nLab group cohomology

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 $V$-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.

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.

Finally one can break this further down into components In

• Special case: Explicit formulas in low degree

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

Definition

Fully general: in homotopy type theory

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, 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 ∞-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 ∞-invariants of $V$.

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 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

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

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$	coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$	trivial representation
dependent product along $\mathbf{B}G \to \ast$	homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum with G-action (naive G-spectrum)

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

Let $G$ be an ordinary group, specifically a group object in a topos $\mathcal{T}$ such that the abelian category $Ab(\mathcal{T})$ has 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 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.

Remark

For $A$ a $G$-module, the 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 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, 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 , 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:

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.

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

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

Remark

Give a normal subgroup $K \hookrightarrow G$ the 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 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

• Structured group cohomology

• Nonabelian group cohomology

Simplicial constructions and explicit formulas in low degree

We unwind the general abstract definition of group cohomology above in terms of constructions on simplicial sets (for cohomology of discrete groups) and simplicial presheaves (for cohomology of general group objects).

$\,$

Let $G$ be a discrete group and $A$ an abelian 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 $A$.

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

By prop. 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 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, 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

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

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 discrete group and regarded as being equipped with the trivial $G$-action.

Proposition

Let $G$ be a discrete group and $A$ an 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

(1)$$c(g_1, g_2) - c(g_1, g_2 \cdot g_3) + c(g_1 \cdot g_2, g_3) - c(g_2, g_3) = 0 \;\;\;\; \in A$$

(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

(2)$$\tilde c(g_1,g_2) = c(g_1,g_2) + (d h)(g_1,g_2)$$

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. .

Proof

By prop. 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.

Definition

A group 2-cocycle $c \colon G \times G \to A$, def. is called normalized if

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

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

The cocycle condition (1) 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

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

Proposition

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

Proof

By lemma 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 (2):

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

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)

If the groups in question are not plain groups (group objects internal to Set) but groups with extra structure, such as topological groups or Lie groups, 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 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 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-spaces. 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 for pointers to the literature.

Nonabelian group cohomology

If the coefficient group $K$ is nonabelian, its higher deloopings $\mathbf{B}^n K$ do not exist. But n-groupoids 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

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))). 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 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]

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]

• Samuel Eilenberg, Saunders MacLane, Relations Between Homology and Homotopy Groups of Spaces, The Annals of Mathematics 46 3 (1945) 480 [doi: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]

• 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]

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)

• Alejandro Adem, R. James Milgram, Cohomology of Finite Groups, Springer 2004

specifically for profinite groups:

• 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, webpage)

specifically for algebraic groups:

• James Milne, Section 15 (16 in the pdf) of: Algebraic Groups, Cambridge University Press 2017 (doi:10.1017/9781316711736, webpage, 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]

Further exposition:

• Alejandro Adem, Lectures on the cohomology of finite groups (pdf)

• Narthana Epa, Platonic 2-groups, 2010 (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)

• Advanced course on classifying spaces and cohomology of groups (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, web)

Cohomology of simplicial groups is discussed for instance in

• Sebastian 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)

For a conceptual discussion of nonabelian group cohomology see

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology (arXiv)

Group cocycles classify group extensions. This is often discussed only for 2-cocycles and extensions by ordinary groups. Higher cocycles classify extensions by 2-groups 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)

On structured group cohomology

In

• Jim Stasheff, Continuous cohomology of groups and classifying spaces Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 (web)

$n$-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 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)

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)

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:

• 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,publisher page)

On various topological groups

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

Compact Lie groups

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

• John Milnor, Jim Stasheff, Characteristic Classes , Princeton University Press and University of Tokyo Press, Princeton, New Jersey, (1974).

• 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)

• Donald Yau, Cohomology of unitary and symplectic groups (pdf)

• D. Benson, John Greenlees, Commutative algebra for cohomology rings of classifying spaces of compact Lie groups (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 classes, one in every even degree.

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

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

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 local coefficients see

• Martin Čadek, The cohomology of $B O(n)$ with twisted integer coefficients, J. Math. Kyoto Univ. 39 (1999), no. 2, 277–286 (Euclid)

• 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)

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)

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)

The exceptional Lie groups

Cohomology of classifying spaces of exceptional Lie groups.

• Akira Kono, Mamoru Mimura, Cohomology mod 3 of the classifying space of the Lie group $E_6$ , Math. Scand. 46 (1980) (pdf)

Loop groups of compact Lie groups

• Daisuke Kishimoto, Akira Kono, Cohomology of the classifying spaces of loop groups (pdf)

Online references

Some of the above materiel is taken from discussion at

• MO, Group cohomology of compact Lie group with integer coeffient