g. [Johansen, section 2.10](#Johansen))

### Relation to Chern roots and the splitting principle
 {#RelationToChernRootsAndSplittingPrinciple}

A group character, hence a group [[homomorphism]] $G \to U(1)$ induces a map of [[classifying spaces]] $B G \to B U(1) \simeq K(\mathbb{Z},2)$. Similarly for the restriction to the [[maximal torus]] [above](#Weights), which induces

$$
  B U(1)^n \simeq B T \to B G \to B U(1)\simeq K(\mathbb{Z},2)
  \,.
$$

Under this identification the [weights](#Weights) $x_i$ of the group character, as above, are the "[[Chern roots]]" as the appear in the [[splitting principle]]. See there for more.

### Characters and fundamental group of tori
 {#CharactersAndFundamentalGroupsOfTori}

Write $S^1$ for the [[circle group]].

Let $T$ be a [[torus]], regarded as an [[abelian group]].
Write $[T,S^1]$ for its character group.

There is a [[bilinear form]]

$$
  \pi_1(T)\otimes [T, S^1] \longrightarrow \mathbb{Z}
$$

on the [[fundamental group]] of the [[torus]] and its character group, given by sending a [[homotopy class]] $[\gamma]$ of a [[continuous map]]

$$
  \gamma \colon S^1 \longrightarrow T
$$

to the [[homotopy class]] $c(\gamma)$ of the [[composition]] with a [[character]] $c \colon T \longrightarrow S^1$

$$
  c(\gamma) \;\colon\; S^1 \stackrel{\gamma}{\longrightarrow}
  T \stackrel{c}{\longrightarrow} S^1
$$

regarded as an element $[c(\gamma)] \in \pi_1(S^1) \simeq \mathbb{Z}$.

This [[bilinear form]] is non-degenerate, and hence constitutes an [[isomorphism]]

$$
  \pi_1(T) \simeq [T,S^1]
  \,.
$$

### Inner product and orthogonality

The complex [[class functions]] on a [[finite group]] $G$ have an [[inner product]] given by

$$
  \langle \alpha, \beta\rangle
  \coloneqq
  \frac{1}{{\vert G \vert}}
  \underset{g \in G}{\sum} \alpha(g) \overline{\beta(g)}
  \,.
$$

The _[[Schur orthogonality relation]]_ is the statement that the [[irreducible representation|irreducible]] group characters $\{\chi_i\}_i$ form an [[orthonormal basis]] of the space of [[class functions]] under this [[inner product]]:

$$
  \langle \chi_i, \chi_j \rangle
  =
  \left\{
    \array{
       1 & if \; i = j
       \\
       0 & otherwise
    }
  \right.
$$

Such properties arise from characters occurring as [[traces]] of [group representations](representation+ring#relation_to_the_character_ring).

### In terms of the classifying space of the group

Consider the [[classifying space]], $B G$, of the group. Then its [[free loop space]], $Map (S^1, B G)$, has as components $G$ modulo [[conjugation]]. Then, the characters of $G$ may be expressed as the zeroth cohomology of this loop space, $H^0(\mathcal{L} B G, \mathbb{C})$. This construction is useful in the generalisation to [[transchromatic characters]].

## Related concepts

* [[Pontryagin duality]], [[Pontryagin dual]]

* [[conjugacy class of subgroups]]

## References

Original articles on character rings/[[representation rings]] of [[compact Lie groups]] include

* [[Graeme Segal]], _The representation ring of a compact Lie group_, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, January 1968, Volume 34, Issue 1, pp 113-128 ([numdam:PMIHES_1968__34__113_0](http://archive.numdam.org/numdam-bin/fitem?id=PMIHES_1968__34__113_0))

* Masaru Tackeuchi, _A remark on the character ring of a compact Lie group_, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 ([euclid:jmsj/1259849785](http://projecteuclid.org/euclid.jmsj/1259849785))

Lecture notes on characters of finite and compact Lie groups include

* {#Johansen} Troels Roussauc Johansen, _Character Theory for Finite Groups and Compact Lie Groups_ [pdf](http://www.math.upb.de/~johansen/character-theory.pdf)

* Andrei Yafaev, _Characters of finite groups_ ([pdf](http://www.ucl.ac.uk/~ucahaya/Characters.pdf))

Discussion for finite groups in the more general context of [[equivariant cohomology|equivariant]] [[complex oriented cohomology theory]] ([[transchromatic character]]) is in

