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group cohomology

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Idea

The group cohomology of a group GG is the cohomology of its delooping BG\mathbf{B}G. This cohomology classifies group extensions of GG.

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

More in detail, if AA is any abelian group then a cocycle in GG-group cohomology with coefficients in AA regarded as equipped with the trivial action is a morphism

c:BGB nA c \colon \mathbf{B}G \to \mathbf{B}^n A

and the cohomology group is the homotopy equivalence classes of this

H Grp n(G,A)π 0H(BG,B nA). H^n_{Grp}(G,A) \simeq \pi_0 \mathbf{H}(\mathbf{B}G,\mathbf{B}^n A) \,.

More generally, AA here may be equipped with a GG-action ρ:A×GA\rho \colon A \times G \to A. There is the the corresponding action groupoid or associated ∞-bundle B nAGBG\mathbf{B}^n A\sslash G \to \mathbf{B}G and now a cocycle is a morphism c:BGB nAG\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^n A\sslash G fitting into a diagram

BG c B nAG BG. \array{ \mathbf{B}G && \stackrel{\mathbf{c}}{\to} && \mathbf{B}^n A \sslash G \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G } \,.

Equivalently this means that the group cohomology of GG with coefficients in an abelian group AA with GG-action ρ\rho is the twisted cohomology of the delooping BG\mathbf{B}G with respect to the local coefficient ∞-bundle B nAG\mathbf{B}^n A \sslash G.

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

BG c VG BG \array{ \mathbf{B}G && \stackrel{\mathbf{c}}{\to} && V \sslash G \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G }

hence a cocycle in the twisted cohomology of BG\mathbf{B}G with coefficients in the local coefficient ∞-bundle given by the universal ρ\rho-associated VV-fiber ∞-bundle.

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

This general definition we discuss below in

The special case where V=B nAV = \mathbf{B}^n A is the nn-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

Finally one can break this further down into components In

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 H\mathbf{H} be an (∞,1)-topos. Let GGrp(H)G \in Grp(\mathbf{H}) be a group object, an ∞-group, in H\mathbf{H}. Write BGH\mathbf{B}G \in \mathbf{H} for its delooping.

An ∞-action ρ:V×GV\rho : V \times G \to V of GG on a VHV \in \mathbf{H} is equivalently, as discussed there, exhibited by a fiber sequence

V VG ρ¯ BG. \array{ V &\to& V \sslash G \\ && \downarrow^{\mathrlap{\bar \rho}} \\ && \mathbf{B}G } \,.

Regarded as an object in the slice (∞,1)-topos H /BG\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:BGV(x):Type. x \colon \mathbf{B}G \;\vdash \; V(x) \colon Type \,.

Also, ρ¯\bar \rho is the local coefficient bundle for GG-group cohomology with coefficients in VV equipped with this GG-∞-action. this means that the group cohomology of GG with coefficients in VV is the hom in the slice (∞,1)-topos over H\mathbf{H} as base (∞,1)-topos

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

where we denote on the right by BG\mathbf{B}G the terminal object in the slice H /BG\mathbf{H}_{/\mathbf{B}G}. Notice that in H\mathbf{H} this is the trivial fiber sequence

* BG id BG \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

( x:BG(*V)):Type. \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 GG on the morphisms *V* \to V of the underlying objects. Since the action on the point is trivial, these are just the ∞-invariants? of VV.

Proposition

In the special case that the GG-∞-action on VV is trivial, the group cohomology is equivalently just the set of connected components of the hom space

H Grp(G,V)π 0H(BG,V). H_{Grp}(G,V) \simeq \pi_0 \mathbf{H}(\mathbf{B}G, V) \,.

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

H Grp n(G,A)H Grp(G,B nA)π 0H(BG,B nA). 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:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}H
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

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

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

GRing \mathbb{Z}G \in Ring

for the group algebra of GG over the integers. Write

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

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

Notice that a module

AGMod A \in \mathbb{Z}G Mod

is equivalently an abelian group equipped with a GG-action. This or rather its nn-fold suspension as a chain complex

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

is the kind of coefficient for the group cohomology of GG to which the following statement applies.

Remark

For AA a GG-module, the invariants? of AA are equivalently the [G]\mathbb{Z}[G]-module homomorphisms from \mathbb{Z} equipped with the trivial module structure

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

By the fully general discussion above, group cohomology of GG with coefficients in some AA is the homotopy-version of the GG-invariants? of AA. In the context of homological algebra and in view of remark 1, this means that it is given by the derived functor of the hom functor out of the trivial GG-module, hence by the Ext-functor:

Definition

For AA an abelian group equipped with a GG-action, the degree-nn group cohomology of GG with coefficients in AA is the nnth-Ext-group

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

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

Remark

By the discussion at projective resolution this means more explicitly the following: let F qiF_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z} be a projective resolution Ch ([G]Mod)Ch_\bullet(\mathbb{Z}[G]Mod) of \mathbb{Z} equipped with the trivial GG-action, hence an exact sequence

