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#######################################################################
#0
#F ControlledSubdivision
##
InstallGlobalFunction(HAP_BaryCentricSubdivisionGComplex,
function(C)
local Cells,coBoundaries,N,i,Dims,pos,j,x,y,w,id,t,k,ck,c,s,a,v,g,b,
Elts,Rep,mult,ListUnion, Chains, IsSameOrbit, AddReturn,
Orbit, Dimension, StabRec, Action, Stabilizer, Boundary,
NChains, BoundaryRec, FinalBoundary;
Elts:=C!.elts;
##################################################################
# If g in Elts return the position of g in the list,
# otherwise, add g to Elts and return the position.
pos:=function(g)
local posit;
posit:=Position(Elts,g);
if posit=fail then
Add(Elts,g);
return Length(Elts);
else
return posit;
fi;
end;
##################################################################
# returns a "canonical" representative of the right coset
# Elts[g]*Stab[i+1][j]
Rep:=function(i,j,g)
return pos(CanonicalRightCountableCosetElement
(C!.stabilizer(i,j),Elts[g]^-1)^-1);
end;
##################################################################
AddReturn:=function(a,g)
local b;
b:=StructuralCopy(a);
Add(b,g);
return b;
end;
##################################################################
mult:=function(L,g)
return List(L,a->[a[1],pos(Elts[g]*Elts[a[2]])]);
end;
##################################################################
ListUnion:=function(x,y)
local a;
for a in y do
if not a in x then
Add(x,a);
fi;
od;
end;
##################################################################
IsSameOrbit:=function(a,b)
local s,w,v,x,y,k,g,i;
for i in [1..Length(a)] do
if not a[i][1]=b[i][1] then
return false;
fi;
od;
x:=List([1..Length(a)],i->[a[i][1],Cells[a[i][1]+1][a[i][2]]]);
y:=List([1..Length(b)],i->[b[i][1],Cells[b[i][1]+1][b[i][2]]]);
for i in [1..Length(x)] do
if not x[i][2][1]=y[i][2][1] then
return false;
fi;
od;
w:=List(C!.stabilizer(x[1][1],x[1][2][1]),i->Elts[y[1][2][2]]*i*Elts[x[1][2][2]]^-1);
for s in [2..Length(x)] do
v:=List(C!.stabilizer(x[s][1],x[s][2][1]),i->Elts[y[s][2][2]]*i*Elts[x[s][2][2]]^-1);
w:=Intersection(w,v);
if IsEmpty(w) then return false;fi;
od;
if not IsEmpty(w) then
return w[1];fi;
end;
##################################################################
Dims:=[];
N:=Length(C); #Added by Graham 02/2024
for i in [0..Length(C)] do
if C!.dimension(i)=0 then N:=i-1; break; fi;
Dims[i+1]:=C!.dimension(i);
od;
Cells:=[];
coBoundaries:=[];
id:=pos(One(C!.group));
for i in [1..N+1] do
Cells[i]:=[];
coBoundaries[i]:=[];
od;
for j in [1..Dims[N+1]] do
Add(Cells[N+1],[j,id]);
od;
# Construct the list of cells and the corresponding coboundary of those cells
i:=N;
while i>0 do
for k in [1..Length(Cells[i+1])] do
x:=Cells[i+1][k];
w:=StructuralCopy(C!.boundary(i,AbsInt(x[1])));
w:=mult(w,x[2]);
w:=List(w,a->[AbsInt(a[1]),Rep(i-1,AbsInt(a[1]),a[2])]);
ListUnion(Cells[i],w);
for y in w do
t:=Position(Cells[i],y);
if not IsBound(coBoundaries[i][t]) then
coBoundaries[i][t]:=[];
fi;
Add(coBoundaries[i][t],k);
od;
od;
i:=i-1;
od;
# Record k-chains as a list
Chains:=[];
# Record the 1-chains
Chains[1]:=[];
for i in [1..1] do
for j in [1..Length(Cells[i])] do
Add(Chains[1],[[i-1,j]]);
od;
od;
# Construct the list of N-chains
for k in [1..(N)] do
Chains[k+1]:=[];
for i in [1..Length(Chains[k])] do
ck:=StructuralCopy(Chains[k][i]);
c:=ck[k];
w:=List(coBoundaries[c[1]+1][c[2]],x->AddReturn(ck,[c[1]+1,x]));
Append(Chains[k+1],w);
od;
od;
NChains:=StructuralCopy(Chains[N+1]);