* [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], _Generalized group characters and complex oriented cohomology theories_, J. Amer. Math. Soc. 13 (2000), 553-594 ([publisher](http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf))

[[!redirects group characters]]

[[!redirects character lattice]]
[[!redirects character lattices]]

[[!redirects cocharacter lattice]]
[[!redirects cocharacter lattices]]

[[!redirects co-character lattice]]
[[!redirects co-character lattices]]

[[!redirects character group]]
[[!redirects character groups]]


group cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _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 _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

$$
  c \colon \mathbf{B}G \to \mathbf{B}^n A
$$

and the [[cohomology group]] is the [[homotopy]] [[equivalence classes]] of this

$$
  H^n_{Grp}(G,A) \simeq \pi_0 \mathbf{H}(\mathbf{B}G,\mathbf{B}^n A)
  \,.
$$

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

$$
  \array{
    \mathbf{B}G && \stackrel{\mathbf{c}}{\longrightarrow}  && \mathbf{B}^n A \sslash G
    \\
    & \searrow &\swArrow& \swarrow
    \\
    && \mathbf{B}G
  }
  \,.
$$

Equivalently this means that the group cohomology of $G$ with coefficients in an abelian group $A$ with $G$-[[action]] $\rho$ is the _[[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

$$
  \array{
    \mathbf{B}G && \stackrel{\mathbf{c}}{\longrightarrow}  && V \sslash G
    \\
    & \searrow &\swArrow& \swarrow
    \\
    && \mathbf{B}G
  }
$$

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

* [Fully general definition in homotopy type theory](#InHomotopyTypeTheory).

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

* [Of an ordinary group with abelian group coefficients - In terms of homological algebra](#InTermsOfHomologicalAlgebra).

Finally one can break this further down into components In

* [Special case: Explicit formulas in low degree](#InLowDegree)

we give some standard formulas for group cohomology in low degree.

## Definition

### Fully general: in homotopy type theory
 {#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]]

$$
  \array{
    V &\to& V \sslash G
    \\
    && \downarrow^{\mathrlap{\bar \rho}}
    \\
    && \mathbf{B}G
  }
  \,.
$$

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]]

$$
  x \colon \mathbf{B}G \;\vdash \; V(x) \colon Type
  \,.
$$

Also, $\bar \rho$ is the _[[local coefficient bundle]]_ for $G$-group cohomology with [[coefficients]] in $V$ equipped with this $G$-[[∞-action]]. this means that the
 _group cohomology_ of $G$ with coefficients in $V$ is the hom in the [[slice (∞,1)-topos]] over $\mathbf{H}$ as [[base (∞,1)-topos]]

$$
  H^1_{Grp}(G,V)
  \coloneqq
  \mathbf{H}_{/\mathbf{B}G}(\mathbf{B}G, V)
  \,,
$$

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]]

$$
  \array{
     * &\to& \mathbf{B}G
     \\
     && \downarrow^{\mathrlap{id}}
     \\
     && \mathbf{B}G
  }
$$

This is the [[categorical semantics]] of what in the [[syntax]] of [[homotopy type theory]] is

$$
  \vdash \; \left(\prod_{x \colon \mathbf{B}G} \left(* \to V \right)\right) \colon Type
  \,.
$$

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 $V$.

+-- {: .num_prop #GroupCohomologyInInfinityToposForTrivialAction}
###### Proposition

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]]

$$
  H_{Grp}(G,V) \simeq \pi_0 \mathbf{H}(\mathbf{B}G, V)
  \,.
$$

In particular if $V = \mathbf{B}^n A$ for $A$ an abelian group, this is

$$
  H^n_{Grp}(G,A) \coloneqq H_{Grp}(G,\mathbf{B}^n A)
  \simeq
  \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)
  \,.
$$

=--

[[!include homotopy type representation theory -- table]]

### For an ordinary group and abelian coefficients: In terms of homological algebra
 {#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

$$
  \mathbb{Z}[G] \in Ring
$$

for the [[group algebra]] of $G$ over the [[integers]].
Write

$$
  \mathcal{A} \coloneqq \mathbb{Z}[G] Mod
$$

for the category $\mathbb{Z}[G]$[[Mod]] of [[modules]] over $\mathbb{Z}[G]$.

Notice that a [[module]]

$$
  A \in \mathbb{Z}[G] Mod
$$

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]]

$$
  A[n] \in Ch_\bullet(\mathbb{Z}[G]Mod)
$$

is the kind of [[coefficient]] for the group cohomology of $G$ to which the following statement applies.

