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| (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 http://citeseer.ist.psu.edu/409869.html. There are 62 in all, and a list 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 HAPcryst library written by Marc Röder. To do this one first saves the list of groups as the file examples7dim.g . Free resolutions for the groups are then computed using the following commands. (These commands use Polymake software.) |
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| gap>
LoadPackage("HAPcryst");; gap> Read("examples7dim.g");; gap> resolutions:=List(HWO7Gr,ResolutionBieberbachGroup);; |
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| 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). ) |
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| 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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