Group cohomology

In

M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of ${\displaystyle G^{n}}$ representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups ${\displaystyle H^{n}(G,M)}$. The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the coefficients.

These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the

. Thus, the group cohomology of ${\displaystyle \mathbb {Z} }$ can be thought of as the singular cohomology of the circle S¹, and similarly for ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ and ${\displaystyle \mathbb {P} ^{\infty }(\mathbb {R} ).}$

A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.

Motivation

A general paradigm in

of M. We will write G multiplicatively and M additively.

Given such a G-module M, it is natural to consider the submodule of

G-invariant

elements:

${\displaystyle M^{G}=\lbrace x\in M\ |\ \forall g\in G:\ gx=x\rbrace .}$

Now, if N is a G-submodule of M (i.e., a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in ${\displaystyle M/N}$ are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The purpose of the first group cohomology ${\displaystyle H^{1}(G,N)}$ is to precisely measure this difference.

The group cohomology functors ${\displaystyle H^{*}}$ in general measure the extent to which taking invariants doesn't respect

long exact sequence

.

Definitions

The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property ${\displaystyle f(gx)=g(f(x))}$ for all g in G and x in M). Sending each module M to the group of invariants ${\displaystyle M^{G}}$ yields a

Their values are abelian groups and they are denoted by ${\displaystyle H^{n}(G,M)}$, "the n-th cohomology group of G with coefficients in M". Furthermore, the group ${\displaystyle H^{0}(G,M)}$ may be identified with ${\displaystyle M^{G}}$.

Cochain complexes

The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful.[1] For ${\displaystyle n\geq 0,}$ let ${\displaystyle C^{n}(G,M)}$ be the group of all functions from ${\displaystyle G^{n}}$ to M (here ${\displaystyle G^{0}}$ means ${\displaystyle \operatorname {id} _{G}}$). This is an abelian group; its elements are called the (inhomogeneous) n-cochains. The coboundary homomorphisms are defined by

${\displaystyle {\begin{cases}d^{n+1}\colon C^{n}(G,M)\to C^{n+1}(G,M)\\\left(d^{n+1}\varphi \right)(g_{1},\ldots ,g_{n+1})=g_{1}\varphi (g_{2},\dots ,g_{n+1})+\sum _{i=1}^{n}(-1)^{i}\varphi \left(g_{1},\ldots ,g_{i-1},g_{i}g_{i+1},\ldots ,g_{n+1}\right)+(-1)^{n+1}\varphi (g_{1},\ldots ,g_{n}).\end{cases}}}$

One may check that ${\displaystyle d^{n+1}\circ d^{n}=0,}$ so this defines a

whose cohomology can be computed. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex

${\displaystyle H^{n}(G,M)=Z^{n}(G,M)/B^{n}(G,M).}$

Here the groups of n-cocycles, and n-coboundaries, respectively, are defined as

${\displaystyle Z^{n}(G,M)=\ker(d^{n+1})}$

${\displaystyle B^{n}(G,M)={\begin{cases}0&n=0\\\operatorname {im} (d^{n})&n\geqslant 1\end{cases}}}$

The functors Extⁿ and formal definition of group cohomology

Interpreting G-modules as modules over the group ring ${\displaystyle \mathbb {Z} [G],}$ one can note that

${\displaystyle H^{0}(G,M)=M^{G}=\operatorname {Hom} _{\mathbb {Z} [G]}(\mathbb {Z} ,M),}$

i.e., the subgroup of G-invariant elements in M is identified with the group of homomorphisms from ${\displaystyle \mathbb {Z} }$, which is treated as the trivial G-module (every element of G acts as the identity) to M.

Therefore, as Ext functors are the derived functors of Hom, there is a natural isomorphism

${\displaystyle H^{n}(G,M)=\operatorname {Ext} _{\mathbb {Z} [G]}^{n}(\mathbb {Z} ,M).}$

These Ext groups can also be computed via a projective resolution of ${\displaystyle \mathbb {Z} }$, the advantage being that such a resolution only depends on G and not on M. We recall the definition of Ext more explicitly for this context. Let F be a projective ${\displaystyle \mathbb {Z} [G]}$-resolution (e.g. a free ${\displaystyle \mathbb {Z} [G]}$-resolution) of the trivial ${\displaystyle \mathbb {Z} [G]}$-module ${\displaystyle \mathbb {Z} }$:

${\displaystyle \cdots \to F_{n}\to F_{n-1}\to \cdots \to F_{0}\to \mathbb {Z} \to 0.}$

e.g., one may always take the resolution of group rings, ${\displaystyle F_{n}=\mathbb {Z} [G^{n+1}],}$ with morphisms

${\displaystyle {\begin{cases}f_{n}:\mathbb {Z} [G^{n+1}]\to \mathbb {Z} [G^{n}]\\(g_{0},g_{1},\ldots ,g_{n})\mapsto \sum _{i=0}^{n}(-1)^{i}\left(g_{0},\ldots ,{\widehat {g_{i}}},\dots ,g_{n}\right)\end{cases}}}$

Recall that for ${\displaystyle \mathbb {Z} [G]}$-modules N and M, Hom_G(N, M) is an abelian group consisting of ${\displaystyle \mathbb {Z} [G]}$-homomorphisms from N to M. Since ${\displaystyle \operatorname {Hom} _{G}(-,M)}$ is a

and reverses the arrows, applying ${\displaystyle \operatorname {Hom} _{G}(-,M)}$ to F termwise and dropping ${\displaystyle \operatorname {Hom} _{G}(\mathbb {Z} ,M)}$ produces a ${\displaystyle \operatorname {Hom} _{G}(-,M)(F,M)}$:

${\displaystyle \cdots \leftarrow \operatorname {Hom} _{G}(F_{n},M)\leftarrow \operatorname {Hom} _{G}(F_{n-1},M)\leftarrow \dots \leftarrow \operatorname {Hom} _{G}(F_{0},M)\leftarrow 0.}$

The cohomology groups ${\displaystyle H^{*}(G,M)}$ of G with coefficients in the module M are defined as the cohomology of the above cochain complex:

${\displaystyle H^{n}(G,M)=H^{n}({\rm {Hom}}_{G}(F,M)),\qquad n\geqslant 0.}$

This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of ${\displaystyle \operatorname {Hom} _{G}(F,M)}$, that is, functions ${\displaystyle \phi _{n}\colon G^{n}\to M}$ that obey

${\displaystyle g\phi _{n}(g_{1},g_{2},\ldots ,g_{n})=\phi _{n}(gg_{1},gg_{2},\ldots ,gg_{n}).}$

The coboundary operator ${\displaystyle \delta \colon C^{n}\to C^{n+1}}$ is now naturally defined by, for example,

${\displaystyle \delta \phi _{2}(g_{1},g_{2},g_{3})=\phi _{2}(g_{2},g_{3})-\phi _{2}(g_{1},g_{3})+\phi _{2}(g_{1},g_{2}).}$

The relation to the coboundary operator d that was defined in the previous section, and which acts on the "inhomogeneous" cochains ${\displaystyle \varphi }$, is given by reparameterizing so that

${\displaystyle {\begin{aligned}\varphi _{2}(g_{1},g_{2})&=\phi _{3}(1,g_{1},g_{1}g_{2})\\\varphi _{3}(g_{1},g_{2},g_{3})&=\phi _{4}(1,g_{1},g_{1}g_{2},g_{1}g_{2}g_{3}),\end{aligned}}}$

and so on. Thus

${\displaystyle {\begin{aligned}d\varphi _{2}(g_{1},g_{2},g_{3})&=\delta \phi _{3}(1,g_{1},g_{1}g_{2},g_{1}g_{2}g_{3})\\&=\phi _{3}(g_{1},g_{1}g_{2},g_{1}g_{2}g_{3})-\phi _{3}(1,g_{1}g_{2},g_{1}g_{2}g_{3})+\phi _{3}(1,g_{1},g_{1}g_{2}g_{3})-\phi _{3}(1,g_{1},g_{1}g_{2})\\&=g_{1}\phi _{3}(1,g_{2},g_{2}g_{3})-\phi _{3}(1,g_{1}g_{2},g_{1}g_{2}g_{3})+\phi _{3}(1,g_{1},g_{1}g_{2}g_{3})-\phi _{3}(1,g_{1},g_{1}g_{2})\\&=g_{1}\varphi _{2}(g_{2},g_{3})-\varphi _{2}(g_{1}g_{2},g_{3})+\varphi _{2}(g_{1},g_{2}g_{3})-\varphi _{2}(g_{1},g_{2}),\end{aligned}}}$

as in the preceding section.

Group homology

Dually to the construction of group cohomology there is the following definition of group homology: given a

${\displaystyle M_{G}:=M/DM,}$

is a

left derived functors

are by definition the group homology

${\displaystyle H_{n}(G,M).}$

The

covariant functor

which assigns M_G to M is isomorphic to the functor which sends M to ${\displaystyle \mathbb {Z} \otimes _{\mathbb {Z} [G]}M,}$ where ${\displaystyle \mathbb {Z} }$ is endowed with the trivial G-action.[b] Hence one also gets an expression for group homology in terms of the Tor functors,

${\displaystyle H_{n}(G,M)=\operatorname {Tor} _{n}^{\mathbb {Z} [G]}(\mathbb {Z} ,M)}$

Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches:

• superscripts correspond to cohomology H* and invariants X^G while
• subscripts correspond to homology H_∗ and coinvariants X_G := X/G.

Specifically, the homology groups H_n(G, M) can be computed as follows. Start with a

F of the trivial ${\displaystyle \mathbb {Z} [G]}$-module ${\displaystyle \mathbb {Z} ,}$ as in the previous section. Apply the covariant functor ${\displaystyle \cdot \otimes _{\mathbb {Z} [G]}M}$ to F termwise to get a chain complex ${\displaystyle F\otimes _{\mathbb {Z} [G]}M}$:

${\displaystyle \cdots \to F_{n}\otimes _{\mathbb {Z} [G]}M\to F_{n-1}\otimes _{\mathbb {Z} [G]}M\to \cdots \to F_{0}\otimes _{\mathbb {Z} [G]}M\to \mathbb {Z} \otimes _{\mathbb {Z} [G]}M.}$

Then H_n(G, M) are the homology groups of this chain complex, ${\displaystyle H_{n}(G,M)=H_{n}(F\otimes _{\mathbb {Z} [G]}M)}$ for n ≥ 0.

Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.

The group homology ${\displaystyle H_{*}(G,k)}$ of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra ${\displaystyle \wedge ^{*}(G\otimes k)}$.^[c]

Low-dimensional cohomology groups

H¹

The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : G → M given by f(g) = gm−m for some fixed m ∈ M. This follows from the definition of cochains above.

If the action of G on M is trivial, then the above boils down to H¹(G,M) = Hom(G, M), the group of group homomorphisms G → M, since the crossed homomorphisms are then just ordinary homomorphisms and the coboundaries (i.e. the principal crossed homomorphisms) must have image identically zero: hence there is only the zero coboundary.