+-- {: .num_remark #InvariantsByHomFunctor}
###### Remark

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

$$
  Invariants(A) \simeq Hom_{\mathbb{Z}[G]}(\mathbb{Z}, A)
  \,.
$$

This equivalence is [[natural isomorphism|natural]] and hence the contravariant [[hom functor]] is equivalently the invariants-functor

$$
  Invariants(-)\simeq Hom_{\mathbb{Z}[G]}(\mathbb{Z}, -)
  \,.
$$

=--

By the fully general discussion [above](#InHomotopyTypeTheory), 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:

+-- {: .num_defn}
###### Definition

For $A$ an [[abelian group]] equipped with a $G$-[[action]], the degree-$n$ _group cohomology_ of $G$ with [[coefficients]] in $A$ is the $n$th-[[Ext]]-group

$$
 H^n_{Grp}(G,A) \coloneqq Ext^n_{\mathbb{Z}[G]}(\mathbb{Z}, A)
  \,,
$$

where on the right $\mathbb{Z} \in \mathbb{Z}[G] Mod$ is regarded as equipped with the trivial $G$-action.

=--

+-- {: .num_remark}
###### Remark

By the discussion at _[[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]]

$$
  \cdots \to F_3 \to F_2 \to F_1 \to F_0 \to \mathbb{Z} \to 0
$$

of $\mathbb{Z}[G]$-[[modules]]. Let

$$
  Hom_{\mathbb{Z}[G]Mod}(F_\bullet, A)
  =
  \left[
    Hom_{\mathbb{Z}[G]Mod}(F_0, A)
      \stackrel{d^0}{\longrightarrow}
    Hom_{\mathbb{Z}[G]Mod}(F_1,A)
      \stackrel{d^1}{\longrightarrow}
    Hom_{\mathbb{Z}[G]Mod}(F_2,A)
      \stackrel{d^2}{\longrightarrow}
      \cdots
  \right]
$$

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

$$
  H^n_{Grp}(G,A)
  \simeq
  H^n(Hom_{\mathbb{Z}[G]Mod}(F_\bullet, A))
  \coloneqq
  ker(d^n)/im(d^{n-1})
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

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 _[[Hochschild-Serre spectral sequence]]_.

=--

## Special aspects and special cases

The fully general definition [above](#InHomotopyTypeTheory) subsumes various cases that are not always discussed on the same footing in traditional literature. For emphasis we highlight these special cases separately.

* [Simplicial constructions and explicit formulas in low degree](#InLowDegree)

* [Structured group cohomology](#StructuredCohomology)

* [Nonabelian group cohomology](#NonabelianGroupCohomology)

### Simplicial constructions and explicit formulas in low degree
 {#InLowDegree}

We unwind the general abstract definition of group cohomology
[above](#InHomotopyTypeTheory) 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 [[simplicial classifying space]], hence  the [[nerve]] of the [[delooping groupoid]] $*\sslash G$;

* $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 $A$.

+-- {: .num_prop #DiscreteGroupCohomologyBySimplicialHomotopy}
###### Proposition

Then the degree-$n$ group cohomology of $G$ with [[coefficients]] in $A$ is the set

$$
  H^n_{Grp}(G,A) \simeq \pi_0 sSet(\overline{W}G, DK(A[n]))
$$

of [[homomorphisms]] of [[simplicial sets]] modulo [[simplicial homotopy]].

=--

+-- {: .proof}
###### Proof

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

$$
  H^n_{Grp}(G,A) \simeq \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)
  \,.
$$

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

$$
  \overline{W}G
  =
  \left(
    \cdots \to G \times G \times G
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    G  \times G
    \stackrel{\longrightarrow}{\longrightarrow}
    G
    \to
    {*}
  \right)
  \,,
$$

which is equivalently the [[nerve]] of the [[groupoid]] $*\sslash G$.

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

=--

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

#### Degree-$1$ group cohomology
 {#Degree1}

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

#### Degree-$2$ group cohomology
 {#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]].

+-- {: .num_prop #2CocyclesAndCoboundaries}
###### Proposition

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 **group 2-[[cocycle]]** on $G$ with coefficients in $A$ is a [[function]]

$$
  c \colon G \times G \to A
$$

such that for all $(g_1, g_2, g_3) \in G \times G \times G$ it satisfies the [[equation]]

\[
  \label{2CocycleConditionForCentralExtension}
  c(g_1, g_2)
  -
  c(g_1, g_2 \cdot g_3)
  +
  c(g_1 \cdot g_2, g_3)
  -
  c(g_2, g_3)
  =
  0
  \;\;\;\; \in A
\]

(called the **group 2-cocycle condition**).

