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|
InstallGlobalFunction(QuotientByTorsionSubcomplex,
function(C,p)
local pTorsionSubcomplex, celldata, quotient, torsionCells,
i, j, k, t, l, currentCell, BoundaryImage, dim0, ReIndexCell,
lnth, dims,n, Elts, Boundary, StabilizerGroups, RotSubGroups, s,
boundaryList, BI, SGN, LstEl, tmp, G, Stabilizer, Action, Dimension;
pTorsionSubcomplex:=GetTorsionSubcomplex(C,p);
celldata:=StructuralCopy(pTorsionSubcomplex!.originalData);
torsionCells:=[];
for i in [1..Length(celldata)] do
torsionCells[i]:=List(pTorsionSubcomplex!.torsionPosition[i],a->a[2]);
Sort(torsionCells[i],function(v,w) return v>w;end);
od;
# Print(torsionCells,"\n");
quotient:=StructuralCopy(celldata);
i:=Length(quotient);
while i > 0 do
if Length(quotient[i])=0 then
Remove(quotient,i);
i:=i-1;
else break;
fi;
od;
# Print(quotient,"\n");
if not Length(torsionCells[1])=0 then
## ** Delete torsion vertices ** ##
for i in torsionCells[1] do
# Print(torsionCells[1],"\n");
Remove(quotient[1],i);
# Print(quotient[1],"\n");
od;
## ** Add a new vertex which is the collapsing point ** ##
Add(quotient[1],rec(TheMatrixStab :=C!.group,
TheRotSubgroup:=C!.group,
BoundaryImage :=rec(
ListIFace:=[],
ListSign:=[],
ListElt:=[])
)
);
# Print(quotient[1],"\n");
dim0:=Length(quotient[1]);
## ** Search for those edges that having torsion vertex and replace
## ** them with the collapsing vertex
for j in torsionCells[2] do
Remove(quotient[2],j);
od;
for i in [1..Length(quotient[2])] do
currentCell:=quotient[2][i];
BoundaryImage:=currentCell.BoundaryImage;
for j in [1..Length(BoundaryImage.ListIFace)] do
if BoundaryImage.ListIFace[j] in torsionCells[1] then
BoundaryImage.ListIFace[j]:=dim0;
BoundaryImage.ListElt[j]:=One(C!.group);
fi;
od;
currentCell.BoundaryImage:=BoundaryImage;
quotient[2][i]:=currentCell;
od;
for i in [3..Length(quotient)] do
## ** Remove the torsion i-cells
for j in torsionCells[i] do
Remove(quotient[i],j);
od;
## ** Search the boundary of an i-cell for the torsion (i-1)-cells
## ** and delete them
for j in [1..Length(quotient[i])] do
currentCell:=quotient[i][j];
BoundaryImage:=currentCell.BoundaryImage;
l:=Length(BoundaryImage.ListIFace);
for k in [1..l] do
t:=l+1-k;
if BoundaryImage.ListIFace[t] in torsionCells[i-1] then
Remove(BoundaryImage.ListIFace,t);
Remove(BoundaryImage.ListSign,t);
Remove(BoundaryImage.ListElt,t);
fi;
od;
currentCell.BoundaryImage:=BoundaryImage;
quotient[i][j]:=currentCell;
od;
od;
fi;
## ** A subfunction which reindexes cells after deleting torsion cells
ReIndexCell:=function(k,currentCell)
local i,j, BoundaryImage, x;
BoundaryImage:=currentCell.BoundaryImage;
for j in [1..Length(BoundaryImage.ListIFace)] do
for i in torsionCells[k] do
if BoundaryImage.ListIFace[j]>i then
BoundaryImage.ListIFace[j]:=BoundaryImage.ListIFace[j]-1;
fi;
od;
od;
currentCell.BoundaryImage:=BoundaryImage;
return currentCell;
end;
##
for i in [2..Length(quotient)] do
for j in [1..Length(quotient[i])] do
quotient[i][j]:=ReIndexCell(i-1,quotient[i][j]);
od;
od;
############Taken from dutour.gi implemented by Graham Ellis#########
lnth:=Length(quotient)-1;
dims:=List([1..lnth+1],n->Length(quotient[n]));
###################
Dimension:=function(n);
if n>lnth then return 0; fi;
return dims[n+1];
end;
###################
Elts:=[Identity(quotient[1][1].TheMatrixStab)];
