It is possible to compute the low degree (co)homology of a finite group or monoid of small order directly from the bar resolution. The following commands take this approach to computing the fifth integral homology
H_5(Q_4, Z) = Z_2⊕ Z_2
of the quaternion group G=Q_4 of order 8.
gap> Q:=QuaternionGroup(8);; gap> B:=BarComplexOfMonoid(Q,6);; gap> C:=ContractedComplex(B);; gap> Homology(C,5); [ 2, 2 ]
However, this approach is of limited applicability since the bar resolution involves |G|^k free generators in degree k. A range of techniques, tailored to specific classes of groups, can be used to compute the (co)homology of larger finite groups.
The following example computes the fourth integral cohomomogy of the Mathieu group M_24.
H^4(M_24, Z) = Z_12
gap> GroupCohomology(MathieuGroup(24),4); [ 4, 3 ]
The following example computes the third integral homology of the Weyl group W=Weyl(E_8), a group of order 696729600.
H_3(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_12
p> L:=SimpleLieAlgebra("E",8,Rationals);; gap> W:=WeylGroup(RootSystem(L));; gap> Order(W); 696729600 gap> GroupHomology(W,3); [ 2, 2, 4, 3 ]
The preceding calculation could be achieved more quickly by noting that W=Weyl(E_8) is a Coxeter group, and by using the associated Coxeter polytope. The following example uses this approach to compute the fourth integral homology of W. It begins by displaying the Coxeter diagram of W, and then computes
H_4(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2.
gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3],[5,3]],[5,[6,3]],[6,[7,3]],[7,[8,3]]];; gap> CoxeterDiagramDisplay(D);
gap> polytope:=CoxeterComplex_alt(D,5);; gap> R:=FreeGResolution(polytope,5); Resolution of length 5 in characteristic 0 for <matrix group with 8 generators> . No contracting homotopy available. gap> C:=TensorWithIntegers(R); Chain complex of length 5 in characteristic 0 . gap> Homology(C,4); [ 2, 2, 2, 2 ]
The following example computes the sixth mod-2 homology of the Sylow 2-subgroup Syl_2(M_24) of the Mathieu group M_24.
H_6(Syl_2(M_24), Z_2) = Z_2^143
gap> GroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
The following example constructs the Poincare polynomial
p(x)=frac1-x^3+3*x^2-3*x+1
for the cohomology H^∗(Syl_2(M_12, F_2). The coefficient of x^n in the expansion of p(x) is equal to the dimension of the vector space H^n(Syl_2(M_12, F_2). The computation involves Singular's Groebner basis algorithms and the Lyndon-Hochschild-Serre spectral sequence.
gap> G:=SylowSubgroup(MathieuGroup(12),2);; gap> PoincareSeriesLHS(G); (1)/(-x_1^3+3*x_1^2-3*x_1+1)
The following example constructs the polynomial
p(x)=fracx^4-x^3+x^2-x+1x^6-x^5+x^4-2*x^3+x^2-x+1
whose coefficient of x^n is equal to the dimension of the vector space H^n(M_11, F_2) for all n in the range 0≤ n≤ 14. The coefficient is not guaranteed correct for n≥ 15.
gap> PoincareSeriesPrimePart(MathieuGroup(11),2,14); (x_1^4-x_1^3+x_1^2-x_1+1)/(x_1^6-x_1^5+x_1^4-2*x_1^3+x_1^2-x_1+1)
The following example computes
H_4(N, Z) = (Z_3)^4 ⊕ Z^84
for the free nilpotent group N of class 2 on four generators.
gap> F:=FreeGroup(4);; N:=NilpotentQuotient(F,2);; gap> GroupHomology(N,4); [ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
The following example computes
H_5(G, Z) = Z_2 ⊕ Z_2
for the 3-dimensional crystallographic space group G with Hermann-Mauguin symbol "P62"
gap> GroupHomology(SpaceGroupBBNWZ("P62"),5); [ 2, 2 ]
The following example computes
H_6(SL_2(cal O, Z) = Z_2
for cal O the ring of integers of the number field Q(sqrt-2).
gap> C:=ContractibleGcomplex("SL(2,O-2)");; gap> R:=FreeGResolution(C,7);; gap> Homology(TensorWithIntegers(R),6); [ 2, 12 ]
The following example computes
H_5(G, Z) = Z_3
for G the classical braid group on eight strings.
gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,3]]];; gap> CoxeterDiagramDisplay(D);;
gap> R:=ResolutionArtinGroup(D,6);; gap> C:=TensorWithIntegers(R);; gap> Homology(C,5); [ 3 ]
The following example computes
H_5(G, Z) = Z_2⊕ Z_2⊕ Z_2 ⊕ Z_2 ⊕ Z_2
for G the graph of groups corresponding to the amalgamated product G=S_5*_S_3S_4 of the symmetric groups S_5 and S_4 over the canonical subgroup S_3.
gap> S5:=SymmetricGroup(5);SetName(S5,"S5"); gap> S4:=SymmetricGroup(4);SetName(S4,"S4"); gap> A:=SymmetricGroup(3);SetName(A,"S3"); gap> AS5:=GroupHomomorphismByFunction(A,S5,x->x); gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x); gap> D:=[S5,S4,[AS5,AS4]]; gap> GraphOfGroupsDisplay(D);
gap> R:=ResolutionGraphOfGroups(D,6);; gap> Homology(TensorWithIntegers(R),5); [ 2, 2, 2, 2, 2 ]
There are various ways to represent a ZG-module A with action G× A → A, (g,a)↦ α(g,a).