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

$$
  h \colon G \to A
$$

such that for all $(g_1,g_2) \in G \times G$ the [[equation]]

\[
  \label{Group2CoboundaryEquation}
  \tilde c(g_1,g_2)
  =
  c(g_1,g_2) + (d h)(g_1,g_2)
\]

holds in $A$, where

$$
  (d h)(g_1, g_2) \coloneqq h(g_1 g_2) - h(g_1) - h(g_2)
$$

is the **group 2-coboundary** encoded by $h$.

The degree-2 **group cohomology** is the set

$$
  H^2_{Grp}(G,A) = 2Cocycles(G,A) / Coboundaries(G,A)
$$

of [[equivalence classes]] of group 2-cocycles modulo group 2-coboundaries. This is itself naturally an [[abelian group]] under pointwise addition of cocycles in $A$

$$
  [c_1] + [c_2] = [c_1 + c_2]
$$

where

$$
  c_1 + c_2 \colon (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)
  \,.
$$

=--

This may be taken as the _definition_ of degree-2 group cohomology (with coefficients in abelian groups and with trivial action). The following proof shows how this _follows_ from the general simplicial presentation of prop. \ref{DiscreteGroupCohomologyBySimplicialHomotopy}.

+-- {: .proof}
###### 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

$$
  (\overline{W}G)_2
  =
  \left\{
  \left.
  \array{
    && {*}
    \\
    & {}^{g_1}\nearrow && \searrow^{g_2}
    \\
    {*}
    &&\stackrel{g_1 g_2}{\longrightarrow}&& {*}
  }
  \right|
  g_1, g_2 \in G
  \right\}
  \,,
$$

and the 3-simplices are

$$
  (\overline{W}G)_3
  =
  \left\{
  \left.
  \array{
    {*} &&\stackrel{g_2}{\longrightarrow}&& {*}
    \\
    \uparrow^{g_1} &&{}^{g_1 g_2}\nearrow&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_1 g_2 g_3}{\longrightarrow}&&
    {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  \array{
    {*} &&\stackrel{g_2}{\longrightarrow}&& {*}
    \\
    \uparrow^{g_1} &&\searrow^{g_2 g_3}&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_1 g_2 g_3}{\longrightarrow}&&
    {*}
  }
  \right|
  g_1, g_2, g_3 \in G
  \right\}
  \,.
$$

Therefore a homomorphism of simplical sets $c \colon \overline{W}G \to DK(A[2])$ is in degree 2 a function

$$
  c_2
  \;\;
  :
  \;\;
  \left(
  \array{
    && {*}
    \\
    & {}^{g_1}\nearrow && \searrow^{g_2}
    \\
    {*}
    &&\stackrel{g_2 g_1}{\to}&& {*}
  }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
  \array{
    && {*}
    \\
    & {}^{{*}}\nearrow &\Downarrow^{c(g_1,g_2)}& \searrow^{{*}}
    \\
    {*}
    &&\stackrel{{*}}{\to}&& {*}
  }
  \right)
$$

i.e. a map $c : G \times G \to K$. To be a simplicial homomorphism this has to extend to 3-[[simplices]] as:

$$
  \begin{aligned}
  c_3
  \;\;\;
  &:
  \;\;\;
  \left(
  \array{
    {*} && \stackrel{g_2}{\longrightarrow} && {*}
    \\
    \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3}
    \\
    {*}
    && \stackrel{g_3 g_2 g_1}{\longrightarrow} && {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  \array{
    {*} &&\stackrel{g_2}{\to}&& {*}
    \\
    \uparrow^{g_1} &&\searrow^{g_3 g_2}&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_3 g_2 g_1}{\to}&&{*}
  }
  \right)
  \\
  & \mapsto
  \left(
  \array{
    {*} &\stackrel{}{\longrightarrow} & &\stackrel{}{\longrightarrow}& {*}
    \\
    \uparrow &\Downarrow^{c(g_1,g_2)}
    &\nearrow&\Downarrow^{c(g_2,g_3)}&
    \downarrow
    \\
    {*}
    &\longrightarrow&&\longrightarrow&{*}
  }
  \;\;\;\;
  \stackrel{}{\Rightarrow}
  \;\;\;\;
  \array{
    {*} &\longrightarrow&&\longrightarrow& {*}
    \\
    \uparrow &\Downarrow^{c(g_1,g_2 g_3)}
    &\searrow &\Downarrow^{c(g_2, g_3)}& \downarrow
    \\
    {*}
    &\longrightarrow&&\longrightarrow&{*}
  }
  \right)
  \end{aligned}
  \,.
$$

