<!
DOCTYPE html PUBLIC
"-//W3C//DTD HTML 4.01 Transitional//EN">
<
html>
<
head>
<
meta http-equiv=
"content-type"
content=
"text/html; charset=ISO-8859-1">
<
title>AboutHap</
title>
</
head>
<
body
style=
"color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
alink=
"#000066" link=
"#000066" vlink=
"#000066">
<
br>
<
table
style=
"text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
border=
"0" cellpadding=
"20" cellspacing=
"10">
<
tbody>
<
tr align=
"center">
<
th style=
"vertical-align: top;">
<
table style=
"width: 100%; text-align: left;" cellpadding=
"2"
cellspacing=
"2">
<
tbody>
<
tr>
<
td style=
"vertical-align: top;"><a
href=
"aboutPeriodic.html"><
small style=
"color: rgb(0, 0, 102);">Previous</
small></a><
br>
</
td>
<
td
style=
"text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><
big><
span
style=
"font-weight: bold;">About HAP: Aspherical presentations<
br>
</
span></
big></
td>
<
td style=
"text-align: right; vertical-align: top;"><a
href=
"aboutTensorSquare.html"><
small style=
"color: rgb(0, 0, 102);">next</
small></a><
br>
</
td>
</
tr>
</
tbody>
</
table>
<
big><
span style=
"font-weight: bold;"></
span></
big><
br>
</
th>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The
2-dimensional
connected CW-space K<
sub>P</
sub><
sub> </
sub>associated to a group
presentation P = < <
span style=
"text-decoration: underline;">x</
span>
| <
span style=
"text-decoration: underline;">r</
span> > is said to
be <
span style=
"font-style: italic;">aspherical</
span> if its second
homotopy group is trivial. In this case the universal cover X<
sub>
</
sub>of K<
sub>P </
sub>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.<
br>
<
br>
A sufficient (but certainly not necessary) condition for K<
sub>P</
sub>
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 <
span
style=
"font-family: helvetica,arial,sans-serif;">IsAspherical()</
span>
applies <a href=
"https://www.math.tu-berlin.de/Polymake">Polymake</a>
software to a subset of these inequalities to test whether K<
sub>P</
sub>
is
aspherical.<
br>
<
br>
The following commands show that this asphericity test is inconclusive
on the standard presentation <
br>
<
br>
<
div style=
"text-align: center;">P=< x, y, z | xyx=yxy,
yzy=zyz, zxz=xzx ><
br>
</
div>
<
br>
of the 4-string affine braid group.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;;<
br>
gap> rels:=[x*y*x*(y*x*y)^-1, y*z*y*(z*y*z)^-1, z*x*z*(x*z*x)^-1];;<
br>
<
br>
gap> IsAspherical(F,rels);<
br>
Presentation is NOT piece-wise Euclidean non-positively curved.<
br>
<
br>
fail<
br>
</
td>
</
tr>
<
tr>
<
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 K<
sub>P</
sub> by applying the above test to
presentations P
' of the affine braid group for which the associated
space K<
sub>P
' is homotopy equivalent to KP.
<
br>
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<
sub>P </
sub>and
K<
sub>P
' are then in fact simple homotopy equivalent. Repeating
this process with b=yz and c=zx yields the presentation<
br>
<
br>
<
div style=
"text-align: center;">P
' = <x,y,z,a,b,c, | a=xy,
b=yz, c=zx, ax=ya, by=zb, cz=xc ><
br>
</
div>
<
br>
of the affine braid group. <
br>
<
br>
The following commands show that K<
sub>P
' is aspherical, and
hence that K<
sub>P</
sub> 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;;<
br>
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];;<
br>
<
br>
gap> IsAspherical(F,rels);<
br>
Presentation is aspherical.<
br>
<
br>
true<
br>
</
td>
</
tr>
<
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 H<
sub>n</
sub>(G,Z)=0
for n>2. The following commands show that H<
sub>1</
sub>(G,Z)=Z and H<
sub>2</
sub>(G,Z)=Z.<
span
style=
"color: rgb(255, 0, 0);"></
span><
br>
</
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;;<
br>
gap> rels:=[x*y*x*(y*x*y)^-1, y*z*y*(z*y*z)^-1, z*x*z*(x*z*x)^-1];;<
br>
<
br>
gap> R:=ResolutionAsphericalPresentation(F,rels);;<
br>
gap>
TR:=TensorWithIntegers(R);;<
br>
gap> Homology(
TR,1);<
br>
[0]<
br>
gap> Homology(
TR,2);<
br>
[0]<
br>
</
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.</span>, 72 (1983), 201-220]. <br>
</td>
</tr>
<tr>
<td style="vertical-align: top;">
<table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
<tbody>
<tr>
<td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutPeriodic.html">Previous
Page</a><br>
</td>
<td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br>
</td>
<td style="text-align: right; vertical-align: top;"><a
href="aboutTensorSquare.html"><span style="color: rgb(0, 0, 102);">Next
page</span><br>
</a> </td>
</tr>
</tbody>
</table>
<a href="aboutTopology.html"><br>
</a> </td>
</tr>
</tbody>
</table>
<br>
<br>
</body>
</html>
