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About HAPcryst:  Betti numbers for orientable
7-dimensional  Hantzsche-Wendt Manifolds
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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 BieberbachExamples.g .  Fundamental domains for the groups are then computed using the following commands. (These commands use Polymake software.)
gap> LoadPackage("HAPcryst");;

gap> Read("BieberbachExamples.g");;

gap> center:=List([1..7],i->0);;
gap> for i in [1..62] do
> S:=HWO7Gr[i];;
> data:=LatticeTransAndPointGroupRepsOnRight(GeneratorsOfGroup(S));;
> orbitpart:=OrbitStabilizerInUnitCubeOnRight(center,data.pointGrpReps).orbit;;
> poly:=FundamentalDomainFromCenterAndOrbitPart(center,orbitpart);;
> fudoms[i]:=poly;;
> od;;
The following commands produce free resolutions for the 62 Bieberbach groups.
gap> for i in [1..62] do
> S:=HWO7Gr[i];;
> data:=LatticeTransAndPointGroupRepsOnRight(GeneratorsOfGroup(S));;
> fl:=FaceLatticeAndBoundaryBieberbachGroup(fudoms[i],data.pointGrpReps);;
> resolutions[i]:=ResolutionFromFLandBoundary(fl,S);;
> od;;
The following commands list the cohomology of the Bieberbach groups. (The groups are Poincare duality groups, so homology Betti numbers are given by Hk(G,Z) = H7-k(G,Z). )
gap> cochaincomplexes:=[];;
gap> for i in [1..62] do
> cochaincomplexes[i]:=HomToIntegers(resolutions[i]);;
> od;;
gap> for j in [1..62] do
> Print(j, ": \n",List([0..6],i->Cohomology(cochaincomplexes[j],i)),"\n",
"-------------------- \n");;
> od;;
1:
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60:
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61:
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62:
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