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

=--

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

+-- {: .num_defn #Normalized2Cocycle}
###### Definition

A group 2-cocycle $c \colon G \times G \to A$, def. \ref{2CocyclesAndCoboundaries} is called **normalized** if

$$
  \forall_{g_0,g_1 \in G}
  \;\;
  \left(g_0 = e \;or\; g_1 = e \right)
  \Rightarrow
  \left( c(g_0,g_1) = e \right)
  \,.
$$

=--

+-- {: .num_lemma #2CocycleOngeisSameasOnee}
###### Lemma

For $c \colon G \times G \to A$ a group 2-cocycle, we have for all $g \in G$
that

$$
  c(e,g) = c(e,e) = c(g,e)
  \,.
$$

=--

+-- {: .proof}
###### Proof

The cocycle condition (eq:2CocycleConditionForCentralExtension) evaluated on

$$
  (g^{-1},  g,  e) \in G^3
$$

says that

$$
 c(g^{-1}, g) + c(e, e) = c(g, e) + c(g^{-1}, g )
$$

hence that

$$
  c(e,e) = c(g, e)
  \,.
$$

Similarly the 2-cocycle condition applied to

$$
  (e,  g,  g^{-1}) \in G^3
$$

says that

$$
  c(e,g) + c(g,g^{-1}) = c(g,g^{-1}) + c(e,e)
$$

hence that

$$
  c(e,g) = c(e,e)
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

Every group 2-cocycle $c \colon G \times G \to A$ is
cohomologous to a normalized one, def. \ref{Normalized2Cocycle}.

=--

+-- {: .proof}
###### 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

$$
  h(g) \coloneqq c(g,g)
  \,.
$$

Then $\tilde c \coloneqq c + d c$ has the desired property, with (eq:Group2CoboundaryEquation):

$$
  \begin{aligned}
    \tilde c(e,e)
    & \coloneqq
    (c + d h)(e,e)
    \\
    & =
    c(e,e)  + c(e \cdot e, e \cdot e) - c(e,e) - c(e,e)
    \\
    & =
    0
  \end{aligned}
  \,.
$$

=--

### Relation with comonoid

1. The simplex category \Delta, whose objects are {1,2,3..n} and
morphisms are nondecreasing functions, is the UNIVERSAL "category
equipped with a monoid".

2. Dually, \Delta^op is the UNIVERSAL "category equipped with a comonoid".

3. For a group G, there's an adjunction between (G-reps) and (Vector
Spaces) by the forgetful functor and the free G-rep functor. The
composition gives you a COMONAD of (G-rep).

4. A comonad is just a categorification of a comonoid. More concretely
a comonad is a comonoid in End( (G-rep), (G-rep) ).

5. Since \Delta^op is has a universal comonoid, we connect both
contexts naturally. No wonder there's nice cohomology theory for
groups!

(ref: Categories-for-the-Working-Mathematician-[Mac-Lane])

### Structured group cohomology (topological groups and Lie groups)
 {#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

$$
  \left(
    \cdots G \times G\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}G \stackrel{\longrightarrow}{\longrightarrow} *
  \right)
$$

to a simplicial object $N (\mathbf{B}^n A)$.

This is what is described [above](#InLowDegree) for _[[discrete groups]]_. But this does **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

$$
  \left(
    \cdots G \times G\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}G \stackrel{\longrightarrow}{\longrightarrow} *
  \right)
$$

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 **not** both cofibrant and fibrant. To that extent plain morphisms out of this object do **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

* _[[Lie group cohomology]]_

See below at _[References - For structured groups](#ReferencesOnStructuredGroupCohomology)_ for pointers to the literature.

### Nonabelian group cohomology
 {#NonabelianGroupCohomology}

If the coefficient group $K$ is nonabelian, its higher [[delooping]]s $\mathbf{B}^n K$ do 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.

