Group cohomology ($H^n(G,M)$)

Group cohomology studies a group through cochain complexes built from its actions on abelian groups. In degree one, $H^1(G,M)$ identifies one-cocycles $c:G\to M$ modulo one-coboundaries.

For a connected CW complex $X$ with contractible universal cover, let $\Gamma ={\pi }_1(X)$ and $R=\mathbb{Z}\Gamma$. Lift each cell once and transport its orientation by deck transformations. Its translates form an $R$-basis of the cellular chains of the cover. The augmentation sends every vertex to one, and contractibility makes the augmented chain complex an exact free resolution of the trivial module $\mathbb{Z}$. An $R$-linear cochain with trivial coefficient action is constant on each orbit of lifts, identifying ${\mathrm{Hom}}_R(C_{\ast }(\tilde{X}),\mathbb{Z})$ with the cellular cohomology complex of $X$. Consequently $H^i(X;\mathbb{Z})={\mathrm{Ext}}_R^i(\mathbb{Z},\mathbb{Z})$. The same argument with singular simplex lifts works for spaces with a contractible universal cover.

The integral cohomological dimension is the least length of a projective resolution of the trivial group ring module $\mathbb{Z}$, equivalently the largest degree in which group cohomology with some module can be nonzero. A group containing torsion has infinite integral cohomological dimension. For ${\mathbb{Z}}^r$ the dimension is $r$, realized by its torus classifying space and detected by its top cohomology.

A group has integral cohomological dimension at most one precisely when it is a free group, including the trivial free group in dimension zero. This result concerns integral coefficients; a rational dimension-one statement has different conclusions.

A Poincare duality group of dimension $n$ has a finite-length resolution of the trivial group ring module $\mathbb{Z}$ by finitely generated projective modules, and $H^k(G,\mathbb{Z}G)$ vanishes except in degree $n$, where it is infinite cyclic as an abelian group. The top module $D$ can have an orientation-sign action. Reversing the dual projective resolution gives $H^k(G,V)\cong {\mathrm{Tor}}_{n-k}^{\mathbb{Z}G}(D,V)$, equivalently $H_{n-k}(G,D\otimes V)$ with the appropriate diagonal action. Untwisted duality requires trivial orientation action. In positive dimension $H^0(G,\mathbb{Z}G)=0$, since a finitely supported invariant element cannot exist for an infinite group.

Let $N$ be a normal finitely presented group inside a dimension-three Poincare duality group $G$, with $G/N$ containing an element of infinite order. Then $N$ is either a free group or a dimension-two Poincare duality group. The Strebel infinite-index subgroup theorem gives $\mathrm{cd}N\le 2$. If the dimension is two, take the preimage of an infinite cyclic subgroup of the quotient. The cyclic-extension shift of group-ring cohomology gives dimension three, forcing this preimage to have finite index by Strebel. Its duality then gives $H^{\ast }(N;\mathbb{Z}N)$ concentrated in degree two with top group $\mathbb{Z}$. The quotient is virtually cyclic, and surface automorphism realization gives a finite-index surface mapping-torus group.

Every integral dimension-two Poincare duality group is the fundamental group of a closed aspherical topological surface. Orientable examples are the torus and closed surfaces of genus at least two. Nonorientable aspherical closed surfaces supply the orientation-twisted examples.

An infinite-index subgroup of an integral dimension-$d$ Poincare duality group has integral cohomological dimension at most $d-1$. No finiteness assumption on the subgroup is needed for this inequality. In dimension three it places every infinite-index subgroup in cohomological dimension at most two.

For an extension with kernel a dimension-$n$ Poincare duality group and quotient a dimension-$q$ Poincare duality group, apply the Lyndon–Hochschild–Serre spectral sequence to the ambient group ring. Kernel cohomology is concentrated in degree $n$ and is a regular quotient module with a rank-one orientation twist. Quotient cohomology is then concentrated in degree $q$ with underlying group $\mathbb{Z}$. The extension resolution supplies finiteness and gives cohomological dimension at most $n+q$, proving duality in that dimension. Orientation actions need not be trivial.

