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nonabelian group cohomology

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This entry largely discusses Schreier theory of nonabelian group extensions – but from the nPOV.

Contents

Idea and Definition

As group cohomology of a group GG is the cohomology of its delooping BG\mathbf{B}G, so nonabelian group cohomology is the corresponding nonabelian cohomology.

By the general abstract definition of cohomology, the abelian group cohomology in degree kk \in \mathbb{N} of a group GG with coefficients in an abelian group KK is the set of equivalence classes of morphisms

H n(G,K)={BGB nK} H^n(G,K) = \{ \mathbf{B}G \to \mathbf{B}^n K \}_\sim

in the (∞,1)-category ∞Grpd, from the delooping BG\mathbf{B}G of GG to the nn-fold delooping B nK\mathbf{B}^n K of KK.

However, if the group KK is not abelian, then its nn-fold delooping does not exist for n2n \geq 2, so accordingly the above does not give a prescription for cohomology of GG with coefficients in a nonabelian group KK in degree greater than 1 (and in degree 1 group cohomology it is not very interesting).

But for nonabelian KK there are higher groupoids that approximate the non-existing higher deloopings. Nonabelian group cohomology is the cohomology of BG\mathbf{B}G with coefficients in such approximations.

More precisely, notice that for n=2n=2 and KK abelian, the nn-fold delooping B 2K\mathbf{B}^2 K is the strict 2-groupoid whose corresponding crossed complex is

[B 2K]=(K**). [\mathbf{B}^2 K] = \left( K \to {*} \stackrel{\to}{\to} {*} \right) \,.

But for every group KK there is also its automorphism 2-group AUT(K)AUT(K). Its delooping corresponds to the crossed complex

[BAUT(K)]=(Kδ=AdAut(K)*), [\mathbf{B} AUT(K)] = \left( K \stackrel{\delta = Ad}{\to} Aut(K) \stackrel{\to}{\to} {*} \right) \,,

where the boundary map δ\delta is the one that sends an element kKk \in K to the automorphism Ad(k):kkkk 1Ad(k) : k' \mapsto k k' k^{-1}.

So this looks much like B 2K\mathbf{B}^2 K (when that exists) only that it has more elements in degree 1.

Accordingly, what is called nonabelian group cohomology of GG with coefficients in KK is the set of equivalence classes of morphisms

H nonab 2(G,K):={BGBAUT(K)} . H^2_{nonab}(G,K) := \{ \mathbf{B}G \to \mathbf{B}AUT(K) \}_\sim \,.

Notice that when KK has nontrivial automorphisms, this differs in general from the ordinary degree 2 abelian group cohomology even if KK is abelian.

It is a familiar fact that abelian group cohomology classifies (shifted) central group extensions. This is really nothing but the statement that to a morphism BGB nK\mathbf{B}G \to \mathbf{B}^n K we may associate its fibration sequence

B n1K BG^ * * BG B nK \array{ \mathbf{B}^{n-1} K& \to&\mathbf{B}\hat G &\to& {*} \\ \downarrow &&\downarrow && \downarrow \\ {*}& \to& \mathbf{B}G &\to& \mathbf{B}^n K }

(where both squares are homotopy pullback squares). In particular, for n=2n = 2 we get ordinary central extensions

BKBG^BB. \mathbf{B}K \to \mathbf{B}\hat G \to \mathbf{B}B \,.

which may be looped to yield exact sequences of morphisms of groups

KG^B. K \to \hat G \to B \,.

In Schreier theory one notices that similarly nonabelian group cohomology in degree 2 classifies nonabelian group extensions, i.e. sequences

KG^G. K \to \hat G \to G \,.

As we shall discuss below, by following the abstract nonsense as described above, nonabelian degree 2 cocycles really classify something slightly richer, namely exact sequences of groupoids

Aut(K)//KAut(K)//G^*//G, Aut(K)//K \to Aut(K)//\hat G \to {*}//G \,,

where the double slashes denote action groupoids (and *//G=BG{*}//G = \mathbf{B}G).

In the existing literature – which does not usually present the picture quite in the way we are doing here – nonabelian group cohomology is rarely considered beyond degree 2. But the picture does straightforwardly generalize. For instance degree 3 nonabelian cohomology of GG with coefficients in KK may be taken to be the cohomology of BG\mathbf{B}G with coefficients in the 3-groupoid BAUT(AUT(K))\mathbf{B}AUT(AUT(K)).