# Recognizing orbits and compute the boundary of cells.
Orbit:=[];
Orbit[N+1]:=[];
Add(Orbit[N+1],NChains[1]);
for i in [2..Length(NChains)] do
for j in [1..Length(Orbit[N+1])] do
c:=0;
if not IsSameOrbit(NChains[i],Orbit[N+1][j])=false then
c:=1;
break;
fi;
od;
if c=0 then Add(Orbit[N+1],NChains[i]);fi;
od;
BoundaryRec:=[];
k:=N+1;
while k>1 do
BoundaryRec[k-1]:=[];
for i in [1..Length(Orbit[k])] do
x:=StructuralCopy(Orbit[k][i]);
b:=[];
for j in [1..Length(x)] do
w:=StructuralCopy(x);
Remove(w,j);
if not IsBound(Orbit[k-1]) then
Orbit[k-1]:=[];
fi;
c:=0;
for s in [1..Length(Orbit[k-1])] do
g:=IsSameOrbit(Orbit[k-1][s],w);
if not g=false then
c:=1;
Add(b,[(-1)^j*s,pos(g)]);
fi;
od;
if c=0 then
Add(Orbit[k-1],w);
Add(b,[(-1)^j*Length(Orbit[k-1]),id]);
fi;
od;
BoundaryRec[k-1][i]:=b;
od;
k:=k-1;
od;
# Find the k-rank
Dimension:=function(k)
if k<0 or k>N then return 0;fi;
return Length(Orbit[k+1]);
end;
# Stabilizer subgroup of the representative of the ith-orbit of (k-1)-cells
StabRec:=[];
for k in [1..(N+1)] do
StabRec[k]:=[];
for i in [1..Dimension(k-1)] do
a:=Orbit[k][i];
x:=List([1..Length(a)],w->[a[w][1],Cells[a[w][1]+1][a[w][2]]]);
w:=ConjugateGroup(C!.stabilizer(a[1][1],x[1][2][1]),Elts[x[1][2][2]]^-1);
for s in [2..Length(x)] do
v:=ConjugateGroup(C!.stabilizer(a[s][1],x[s][2][1]),Elts[x[s][2][2]]^-1);
w:=Intersection(w,v);
od;
StabRec[k][i]:=w;
od;
od;
Stabilizer:=function(k,i)
return StabRec[k+1][i];
end;
# The cell structure is rigid under the action of G then Action(k,i,j) always be 1.
Action:=function(k,i,j)
return 1;
end;
# Calculate the boundary of the representative of the ith-orbit of k-cells
Boundary:=function(n,k)
if k>0 then
return BoundaryRec[n][k];
else
return NegateWord(BoundaryRec[n][AbsInt(k)]);
fi;
end;
##################################################################
return Objectify(HapNonFreeResolution,
rec(
dimension:=Dimension,
Orbit:=Orbit,
Cells:=Cells,
Chains:=Chains,
boundary:=Boundary,
coBoundaries:=coBoundaries,
IsSameOrbit:=IsSameOrbit,
homotopy:=fail,
elts:=Elts,
group:=C!.group,
stabilizer:=Stabilizer,
action:=Action,
properties:=
[["length",Maximum(1000,N)],
["characteristic",0],
["type","resolution"]] ));
end);
################### end of ControlledSubdivision ############################
#########################################################
InstallMethod(BarycentricSubdivision,
"for non-free resolutions",
[IsHapNonFreeResolution],
function(R);
return HAP_BaryCentricSubdivisionGComplex(R);
end);
#########################################################
#########################################################
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