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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Aspherical presentations 
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style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The
2-dimensional
connected CW-space KP 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 KP is a contractible 2-dimensional CW-space
admitting a free cellular action of the group G determined by the
presentation. The cellular chain complex of X is thus a free
ZG-resolution of Z. 
 
A sufficient (but certainly not necessary) condition for KP
to be aspherical is that it admits a non-positively curved metric which
restrict to a Euclidean metric on each 2-cell. This sufficient
condition can be expressed as a set of inequalities. The function IsAspherical()
applies <a href="https://www.math.tu-berlin.de/Polymake">Polymake</a>
software to a subset of these inequalities to test whether KP
is
aspherical. 
 
The following commands show that this asphericity test is inconclusive
on the standard presentation 
 
 <div style="text-align: center;">P=< x, y, z | xyx=yxy,
yzy=zyz, zxz=xzx > 
 </div>
 
of the 4-string affine braid group. 
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style="background-color: rgb(255, 255, 204); vertical-align: top;">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 piece-wise Euclidean non-positively curved. 
 
fail 
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 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">Asphericity
is obviously a homotopy invariant. So we can continue to test the
asphericity of KP by applying the above test to
presentations P' of the affine braid group for which the associated
space KP' 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 KP and
KP' are then in fact simple homotopy equivalent. Repeating
this process with b=yz and c=zx yields the presentation 
 
 <div style="text-align: center;">P' = <x,y,z,a,b,c, | a=xy,
b=yz, c=zx, ax=ya, by=zb, cz=xc > 
 </div>
 
of the affine braid group. 
 
The following commands show that KP' is aspherical, and
hence that KP is also aspherical. </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">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 
 </td>
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 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">The
4-string affine braid group thus has integral homology Hn(G,Z)=0
for n>2. The following commands show that H1(G,Z)=Z and H2(G,Z)=Z. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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), 201-220]. 
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