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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Cohomology of some Artin groups 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCoxeter.html">next</a> 
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 <big></big> 
 </th>
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">So
far we have computed the homology of mainly finite groups or infinite
nilpotent groups. We now
turn to infinite groups such as the braid group on n+1 strings. This is
an example
of an Artin group, the
definition of which we now recall. 
 
A Coxeter diagram is a
finite graph D with at most one edge joining any pair of vertices, and
with each edge labelled by an integer n>2 or n=infinity. The
label n=3 occurs frequently and so, in pictures of D, we denote it by
an unmarked edge. For
typographical reasons, when the label is infinity we shall denote it by
the symbol 0 (but treat it as infinity in any mathematical discussion).
Here are three examples. 
 <div style="text-align: center;"><img alt="" src="coxeter.jpg"
style="width: 224px; height: 286px;"> 
 <div style="text-align: left;"> 
 We can succinctly represent a Coxeter
graph by numbering its vertices and recording a list D = [L1,
... Lt] in which each term is itself a list Lk =
[vk,
[uk1,nk1], [uk2,nk2], ... [ukm,nkm]] such that
vertex vk is connected to vertex ukj by an edge
with label nkj. It is sufficient to record just those
vertices ukj > vk. We set nkj=0 when the edge label is infinity. 
 
The above three diagrams are encoded by the following commands. 
 </div>
 </div>
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
D1:=[ [1,[2,3]], [2,[3,3]], [3,[4,4]] ];; 
 
gap> D2:=[ [1,[2,3],[4,3]], [2,[3,3]], [3,[4,3]] ];; 
 
gap> D3:=[ [1,[2,3],[4,3]], [2,[3,3],[5,0]], [3,[4,4]],
[5,[6,4],[7,4]] ];; 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
Coxeter diagram D can be viewed as a .gif picture using the function CoxeterDiagramDisplay().
For example, the following command displays the above diagram D3. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
CoxeterDiagramDisplay(D3,"mozilla");; 
 
 <div style="text-align: center;"><img alt="" src="D3.gif"
style="width: 320px; height: 269px;"> 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">An Artin group is a finitely presented
group AD associated to a Coxeter diagram D as follows: 
 <ul>
 <li>There is one generator x for each vertex in D.</li>
 <li>There is one relator (xy)n = (yx)n
for each pair of vertices not connected by an infinity edge in D where:
if the vertices x,y are connected by no edge then n=2, otherwise n is
the edge label; the word (xy)n = xyxyx... is a
product of precisely n generators.</li>
 </ul>
Also associated to the diagram D is the finitely presented Coxeter group WD. This
is the quotient of AD obtained by imposing the additional
relations x2 = 1 for each generator x. 
 
The following commands give the Artin group associated the first of the
above diagrams, and the Coxeter group associated to the second. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
CoxeterDiagramFpArtinGroup(D1); 
[ <free group on the generators [ f1, f2, f3, f4 ]>, 
 [ f1*f2*f1*f2^-1*f1^-1*f2^-1, f1*f3*f1^-1*f3^-1,
f1*f4*f1^-1*f4^-1, 
 f2*f3*f2*f3^-1*f2^-1*f3^-1,
f2*f4*f2^-1*f4^-1, 
 f3*f4*f3*f4^-1*f3^-1*f4^-1 ] ] 
 
gap> CoxeterDiagramFpCoxeterGroup(D2); 
[ <free group on the generators [ f1, f2, f3, f4 ]>, 
 [ f1*f2*f1*f2^-1*f1^-1*f2^-1, f1*f3*f1^-1*f3^-1, 
 f1*f4*f1*f4^-1*f1^-1*f4^-1,
f2*f3*f2*f3^-1*f2^-1*f3^-1, 
 f2*f4*f2^-1*f4^-1,
f3*f4*f3*f4^-1*f3^-1*f4^-1, f1^2, f2^2, f3^2, f4^2 ] ] 
 </td>
 </tr>
 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">An
Artin group AD is said to be spherical if the associated Coxeter
group is finite. The following commands show that the first of the
above diagrams yields a spherical Artin group, whereas the second and
third diagrams both yield non-spherical groups. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
CoxeterDiagramIsSpherical(D1); 
true 
 
gap> CoxeterDiagramIsSpherical(D2); 
false 
 
gap> CoxeterDiagramIsSpherical(D3); 
false 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Some
years ago Craig Squier discovered a resolution R for spherical Artin
groups AD. The n-dimensional generators of R correspond to
the
subsets of vertices of D of size n. Thus Rn=0 for n greater
than the number of vertices in D.
This result was only published more recently in [ C.C. Squier, "The
homological algebra of Artin groups", Math.
Scand., 75 no. 1 (1994), 5-43]. The resolution was independently
re-discovered by M. Salvetti [M. Salvetti, "The homotopy type of Artin
groups, Math. Res. Lett., 1
no. 5 (1994),
565-577]. 
 