On the other hand, consider the case of ${\displaystyle H^{1}(\mathbb {Z} /2,\mathbb {Z} _{-}),}$ where ${\displaystyle \mathbb {Z} _{-}}$ denotes the non-trivial ${\displaystyle \mathbb {Z} /2}$-structure on the additive group of integers, which sends a to -a for every ${\displaystyle a\in \mathbb {Z} }$; and where we regard ${\displaystyle \mathbb {Z} /2}$ as the group ${\displaystyle \{\pm 1\}}$. By considering all possible cases for the images of ${\displaystyle \{1,-1\}}$, it may be seen that crossed homomorphisms constitute all maps ${\displaystyle f_{t}:\{\pm 1\}\to \mathbb {Z} }$ satisfying ${\displaystyle f_{t}(1)=0}$ and ${\displaystyle f_{t}(-1)=t}$ for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy ${\displaystyle f_{t}(-1)=(-1)*m-m=-2m}$ for some integer m: hence every crossed homomorphism ${\displaystyle f_{t}}$ sending -1 to an even integer ${\displaystyle t=-2m}$ is principal, and therefore:

${\displaystyle H^{1}(\mathbb {Z} /2,\mathbb {Z} _{-})\cong \mathbb {Z} /2={\rm {\ (say)\ {\it {}}}}\langle f:f(1)=0,f(-1)=1\rangle ,}$

with the group operation being pointwise addition: ${\displaystyle (f_{s}+f_{t})(x)=f_{s}(x)+f_{t}(x)=f_{s+t}(x)}$, noting that ${\displaystyle f_{0}}$ is the identity element.

H²

If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H²(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H²(G,M) classifies the isomorphism classes of all extensions ${\displaystyle 0\to M\to E\to G\to 0}$ of G by M, in which the action of G on E (by inner automorphisms), endows (the image of) M with an isomorphic G-module structure.

In the example from the section on ${\displaystyle H^{1}}$ immediately above, ${\displaystyle H^{2}(\mathbb {Z} /2,\mathbb {Z} _{-})=0,}$ as the only extension of ${\displaystyle \mathbb {Z} /2}$ by ${\displaystyle \mathbb {Z} }$ with the given nontrivial action is the

and so trivial inside the ${\displaystyle H^{2}}$ group. This is in fact the significance in group-theoretical terms of the unique non-trivial element of ${\displaystyle H^{1}(\mathbb {Z} /2,\mathbb {Z} _{-}),}$.

An example of a second cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:

${\displaystyle H^{2}\left(\mathrm {Gal} (k),(k^{\mathrm {sep} })^{\times }\right).}$

See also [1].

Basic examples

Group cohomology of a finite cyclic group

For the finite cyclic group ${\displaystyle G=C_{m}}$ of order ${\displaystyle m}$ with generator ${\displaystyle \sigma }$, the element ${\displaystyle \sigma -1\in \mathbb {Z} [G]}$ in the associated group ring is a divisor of zero because its product with ${\displaystyle N}$, given by

${\displaystyle N=1+\sigma +\sigma ^{2}+\cdots +\sigma ^{m-1}\in \mathbb {Z} [G],}$

gives

${\displaystyle {\begin{aligned}N(1-\sigma )&=1+\sigma +\cdots +\sigma ^{m-1}\\&\quad -\sigma -\sigma ^{2}-\cdots -\sigma ^{m}\\&=1-\sigma ^{m}\\&=0.\end{aligned}}}$

This property can be used to construct the resolution^[2][3] of the trivial ${\displaystyle \mathbb {Z} [G]}$-module ${\displaystyle \mathbb {Z} }$ via the complex

${\displaystyle \cdots \xrightarrow {\sigma -1} \mathbb {Z} [G]\xrightarrow {N} \mathbb {Z} [G]\xrightarrow {\sigma -1} \mathbb {Z} [G]\xrightarrow {\text{aug}} \mathbb {Z} \to 0}$

giving the group cohomology computation for any ${\displaystyle \mathbb {Z} [G]}$-module ${\displaystyle A}$. Note the augmentation map gives the trivial module ${\displaystyle \mathbb {Z} }$ its ${\displaystyle \mathbb {Z} [G]}$-structure by

${\displaystyle {\text{aug}}\left(\sum _{g\in G}a_{g}g\right)=\sum _{g\in G}a_{g}}$

This resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groups

${\displaystyle H^{k}(G,A)\cong {\text{Ext}}_{\mathbb {Z} [G]}^{k}(\mathbb {Z} ,A)}$

showing that applying the functor ${\displaystyle {\text{Hom}}_{\mathbb {Z} [G]}(-,A)}$ to the complex above (with ${\displaystyle \mathbb {Z} }$ removed since this resolution is a quasi-isomorphism), gives the computation

${\displaystyle H^{k}(G,A)={\begin{cases}A^{G}/NA&k{\text{ even}},k\geq 2\\{}_{N}A/(\sigma -1)A&k{\text{ odd}},k\geq 1\end{cases}}}$

for

${\displaystyle {}_{N}A=\{a\in A:Na=0\}}$

For example, if ${\displaystyle A=\mathbb {Z} }$, the trivial module, then ${\displaystyle \mathbb {Z} ^{G}=\mathbb {Z} }$, ${\displaystyle N\mathbb {Z} ={\text{aug}}(N)\mathbb {Z} =m\mathbb {Z} }$, and ${\displaystyle {}_{N}\mathbb {Z} =0}$, hence

${\displaystyle H^{k}(C_{m},\mathbb {Z} )={\begin{cases}\mathbb {Z} /m\mathbb {Z} &k{\text{ even}},k\geq 2\\0&k{\text{ odd}},k\geq 1\end{cases}}}$