## Examples

### Specific examples

#### Simple examples

* [[carrying]]

#### Cohomology of $\mathbb{Z}/2\mathbb{Z}$

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

#### Cohomology of $\mathbb{Z}_2^n$

The cohomology of $\mathbb{Z}_2^n$ with coefficients in $\mathbb{Z}_2$ admits a very simple description (see ([p.66 of Adem & Milgram (2004)](#AM04finite))). There exists a ring isomorphism
$$
H^{\bullet} (\mathbb{Z}_2^n ,\mathbb{Z}_2 ) \cong \mathbb{Z}_2 [e_1 ,\cdots,e_n ]
$$
of the cohomology ring $H^n (\mathbb{Z}_2^n,\mathbb{Z}_2 )$ to the polynomial ring of $n$ commuting variables with coefficients in $\mathbb{Z}_2$. Furthermore, each cohomology group $H^q (\mathbb{Z}_2^n,\mathbb{Z}_2 )$ is generated by the classes of the functions:
$$
(x^{(1)},\cdots, x^{(q)} )\mapsto x^{(1)}_{i_1}\cdots x^{(q)}_{i_q}
$$
with $i_1 \leq \cdots \leq i_q$, where each argument $x^{(j)}\in \mathbb{Z}_2^n$ is an $n$-tuple $(x^{(j)}_1 ,\cdots, x^{(j)}_n )$.

#### Cohomology of $U(n)$, $O(n)$, etc.

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

* the [[unitary group]] $U(n)$;

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

* the [[symplectic group]] $Sp(n)$

the corresponding group cohomology in terms of the cohomology of the [[classifying space]]/[[delooping]] $B G$.

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

$$
\begin{aligned}
 H^\bullet(BU(n);\mathbb{Z}) &= \mathbb{Z}[c_1,\cdots,c_n]
 \\
 H^\bullet(BSU(n);\mathbb{Z}) &= \mathbb{Z}[c_2,\cdots,c_n]
 \\
 H^\bullet(BSp(n);\mathbb{Z}) &= \mathbb{Z}[p_1,\cdots,p_n]
\end{aligned}
$$

where $c_i\in H^{2i}$ and $p_i\in H^{4i}$.

#### Heisenberg cocycle

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

$$
  (a,b) + (a',b') = (a + a' , b + b')
$$

carries a degree-2 group cocycle $\omega$ with values in $\mathbb{R}$ given by

$$
  \omega : ((a_1,b_1), (a_2,b_2)) \mapsto a_1 \cdot b_2
  \,.
$$

The cocycle condition for this is the identity

$$
  a_1 \cdot (b_2 + b_3) + a_2 \cdot b_3
  =
  a_1 \cdot b_2 + (a_1 + a_2) \cdot b_3
$$

The [[group extension]] classified by this cocycle is the [[Heisenberg group]].

#### Galileo 2-cocycle

* [[Galileo 2-cocycle]]

### Classes of examples

#### Galois cohomology

The group cohomology of [[Galois groups]] is called _[[Galois cohomology]]_. See there for more details.

#### 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.

## Related concepts

* **group cohomology**

  * [[nonabelian group cohomology]], [[groupoid cohomology]]

  * [[Galois cohomology]]

* [[group homology]]

* [[crossed homomorphism]]

* [[transgression in group cohomology]]

* [[group extension]]

  * [[Baer sum]]

* [[Lie group cohomology]]

  * [[∞-group cohomology]]

  * [[∞-Lie groupoid cohomology]]

* [[equivariant cohomology]]

* [[Friedlander-Milnor isomorphism conjecture]]

## References

### Original references

* [[Samuel Eilenberg]], [[Saunders MacLane]], *Cohomology Theory in Abstract Groups. I*, Annals of Mathematics, Second Series **48** 1 (1947) 51-78 [[doi:10.2307/1969215](https://doi.org/10.2307/1969215)]

See also:

* [[S. Eilenberg]], [[S. MacLane]], _Relations between Homology and Homotopy Groups_.  Proceedings of the National Academy of Sciences **29** 5 (1943) 155–158.  [[doi:10.1073/pnas.29.5.155](https://doi.org/10.1073/pnas.29.5.155)]

* [[Samuel Eilenberg]], [[Saunders MacLane]], _Relations Between Homology and Homotopy Groups of Spaces_,  The Annals of Mathematics **46** 3 (1945) 480  [[doi:10.2307/1969165](https://doi.org/10.2307/1969165)]

* [[Samuel Eilenberg]], [[Saunders MacLane]], _Relations Between Homology and Homotopy Groups of Spaces. II_, The Annals of Mathematics **51** 3 (1950) 514.  [[doi:10.2307/1969365](https://doi.org/10.2307/1969365)]

* [[Dmitri Faddeev]], _On factor-systems in Abelian groups with operators_. (Russian), Doklady Akad. Nauk SSSR (N. S.) 58, (1947). 361–364.