H nonab 3(G,K)={BGBAUT(AUT(K))} . H^3_{nonab}(G,K) = \{\mathbf{B}G \to \mathbf{B}AUT(AUT(K))\}_\sim \,.

And so on.

Details

We work out in detail what nonabelian group cocycles, such as morphisms

BGBAUT(K) \mathbf{B}G \to \mathbf{B}AUT(K)

correspond to in terms of claassical group data, using the relation between strict 2-groups and crossed modules that is spelled out in detail at strict 2-group – in terms of crossed modules.

For making the translation we follow the convention LB there.

Degree 2 cocycles

Proposition

Degree 2 cocycles of nonabelian group cohomology on GG with coefficients in KK are given by the following data:

  • a map ψ:GAut(K)\psi : G \to Aut(K);

  • a map χ:G×GK\chi : G \times G \to K

  • subject to the constraint that for all g 1,g 2Gg_1, g_2 \in G we have

    ψ(g 1g 2)=Ad(χ(g 1,g 2))ψ(g 2)ψ(g 1). \psi(g_1 g_2) = Ad(\chi(g_1, g_2)) \psi(g_2) \psi(g_1) \,.
  • and subject to the cocycle condition that for all g 1,g 2,g 3Gg_1, g_2, g_3 \in G we have

    χ(g 1g 2,g 3)ψ(g 3)(ξ(g 1,g 2))=χ(g 1,g 2g 3)χ(g 2,g 3) \chi(g_1 g_2, g_3) \psi(g_3)(\xi(g_1,g_2)) = \chi(g_1, g_2 g_3) \chi(g_2, g_3)
Proof

Use the identification of BAUT(K)\mathbf{B}AUT(K) with its crossed module (AAdAut(K))(A \stackrel{Ad}{\to} Aut(K)) in the convention L B as described at strict 2-group – in terms of crossed modules to translate the relevant diagrams – which are of the sort spelled out in great detail at group cohomology: the first three items of the above describe the maps

(ψ,χ):( g 1 = g 2 g 1g 2 )( ψ(g 1) χ(g 1,g 2) ψ(g 2) ψ(g 1g 2) ). (\psi, \chi) : \left( \array{ && \bullet \\ & {}^{\mathllap{g_1}}\nearrow & \Downarrow^{\mathrlap{=}}& \searrow^{\mathrlap{g_2}} \\ \bullet &&\stackrel{g_1 g_2}{\to} && \bullet } \right) \;\;\; \mapsto \;\;\; \left( \array{ && \bullet \\ & {}^{\mathllap{\psi(g_1)}}\nearrow & \Downarrow^{\mathrlap{\chi(g_1,g_2)}}& \searrow^{\mathrlap{\psi(g_2)}} \\ \bullet &&\stackrel{\psi(g_1 g_2)}{\to} && \bullet } \right) \,.

The cocycle condition is the fact that this assignment has to make all tetrahedras commute (since there are only trivial k-morphisms with k3k \geq 3 in BAUT(K)\mathbf{B}AUT(K)):

ψ(g 2) χ(g 1,g 2) ψ(g 1) ψ(g 1g 2) ψ(g 3) χ(g 1g 2,g 2) ψ(g 3) = ψ(g 2) χ(g 2,g 3) ψ(g 1) ψ(g 2g 3) ψ(g 3) χ(g 1,g 2g 3) ψ(g 3) \array{ \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet \\ \uparrow & \Downarrow{}^{\mathrlap{\chi(g_1, g_2)}} &&& \downarrow \\ {}^{\mathllap{\psi(g_1)}}\uparrow &&{}^{\mathllap{\psi(g_1 g_2)}}\nearrow&& \downarrow^{\mathrlap{\psi(g_3)}} \\ \uparrow &&& {}^{\mathllap{\chi(g_1 g_2, g_2)}}\Downarrow & \downarrow \\ \bullet &&\stackrel{\psi(g_3)}{\to}&& \bullet } \;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\; \array{ \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet \\ \uparrow &&& {}^{\mathllap{\chi(g_2, g_3)}} \Downarrow & \downarrow \\ {}^{\mathllap{\psi(g_1)}}\uparrow &&\searrow^{\mathrlap{\psi(g_2 g_3)}}&& \downarrow^{\mathrlap{\psi(g_3)}} \\ \uparrow & \Downarrow{}^{\mathrlap{\chi(g_1 , g_2 g_3)}} &&& \downarrow \\ \bullet &&\stackrel{\psi(g_3)}{\to}&& \bullet }
Remark

Precisely the same kind of “twisted” cocycles appear as the cocycles of nonabelian gerbes and principal 2-bundles: for a KK-gerbe these are cocycles with coefficients in BAUT(K)\mathbf{B}AUT(K) but on a domain that is the discrete groupoid given by the given base space.