One possibility is to use the data type of a G-Outer Group which involves three components: an ActedGroup A; an Acting Group G; a Mapping (g,a)↦ α(g,a). The following example uses this data type to compute the cohomology H^4(G,A) = Z_5 ⊕ Z_10 of the symmetric group G=S_6 with coefficients in the integers A= Z where odd permutations act non-trivially on A.
gap> G:=SymmetricGroup(6);; gap> A:=AbelianPcpGroup([0]);; gap> alpha:=function(g,a); return a^SignPerm(g); end;; gap> A:=GModuleAsGOuterGroup(G,A,alpha); ZG-module with abelian invariants [ 0 ] and G= SymmetricGroup( [ 1 .. 6 ] ) gap> R:=ResolutionFiniteGroup(G,5);; gap> C:=HomToGModule(R,A); G-cocomplex of length 5 . gap> Cohomology(C,4); [ 2, 2, 5 ]
If A= Z^n and G acts as
G× A → A, (g, (x_1,x_2,...,x_n)) ↦ (x_π(g)^-1(1), x_π(g)^-1(2), ..., x_π(g)^-1(n))
where π: G→ S_n is a (not necessarily faithful) permutation representation of degree n then we can avoid the use of G-outer groups and use just the homomorphism π instead. The following example uses this data type to compute the cohomology
H^6(G,A) = Z_2 ⊕ Z_6
and the homology
H_6(G,A) = Z_2
of the alternating group G=A_6 with coefficients in A= Z^5 where elements of G act on Z^5 via the canonical permutation of basis elements.
gap> G:=AlternatingGroup(5);; gap> pi:=PermToMatrixGroup(SymmetricGroup(5),5);; gap> R:=ResolutionFiniteGroup(G,7);; gap> C:=HomToIntegralModule(R,pi);; gap> Cohomology(C,6); [ 2, 6 ] gap> D:=TensorWithIntegralModule(R,pi);; gap> Homology(D,6); [ 2 ]
A short exact sequence of ZG-modules A ↣ B ↠ C induces a long exact sequence of cohomology groups
→ H^n(G,A) → H^n(G,B) → H^n(G,C) → H^n+1(G,A) → .
Consider the symmetric group G=S_4 and the sequence Z_4 ↣ Z_8 ↠ Z_2 of trivial ZG-modules. The following commands compute the induced cohomology homomorphism
f: H^3(S_4, Z_4) → H^3(S_4, Z_8)
and determine that the image of this induced homomorphism has order 8 and that its kernel has order 2.
gap> G:=SymmetricGroup(4);; gap> x:=(1,2,3,4,5,6,7,8);; gap> a:=Group(x^2);; gap> b:=Group(x);; gap> ahomb:=GroupHomomorphismByFunction(a,b,y->y);; gap> A:=TrivialGModuleAsGOuterGroup(G,a);; gap> B:=TrivialGModuleAsGOuterGroup(G,b);; gap> phi:=GOuterGroupHomomorphism();; gap> phi!.Source:=A;; gap> phi!.Target:=B;; gap> phi!.Mapping:=ahomb;; gap> Hphi:=CohomologyHomomorphism(phi,3);; gap> Size(ImageOfGOuterGroupHomomorphism(Hphi)); 8 gap> Size(KernelOfGOuterGroupHomomorphism(Hphi)); 2
The following commands then compute the homomorphism
H^3(S_4, Z_8) → H^3(S_4, Z_2)
induced by Z_4 ↣ Z_8 ↠ Z_2, and determine that the kernel of this homomorphsim has order 8.
gap> bhomc:=NaturalHomomorphismByNormalSubgroup(b,a); gap> B:=TrivialGModuleAsGOuterGroup(G,b); gap> C:=TrivialGModuleAsGOuterGroup(G,Image(bhomc)); gap> psi:=GOuterGroupHomomorphism(); gap> psi!.Source:=B; gap> psi!.Target:=C; gap> psi!.Mapping:=bhomc; gap> Hpsi:=CohomologyHomomorphism(psi,3); gap> Size(KernelOfGOuterGroupHomomorphism(Hpsi)); 8
The following commands then compute the connecting homomorphism
H^2(S_4, Z_2) → H^3(S_4, Z_4)
and determine that the image of this homomorphism has order 2.
gap> delta:=ConnectingCohomologyHomomorphism(psi,2);; gap> Size(ImageOfGOuterGroupHomomorphism(delta));
Note that the various orders are consistent with exactness of the sequence
H^2(S_4, Z_2) → H^3(S_4, Z_4) → H^3(S_4, Z_8) → H^3(S_4, Z_2) .