F 3F 2F 1F 00 \cdots \to F_3 \to F_2 \to F_1 \to F_0 \to \mathbb{Z} \to 0

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

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

be the corresponding cochain complex. Then the degree-nn group cohomology of GG with coefficient in AA is the degree-nn cochain cohomology of this complex

H Grp n(G,A)H n(Hom [G]Mod(F 0,A))ker(d n)/im(d n1). H^n_{Grp}(G,A) \simeq H^n(Hom_{\mathbb{Z}[G]Mod}(F_0, A)) \coloneqq ker(d^n)/im(d^{n-1}) \,.
Remark

Give a normal subgroup KGK \hookrightarrow G the invariants?-functor may be decomposed as a composition of the functor that forms KK-invariants with that which forms (G/K)(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

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 GG be a discrete group and AA an abelian discrete group, regarded as equipped with the trivial GG-action. Let nn \in \mathbb{N}.

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

Proposition

Then the degree-nn group cohomology of GG with coefficients in AA is the set

H Grp n(G,A)π 0sSet(W¯G,DK(A[n])) 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. 1 the group cohomology is

H Grp n(G,A)π 0H(BG,B nA). 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 H=\mathbf{H} = ∞Grpd, which for emphasis we might write “Disc∞Grpd”. This is presented by the standard model structure on simplicial sets, DiscGrpdL whesSetDisc\infty Grpd \simeq L_{whe} sSet.

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

W¯G=(G×G×GG×GG*), \overline{W}G = \left( \cdots \to G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \times G \stackrel{\to}{\to} G \to {*} \right) \,,

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

Moreover, by the discussion at Dold-Kan correspondence a presentation of the Eilenberg-MacLane object B nA\mathbf{B}^n A is DK(A[n])sSetDK(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(W¯G,DK(A[n]))sSet(\overline{W}G, DK(A[n])) is a Kan complex which presents the required hom-\infty-groupoid.

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

Degree-11 group cohomology

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

Degree-22 group cohomology

We discuss here in detail and in components the special case of degree-2 group cohomology of a discrete group GG with coefficients in AA an abelian discrete group and regarded as being equipped with the trivial GG-action.

Proposition

Let GG be a discrete group and AA an abelian discrete group, regarded as being equipped with the trivial GG-action.

Then a group 2-cocycle on GG with coefficients in AA is a function

c:G×GA c \colon G \times G \to A

such that for all (g 1,g 2,g 3)G×G×G(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 2g 3)+c(g 1g 2,g 3)c(g 2,g 3)=0A 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,c˜c, \tilde c two such cocycles, a coboundary h:cc˜h \colon c \to \tilde c between them is a function

h:GA h \colon G \to A

such that for all (g 1,g 2)G×G(g_1,g_2) \in G \times G the equation

(2)c˜(g 1,g 2)=c(g 1,g 2)+(dh)(g 1,g 2) \tilde c(g_1,g_2) = c(g_1,g_2) + (d h)(g_1,g_2)

holds in AA, where

(dh)(g 1,g 2)h(g 1g 2)h(g 1)h(g 2) (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 hh.

The degree-2 group cohomology is the set

H Grp 2(G,A)=2Cocycles(G,A)/Coboundaries(G,A) 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 AA

[c 1]+[c 2]=[c 1+c 2] [c_1] + [c_2] = [c_1 + c_2]

where

c 1+c 2:(g 1,g 2)c 1(g 1,g 2)+c 2(g 1,g 2). 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. 2.

Proof

By prop. 2 we have H Grp 2(G,A)π 0sSet(W¯G,DK(A[2]))H^2_{Grp}(G,A) \simeq \pi_0 sSet(\overline{W}G, DK(A[2])).

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

(W¯G) 2={ * g 1 g 2 * g 1g 2 *|g 1,g 2G}, (\overline{W}G)_2 = \left\{ \left. \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_1 g_2}{\to}&& {*} } \right| g_1, g_2 \in G \right\} \,,

and the 3-simplices are

(W¯G) 3={* g 2 * g 1 g 1g 2 g 3 * g 1g 2g 3 ** g 2 * g 1 g 2g 3 g 3 * g 1g 2g 3 *|g 1,g 2,g 3G}. (\overline{W}G)_3 = \left\{ \left. \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&{}^{g_1 g_2}\nearrow&& \downarrow^{g_3} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& {*} } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&\searrow^{g_2 g_3}&& \downarrow^{g_3} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& {*} } \right| g_1, g_2, g_3 \in G \right\} \,.