### General

Textbook accounts:

* [[Kenneth Brown]],  *Cohomology of Groups*, Graduate Texts in Mathematics, **87**, Springer 1982 [[doi:10.1007/978-1-4684-9327-6](https://link.springer.com/book/10.1007/978-1-4684-9327-6)]

specifically with an eye towards cohomology of [[finite groups]]:

* [[Michael Atiyah]], _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, 9 (1961), p. 23-64  ([Numdam](http://www.numdam.org/item?id=PMIHES_1961__9__23_0))

*  {#AM04finite} [[Alejandro Adem]], [[R. James Milgram]], _Cohomology of Finite Groups_, Springer 2004

specifically for [[profinite groups]]:

* {#NSW08} [[Jürgen Neukirch]], Alexander Schmidt, Kay Wingberg, Chapter 1 of: *Cohomology of Number Fields*, Springer Grundlehren der mathematischen Wissenschaften **323**, Springer 2008 ([doi:10.1007/978-3-540-37889-1](https://link.springer.com/book/10.1007/978-3-540-37889-1), [webpage](https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/))

specifically for [[algebraic groups]]:

* {#Milne17} [[James Milne]], Section 15 (16 in the pdf) of: *Algebraic Groups*, Cambridge University Press 2017  ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))

On [[homotopy theory]]-methods:

* [[William G. Dwyer]], [[Hans-Werner Henn]], *Homotopy Theoretic Methods in Group Cohomology*, Advanced Courses in Mathematics - CRM Barcelona, Springer (2001) [[doi:10.1007/978-3-0348-8356-6](https://doi.org/10.1007/978-3-0348-8356-6)]

Further exposition:

* [[Alejandro Adem]], _Lectures on the cohomology of finite groups_ ([pdf](http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf))

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf))

An introduction to group cohomology of a group $G$ as the cohomology of the [[classifying space]] $B G$:

* Joshua Roberts, _Group cohomology: a classifying space perspective_ ([pdf](http://www.piedmont.edu/math/jroberts/notes/qualifyingtalk.pdf))

* _Advanced course on classifying spaces and cohomology of groups_ ([ps](www-irma.u-strasbg.fr/~henn/notes.ps))

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

* R. Brown, P. Higgins, R. Sivera, _Nonabelian algebraic topology_ ([pdf](http://www.bangor.ac.uk/%7Emas010/pdffiles/rbrsbookb-e040609.pdf), [web](http://www.bangor.ac.uk/~mas010/nonab-a-t.html))

Cohomology of [[simplicial groups]] is discussed for instance in

* Sebastian Thomas, _On the second cohomology group of a simplicial group_ ([pdf](http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/Thomas_On_the_second_cohomology_group_of_a_simplicial_group.pdf))

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

* Roggenkamp, Scott, _Automorphisms and nonabelian cohomology_ ([pdf](http://www.math.virginia.edu/~lls2l/automorphisms_and_nonabelian.pdf))

For a conceptual discussion of nonabelian group cohomology see

* [[John Baez]], [[Mike Shulman]], _[[Lectures on n-Categories and Cohomology]]_ ([arXiv](http://arxiv.org/abs/math/0608420))

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 _strict_ $\infty$-groups, this is discussed for instance in

* [[Johannes Huebschmann]], _Crossed $n$-fold extensions and group cohomology_ ([web](http://dx.doi.org/10.1007/BF02566688))

### On structured group cohomology
 {#ReferencesOnStructuredGroupCohomology}

In

* [[Jim Stasheff]], _Continuous cohomology of groups and classifying spaces_  Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 ([web](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540920))

$n$-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

* David Wigner, _Algebraic cohomology of topological groups_ Transactions of the American Mathematical Society, volume 178 (1973)([pdf](http://egg.epfl.ch/~nmonod/bonn/Wigner_1973.pdf))

A classical reference that considers the cohomology of Lie groups as topological spaces is

* [[Armand Borel]], _Homology and cohomology of compact connected Lie groups_ ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063923/pdf/pnas01596-0040.pdf))

A corrected definition of topological group cohomology has been given by Segal

* [[Graeme Segal]], _Cohomology of topological groups_ In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377{387. Academic Press, London, (1970).