A spherical 3-manifold is a 3-manifold arising as the quotient S^3/Γ of the 3-sphere S^3 by a finite subgroup Γ of SO(4) acting freely as rotations. The geometrization conjecture, proved by Grigori Perelman, implies that every closed connected 3-manifold with a finite fundamental group is homeomorphic to a spherical 3-manifold.
A spherical 3-manifold S^3/Γ has finite fundamental group isomorphic to Γ. This fundamental group is one of:
Γ=C_m=⟨ x | x^m⟩ (cyclic fundamental group)
Γ=C_m× ⟨ x,y | xyx^-1=y^-1, x^2^k=y^n ⟩ for integers k, m≥ 1, n≥ 2 and m coprime to 2n (prism manifold case)
Γ= C_m× ⟨ x,y, z | (xy)^2=x^2=y^2, zxz^-1=y, zyz^-1=xy, z^3^k=1⟩ for integers k,m≥ 1 and m coprime to 6 (tetrahedral case)
Γ=C_m×⟨ x,y | (xy)^2=x^3=y^4⟩ for m≥ 1 coprime to 6 (octahedral case)
Γ=C_m× ⟨ x,y | (xy)^2=x^3=y^5⟩ for m≥ 1 coprime to 30 (icosahedral case).
This list of cases is taken from the Wikipedia pages. The group Γ has periodic cohomology since it acts on a sphere. The cyclic group has period 2 and in the other four cases it has period 4. (Recall that in general a finite group G has periodic cohomology of period n if there is an element u∈ H^n(G, Z) such that the cup product - ∪ u: H^k(G, Z) → H^k+n(G, Z) is an isomorphism for all k≥ 1. It can be shown that G has periodic cohomology of period n if and only if H^n(G, Z)= Z_|G|.)
The cohomology of the cyclic group is well-known, and the cohomology of a direct product can be obtained from that of the factors using the Kunneth formula.
In the icosahedral case with m=1 the following commands yield $$H^\ast(\Gamma,\mathbb Z)=Z[t]/(120t=0)$$ with generator t of degree 4. The final command demonstrates that a periodic resolution is used in the computation.
gap> F:=FreeGroup(2);;x:=F.1;;y:=F.2;; gap> G:=F/[(x*y)^2*x^-3, x^3*y^-5];; gap> Order(G); 120 gap> R:=ResolutionSmallGroup(G,5);; gap> n:=0;;Cohomology(HomToIntegers(R),n); [ 0 ] gap> n:=1;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=2;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=3;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=4;;Cohomology(HomToIntegers(R),n); [ 120 ] gap> List([0..5],k->R!.dimension(k)); [ 1, 2, 2, 1, 1, 2 ]
In the octahedral case with m=1 we obtain $$H^\ast(\Gamma,\mathbb Z) = \mathbb Z[s,t]/(s^2=24t, 2s=0, 48t=0)$$ where s has degree 2 and t has degree 4, from the following commands.
gap> F:=FreeGroup(2);;x:=F.1;;y:=F.2;; gap> G:=F/[(x*y)^2*x^-3, x^3*y^-4];; gap> Order(G); 48 gap> R:=ResolutionFiniteGroup(G,5);; gap> n:=0;;Cohomology(HomToIntegers(R),n); [ 0 ] gap> n:=1;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=2;;Cohomology(HomToIntegers(R),n); [ 2 ] gap> n:=3;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=4;;Cohomology(HomToIntegers(R),n); [ 48 ] gap> IntegralCupProduct(R,[1],[1],2,2); [ 24 ]
In the tetrahedral case with m=1 we obtain $$H^\ast(\Gamma,\mathbb Z) = \mathbb Z[s,t]/(s^2=16t, 3s=0, 24t=0)$$ where s has degree 2 and t has degree 4, from the following commands.
gap> F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;; gap> G:=F/[(x*y)^2*x^-2, x^2*y^-2, z*x*z^-1*y^-1, z*y*z^-1*y^-1*x^-1,z^3];; gap> Order(G); 24 gap> R:=ResolutionFiniteGroup(G,5);; gap> n:=1;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=2;;Cohomology(HomToIntegers(R),n); [ 3 ] gap> n:=3;;Cohomology(HomToIntegers(R),n); [ ] gap> n:=4;;Cohomology(HomToIntegers(R),n); [ 24 ] gap> IntegralCupProduct(R,[1],[1],2,2); [ 16 ]
A theoretical calculation of the integral and mod-p cohomology rings of all of these fundamental groups of spherical 3-manifolds is given in [TZ08].
generated by GAPDoc2HTML