Explicit cocycles

Cocycles for the group cohomology of a cyclic group can be given explicitly^[4] using the Bar resolution. We get a complete set of generators of ${\displaystyle l}$-cocycles for ${\displaystyle l}$ odd as the maps

${\displaystyle \omega _{a}:B_{l}\to k^{*}}$

given by

${\displaystyle [g^{i_{1}},\ldots ,g^{i_{l}}]\mapsto \zeta _{m}^{ai_{1}\left[{\frac {i_{2}+i_{3}}{m}}\right]\cdots \left[{\frac {i_{l-1}+i_{l}}{m}}\right]}}$

for ${\displaystyle l}$ odd, ${\displaystyle 0\leq a\leq m-1}$, ${\displaystyle \zeta _{m}}$ a primitive ${\displaystyle m}$-th root of unity, ${\displaystyle k}$ a field containing ${\displaystyle m}$-th roots of unity, and

${\displaystyle \left[{\frac {a}{b}}\right]}$

for a rational number ${\displaystyle a/b}$ denoting the largest integer not greater than ${\displaystyle a/b}$. Also, we are using the notation

${\displaystyle B_{l}=\bigoplus _{0\leq i_{1},\ldots ,i_{l}\leq m-1}\mathbb {Z} G\cdot [g^{i_{1}},\ldots ,g^{i_{l}}]}$

where ${\displaystyle g}$ is a generator for ${\displaystyle G=C_{m}}$. Note that for ${\displaystyle l}$ non-zero even indices the cohomology groups are trivial.

Cohomology of free groups

Using a resolution

Given a set ${\displaystyle S}$ the associated free group ${\displaystyle G={\text{Free}}(S)={\underset {s\in S}{*}}\mathbb {Z} }$ has an explicit resolution^[5] of the trivial module ${\displaystyle \mathbb {Z} _{\text{triv}}}$ which can be easily computed. Notice the augmentation map

${\displaystyle {\text{aug}}:\mathbb {Z} [G]\to \mathbb {Z} _{\text{triv}}}$

has kernel given by the free submodule ${\displaystyle I_{S}}$ generated by the set ${\displaystyle \{s-1:s\in S\}}$, so

${\displaystyle I_{S}=\bigoplus _{s\in S}\mathbb {Z} [G]\cdot (s-1)}$.

Because this object is free, this gives a resolution

${\displaystyle 0\to I_{S}\to \mathbb {Z} [G]\to \mathbb {Z} _{\text{triv}}\to 0}$

hence the group cohomology of ${\displaystyle G}$ with coefficients in ${\displaystyle \mathbb {Z} _{\text{triv}}}$ can be computed by applying the functor ${\displaystyle {\text{Hom}}_{\mathbb {Z} [G]}(-,\mathbb {Z} )}$ to the complex ${\displaystyle 0\to I_{S}\to \mathbb {Z} [G]\to 0}$, giving

${\displaystyle H^{k}(G,\mathbb {Z} _{\text{triv}})={\begin{cases}\mathbb {Z} &k=0\\\bigoplus _{s\in S}\mathbb {Z} &k=1\\0&k\geq 2\end{cases}}}$

this is because the dual map

${\displaystyle {\text{Hom}}_{\mathbb {Z} [G]}(\mathbb {Z} [G],\mathbb {Z} _{\text{triv}})\to {\text{Hom}}_{\mathbb {Z} [G]}(I_{S},\mathbb {Z} _{\text{triv}})}$

sends any ${\displaystyle \mathbb {Z} [G]}$-module morphism

${\displaystyle \phi :\mathbb {Z} [G]\to \mathbb {Z} _{\text{triv}}}$

to the induced morphism on ${\displaystyle I_{S}}$ by composing the inclusion. The only maps which are sent to ${\displaystyle 0}$ are ${\displaystyle \mathbb {Z} }$-multiples of the augmentation map, giving the first cohomology group. The second can be found by noticing the only other maps

${\displaystyle \psi \in {\text{Hom}}_{\mathbb {Z} [G]}(I_{S},\mathbb {Z} _{\text{triv}})}$

can be generated by the ${\displaystyle \mathbb {Z} }$-basis of maps sending ${\displaystyle (s-1)\mapsto 1}$ for a fixed ${\displaystyle s\in S}$, and sending ${\displaystyle (s'-1)\mapsto 0}$ for any ${\displaystyle s'\in S-\{s\}}$.

Using topology

The group cohomology of free groups ${\displaystyle \mathbb {Z} *\mathbb {Z} *\cdots *\mathbb {Z} }$ generated by ${\displaystyle n}$ letters can be readily computed by comparing group cohomology with its interpretation in topology. Recall that for every group ${\displaystyle G}$ there is a topological space ${\displaystyle BG}$, called the classifying space of the group, which has the property

${\displaystyle \pi _{1}(BG)=G{\text{ and }}\pi _{k}(BG)=0{\text{ for }}k\geq 2}$

In addition, it has the property that its topological cohomology is isomorphic to group cohomology

${\displaystyle H^{k}(BG,\mathbb {Z} )\cong H^{k}(G,\mathbb {Z} )}$

giving a way to compute some group cohomology groups. Note ${\displaystyle \mathbb {Z} }$ could be replaced by any local system ${\displaystyle {\mathcal {L}}}$ which is determined by a map

${\displaystyle \pi _{1}(G)\to GL(V)}$

for some abelian group ${\displaystyle V}$. In the case of ${\displaystyle B(\mathbb {Z} *\cdots *\mathbb {Z} )}$ for ${\displaystyle n}$ letters, this is represented by a wedge sum of ${\displaystyle n}$ circles ${\displaystyle S^{1}\vee \cdots \vee S^{1}}$

