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 <td style="vertical-align: top;"><a href="aboutCoxeter.html">Previous</a> 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Integral Cohomology Rings 
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href="aboutModPRings.html">next</a> 
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The
integral cohomology Hn(G,Z) of a group G is by definition
the quotient 
 
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 Hn(G,Z) = 
 </td>
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style="text-align: center; vertical-align: top; background-color: rgb(204, 255, 255);">
 <div style="text-align: center;"> 
Ker( HomZG(Rn,Z) → HomZG(Rn+1,Z) ) 
 <hr style="width: 100%; height: 2px;"> </div>
 <div style="text-align: center;">Image( HomZG(Rn-1,Z) → HomZG(Rn,Z) 
 
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where HomZG(Rn,Z)
denotes the abelian group of G-equivariant homomorphisms from the n-th
term of a free ZG-resolution R to the trivial G-module Z. The
ZG-homomorphisms in 
 
 
 <div style="text-align: center;">Zn(R,Z) = Ker( HomZG(Rn,Z)
→ HomZG(Rn+1,Z)
) 
 </div>
 
are called n-cocycles. Those
cocycles lying in 
 
 
 <div style="text-align: center;">Bn(R,Z) = Image( HomZG(Rn-1,Z) → HomZG(Rn,Z) 
 </div>
 
are called n-coboundaries. 
 
For a finite group G the Universal Coefficient Theorem and the lack of
torsion in Z imply an isomorphism Hn+1(G,Z) = Hn(G,Z) for n>0.
The isomorphism does not hold in general for infinite groups. 
 
The following commands calculate the integral cohomology groups
of the Artin group
defined by the diagram 
 
 <div style="text-align: center;"><img alt="" src="coxeter2.jpg"
style="width: 207px; height: 114px;"> 
 
 <div style="text-align: left;">and show that they are different to its
integral homology groups calculated on the previous page. 
 </div>
 </div>
 </td>
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 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
D:=[ [1,[2,3],[4,3]], [2,[3,3],[5,0]], [3,[4,4]], [5,[6,4],[7,4]] ];; 
 
gap> R:=ResolutionArtinGroup(D,8);; 
 
gap> TR:=HomToIntegers(R);; 
 
gap> for i in [1..7] do 
> Print(Cohomology(TR,i),"\n"); 
> od; 
[ 0, 0, 0, 0 ] 
[ 0, 0, 0, 0, 0, 0, 0, 0 ] 
[ 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
[ 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
[ 2, 2, 0, 0, 0, 0 ] 
[ ] 
[ ] 
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Cohomology
is a contravariant functor, meaning
that a group homomorphism f:G→Q induces a cohomology homomorphism Hn(f):Hn(Q,Z)
→ Hn(G,Z) for all n. The following commands compute the
orders of the kernel and image of H3(f) for the quotient
homomorphism f:B5 → S5 from the 5-string braid
group to the symmetric group of degree 5. 
 </td>
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]]];; 
gap> R:=ResolutionArtinGroup(D,4);; 
gap> TR:=HomToIntegers(R);; 
 
gap> for i in [1..3]
do
#These seemingly unnecessary commands are needed to 
gap> x:=Cohomology(TR,i);; od;;
#construct the boundary homomorphisms in R. 
 
gap> B5:=R!.group;; B5gens:=GeneratorsOfGroup(B5);; 
gap> S5:=SymmetricGroup(5);; S5gens:=[(1,2),(2,3),(3,4),(4,5)];; 
gap> S:=ResolutionFiniteGroup(S5,4);; 
 
gap> f:=GroupHomomorphismByImages(B5,S5,B5gens,S5gens);; 
gap> eqchmap:=EquivariantChainMap(R,S,f);; 
gap> chmap:=HomToIntegers(eqchmap);; 
gap> Hf:=Cohomology(chmap,3);; 
 
gap> Order(Kernel(Hf)); 
1 
gap> Order(Image(Hf)); 
2 
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style="background-color: rgb(255, 255, 255); vertical-align: top;">For any group G there is a
bilinear mapping 
 
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 Hp(G,Z)
× Hq(G,Z) → Hp+q(G,Z),
(u,v) →u·v 
 
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 </tr>
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called the cup product.
The product is associative, and u·v = (-1)pq
v·u . 
 