For a torsion-free group $G$ and finite-index subgroup $H$, $G$ is a Poincare duality group of dimension $n$ if and only if $H$ is. Finite-index induction and coinduction of $\mathbb{Z}H$ coincide and give $\mathbb{Z}G$, so Shapiro's lemma identifies their group-ring cohomology as abelian groups. The finite-index finiteness and cohomological-dimension lemmas supply finite projective resolutions. Without torsion-freeness of $G$ the converse fails: $\mathbb{Z}\times C_2$ contains the index-two $P{D}^1$ subgroup $\mathbb{Z}$ but has infinite integral cohomological dimension.

For a prime $p$, a classifying Eilenberg–MacLane space for $C_p$ is the infinite lens space $S^{\infty }/C_p$, using scalar multiplication by a primitive $p$th root on the complex-coordinate sphere. Its contractible universal cover is $S^{\infty }$. The standard two-periodic free resolution has differentials $g-1$ and $1+g+\cdots +g^{p-1}$, both zero after applying cochains with trivial ${\mathbb{F}}_p$ coefficients. Thus every cohomology degree has dimension one. If $p=2$, the ring is ${\mathbb{F}}_2[t]$, $|t|=1$. If $p$ is odd, it is $\Lambda (a)\otimes {\mathbb{F}}_p[b]$, with $|a|=1$, $|b|=2$: graded commutativity gives $a^2=0$ and the periodicity class $b$ induces the degree-two isomorphisms. The Künneth theorem gives two copies of these generators for $C_p\times C_p$; the dimension in degree $d$ is $d+1$ for every $p$.

Embed a module $M$ over the group ring $\mathbb{Z}[G]$ in a module $I$ with vanishing positive group cohomology, and write $M^{\prime }=I/M$. The connecting map in the long exact group cohomology sequence of $0\to M\to I\to M^{\prime }\to 0$ is an isomorphism $H^r(G,M^{\prime })\to H^{r+1}(G,M)$ for $r\ge 1$, since the groups involving $I$ on both sides vanish. For a finite group and a module $I$ with vanishing Tate cohomology of a finite group in every degree, the same argument shifts every Tate degree. Repeating this construction transports computations to a convenient resolution.

For a finite group $G$, Tate cohomology joins group cohomology and group homology through the norm map $N_G={\sum }_{g\in G}g$. For $r\ge 1$ it equals ordinary group cohomology; for $r\le -2$ it equals $H_{-r-1}(G,M)$. The middle terms are ${\hat{H}}^0(G,M)=M^G/N_{G}M$ and ${\hat{H}}^{-1}(G,M)=\ker (N_G)/I_{G}M$, where $I_{G}M$ is generated by $gm-m$. For the trivial module $\mathbb{Z}$, ${\hat{H}}^{-2}(G,\mathbb{Z})=G^{\mathrm{ab}}$. With multiplicative coefficients the norm is the product of the translates.

For a topological group module, use continuous maps with pointwise addition and the inhomogeneous group-cohomology differential. The kernel modulo the image defines continuous group cohomology as an abelian group. With general topological coefficients the quotient topology need not be Hausdorff; the cohomology here is the algebraic quotient of continuous cocycles by continuous coboundaries. With trivial discrete coefficients in a prime field, its first cohomology is the space of continuous group homomorphisms to that field's additive group.

For $G=⟨\sigma ⟩$, set $D=\sigma -1$ and $N={\sum }_{g\in G}g$. The two periodic groups are $M^G/NM$ and $\ker N/DM$. A short exact sequence of modules gives a six-term periodic exact sequence. For multiplicative modules the norm operator is a product and $D(x)=\sigma (x)/x$.

Let $G$ be cyclic and let $R[G]$ carry the regular permutation action. Invariants are constant coefficient vectors, which are the images of the norm operator. A vector with coefficient sum zero is the difference of a vector and its cyclic translate, by successively solving the coordinate differences. Thus the norm group kernel equals $(\sigma -1)R[G]$. Both cyclic Tate cohomology groups vanish. This applies to $R=\mathbb{Z}$ and $R={\mathbb{Z}}_p$, and to finite direct sums of regular lattices.