${\displaystyle H^{k}(\mathbb {Z} *\cdots *\mathbb {Z} )={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} ^{n}&k=1\\0&k\geq 2\end{cases}}}$

Group cohomology of an integral lattice

For an integral lattice ${\displaystyle \Lambda }$ of rank ${\displaystyle n}$ (hence isomorphic to ${\displaystyle \mathbb {Z} ^{n}}$), its group cohomology can be computed with relative ease. First, because ${\displaystyle B\mathbb {Z} \cong S^{1}}$, and ${\displaystyle B\mathbb {Z} \times B\mathbb {Z} }$ has ${\displaystyle \pi _{1}\cong \mathbb {Z} \times \mathbb {Z} }$, which as abelian groups are isomorphic to ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} }$, the group cohomology has the isomorphism

${\displaystyle H^{k}(\Lambda ,\mathbb {Z} _{\text{triv}})\cong H^{k}(\mathbb {R} ^{n}/\mathbb {Z} ^{n},\mathbb {Z} )}$

with the integral cohomology of a torus of rank ${\displaystyle n}$.

Properties

In the following, let M be a G-module.

Long exact sequence of cohomology

In practice, one often computes the cohomology groups using the following fact: if

${\displaystyle 0\to L\to M\to N\to 0}$

is a

short exact sequence

of G-modules, then a long exact sequence is induced:

${\displaystyle 0\longrightarrow L^{G}\longrightarrow M^{G}\longrightarrow N^{G}{\overset {\delta ^{0}}{\longrightarrow }}H^{1}(G,L)\longrightarrow H^{1}(G,M)\longrightarrow H^{1}(G,N){\overset {\delta ^{1}}{\longrightarrow }}H^{2}(G,L)\longrightarrow \cdots }$

The so-called

connecting homomorphisms

,

${\displaystyle \delta ^{n}:H^{n}(G,N)\to H^{n+1}(G,L)}$

can be described in terms of inhomogeneous cochains as follows.[8] If ${\displaystyle c\in H^{n}(G,N)}$ is represented by an n-cocycle ${\displaystyle \phi :G^{n}\to N,}$ then ${\displaystyle \delta ^{n}(c)}$ is represented by ${\displaystyle d^{n}(\psi ),}$ where ${\displaystyle \psi }$ is an n-cochain ${\displaystyle G^{n}\to M}$ "lifting" ${\displaystyle \phi }$ (i.e. ${\displaystyle \phi }$ is the composition of ${\displaystyle \psi }$ with the surjective map M → N).

Functoriality

Group cohomology depends contravariantly on the group G, in the following sense: if f : H → G is a

${\displaystyle cor_{H}^{G}:H^{n}(H,M)\to H^{n}(G,M).}$

In degree 0, it is given by the map

${\displaystyle {\begin{cases}M^{H}\to M^{G}\\m\mapsto \sum _{g\in G/H}gm\end{cases}}}$

Given a morphism of G-modules M → N, one gets a morphism of cohomology groups in the Hⁿ(G, M) → Hⁿ(G, N).

Products

Similarly to other cohomology theories in topology and geometry, such as

, group cohomology enjoys a product structure: there is a natural map called cup product:

${\displaystyle H^{n}(G,N)\otimes H^{m}(G,M)\to H^{n+m}(G,M\otimes N)}$

for any two G-modules M and N. This yields a graded anti-commutative ring structure on ${\displaystyle \oplus _{n\geqslant 0}H^{n}(G,R),}$ where R is a ring such as ${\displaystyle \mathbb {Z} }$ or ${\displaystyle \mathbb {Z} /p.}$ For a finite group G, the even part of this cohomology ring in characteristic p, ${\displaystyle \oplus _{n\geqslant 0}H^{2n}(G,\mathbb {Z} /p)}$ carries a lot of information about the group the structure of G, for example the Krull dimension of this ring equals the maximal rank of an abelian subgroup ${\displaystyle (\mathbb {Z} /p)^{r}}$.^[10]

For example, let G be the group with two elements, under the discrete topology. The real projective space ${\displaystyle \mathbb {P} ^{\infty }(\mathbb {R} )}$ is a classifying space for G. Let ${\displaystyle k=\mathbb {F} _{2},}$ the field of two elements. Then

${\displaystyle H^{*}(G;k)\cong k[x],}$

a polynomial k-algebra on a single generator, since this is the

ring of ${\displaystyle \mathbb {P} ^{\infty }(\mathbb {R} ).}$

Künneth formula

If, M = k is a field, then H*(G; k) is a graded k-algebra and the cohomology of a product of groups is related to the ones of the individual groups by a

:

${\displaystyle H^{*}(G_{1}\times G_{2};k)\cong H^{*}(G_{1};k)\otimes H^{*}(G_{2};k).}$

For example, if G is an elementary abelian 2-group of rank r, and ${\displaystyle k=\mathbb {F} _{2},}$ then the Künneth formula shows that the cohomology of G is a polynomial k-algebra generated by r classes in H¹(G; k).,

${\displaystyle H^{*}(G;k)\cong k[x_{1},\ldots ,x_{r}].}$

Homology vs. cohomology

As for other cohomology theories, such as

${\displaystyle 0\to \mathrm {Ext} _{\mathbb {Z} }^{1}\left(H_{n-1}(G,\mathbb {Z} ),A\right)\to H^{n}(G,A)\to \mathrm {Hom} \left(H_{n}(G,\mathbb {Z} ),A\right)\to 0,}$

where A is endowed with the trivial G-action and the term at the left is the first Ext group.