StabilizerGroups:=[];
RotSubGroups:=[];
boundaryList:=[];
#######
for n in [1..lnth+1] do
boundaryList[n]:=[];
StabilizerGroups[n]:=[];
RotSubGroups[n]:=[];
for k in [1..Dimension(n-1)] do
#if not name in InfGrps then
# Append(Elts,Elements(quotient[n][k].TheMatrixStab));
#fi;
Add(StabilizerGroups[n],quotient[n][k].TheMatrixStab);
Add(RotSubGroups[n],quotient[n][k].TheRotSubgroup);
od;
od;
####
Elts:=SSortedList(Elts);
#######
for n in [1..lnth+1] do
boundaryList[n]:=[];
for k in [1..Dimension(n-1)] do
tmp:=quotient[n][k].BoundaryImage;
BI:=tmp.ListIFace;
SGN:=tmp.ListSign;
LstEl:=StructuralCopy(tmp.ListElt);
Append(Elts,Difference(LstEl,Elts));
for s in [1..Length(BI)] do
BI[s]:=[SGN[s]*BI[s],Position(Elts,LstEl[s])];
od;
Add(boundaryList[n],BI);
od;
od;
####
G:=C!.group;
####################
Boundary:=function(n,k);
if k>0 then
return boundaryList[n+1][k];
else
return NegateWord(boundaryList[n+1][-k]);
fi;
end;
####################
####################
Stabilizer:=function(n,k);
return StabilizerGroups[n+1][k];
end;
####################
####################
Action:=function(n,k,g)
return 1;
end;
##################################################################
return Objectify(HapNonFreeResolution,
rec(
quotientCells:=quotient,
stabilizer:=Stabilizer,
elts:=Elts,
group:=G,
length:=lnth,
boundary:=Boundary,
homotopy:=fail,
action:=Action,
dimension:=Dimension,
properties:=
[["length",Maximum(1000,lnth)],
["characteristic",0],
["type","resolution"]] ));
end);
################### end of ControlledSubdivision ############################
InstallGlobalFunction( "ExtractTorsionSubcomplex", function(C, p)
#####################################################################
## Here, p is the prime for which to take the torsion subcomplex. ##
## We extract the cells the stabilizer of which contains p-torsion.##
#####################################################################
local vcd, stabilizerCardinalities, celldata, data, torsionPosition,
torsionCells, numberOfTorsionCells, n, j, returnedData, warned,
groupname, admissibilityCheck, x, i, b, tmpCell, cell, boundary, groupName,
lnth, dims, Elts, Boundary, StabilizerGroups, RotSubGroups, s,k,
boundaryList, BI, SGN, LstEl, tmp, G, Stabilizer, Action, Dimension;
admissibilityCheck := function(celldata)
#########################################################
## A cell complex is admissible in the sense of Brown, ##
## if each cell stabilizer fixes its cell pointwise. ##
## Additionally, ##
## we gather the cardinalities of the stabilizers. ##
#########################################################
local stabilizerCardinalities, G, card, n, j, R, vcd, warned;
warned := false;
stabilizerCardinalities := [];
vcd := Length(celldata)-1;
for n in [0..vcd] do
stabilizerCardinalities[n+1] := [];
for j in [1..Length(celldata[n+1])] do
G := celldata[n+1][j]!.TheMatrixStab;
if IsFinite(G) then
card := Order(G);
stabilizerCardinalities[n+1][j] := card;
## *** Now we have to compare *** ##
## *** with the order of "TheRotSubgroup" *** ##
R := celldata[n+1][j]!.TheRotSubgroup;
if card > Order(R) and warned = false then
Print("****Warning: cell complex not admissible ",
"in the sense of Brown!****\n",
" Torsion subcomplex reduction requires cell subdivision.\n");
warned := true;
fi;
fi;
od;
od;
return [stabilizerCardinalities, warned];
end;