When the two indicated groups are finite, their size ratio is multiplicative on short exact sequences. It is one for finite modules and therefore unchanged by finite-index modifications. A trivial integral module $\mathbb{Z}$ has quotient $|G|$, while an induced regular lattice ${\mathbb{Z}}_p[G]$ has both Tate groups zero and quotient one. These facts permit norm-index calculations without explicitly finding every norm.

For a finite module acted on by a cyclic group, write $D=\sigma -1$ and $N=1+\sigma +\cdots +{\sigma }^{n-1}$. The two Tate cohomology group sizes are $|\ker D|/|NM|$ and $|\ker N|/|DM|$. The equalities $|NM|=|M|/|\ker N|$ and $|DM|=|M|/|\ker D|$ show that these sizes are equal. The exact-sequence multiplication rule for the Herbrand quotient consequently makes it unchanged by finite-index modifications of lattices.

For a cyclic extension of p-adic fields, sufficiently deep principal units are equivariantly isomorphic to an additive lattice by the p-adic logarithm. The normal basis theorem makes its Herbrand quotient one. Passing across finite unit quotients preserves it. The valuation exact sequence with quotient $\mathbb{Z}$ therefore gives the displayed formula.

For an open finite-index subgroup $U$ of a profinite group $G$ and a discrete $G$-module $M$, cohomological transfer gives ${\mathrm{Cor}}_U^G:H^q(U,M)\to H^q(G,M)$. In degree zero it is the sum over cosets, or the norm for multiplicative coefficients. It is transitive and satisfies $\mathrm{Cor}\mathrm{Res}=[G:U]$ and the cup-product projection formula.

For a profinite group and a discrete module with continuous action, continuous cohomology is computed by continuous cochains. On a compact domain these cochains have finite image. Positive-degree classes become zero on a sufficiently small open subgroup and hence are torsion by restriction followed by corestriction.

For a closed subgroup $P$ of a profinite group $G$ and a discrete $G$-module $M$, restriction gives $H^q(P,M)={\underrightarrow{\lim }}_{U\supseteq P,U\mathrm{open}}{H}^q(U,M)$. Thus a class restricting to zero on $P$ already restricts to zero on one such open subgroup. The property follows by extending continuous finite-image cochains and their identities to a sufficiently small open neighborhood of $P$.

The value ${\mathrm{cd}}_p(G)$ is the least $n$ such that all cohomology in degrees greater than $n$ vanishes for every discrete $p$-primary torsion module. Vanishing in degree $n+1$ for all such modules suffices by dimension shifting. Dimension does not increase on closed subgroups. For a field, use its absolute Galois group.

For a pro-p group $P$, vanishing of $H^{n+1}(P,{\mathbb{F}}_p)$ implies vanishing in that degree for every discrete $p$-primary module. Finite such modules have composition factors equal to the trivial module ${\mathbb{F}}_p$, so the long exact sequence gives the result by induction. General modules are unions of finite stable submodules, and continuous cohomology commutes with filtered unions. Dimension shifting gives ${\mathrm{cd}}_p(P)\le n$. The converse is immediate from the definition.

A field of characteristic $p$ satisfies ${\mathrm{cd}}_p(K)\le 1$. The Artin-Schreier sequence and vanishing of positive-degree cohomology of the additive separable-closure module yield the bound. It concerns discrete $p$-primary modules; it does not assert that the $p$-primary Brauer group vanishes.

For a group $G$ acting by automorphisms on a possibly noncommutative group $B$, $H^1(G,B)$ is the pointed set of maps satisfying $c_{\sigma \tau }=c_{\sigma }\sigma (c_{\tau })$ modulo $c_{\sigma }\mapsto b{c}_{\sigma }\sigma (b{)}^{-1}$. It classifies twisting of a reference action. It need not be an abelian group.

For a finite group, the usual integral notion asks for a class $\Delta \in H^n(G;\mathbb{Z})$ whose cup product gives the displayed isomorphisms for every coefficient module $M$ and every $i>0$. The analogous claim for just trivial coefficients over a fixed field is weaker and should be stated as such. A free cohomologically orientation-preserving sphere action supplies that field-coefficient version through periodic group cohomology from a free sphere action.