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

For [[reductive algebraic groups]]:

* {#HKLV98} [[Hélène Esnault]], [[Bruno Kahn]], [[Marc Levine]], [[Eckart Viehweg]], _The Arason invariant and mod 2 algebraic cycles_,  J. Amer. Math. Soc. 11 (1998), 73-118  ([pdf](https://www.uni-due.de/~bm0032/publ/Arason.pdf),[publisher page](http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00248-3/))

### On various topological groups
 {#OnTopologicalGroups}

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

#### Compact Lie groups

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

* [[John Milnor]], [[Jim Stasheff]], _Characteristic Classes_ , Princeton University Press and University of Tokyo Press, Princeton, New Jersey, (1974).
 {#MilnorStasheff}

* Mark Feshbach, _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 ([JSTOR](http://www.jstor.org/stable/1998409))

* [[Donald Yau]], _Cohomology of unitary and symplectic groups_ ([pdf](http://www.mathnet.ru/links/c8897bc54ec0a1a49d03ca681d53db7b/ljm147.pdf))

* D. Benson, [[John Greenlees]], _Commutative algebra for cohomology rings of classifying spaces of compact Lie groups_ ([pdf](http://hopf.math.purdue.edu/Benson-Greenlees/Liegroupca.pdf))

* [[Eric Friedlander]], Guido Mislin, _Cohomology of classifying spaces of complex Lie groups and related discrete groups_ Commentarii Mathematici Helvetici
Volume 59, Number 1, 347-361,

#### The unitary groups $U(n)$

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.

#### The (special) orthogonal groups $O(n)$, $SO(n)$

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

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

* Emery Thomas , _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.

The [[ring]]-structure on the cohomology with integer coefficients was given in

* E. Brown (Jr.), _The cohomology of $B SO(n)$ and $B O(n)$ with integer coefficients_ Proc. AMS Soc. 85 (1982)

* Mark Feshbach, _The integral cohomology rings of the classifying spaces of $O(n)$ and $SO(n)$, Indiana Univ. Math. J. 32 (1983), 511–516.

For [[twisted cohomology|local coefficients]] see

* {#Cadek99} [[Martin Čadek]], _The cohomology of $B O(n)$ with twisted integer coefficients_, J. Math. Kyoto Univ. 39 (1999), no. 2, 277–286 ([Euclid](http://projecteuclid.org/euclid.kjm/1250517912))

* Richard Lastovecki, _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 ([pdf](http://www.emis.de/journals/CMUC/pdf/cmuc0501/lastovec.pdf))

#### For spin-group, string-group, ...

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

* Emery Thomas, _On the cohomology groups of the classifying space for the stable spinor group_ , Bol. Soc. Mex. (1962), 57 - 69

* Masana Harada, Akira Kono, _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) ([web](http://dl.acm.org/citation.cfm?id=23393))

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

* _On the integral cohomology of the seven-connective cover of B O_ Bull. Austral. Math. Soc. Vol 38 (1988) ([pdf](http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ38_01%2FS0004972700027179a.pdf&code=8374f17b7aa6ffb83f42f347c677729f))

#### The exceptional Lie groups

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

* Akira Kono, Mamoru Mimura, _Cohomology mod 3 of the classifying space of the Lie group $E_6$_ , Math. Scand. 46 (1980) ([pdf](http://www.mscand.dk/article.php?id=2533))

#### Loop groups of compact Lie groups

* Daisuke Kishimoto, Akira Kono, _Cohomology of the classifying spaces of loop groups_ ([pdf](http://www.math.kyoto-u.ac.jp/preprint/2004/13kishimoto.pdf))

### Online references

Some of the above materiel is taken from discussion at

* MO, _[Group cohomology of compact Lie group with integer coeffient ](http://mathoverflow.net/questions/75389/group-cohomology-of-compact-lie-group-with-integer-coeffient)_

[[!redirects group cocycle]]
[[!redirects group cocycles]]


group commutator

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#### Group Theory
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#Contents#
* table of contents
{:toc}

## Definition

For $g,h \in G$ two elements of a [[group]], their **group commutator** is defined as

$$
  [g,h] \coloneqq g^{-1} h^{-1} g h
  \,.
$$

This definition Is also valid for [[invertible semigroups]].

## Related concepts

* [[commutator subgroup]]

* [[commutator]]

* [[Lie bracket]]

[[!redirects group commutators]]