# Case 1: the input is a group name
if IsString(C) then
groupName:=C;
groupname := Filtered( C, function ( x )
return not (x = '(' or x = ')' or x = ',' or x = '[' or x = ']');
end );
Read(Concatenation( DirectoriesPackageLibrary("HAP")[1]![1],
"Perturbations/Gcomplexes/",groupname));
celldata := StructuralCopy(HAP_GCOMPLEX_LIST);
# Case 2: the input is a variable
else
groupName:=fail;
celldata:=[];
i:=0;
while C!.dimension(i) > 0 do
cell:=[];
for j in [1..C!.dimension(i)] do
if not i=0 then
boundary:=C!.boundary(i,j);
Add(cell,rec(TheMatrixStab :=C!.stabilizer(i,j),
TheRotSubgroup:=C!.stabilizer(i,j),
BoundaryImage :=rec(
ListIFace:=List(boundary,w->AbsInt(w[1])),
ListSign:=List(boundary,w->SignInt(w[1])),
ListElt:=List(boundary,w->C!.elts[w[2]])
)
)
);
else
Add(cell,rec(TheMatrixStab :=C!.stabilizer(i,j),
TheRotSubgroup:=C!.stabilizer(i,j),
BoundaryImage :=rec(
ListIFace:=[],
ListSign:=[],
ListElt:=[]
)
)
);
fi;
od;
Add(celldata,cell);
i:=i+1;
od;
fi;
vcd := Length(celldata) -1;
# Print("Extracting the ",p,"-torsion subcomplex of the ",
# vcd,"-dimensional ",groupName,"-cell complex ... \n");
returnedData := admissibilityCheck(celldata);
stabilizerCardinalities := returnedData[1];
warned := returnedData[2];
torsionCells := [];
numberOfTorsionCells := [];
for n in [0..vcd] do
torsionCells[n+1] := [];
numberOfTorsionCells[n+1] := 0;
for j in [1..Length(celldata[n+1])] do
## Check if the stabilizer contains p-torsion ##
if stabilizerCardinalities[n+1][j] mod p = 0 then
# Print("Extracted ",n,"-cell numero ",j,
# " of stabilizer cardinality ",
# stabilizerCardinalities[n+1][j],".\n");
numberOfTorsionCells[n+1]
:= numberOfTorsionCells[n+1]+1;
torsionCells[n+1][numberOfTorsionCells[n+1]]
:=[n, j];
fi;
od;
od;
# return
# [torsionCells, numberOfTorsionCells, celldata, stabilizerCardinalities, warned];
data:=[];
for i in [1..Length(torsionCells)] do
data[i]:=[];
for x in torsionCells[i] do
Add(data[i],celldata[i][x[2]]);
od;
od;
for j in [2..Size(data)] do
for i in [1..Size(data[j])] do
tmpCell:=StructuralCopy(data[j][i]!.BoundaryImage);
b:=List(tmpCell!.ListIFace,w->Position(torsionCells[j-1], [j-2,w]));
tmpCell!.ListIFace:=b;
data[j][i]!.BoundaryImage:=tmpCell;
od;
od;
torsionPosition:=torsionCells;
torsionCells:=[];
for i in [1..Size(data)] do
torsionCells[i]:=[];
for j in [1..Size(data[i])] do
torsionCells[i][j]:=[i-1,j];
od;
od;
############################
############Taken from dutour.gi implemented by Graham Ellis#########
lnth:=Length(data)-1;
dims:=List([1..lnth+1],n->Length(data[n]));
###################
Dimension:=function(n);
if n>lnth then return 0; fi;
return dims[n+1];
end;
###################
Elts:=[Identity(data[1][1].TheMatrixStab)];
StabilizerGroups:=[];
RotSubGroups:=[];
boundaryList:=[];
#######
for n in [1..lnth+1] do
boundaryList[n]:=[];
StabilizerGroups[n]:=[];
RotSubGroups[n]:=[];
for k in [1..Dimension(n-1)] do
#if not name in InfGrps then
Append(Elts,Elements(data[n][k].TheMatrixStab));
#fi;
Add(StabilizerGroups[n],data[n][k].TheMatrixStab);
Add(RotSubGroups[n],data[n][k].TheRotSubgroup);
od;
od;
####
Elts:=SSortedList(Elts);
#######
for n in [1..lnth+1] do
boundaryList[n]:=[];
for k in [1..Dimension(n-1)] do
tmp:=data[n][k].BoundaryImage;
BI:=tmp.ListIFace;
SGN:=tmp.ListSign;
LstEl:=StructuralCopy(tmp.ListElt);
Append(Elts,Difference(LstEl,Elts));
for s in [1..Length(BI)] do