Extensions classified by degree 2-cocycles

The following statement is classically the central statement of Schreier theory. We state and prove it in the abstract nonsense context of general cohomology, where the things classified by a cocycle are nothing but its homotopy fibers.

Proposition

Cohomology classes of nonabelian 2-cocycles (ψ,χ):BGBAUT(K)(\psi, \chi) : \mathbf{B}G \to \mathbf{B}AUT(K) are in bijection with equivalence classes of extensions

KG^G K \to \hat G \to G
Proof

In fact, we claim a bit more: we claim that the fibration sequence to the left defined by the cocycle (ψ,χ):BGBAUT(K)(\psi, \chi) : \mathbf{B}G \to \mathbf{B}AUT(K) is

Aut(K)Aut(K)//KAut(K)//G^BG(ψ,ξ)BAUT(K), \cdots \to Aut(K) \to Aut(K)//K \to Aut(K)//\hat G \to \mathbf{B}G \stackrel{(\psi,\xi)}{\to} \mathbf{B}AUT(K) \,,

where

G^:=K× (ψ,χ)G \hat G := K \times_{(\psi,\chi)} G

is the twisted product of KK with GG, using the maps χ\chi and ψ\psi, i.e. the group whose underlying set is the cartesian product K×GK \times G with multiplication given by

(k 1,g 1)(k 2,g 2)=(χ(g 1,g 2)ψ(g 2)(k 1)k 2,g 1g 2). (k_1, g_1) (k_2, g_2) = \left( \chi(g_1,g_2) \psi(g_2)(k_1) k_2 \;\; , \;\; g_1 g_2 \right) \,.

To see this, we compute the homotopy pullback

Aut(K)//G^ * BG (ψ,χ) BAUT(K) \array{ Aut(K)//\hat G & \to & {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}& \mathbf{B}AUT(K) }

as the ordinary pullback

Aut(K)//G^ EAUT(K) BG (ψ,χ) BAUT(K) \array{ Aut(K)//\hat G & \to & \mathbf{E}AUT(K) \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}& \mathbf{B}AUT(K) }

as described at generalized universal bundle. (EAUT(K)\mathbf{E}AUT(K) is the universal AUT(K)AUT(K)-principal 2-bundle).

Recall from the discussion there that a morphism in AUT(K)\mathbf{AUT}(K) is a triangle

α k β γ \array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow &{}^\mathrlap{k}\swArrow& \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{\gamma}{\to}&& \bullet }

in BAUT(K)\mathbf{B}AUT(K), and composition of morphisms is pasting of these triangles along their vertical edges. 2-morphisms in EAUT(K)\mathbf{E}AUT(K) are given by paper-cup pasting diagrams of such triangles in BAUT(K)\mathbf{B}AUT(K)

Accordingly, the pullback BG× (ψ,ξ)EAUT(K)\mathbf{B}G \times_{(\psi,\xi)} \mathbf{E}AUT(K) has

  • objects are elements of Aut(K)Aut(K) (this is the bit not seen in the classical picture of Schreier theory, as that doesn’t know about groupoids);

  • morphisms are pairs

    (k,g):=( α k β AUT(K) ψ(g) g BG) (k,g) \;\;\; := \left( \array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow &{}^\mathrlap{k}\swArrow& \searrow^{\mathrlap{\beta}} &&&&& \in \mathbf{AUT(K)} \\ \bullet &&\stackrel{\psi(g)}{\to}&& \bullet \\ \\ \bullet &&\stackrel{g}{\to}&& \bullet &&&& \in \mathbf{B}G } \right)
  • 2-morphisms (thought of as 2-simplexes) take two such triangles (k 1,g 1)(k_1, g_1) and (k 2,g 2)(k_2, g_2) to the pair (k,g 1,g 2)(k', g_1, g_2), where kk' is given by the pasting diagram

    k 1 k 2 χ(g 1,g 2) . \array{ && \bullet \\ &\swarrow& \downarrow & \searrow \\ \downarrow &\Downarrow^{\mathrlap{k_1}}& \bullet &{}^{\mathllap{k_2}}\Downarrow& \downarrow \\ \downarrow & \nearrow &\Downarrow^{\mathrlap{\chi(g_1, g_2)}} & \searrow & \downarrow \\ \bullet && \stackrel{}{\to} && \bullet } \,.