Amalgamated products

Given a group A which is the subgroup of two groups G₁ and G₂, the homology of the

${\displaystyle G:=G_{1}\star _{A}G_{2}}$ (with integer coefficients) lies in a long exact sequence

${\displaystyle \cdots \to H_{n}(A)\to H_{n}(G_{1})\oplus H_{n}(G_{2})\to H_{n}(G)\to H_{n-1}(A)\to \cdots }$

The homology of ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )=\mathbb {Z} /4\star _{\mathbb {Z} /2}\mathbb {Z} /6}$ can be computed using this:

${\displaystyle H_{n}(\mathrm {SL} _{2}(\mathbb {Z} ))={\begin{cases}\mathbb {Z} &n=0\\\mathbb {Z} /12&{\text{odd degrees}}\\0&{\text{otherwise}}\end{cases}}}$

This exact sequence can also be applied to show that the homology of the ${\displaystyle \mathrm {SL} _{2}(k[t])}$ and the special linear group ${\displaystyle \mathrm {SL} _{2}(k)}$ agree for an infinite field k.^[12]

Change of group

The

.

Cohomology of the classifying space

Group cohomology is closely related to topological cohomology theories such as sheaf cohomology, by means of an isomorphism^[13]

${\displaystyle H^{n}(BG,\mathbb {Z} )\cong H^{n}(G,\mathbb {Z} ).}$

The expression ${\displaystyle BG}$ at the left is a classifying space for ${\displaystyle G}$. It is an Eilenberg–MacLane space ${\displaystyle K(G,1)}$, i.e., a space whose fundamental group is ${\displaystyle G}$ and whose higher homotopy groups vanish).^[d] Classifying spaces for ${\displaystyle \mathbb {Z} ,\mathbb {Z} /2}$ and ${\displaystyle \mathbb {Z} /n}$ are the

${\displaystyle \mathbb {P} ^{\infty }(\mathbb {R} )=\cup _{n}\mathbb {P} ^{n}(\mathbb {R} ),}$ and lens spaces, respectively. In general, ${\displaystyle BG}$ can be constructed as the quotient ${\displaystyle EG/G}$, where ${\displaystyle EG}$ is a contractible space on which ${\displaystyle G}$ acts freely. However, ${\displaystyle BG}$ does not usually have an easily amenable geometric description.

More generally, one can attach to any ${\displaystyle G}$-module ${\displaystyle M}$ a local coefficient system on ${\displaystyle BG}$ and the above isomorphism generalizes to an isomorphism^[14]

${\displaystyle H^{n}(BG,M)=H^{n}(G,M).}$

Further examples

Semi-direct products of groups

There is a way to compute the semi-direct product of groups using the topology of fibrations and properties of Eilenberg-Maclane spaces. Recall that for a semi-direct product of groups ${\displaystyle G=N\rtimes H}$ there is an associated short exact sequence of groups

${\displaystyle 1\to N\to N\rtimes H\to H\to 1}$

Using the associated Eilenberg-Maclane spaces there is a

${\displaystyle K(N,1)\to K(G,1)\to K(H,1)}$

which can be put through a Serre spectral sequence. This gives an ${\displaystyle E_{2}}$-page

${\displaystyle E_{2}^{p,q}=H^{p}(K(H,1),H^{q}(K(N,1)))\Rightarrow H^{p+q}(K(G,1))}$

which gives information about the group cohomology of ${\displaystyle G}$ from the group cohomology groups of ${\displaystyle H,N}$. Note this formalism can be applied in a purely group-theoretic manner using the Lyndon–Hochschild–Serre spectral sequence.

Cohomology of finite groups

Higher cohomology groups are torsion

The cohomology groups Hⁿ(G, M) of finite groups G are all torsion for all n≥1. Indeed, by

to its base field and hence has trivial cohomology.

If the order of G is invertible in a G-module M (for example, if M is a ${\displaystyle \mathbb {Q} }$-vector space), the transfer map can be used to show that ${\displaystyle H^{n}(G,M)=0}$ for ${\displaystyle n\geqslant 1.}$ A typical application of this fact is as follows: the long exact cohomology sequence of the short exact sequence (where all three groups have a trivial G-action)

${\displaystyle 0\to \mathbb {Z} \to \mathbb {Q} \to \mathbb {Q} /\mathbb {Z} \to 0}$

yields an isomorphism

${\displaystyle \mathrm {Hom} (G,\mathbb {Q} /\mathbb {Z} )=H^{1}(G,\mathbb {Q} /\mathbb {Z} )\cong H^{2}(G,\mathbb {Z} ).}$

Tate cohomology

groups combine both homology and cohomology of a finite group G:

${\displaystyle {\widehat {H}}^{n}(G,M):={\begin{cases}H^{n}(G,M)&n\geqslant 1\\\operatorname {coker} N&n=0\\\ker N&n=-1\\H_{-n-1}(G,M)&n\leqslant -2,\end{cases}}}$

where ${\displaystyle N:M_{G}\to M^{G}}$ is induced by the norm map:

${\displaystyle {\begin{cases}M\to M\\m\mapsto \sum _{g\in G}gm\end{cases}}}$

Tate cohomology enjoys similar features, such as long exact sequences, product structures. An important application is in class field theory, see class formation.

Tate cohomology of finite cyclic groups, ${\displaystyle G=\mathbb {Z} /n,}$ is 2-periodic in the sense that there are isomorphisms

${\displaystyle {\widehat {H}}^{m}(G,M)\cong {\widehat {H}}^{m+2}(G,M)\qquad {\text{for all }}m\in \mathbb {Z} .}$

A necessary and sufficient criterion for a d-periodic cohomology is that the only abelian subgroups of G are cyclic.

