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products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 19.6.2025 mit Größe 5 kB

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products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/bieberbach.html

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 <title>AboutHap</title>
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style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
alink="#000066" link="#000066" vlink="#000066">
 
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style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
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 <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="aboutPolytopes.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAPcryst: Betti numbers for
orientable 
7-dimensional Hantzsche-Wendt Manifolds 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="3dflatmanifolds.html">next</a> 
 </td>
 </tr>
 </tbody>
 </table>
 <big></big> 
 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">(Thanks
to Marc Röder
for supplying details for this page.) 
 
Polytopes can also be used to calculate the cohomology of some infinite
groups. In particular, Bieberbach groups with
point group (C2)6 which arise as the fundamental
groups of orientable, aspherical, 7-dimensional Hantsche-Wendt
manifolds have been classified in <big><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></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 examples7dim.g
. Free resolutions for the groups are then computed using the
following commands. (These commands use Polymake software.) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
LoadPackage("HAPcryst");; 
 
gap> Read("examples7dim.g");; 
 
gap> resolutions:=List(HWO7Gr,ResolutionBieberbachGroup);;
</td>
 </tr>

 <tr>
 <td
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 Hk(G,Z) = H7-k(G,Z). ) 
 </td>
 </tr>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);"
>gap> chaincomplexes:=List(resolutions,r->TensorWithIntegers(r));; 
gap> hGrps:=List(chaincomplexes,i->List([0..6],j->Homology(i,j)));; 
gap> indexlist:=List(hGrps,g->Filtered([1..Size(HWO7Gr)],j->hGrps[j]=g));; 
gap> for s in Set(indexlist,i->[hGrps[i[1]],i]) 
> do 
> Print(s[2],":\n",s[1],"\n\n"); 
> od; 
[ 8, 9, 12, 14, 18, 21, 26, 28, 29, 41, 45, 46, 49, 51, 54 ]: 
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 2, 2, 2, 2, 2, 2, 2, 2, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 2, 2, 2, 2, 2, 2 ], [ ] ] 

[ 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 ]: 
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 2, 2, 2, 2, 4, 4, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 2, 2, 2, 2, 2, 2 ], [ ] ] 

[ 1, 30, 44, 60 ]: 
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 4, 4, 4, 4, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2 ], 
 [ ] ] 

[ 2, 3, 5, 6, 11, 13, 15, 17, 23, 24, 35, 39 ]: 
[ [ 0 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ], 
 [ 2, 2, 2, 2, 2, 2 ], [ ] ] 

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style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
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style="color: rgb(0, 0, 102);" href="aboutPolytopes.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
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 <td style="text-align: right; vertical-align: top;"><a
href="3dflatmanifolds.html">Next
page 
 </a> </td>
 </tr>
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 <a href="aboutTopology.html"> 
 </a> </td>
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