(One construction of the cup product is as follows. Let R be a
ZG-resolution of Z. The cohomology class u is represented by a cocycle u : Rp → Z which
induces a chain mapping un
: Rn→ Rn-p
(for n > p-1). The composition of up+q with the
cocycle v: Rq
→ Z is a cocycle representing a cohomology class u·v in Hp+q(G,Z).) 
 
 
We need some notation for the HAP function IntegralCupProduct(R,u,v,p,q)
 . 
 <ul>
 <li>Let ai be the i-th canonical generator of the
d-generator
abelian group Hn(G,Z). A cohomology class n1a1
+ ... +ndad is represented by the integer
vector
u=[n1, ..., nd].</li>
 </ul>
The following commands illustrate the cup product H4(S4,Z) × H4(S4,Z)
→ H8(S4,Z) in the integral cohomology of the
symmetric group S4 . </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
R:=ResolutionFiniteGroup(SymmetricGroup(4), 9 );; 
 
gap> TR:=HomToIntegers(R);; 
 
gap> Cohomology(TR,4); 
[ 2, 12 ] 
 
gap> Cohomology(TR,8); 
[ 2, 2, 12 ] 
 
gap> u:=[1,0];; v:=[0,1];; p:=4;; q:=4;; 
 
gap> IntegralCupProduct(R,u,v,p,q); 
[ 1, 0, 6 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
above command can be used to compute the complete cohomology ring H*(G,Z)
for finite index subgroups of Artin groups where the K(pi,1) conjecture
is known to hold (once I've implemented the contracting homotopy on the
associated resolution). For example, the four generator affine braid
group, which was considered on the previous page, has cohomology ring 
 
 <div style="text-align: center;">H*(AD,Z) =
Z[s,t,u1,u2,u3,u4] / (st, 2u3,
2u4, J>3) 
 <div style="text-align: left;"> 
 </div>
 </div>
where deg(s)=1, deg(t)=2, deg(ui)=3. 
 
The integral cohomology ring was computed for all 4-generator Artin
groups with compact hyperbolic Coxeter group in the preprint [G. Ellis
& E. Sköldberg, "Cohomology rings for some non-spherical Artin
groups", http://hamilton.nuigalway.ie] using Haskell code written
by Emil Sköldberg. Previously, the integral cohomology rings of
all the spherical Artin groups has been calculated by theoretical
methods in the papers: 
 <ul>
 <li>F.V. Vainshtein, "The cohomology of braid groups
(Russian)", Funktsional. Anal. i
Prilozhen, 12 no. 2 (1978), 72-73.</li>
 <li>V.V. Gorjunov, "The cohomology of braid groups of series C
and D and certain stratifications (Russian)",
style="font-style: italic;">Funktsional. Anal. i Prilozhen, 12
no. 2 (1978), 76-77.</li>
 <li> C. Landi, "Cohomology rings of Artin groups", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat.
Natur. Rend. Lincei (9) Mat. Appl., 11 no. 1 (2000), 41-65.</li>
 </ul>
 </td>
 </tr>
 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">Given a ZG-resolution R the
function IntegralRingGenerators(R,n)
returns a list of cohomology classes in Hn(G,Z)
which, together with all cup products of classes of lower degrees,
generates the group Hn(G,Z). 
 
 <a name="rankthree"></a>The following commands show that, for the
free nilpotent group G of
class 2 on three generators, the cohomology H*(G,Z) is
generated as a ring
by three classes in dimension 1, eight classes in dimension 2 and
six classes in dimension 3. They also show that, as an abelian
group, the ring H*(G,Z) is free abelian of rank 35. 
 </td>
 </tr>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FreeGroup(3);;G:=NilpotentQuotient(F,2);; 
gap> R:=ResolutionNilpotentGroup(G,10);; 
 
gap> for n in [1..9] do 
> Print("Cohomology group in dimension ",n," =
",Cohomology(HomToIntegers(R),n),"\n");