${\displaystyle \mathbb {Z} /n\rtimes \mathbb {Z} /m}$ has this property for coprime integers n and m.

Applications

Algebraic K-theory and homology of linear groups

Algebraic K-theory is closely related to group cohomology: in Quillen's +-construction of K-theory, K-theory of a ring R is defined as the homotopy groups of a space ${\displaystyle \mathrm {BGL} (R)^{+}.}$ Here ${\displaystyle \mathrm {GL} (R)=\cup _{n\geq 1}\mathrm {GL} _{n}(R)}$ is the infinite general linear group. The space ${\displaystyle \mathrm {BGL} (R)^{+}}$ has the same homology as ${\displaystyle \mathrm {BGL} (R),}$ i.e., the group homology of GL(R). In some cases, stability results assert that the sequence of cohomology groups

${\displaystyle \dots \to H_{m}\left(\mathrm {GL} _{n}(R)\right)\to H_{m}\left(\mathrm {GL} _{n+1}(R)\right)\to \cdots }$

becomes stationary for large enough n, hence reducing the computation of the cohomology of the infinite general linear group to the one of some ${\displaystyle \mathrm {GL} _{n}(R)}$. Such results have been established when R is a field

The phenomenon that group homology of a series of groups ${\displaystyle G_{n}}$ stabilizes is referred to as homological stability. In addition to the case ${\displaystyle G_{n}=\mathrm {GL} _{n}(R)}$ just mentioned, this applies to various other groups such as symmetric groups or mapping class groups.

Projective representations and group extensions

In quantum mechanics we often have systems with a symmetry group ${\displaystyle G.}$ We expect an action of ${\displaystyle G}$ on the Hilbert space ${\displaystyle {\mathcal {H}}}$ by unitary matrices ${\displaystyle U(g).}$ We might expect ${\displaystyle U(g_{1})U(g_{2})=U(g_{1}g_{2}),}$ but the rules of quantum mechanics only require

${\displaystyle U(g_{1})U(g_{2})=\exp\{2\pi i\omega (g_{1},g_{2})\}U(g_{1}g_{2}),}$

where ${\displaystyle \exp\{2\pi i\omega (g_{1},g_{2})\}\in {\rm {U}}(1)}$ is a phase. This projective representation of ${\displaystyle G}$ can also be thought of as a conventional representation of a group extension ${\displaystyle {\tilde {G}}}$ of ${\displaystyle G}$ by ${\displaystyle \mathrm {U} (1),}$ as described by the exact sequence

${\displaystyle 1\to {\rm {U}}(1)\to {\tilde {G}}\to G\to 1.}$

Requiring associativity

${\displaystyle U(g_{1})[U(g_{2})U(g_{3})]=[U(g_{1})U(g_{2})]U(g_{3})}$

leads to

${\displaystyle \omega (g_{2},g_{3})-\omega (g_{1}g_{2},g_{3})+\omega (g_{1},g_{2}g_{3})-\omega (g_{1},g_{2})=0,}$

which we recognise as the statement that ${\displaystyle d\omega (g_{1},g_{2},g_{3})=0,}$ i.e. that ${\displaystyle \omega }$ is a cocycle taking values in ${\displaystyle \mathbb {R} /\mathbb {Z} \simeq {\rm {U}}(1).}$ We can ask whether we can eliminate the phases by redefining

${\displaystyle U(g)\to \exp\{2\pi i\eta (g)\}U(g)}$

which changes

${\displaystyle \omega (g_{1},g_{2})\to \omega (g_{1},g_{2})+\eta (g_{2})-\eta (g_{1}g_{2})+\eta (g_{1}).}$

This we recognise as shifting ${\displaystyle \omega }$ by a coboundary ${\displaystyle \omega \to \omega +d\eta .}$ The distinct projective representations are therefore classified by ${\displaystyle H^{2}(G,\mathbb {R} /\mathbb {Z} ).}$ Note that if we allow the phases themselves to be acted on by the group (for example, time reversal would complex-conjugate the phase), then the first term in each of the coboundary operations will have a ${\displaystyle g_{1}}$ acting on it as in the general definitions of coboundary in the previous sections. For example, ${\displaystyle d\eta (g_{1},g_{2})\to g_{1}\eta (g_{2})-\eta (g_{1}g_{2})+\eta (g_{1}).}$

Extensions

Cohomology of topological groups

Given a topological group G, i.e., a group equipped with a topology such that product and inverse are continuous maps, it is natural to consider continuous G-modules, i.e., requiring that the action

${\displaystyle G\times M\to M}$

is a continuous map. For such modules, one can again consider the derived functor of ${\displaystyle M\mapsto M^{G}}$. A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field. The resulting cohomology is called Galois cohomology.

Non-abelian group cohomology

Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.

The zeroth cohomology of G with coefficients in A is defined to be the subgroup

${\displaystyle H^{0}(G,A)=A^{G},}$

of elements of A fixed by G.

The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map ${\displaystyle \varphi }$ to be a 1-cocycle is that ${\displaystyle \varphi (gh)=\varphi (g)[g\varphi (h)]}$ and ${\displaystyle \ \varphi \sim \varphi '}$ if there is an a in A such that ${\displaystyle \ a\varphi '(g)=\varphi (g)\cdot (ga)}$. In general, ${\displaystyle H^{1}(G,A)}$ is not a group when A is non-abelian. It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group, ${\displaystyle \ \pi _{0}(X;x)}$ which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point.

Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let

${\displaystyle 1\to A\to B\to C\to 1\,}$

be a short exact sequence of G-groups, then there is an exact sequence of pointed sets

${\displaystyle 1\to A^{G}\to B^{G}\to C^{G}\to H^{1}(G,A)\to H^{1}(G,B)\to H^{1}(G,C).\,}$

History and relation to other fields

The low-dimensional cohomology of a group was classically studied in other guises, well before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as Hilbert's Theorem 90 in 1897; this was recast into Emmy Noether's equations in Galois theory (an appearance of cocycles for ${\displaystyle H^{1}}$). The idea of

for groups (connected with ${\displaystyle H^{2}}$) arose in the work of

simple algebras and the Brauer group. A fuller discussion of this history may be found in (Weibel 1999

, pp. 806–811).

In 1941, while studying ${\displaystyle H^{2}(G,\mathbb {Z} )}$ (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula (Hopf 1942), which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:

${\displaystyle H_{2}(G,\mathbb {Z} )\cong (R\cap [F,F])/[F,R],}$

where ${\displaystyle G\cong F/R}$ and F is a free group.

Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: Samuel Eilenberg and Saunders Mac Lane in the United States (Rotman 1995, p. 358); Hopf and Beno Eckmann in Switzerland; Hans Freudenthal in the Netherlands (Weibel 1999, p. 807); and Dmitry Faddeev in the Soviet Union (Arslanov 2011, p. 29, Faddeev 1947). The situation was chaotic because communication between these countries was difficult during World War II.

From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological classifying space BG as discussed above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions. From a module-theoretic point of view this was integrated into the Cartan–Eilenberg theory of homological algebra in the early 1950s.

The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) (Weibel 1999, p. 822). Some refinements in the theory post-1960 have been made, such as continuous cocycles and John Tate's redefinition, but the basic outlines remain the same. This is a large field, and now basic in the theories of algebraic groups.

The analogous theory for Lie algebras, called Lie algebra cohomology, was first developed in the late 1940s, by Claude Chevalley and Eilenberg, and Jean-Louis Koszul (Weibel 1999, p. 810). It is formally similar, using the corresponding definition of invariant for the action of a Lie algebra. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics.

Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of spontaneous symmetry breaking phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter—short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as symmetry-protected topological states.^{[18] [19]}

See also

• Lyndon–Hochschild–Serre spectral sequence
• N-group (category theory)
• Postnikov tower

Notes

1. ^ This uses that the category of G-modules has enough injectives, since it is isomorphic to the category of all modules over the group ring ${\displaystyle \mathbb {Z} [G].}$
2. ^ Recall that the tensor product ${\displaystyle N\otimes _{\mathbb {Z} [G]}M}$ is defined whenever N is a right ${\displaystyle \mathbb {Z} [G]}$-module and M is a left ${\displaystyle \mathbb {Z} [G]}$-module. If N is a left ${\displaystyle \mathbb {Z} [G]}$-module, we turn it into a right ${\displaystyle \mathbb {Z} [G]}$-module by setting ag = g⁻¹a for every g ∈ G and every a ∈ N. This convention allows to define the tensor product ${\displaystyle N\otimes _{\mathbb {Z} [G]}M}$ in the case where both M and N are left ${\displaystyle \mathbb {Z} [G]}$-modules.
3. ^ For example, the two are isomorphic if all primes p such that G has p-torsion are invertible in k. See (Knudson 2001), Theorem A.1.19 for the precise statement.
4. ^ For this, G is assumed to be discrete. For general topological groups, ${\displaystyle \pi _{n}(BG)=\pi _{n-1}(G)}$.

References

1. ^ Page 62 of Milne 2008 or section VII.3 of Serre 1979
2. OCLC 52559229
   .
3. OCLC 853269200
   .
4. MR 3228486
   . See Proposition 2.3.
5. OCLC 23732584
   .
6. OCLC 45420394
   .
7. ^ Webb, Peter. "An Introduction to the Cohomology of Groups" (PDF). Archived (PDF) from the original on 6 May 2020.
8. ^ Remark II.1.21 of Milne 2008
9. ^ (Brown 1972), §III.9
10. ^ Quillen, Daniel. The spectrum of an equivariant cohomology ring. I. II. Ann. Math. (2) 94, 549-572, 573-602 (1971).
11. ^ (Brown 1972), Exercise III.1.3
12. ^ (Knudson 2001), Chapter 4
13. ISSN 0002-9904
    .
14. ^ (Adem & Milgram 2004), Chapter II.
15. ^ (Brown 1972), §VI.9
16. ^ Suslin, Andrei A. (1984), "Homology of ${\displaystyle \operatorname {GL} _{n}}$, characteristic classes and Milnor K-theory", Algebraic K-theory, number theory, geometry and analysis, Lecture Notes in Mathematics, vol. 1046, Springer, pp. 357–375
17. doi:10.24033/asens.1269
    .
18. S2CID 2370407
    .
19. S2CID 13950401
    .

Works cited

• Zbl 1061.20044
• Arslanov, M. M. (2011), Математическая жизнь в Казани в годы войны,
  ISBN 978-5-94057-741-6
• MR 0672956
• ISSN 0002-3264
• Zbl 0027.09503
• Knudson, Kevin P. (2001), Homology of Linear Groups, Progress in Mathematics, vol. 193, Birkhäuser Verlag,
  Zbl 0997.20045
• Milne, James (2013), "Chapter II: The Cohomology of Groups", Class Field Theory, vol. v4.02
• MR 1307623
• Zbl 0423.12016
  .
• Weibel, Charles A. (1999), "History of homological algebra", History of Topology, Cambridge University Press, pp. 797–836,
  MR 1721123

Further reading

• MR 1324577
• Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton, NJ: Princeton University Press,
  MR 0347778
• Chapter 6 of
  OCLC 36131259
  .