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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. 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.
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]] ];;
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.
gap> CoxeterDiagramDisplay(D3,"mozilla");;
An Artin group is a finitely presented group AD associated to a Coxeter diagram D as follows: There is one generator x for each vertex in D. 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. 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.
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 ] ]
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.
gap> CoxeterDiagramIsSpherical(D1); true gap> CoxeterDiagramIsSpherical(D2); false gap> CoxeterDiagramIsSpherical(D3); false
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. 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.
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 ]
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.)
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 ] [ ]
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.
Suppose that XT is contractible for every connected full subgraph T of D where T involves no infinity edges. Then XD is contractible.
(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.
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 1/nxy + 1/nyz + 1/nxz < 1 or = 1.
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.
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.)
The following commands completely determine the additive structure of the integral homology of the affine braid groups on six, seven, eight and nine generators.
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 ] [ ]
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