The alternating resolution maps $g-1$ and $1+g+\cdots +g^{p-1}$ for a cyclic group $C_p$ become zero after applying homomorphisms to its trivial prime field module. Thus there is one class in every degree. A two-step shift of the resolution represents a degree-two periodicity class $u$. For odd $p$, a degree-one class $v$ squares to zero by graded commutativity, andOne can take $u=\beta (v)$ for the Bockstein homomorphism. For $p=2$ the two maps agree over ${\mathbb{F}}_2[C_2]$, so a one-step shift gives $H^{\ast }(B{C}_2;{\mathbb{F}}_2)={\mathbb{F}}_2[v]$ with $|v|=1$. The Künneth theorem then gives dimension $i+1$ in degree $i$ for $B(C_p\times C_p)$, obstructing periodic group cohomology and hence such a free sphere action.

Suppose a finite group acts freely on $S^{n-1}$ and trivially on its integral cohomology. The Borel construction gives a sphere fibration over $BG$ whose total space is homotopy equivalent to the $(n-1)$-dimensional quotient. Its Serre spectral sequence has just two rows with trivial local coefficients. Transgressing the fibre generator gives $\Delta$; the sole differential is multiplication by it up to sign. For $i>0$, both the source and target terms have total degrees above $n-1$, so their surviving kernel and cokernel must vanish. Hence multiplication by $\Delta$ gives period $n$ in positive-degree cohomology over any field.

Choose a projective resolution $P_{\bullet }\to \mathbb{Z}$ of the trivial $\mathbb{Z}G$-module. For a $\mathbb{Z}G$-module $M$, group cohomology isDifferent projective resolutions give naturally isomorphic groups.

For $\mathbb{Z}G$-modules $M_1,M_2$,because the Hom functor into a finite direct sum splits degree by degree and taking cohomology preserves that splitting.

For $G=⟨x,y\mid xy=yx⟩$, the trivial $\mathbb{Z}G$-module has the free resolution

If $G\cong {\mathbb{Z}}^2$, $I$ is the augmentation ideal, and $M=\mathbb{Z}G/I^2$, thenThe quotient map $M\to \mathbb{Z}G/I\cong \mathbb{Z}$ induces an isomorphism in second cohomology.

For a subgroup $K\le G$ and a $\mathbb{Z}K$-module $X$, the coinduced module is

Shapiro's lemma gives a natural isomorphismIt follows by applying the Hom functor adjunction for a coinduced module to a projective resolution and observing that restriction from $G$ to $K$ preserves projective modules.

For a $\mathbb{Z}G$-module $P$ and a $\mathbb{Z}K$-module $X$,Evaluation at $1\in G$ gives the forward map. If $a:P\to X$ is $\mathbb{Z}K$-linear, its inverse sends $a$ to $p\mapsto (r\mapsto a(rp))$.

The group ring $\mathbb{Z}G$ becomes a $\mathbb{Z}G$-module under the conjugation action $g\cdot x=gx{g}^{-1}$. Its basis splits into conjugacy classes, and the span of the class of $x$ is the permutation module on $G/C_G(x)$.

For a finite group $G$ and representatives $g_i$ of its conjugacy classes, Shapiro's lemma gives

For the trivial action of the finite cyclic group $C_m$ on $\mathbb{Z}$,for $k\ge 1$. The periodic resolution of a finite cyclic group proves this directly.

If $C_m=⟨t⟩$ and $N=1+t+\cdots +t^{m-1}$, a free resolution of the trivial $\mathbb{Z}{C}_m$-module alternates multiplication by $t-1$ and $N$. Applying ${\mathrm{Hom}}_{\mathbb{Z}{C}_m}(-,\mathbb{Z})$ for the trivial action alternates the zero map and multiplication by $m$.

A group extension of $G$ by an abelian $G$-module $A$ is a short exact sequencewhose conjugation action on $A$ is the specified $G$-action.