The resolution for a spherical Artin group AD can be
obtained as the cellular chain complex of an easily constructed
cellular space XD. For the construction we note that the
finite Coxeter group WD is isomorphic to a group generated
by d reflections in Euclidean space Rd, where d is the
number of vertices in the diagram D. Choose any vector v in Rd
which is fixed by no reflection in WD. The convex hull of
the orbit of v under the action of WD is then a polytope
whose 1-skeleton can be viewed as the cayley graph of WD.
The edges of faces in the polytope are thus labelled by the generating
reflections in WD. Let YD be the space obtained
from this polytope by identifying all similary labelled faces (in all
dimensions <d). The space XD is the universal cover of YD. 
 
As an example, the space YD for the 3-generator braid group
is obtained by identifying similarly labelled faces of the following
3-dimensional polytope. 
 <div style="text-align: center;"> <img alt="" src="s4.jpg"
style="width: 408px; height: 278px;"> 
 </div>
The following commands use this resolution to show that the Artin group
corresponding to the first of the three diagrams above has integral
homology H1(AD,Z)=Z+Z,
H2(AD,Z)=Z2+Z+Z, H3(AD,Z)=Z+Z,
H4(AD,Z)=Z and Hn(AD,Z)=0
for n>4. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionArtinGroup(D1,5);; 
 
gap> TR:=TensorWithIntegers(R);; 
 
gap> Homology(TR,1); 
[ 0, 0 ] 
 
gap> Homology(TR,2); 
[ 2, 0, 0 ] 
 
gap> Homology(TR,3); 
[ 0, 0 ] 
 
gap> TRHomology(R,4); 
[ 0 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
can, in principle, use a ZG-resolution R to compute the homology of a
finite index subgroup K<G. The command ResolutionSubgroup(R,K)
can be used for this. 
 
For example, any Artin group AD has a normal subgroup 2AD
 , the even subgroup,
consisting of all products of an even number of generators of AD.
The following commands show that the even subgroup of the 5-string
braid group has integral homology H1(2AD,Z)=Z, H2(2AD,Z)=Z2+Z2,
H3(2AD,Z)=Z5, Hn(2AD,Z)=0
for n>3. 
 
(As a curiosity, we note that similar commands can be
used to show that, for certain Coxeter diagrams such as D=E7,
the Artin group has the same integral homology as its even subgroup.) 
 </td>
 </tr>
 <tr>
 <td
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,5);; 
 
gap> A_D:=R!.group;; 
 
gap> 2A_D:=EvenSubgroup(A_D);; 
 
gap> S:=ResolutionSubgroup(R,2A_D);; 
 
gap> TS:=TensorWithIntegers(S);; 
 
gap> for i in [1..4] do 
> Print(Homology(TS,i),"\n"); 
> od; 
[ 0 ] 
[ 2, 2 ] 
[ 5 ] 
[ ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Squier's
resolution for Artin groups can be viewed as the cellular chain complex
of a
contractible
space XD on which AD acts freely. The space XD
exists even
when AD is not spherical and its n-cells correspond to those
subsets S of the vertices of D such that |S|=n and the image of S
generates a finite subgroup in WD. It is conjectured that XD
is
always contractible. In those cases where the conjecture is known to
hold the command ResolutionArtin(D)
can be used to construct a free ZAD-resolution R. (In
all cases one can view the output R of this command as a free ZMD-resolution
where MD
is the Artin monoid defined by D.) 
 