>od; 
Cohomology group in dimension 1 = [ 0, 0, 0 ] 
Cohomology group in dimension 2 = [ 0, 0, 0, 0, 0, 0, 0, 0 ] 
Cohomology group in dimension 3 = [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
Cohomology group in dimension 4 = [ 0, 0, 0, 0, 0, 0, 0, 0 ] 
Cohomology group in dimension 5 = [ 0, 0, 0 ] 
Cohomology group in dimension 6 = [ 0 ] 
Cohomology group in dimension 7 = [ ] 
Cohomology group in dimension 8 = [ ] 
Cohomology group in dimension 9 = [ ] 
 
gap> for n in [1..9] do 
> Print("Ring generators in dimension ",n, " =
",IntegralRingGenerators(R,n),"\n");

> od; 
Ring generators in dimension 1 = [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1
] ] 
Ring generators in dimension 2 = 
[ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], 
 [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], 
 [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], 
 [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ] ] 
Ring generators in dimension 3 = 
[ [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], 
 [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], 
 [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], 
 [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], 
 [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], 
 [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ] 
Ring generators in dimension 4 = [ ] 
Ring generators in dimension 5 = [ ] 
Ring generators in dimension 6 = [ ] 
Ring generators in dimension 7 = [ ] 
Ring generators in dimension 8 = [ ] 
Ring generators in dimension 9 = [ ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Similar
commands can be used to obtain partial information on the cohomology
rings of finite groups. For instance, the following commands show that
the first fifteen
degrees of the (infinite dimensional) graded ring H*(S4,Z)
are generated
by 
 <ul>
 <li>one class of order 2 in degree 2,</li>
 <li>one class of order 2 in degree 3,</li>
 <li>and one class of order 12 in degree 4.</li>
 </ul>
One would have to use theoretical arguments to prove that H*(S4,Z)
has no other generators in higher degrees. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionFiniteGroup(SymmetricGroup(4),16);; 
gap> TR:=HomToIntegers(R);; 
gap> for n in [1..15] do 
> Print("The cohomology in degree ", n," is ",
Cohomology(TR,n),"\n"); 
> Print("The cohomology ring generators in degree ", n, " are
",

> IntegralRingGenerators(R,n), "\n"); 
> od; 
The cohomology in degree 1 is [ ] 
The cohomology ring generators in degree 1 are [ ] 
The cohomology in degree 2 is [ 2 ] 
The cohomology ring generators in degree 2 are [ [ 1 ] ] 
The cohomology in degree 3 is [ 2 ] 
The cohomology ring generators in degree 3 are [ [ 1 ] ] 
The cohomology in degree 4 is [ 2, 12 ] 
The cohomology ring generators in degree 4 are [ [ 0, 1 ] ] 
The cohomology in degree 5 is [ 2 ] 
The cohomology ring generators in degree 5 are [ ] 
The cohomology in degree 6 is [ 2, 2, 2 ] 
The cohomology ring generators in degree 6 are [ ] 
The cohomology in degree 7 is [ 2, 2 ] 
The cohomology ring generators in degree 7 are [ ] 
The cohomology in degree 8 is [ 2, 2, 12 ] 
The cohomology ring generators in degree 8 are [ ] 
The cohomology in degree 9 is [ 2, 2, 2 ] 
The cohomology ring generators in degree 9 are [ ] 
The cohomology in degree 10 is [ 2, 2, 2, 2 ] 
The cohomology ring generators in degree 10 are [ ] 
The cohomology in degree 11 is [ 2, 2, 2 ] 
The cohomology ring generators in degree 11 are [ ] 
The cohomology in degree 12 is [ 2, 2, 2, 2, 12 ] 
The cohomology ring generators in degree 12 are [ ] 
The cohomology in degree 13 is [ 2, 2, 2, 2 ] 
The cohomology ring generators in degree 13 are [ ] 
The cohomology in degree 14 is [ 2, 2, 2, 2, 2 ] 
The cohomology ring generators in degree 14 are [ ] 
The cohomology in degree 15 is [ 2, 2, 2, 2, 2 ] 
The cohomology ring generators in degree 15 are [ ] 
 </td>
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