For a specified finite group action on $A={\mathbb{Z}}^n$, extensions are classified by $e\in H^2(Q,A)$. For an element $q$ of prime order $p$, a lift $g$ satisfies $g^p=a\in A^q$; changing the lift changes $a$ by $N_{q}v=(1+q+\cdots +q^{p-1})v$. Thus the extension is torsion-free precisely when every restriction to a prime-order cyclic subgroup is nonzero in $A^q/N_{q}A$. Any finite-order element has a prime-order power, so this tests all torsion. An arbitrary finite quotient must not be confused with faithful holonomy on a maximal translation lattice.

A group extension is split when its quotient homomorphism admits a group-homomorphic section. Equivalently, the middle group is a semidirect product of the kernel by the quotient.

If a group homomorphism section $i:G/N\to G$ satisfies $\pi i=\mathrm{id}$, then each $g\in G$ has a unique form $i(q)n$ with $q\in G/N$ and $n\in N$: take $q=\pi (g)$ and $n=i(q{)}^{-1}g$. Thus $(q,n)\mapsto i(q)n$ is a bijection. It is a homomorphism for the ordinary direct product of groups exactly when the section image centralizes $N$. In general it describes a semidirect product with a nontrivial conjugation action.

Two extensions of $G$ by the same $G$-module $M$ are equivalent when an isomorphism between their middle groups commutes with the inclusions of $M$ and the projections to $G$. Splitness is preserved under this equivalence.

For homomorphisms $E\to H$ and $G\to H$, their fiber product isIt is a subgroup of $E\times G$. Pulling a group extension back along $G\to H$ uses this fiber product as its middle group.

For a finite group presentation $H=⟨x_1,\dots ,x_m\mid r_1,\dots ,r_n⟩$, let $F$ be the free group on its generators and $\pi :F\to H$. The corresponding fiber product of groups is generated by $(x_i,x_i)$ and $(1,r_j)$. Conjugation by the diagonal pairs supplies every pair $(1,r)$ with $r$ in the normal closure of the relators, and $(u,v)=(u,u)(1,u^{-1}v)$ proves the generating claim. Membership of $(1,w)$ is equivalent to $w=1$ in $H$, so this finitely generated subgroup can have undecidable membership even though $F\times F$ has soluble word problem.

Equivalence classes of group extensions of $G$ by an abelian $G$-module $A$ correspond to $H^2(G,A)$. A section $s:G\to E$ produces the extension cocycleand changing the section changes $c$ by a group coboundary.

With a fixed identification of the kernel with $\mathbb{Z}$ and quotient with ${\mathbb{Z}}^2$, these central group extensions are classified by $n\in \mathbb{Z}$. Choose lifts $x,y$ of the quotient generators; their commutator is $z^n$ and is unchanged by central changes of lifts. Every element has unique normal form $z^m{y}^b{x}^a$, proving this invariant is complete. A cocycle representative is $c((a,b),(c,d))=nad$, whose cocycle identity follows from bilinearity. Its extension multiplication is $(a,b,m)(c,d,l)=(a+c,b+d,m+l+nad)$. The zero parameter gives ${\mathbb{Z}}^3$ and parameter one gives the integer Heisenberg group. The Koszul resolution for a rank-two free abelian group independently computes $H^2({\mathbb{Z}}^2,\mathbb{Z})\cong \mathbb{Z}$.

An extension cocycle is the two-cocycle obtained from a section of a group extension. It measures the failure of the section to be a group homomorphism.

A group cocycle is an element of the kernel of the coboundary map in the standard cochain complex computing group cohomology.

For a $G$-module $M$, a one-cocycle is a map $f:G\to M$ satisfying $f(gh)=f(g)+g\cdot f(h)$.

A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma \mapsto \sigma u-u$ form $H^1(K,M)$ in Galois cohomology. For an elliptic curve, choosing $mQ=P$ gives the cocycle $\sigma \mapsto \sigma Q-Q$ in $E[m]$, which represents the Kummer map of an elliptic curve.

A crossed homomorphism from $G$ to a $G$-module $M$ is a map $f:G\to M$ satisfyingThus crossed homomorphisms are precisely one-cocycles.

