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style= "font-weight: bold;">About HAPcryst: Betti numbers for
orientable < br>
7-dimensional Hantzsche-Wendt Manifolds < br>
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style= "vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">(Th anks
to Marc Röder
for supplying details for this page.)<br>
<br>
Polytopes can also be used to calculate the cohomology of some infinite
groups. In particular, Bieberbach groups with
point group (C<sub>2</sub>)<sup>6</sup> which arise as the fundamental
groups of orientable, aspherical, 7-dimensional Hantsche-Wendt
manifolds have been classified in <big><font size="2"><big><a
href="https://staffmail.nuigalway.ie/exchweb/bin/redir.asp?URL=http://citeseer.ist.psu.edu/409869.html"
target="_blank">http://citeseer.ist.psu.edu/409869.html</a></big></font></big>.
There are 62 in all, and a <a href="examples7dim.g">list</a> of
these in GAP format has been provided by Bartosz Putrycz. The
integral homology of
these groups (i.e. the Betti numbers of the corresponding manifolds)
can be calculated using the <a
href="http://hamilton.nuigalway.ie/CHA/HAPcryst/HAPcrystindex.shtml">HAPcryst</a>
library written by Marc Röder. To do this one
first saves the <a href="examples7dim.g">list</a> of groups as the
file <span style="font-family: monospace;">examples7dim.g</span>
. Free resolutions for the groups are then computed using the
following commands. (These commands use Polymake software.)<br>
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
LoadPackage("HAPcryst");;<br>
<br>
gap> Read("examples7dim.g");;<br>
<br>
gap> resolutions:=List(HWO7Gr,ResolutionBieberbachGroup);;
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style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands list the homology of the Bieberbach groups. (The
groups are Poincare duality groups, so cohomology Betti numbers are given
by H<sub>k</sub>(G,Z) = H<sup>7-k</sup>(G,Z). )<br>
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>gap> chaincomplexes:=List(resolutions,r->TensorWithIntegers(r));;<br/>
gap> hGrps:=List(chaincomplexes,i->List([0..6],j->Homology(i,j)));;<br/>
gap> indexlist:=List(hGrps,g->Filtered([1..Size(HWO7Gr)],j->hGrps[j]=g));;<br/>
gap> for s in Set(indexlist,i->[hGrps[i[1]],i])<br/>
> do<br/>
> Print(s[2],":\n",s[1],"\n\n");<br/>
> od;<br/>
[ 8, 9, 12, 14, 18, 21, 26, 28, 29, 41, 45, 46, 49, 51, 54 ]:<br/>
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 2, 2, 2, 2, 2, 2, 2, 2, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 2, 2, 2, 2, 2, 2 ], [ ] ]<br/><br/>
[ 4, 7, 10, 16, 19, 20, 22, 25, 27, 31, 32, 33, 34, 36, 37, 38, 40, 42,
43, 47, 48, 50, 52, 53, 55, 56, 57, 58, 59, 61, 62 ]:<br/>
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 2, 2, 2, 2, 4, 4, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 2, 2, 2, 2, 2, 2 ], [ ] ]<br/><br/>
[ 1, 30, 44, 60 ]:<br/>
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 4, 4, 4, 4, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2 ],<br/>
[ ] ]<br/><br/>
[ 2, 3, 5, 6, 11, 13, 15, 17, 23, 24, 35, 39 ]:<br/>
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ],<br/>
[ 2, 2, 2, 2, 2, 2 ], [ ] ]<br/>
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