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

c 2:( * g 1 g 2 * g 2g 1 *)( * * c(g 1,g 2) * * * *) 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×GKc : G \times G \to K. To be a simplicial homomorphism this has to extend to 3-simplices as:

c 3 :(* g 2 * g 1 g 2g 1 g 3 * g 3g 2g 1 ** g 2 * g 1 g 3g 2 g 3 * g 3g 2g 1 *) (* * * * c(g 1,g 2) * c(g 2,g 3) * * * ** * * * c(g 1,g 2g 3) * c(g 2,g 3) * * * *). \begin{aligned} c_3 \;\;\; &: \;\;\; \left( \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3} \\ {*} &&\stackrel{g_3 g_2 g_1}{\to}&&{*} } \;\;\;\; \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{{*}}{\to}&& {*} \\ \uparrow^{{*}} &\Downarrow^{c(g_1,g_2)} &{}^{{*}}\nearrow&\Downarrow^{c(g_2,g_3)}& \downarrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&&{*} } \;\;\;\; \stackrel{}{\Rightarrow} \;\;\;\; \array{ {*} &&\stackrel{{*}}{\to}&& {*} \\ \uparrow^{{*}} &\Downarrow^{c(g_1,g_2 g_3)} &\searrow^{{*}}&\Downarrow^{c(g_2, g_3)}& \downarrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&&{*} } \right) \end{aligned} \,.

Since there is a unique 3-cell in DK(A[2])DK(A[2]) whenever the oriented su, of the AA-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 Grp 2(G,A)H^2_{Grp}(G,A) one may concentrate on the normalized cocycles.

Definition

A group 2-cocycle c:G×GAc \colon G \times G \to A, def. 3 is called normalized if

g 0,g 1G(g 0=eorg 1=e)(c(g 0,g 1)=e). \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:G×GAc \colon G \times G \to A a group 2-cocycle, we have for all gGg \in G that

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

The cocycle condition (1) evaluated on

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

says that

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

hence that

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

Similarly the 2-cocycle condition applied to

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

says that

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

hence that

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

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

Proof

By lemma 1 it is sufficient to show that cc is cohomologous to a cocycle c˜\tilde c satisfying c˜(e,e)=e\tilde c(e,e) = e. Now given cc, Let h:GAh \colon G \to A be given by

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

Then c˜c+dc\tilde c \coloneqq c + d c has the desired property, with (2):

c˜(e,e) (c+dh)(e,e) =c(e,e)+c(ee,ee)c(e,e)c(e,e) =0. \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} \,.

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 GG is defined in direct generalization of the formulas above as homotopy classes of morphisms from the simplicial object

(G×GG*) \left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right)

to a simplicial object N(B nA)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 W¯G=G × \overline{W}G = G^{\times^\bullet} is cofibrant in the relevant model category presenting the ambient (∞,1)-topos Disc∞Grpd, for GG 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 GG, the object

(G×GG*) \left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \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

See below at References - For structured groups for pointers to the literature.

Nonabelian group cohomology

If the coefficient group KK is nonabelian, its higher deloopings B nK\mathbf{B}^n K to not exist. But n-groupoids approximating this non-existant delooping do exists. Cohomology of BG\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

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

For group cohomology of the group of order 2 2=/2\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z} see at Groupprops, Group cohomology of cyclic group Z2

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

We consider for GG a topological group such as

the corresponding group cohomology in terms of the cohomology of the classifying space/delooping BGB G.

For all nn \in \mathbb{N} we have

H (BU(n);) =[c 1,,c n] H (BSU(n);) =[c 2,,c n] H (BSp(n);) =[p 1,,p n] \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 iH 2ic_i\in H^{2i} and p iH 4ip_i\in H^{4i}.

Heisenberg cocycle

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

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

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

ω:((a 1,b 1),(a 2,b 2))a 1b 2. \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(b 2+b 3)+a 2b 3=a 1b 2+(a 1+a 2)b 3 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

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.

References

General

Standard textbook references on group cohomology include

  • Adem, Milgram, Cohomology of finite groups

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

  • Kenneth Brown, Cohomology of Groups , Graduate Texts in Mathematics, 87 (1972)

An introduction to group cohomology of a group GG as the cohomology of the classifying space BGB G is for instance in

  • 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)AUT(H) of some possibly nonabelian group HH.

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

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

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)

nn-cocycles on a topological group GG with values in a topological abelian group AA are considered as continuous maps G ×nAG^{\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:

On various topological groups

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

Compact Lie groups

Cohomology of the classifying space BGB G for GG 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)U(n)

For UU the unitary group, the integral cohomology of the classifying space BU(n)B U(n) consists of the Chern classes, one in every even degree.

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

The cohomology of BO(n)B O(n) (orthogonal group) and BSO(n)B SO(n) (special orthogonal group) with coefficients in 2\mathbb{Z}_2 is discussed in (MilnorStasheff, 1974).

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

  • Emery Thomas , On the cohomology of the real Grassman complexes and the characteristic classes of the nn-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 BSO(n)B SO(n) and BO(n)B O(n) with integer coefficients Proc. AMS Soc. 85 (1982)

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

For local coefficients see

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

  • Richard Lastovecki, Cohomology of BO(n 1)××BO(n m)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 BSpinB 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)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 BStringB 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 6E_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

Revised on December 20, 2014 15:29:10 by David Corfield (127.0.0.1)