For $m\in M$, the map $g\mapsto g\cdot m-m$ is a principal crossed homomorphism. It is the group coboundary of the zero-cochain $m$, so first group cohomology is the quotient of crossed homomorphisms by principal crossed homomorphisms.

For a $G$-module $M$, a two-cocycle is a map $c:G\times G\to M$ satisfying

A two-cocycle is normalized when $c(1,g)=c(g,1)=0$ in additive notation, or $c(1,g)=c(g,1)=1$ in multiplicative notation.

A group coboundary is a cochain in the image of the preceding coboundary map. Quotienting group cocycles by group coboundaries gives group cohomology.

An inhomogeneous group $n$-cochain with values in a $G$-module $M$ is a function $G^n\to M$. The standard alternating group coboundary turns these groups into the cochain complex whose cohomology is $H^n(G,M)$.

A short exact sequence $0\to M_1\to M_2\to M_3\to 0$ of $G$-modules induces a natural long exact sequenceIt follows by applying the cochain functor and the long exact cohomology sequence of a short exact sequence of cochain complexes.

If $G$ is finite and $n>0$, multiplication by $|G|$ annihilates $H^n(G,M)$. Indeed, restriction to the trivial subgroup is zero in positive degree, while the restriction-corestriction composite is multiplication by $|G|$.

For a subgroup $K\le G$, restriction followed by corestriction acts on group cohomology as the sum over coset translates. For $K$ trivial, inner automorphisms act trivially on cohomology, so this sum is multiplication by $|G|$.

For a free presentation $G=F/R$ and a $\mathbb{Z}G$-module $M$, restriction of derivations to $R$ gives an exact sequenceIt follows by applying ${\mathrm{Hom}}_{\mathbb{Z}G}(-,M)$ to the presentation relation sequence.

For a normal subgroup $H◃G$, quotient $Q=G/H$, and a $G$-module $M$, the Lyndon–Hochschild–Serre spectral sequence has

If $N$ has a finite resolution by finitely generated projective modules, restriction of the semidirect-product group ring gives a direct sum of copies of $\mathbb{Z}N$, indexed by the integers. On its kernel cohomology the cyclic generator shifts those copies, possibly twisting each copy by an automorphism. Invariants are zero because elements have finite support; coinvariants identify with a single copy. The two-column Lyndon–Hochschild–Serre spectral sequence therefore gives the displayed isomorphism of abelian groups. A group with such a finite resolution and cohomological dimension $d$ has $H^d(N;\mathbb{Z}N)\ne 0$, so the cyclic extension has dimension $d+1$.

The low-degree edge maps of the Lyndon–Hochschild–Serre spectral sequence give

For a normal subgroup $N$ of $G$ and a $G$-module $M$, the low-degree exact sequence begins $0\to H^1(G/N,M^N)\to H^1(G,M)\to H^1(N,M{)}^{G/N}\to H^2(G/N,M^N)$. The first map inflates cocycles along $G\to G/N$; the second restricts them. If the restriction is trivial, subtract a coboundary to make the cocycle vanish on $N$ and descend to the quotient. Thus for a finite Galois extension, the kernel of restriction on Galois cohomology with finite coefficients is finite.

Inflation composes a cocycle on the quotient $Q=G/H$ with the quotient map $G\to Q$ and regards its values in the invariant submodule $M^H$ as values in $M$.

Restriction sends a cocycle on $G$ to its restriction to a subgroup $H$. If $H$ is normal, its image in $H^1(H,M)$ is fixed by the induced $G/H$-action.

The transgression $d_2:H^1(H,M{)}^Q\to H^2(Q,M^H)$ is the first differential crossing from the vertical to the horizontal edge of the Lyndon–Hochschild–Serre spectral sequence. If a representative on $H$ is extended to a cochain on $G$, its coboundary descends to the representing two-cocycle on $Q$.

For the dihedral group $D_{10}=C_5⋊C_2$,The Lyndon–Hochschild–Serre spectral sequence proves this from the inversion action of $C_2$ on $H^2(C_5,\mathbb{Z})$.