Translating these diagrams into forumas using the convention LB as described at strict 2-group – in terms of crossed modules yields the given formulas.

Homotopies between 2-cocycles

Given two 2-cocycles

(ψ 1,χ 1),(ψ 2,χ 2):BGBAUT(K) (\psi_1, \chi_1), (\psi_2, \chi_2) : \mathbf{B}G \to \mathbf{B}AUT(K)

a homotopy (coboundary) between them is a transformation

λ:(ψ 1,χ 1)(ψ 2,χ 2). \lambda : (\psi_1, \chi_1) \Rightarrow (\psi_2, \chi_2) \,.

Its components

λ:(g)( ψ 1(g) λ() λ(g) λ() ψ 2(g) ) \lambda : (\bullet \stackrel{g}{\to} \bullet) \;\; \mapsto \;\; \left( \array{ \bullet &\stackrel{\psi_1(g)}{\to}& \bullet \\ {}^{\mathllap{\lambda(\bullet)}} \downarrow &{}^{\mathllap{\lambda(g)}}\swArrow& \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &\stackrel{\psi_2(g)}{\to}& \bullet } \right)

are given in terms of group elements by

  • λ()Aut(K)\lambda(\bullet) \in Aut(K)

  • {λ(g)K|gG}\{\lambda(g) \in K | g \in G\}

such that

(1)λ()ψ 1(g)=Ad(λ(g))ψ 2(g)λ(). \lambda(\bullet) \psi_1(g) = Ad(\lambda(g)) \psi_2(g) \lambda(\bullet) \,.

The naturality condition on this datat is that for all g 1,g 2Gg_1, g_2 \in G we have

ψ 1(g 1) χ 1(g 1,g 2) ψ 1(g 2) ψ 1(g 2g 1) λ() λ(g 1,g 2) λ() ψ 2(g 2g 1) = ψ 1(g 1) ψ 1(g 1) λ(g 2) λ() λ(g 2) λ() ψ 2(g 1) χ 2(g 1,g 2) ψ 2(g 2) λ() ψ 2(g2g 1) \array{ && \bullet \\ & {}^{\mathllap{\psi_1(g_1)}}\nearrow &\Downarrow^{\chi_1(g_1,g_2)}& \searrow^{\mathrlap{\psi_1(g_2)}} \\ \bullet &&\stackrel{\psi_1(g_2 g_1)}{\to}&& \bullet \\ {}^{\mathllap{\lambda(\bullet)}}\downarrow && {}^{\mathllap{\lambda(g_1, g_2)}}\swArrow && \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &&\stackrel{\psi_2(g_2 g_1)}{\to}&& \bullet } \;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\; \array{ & {}^{\mathllap{\psi_1(g_1)}}\nearrow &\downarrow& \searrow^{\mathrlap{\psi_1(g_1)}} \\ \downarrow &{}^{\lambda(g_2)}\swArrow & \downarrow^{\mathrlap{\lambda(\bullet)}} &{}^{\lambda(g_2)}\swArrow& \downarrow \\ {}^{\mathllap{\lambda(\bullet)}}\downarrow & {}^{\mathllap{\psi_2(g_1)}} \nearrow & \Downarrow^{\chi_2(g_1,g_2)} & \searrow^{\mathrlap{\psi_2(g_2)}} & \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &&\stackrel{\psi_2(g2 g_1)}{\to}&& \bullet }

In terms of the conventionl LB at strict 2-group – in terms of crossed modules, this is equivalent to the equation

(2)λ(g 2g 1)ρ(λ())(χ 1(g 1,g 2))=χ 2(g 1,g 2)ρ(ψ 1(g 2))(λ(g 2))λ(g 2). \lambda(g_2 g_1) \; \rho(\lambda(\bullet))(\chi_1(g_1,g_2)) = \chi_2(g_1, g_2) \; \rho(\psi_1(g_2))(\lambda(g_2)) \; \lambda(g_2) \,.

Compare this to the discussion of 2-coboundaries of extensions at group extension.

Nonabelian Lie algebra cohomology

When the groups in question are Lie groups, there is an infinitesimal version of nonabelian group cohomology:

See there for details.

Revised on October 4, 2012 13:08:58 by Tim Porter (95.147.236.151)