


The
2dimensional
connected CWspace K_{P}_{ }associated to a group
presentation P = < x
 r > is said to
be aspherical if its second
homotopy group is trivial. In this case the universal cover X_{
}of K_{P }is a contractible 2dimensional CWspace
admitting a free cellular action of the group G determined by the
presentation. The cellular chain complex of X is thus a free
ZGresolution of Z. A sufficient (but certainly not necessary) condition for K_{P} to be aspherical is that it admits a nonpositively curved metric which restrict to a Euclidean metric on each 2cell. This sufficient condition can be expressed as a set of inequalities. The function IsAspherical() applies Polymake software to a subset of these inequalities to test whether K_{P} is aspherical. The following commands show that this asphericity test is inconclusive on the standard presentation P=< x, y, z  xyx=yxy,
yzy=zyz, zxz=xzx >
of the 4string affine braid group. 

gap>
F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;; gap> rels:=[x*y*x*(y*x*y)^1, y*z*y*(z*y*z)^1, z*x*z*(x*z*x)^1];; gap> IsAspherical(F,rels); Presentation is NOT piecewise Euclidean nonpositively curved. fail 

Asphericity
is obviously a homotopy invariant. So we can continue to test the
asphericity of K_{P} by applying the above test to
presentations P' of the affine braid group for which the associated
space K_{P'} is homotopy equivalent to K_{P}. One way to construct a suitable presentation P' is to add to the presentation P one generator a and one relation a=xy, and replace by a all occurences of xy in the relators. The resulting spaces K_{P }and K_{P'} are then in fact simple homotopy equivalent. Repeating this process with b=yz and c=zx yields the presentation P' = <x,y,z,a,b,c,  a=xy,
b=yz, c=zx, ax=ya, by=zb, cz=xc >
of the affine braid group. The following commands show that K_{P'} is aspherical, and hence that K_{P} is also aspherical. 

gap>
F:=FreeGroup(6);;x:=F.1;;y:=F.2;;z:=F.3;;a:=F.4;;b:=F.5;;c:=F.6;; gap> rels:=[a^1*x*y, b^1*y*z, c^1*z*x, a*x*(y*a)^1, b*y*(z*b)^1, c*z*(x*c)^1];; gap> IsAspherical(F,rels); Presentation is aspherical. true 

The
4string affine braid group thus has integral homology H_{n}(G,Z)=0
for n>2. The following commands show that H_{1}(G,Z)=Z and H_{2}(G,Z)=Z. 

gap>
F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;; gap> rels:=[x*y*x*(y*x*y)^1, y*z*y*(z*y*z)^1, z*x*z*(x*z*x)^1];; gap> R:=ResolutionAsphericalPresentation(F,rels);; gap> TR:=TensorWithIntegers(R);; gap> Homology(TR,1); [0] gap> Homology(TR,2); [0] 

We
should remark that the asphericity of the above presentation P can be
derived from a lemma in [K.J. Appel and P.E. Schupp, "Artin
groups and infinite Coxeter groups", Invent.
Math., 72 (1983), 201220]. 