The conjecture has been studied by many people. It is
discussed in [G. Ellis & E. Sköldberg,"The K(pi,1) conjecture
for a class of Artin groups. Comment. Math. Helv.,
85, no. 2, 409--415 (2010)]. That preprint gives a short proof of
the following result. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">Suppose that XT is contractible for every connected
full subgraph T of D where T involves no infinity edges. Then XD is contractible. 
 </td>
 </tr>
 </tbody>
 </table>
 
(A special case of the above result, in which each AT is
assumed to be spherical, was proved in [R. Charney and M.W. Davis, "The
K(\pi,1) problem for hyperplane complements associated to infinite
reflection groups", Journal Amer.
Math. Soc., vol. 8, issue 3 (1995), 597-627].) 
 
The paper explains
how a lemma in [ K.J. Appel and P.E. Schupp, "Artin groups and
infinite Coxeter groups", Invent.
Math., 72 (1983), 201-220] implies the following result. 
 
 <table
style="background-color: rgb(204, 255, 255); margin-left: auto; margin-right: auto; text-align: left; width: 568px; height: 101px;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="width: 80%; vertical-align: top;">Let x, y, z be an arbitrary triple
of vertices in D. Let nxy denote the label on the edge
joining x and y when such an edge exists; otherwise let nxy=2.
The space XD is contractible if 
 
 <div style="text-align: center;">1/nxy + 1/nyz +
1/nxz < 1 or = 1. 
 </div>
 </td>
 </tr>
 </tbody>
 </table>
 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
further case when XD is known to be contractible is proved
in [R. Charney & D. Peifer, "The $K(\pi,1)$-conjecture for the
affine braid groups", Comment. Math.
Helv., 78 no. 3 (2003), 584--600.] Their result is
the following. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">The space XD is contractible
when the diagram D is an n-sided polygon with each side labelled
by n=3. (The corresponding group AD is called the affine braid group.) 
 </td>
 </tr>
 </tbody>
 </table>
 
The following commands completely determine the additive structure of
the integral homology of the affine braid groups on six, seven, eight
and nine generators. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
D6gens:=[[1,[2,3],[6,3]], [2,[3,3]], [3,[4,3]], [4,[5,3]], [5,[6,3]] ];; 
gap> R:=ResolutionArtinGroup(D6gens,7);; 
gap> TR:=TensorWithIntegers(R);; 
gap> for n in [1..6] do 
> Print(Homology(TR,n),"\n"); 
> od; 
[ 0 ] 
[ 2, 0 ] 
[ 2, 2, 0 ] 
[ 3, 3, 3, 3, 0 ] 
[ 0, 0 ] 
[ ] 
 
 
gap> D7gens:=[ [1,[2,3],[7,3]], [2,[3,3]], [3,[4,3]], [4,[5,3]],
[5,[6,3]], [6,[7,3]] ];; 
gap> R:=ResolutionArtinGroup(D7gens,8);; 
gap> TR:=TensorWithIntegers(R);; 
gap> for n in [1..7] do 
> Print(Homology(TR,n),"\n"); 
> od; 
[ 0 ] 
[ 2, 0 ] 
[ 2, 0 ] 
[ 6, 0 ] 
[ 0 ] 
[ 0 ] 
[ ] 
 
 
gap> D8gens:=[ [1,[2,3],[8,3]], [2,[3,3]], [3,[4,3]], [4,[5,3]],
[5,[6,3]], [6,[7,3]], [7,[8,3]] ];; 
gap> R:=ResolutionArtinGroup(D8gens,9);; 
gap> TR:=TensorWithIntegers(R);; 
gap> for n in [1..8] do 
> Print(Homology(TR,n),"\n"); 
> od; 
[ 0 ] 
[ 2, 0 ] 
[ 2, 0 ] 
[ 2, 2, 6, 0 ] 
[ 3, 3, 0 ] 
[ 2, 2, 2, 2, 4, 4, 0 ] 
[ 0, 0 ] 
[ ] 
 
 
gap> D9gens:=[ [1,[2,3],[9,3]], [2,[3,3]], [3,[4,3]], [4,[5,3]],
[5,[6,3]], [6,[7,3]], [7,[8,3]], [8,[9,3]] ];; 
gap> R:=ResolutionArtinGroup(D9gens,10);; 
gap> TR:=TensorWithIntegers(R);; 
gap> for n in [1..9] do 
> Print(Homology(TR,n),"\n"); 
> od; 
[ 0 ] 
[ 2, 0 ] 
[ 2, 0 ] 
[ 2, 6, 0 ] 
[ 6, 0 ] 
[ 2, 6, 0 ] 
[ 0 ] 
[ 0 ] 
[ ] 
 </td>
 </tr>
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