BI[s]:=[SGN[s]*BI[s],Position(Elts,LstEl[s])];
od;
Add(boundaryList[n],BI);
od;
od;
####
G:=C!.group;
####################
Boundary:=function(n,k);
if k>0 then
return boundaryList[n+1][k];
else
return NegateWord(boundaryList[n+1][-k]);
fi;
end;
####################
####################
Stabilizer:=function(n,k);
return StabilizerGroups[n+1][k];
end;
####################
####################
Action:=function(n,k,g)
return 1;
end;
#############################
return Objectify(HapNonFreeResolution,
rec(
stabilizer:=Stabilizer,
elts:=Elts,
group:=G,
length:=lnth,
boundary:=Boundary,
homotopy:=fail,
action:=Action,
dimension:=Dimension,
torsion:=p,
groupname:=groupName,
torsionCells:=torsionCells,
torsionPosition:=torsionPosition,
originalData:=celldata,
celldata:= data,
numberOfTorsionCells:= numberOfTorsionCells,
stabilizerCardinalities:= stabilizerCardinalities,
warned:= warned,
properties:=
[["length",Maximum(1000,lnth)],
["characteristic",0],
["type","resolution"]] ));
end);
###########################################################################
DeclareGlobalFunction("PrintTorsionSubcomplex");
InstallGlobalFunction("PrintTorsionSubcomplex",
function(R)
local N, reducedTorsionCells,J,n,j,G,B,b, ReduceModP, p, celldata, Stab,freq,
t, StabList, StabListStr;
reducedTorsionCells:=R!.torsionCells;
celldata:=R!.celldata;
p:=R!.torsion;
#########
ReduceModP := function( G, p)
local K, h, H, Q, P;
Q := G;
K := ConjugacyClassesSubgroups(G);
for h in K do
if Size(h)=1 then
H := Representative(h);
if Size(H) > 1 then
if Gcd(Size(H),p) = 1 then
# We pass to the quotient Q in the extension
# 1 -> H -> G -> Q -> 1,
# because H has trivial mod p cohomology.
# Our loop ends up using the normal subgroup H of maximal size,
# because the list K is ordered from small to large.
Q := Image( NaturalHomomorphismByNormalSubgroup( G,H ));
fi;
fi;
fi;
od;
P:=Q;
if IsPNormal(Q,p) then P:=Normalizer(Q,Center(SylowSubgroup(Q,p)));fi;
return P;
end;
#########
freq:=[];
StabList:=[];
for N in [1..Size(reducedTorsionCells)] do
freq[N]:=[];
for J in reducedTorsionCells[N] do
n := J[1];
j := J[2];
G := celldata[n+1][j]!.TheMatrixStab;;
Stab:=IdSmallGroup(ReduceModP(G,p));
t:=Position(StabList,Stab);
if t=fail then
Add(StabList,Stab);
freq[N][Length(StabList)]:=1;
else
if IsBound(freq[N][t]) then
freq[N][t]:=freq[N][t]+1;
else
freq[N][t]:=1;
fi;
fi;
od;
od;
StabListStr:=StructuralCopy(StabList);
for J in [1..Length(StabList)] do
StabList[J]:=StructureDescription(SmallGroup(StabList[J]));
od;
for N in [1..Size(reducedTorsionCells)] do
for J in [1..Length(StabList)] do
if not IsBound(freq[N][J]) then
freq[N][J]:=0;
fi;
od;
od;
return [StabListStr,StabList,freq];
end);
#################################
DeclareGlobalFunction("ConvertTorsionComplexToGcomplex");
InstallGlobalFunction("ConvertTorsionComplexToGcomplex",
function(C)
local data, Elts,n,lnth,dims,Dimension, StabilizerGroups,
RotSubGroups,boundaryList,k, tmp, BI, SGN, LstEl,
s, G, Boundary, Stabilizer, Action;
data:=StructuralCopy(C!.celldata);
lnth:=Length(data)-1;
dims:=List([1..lnth+1],n->Length(data[n]));
###################
Dimension:=function(n);
if n>lnth then return 0; fi;
return dims[n+1];
end;
###################
Elts:=[Identity(data[1][1].TheMatrixStab)];
StabilizerGroups:=[];
RotSubGroups:=[];
boundaryList:=[];
#######
for n in [1..lnth+1] do
boundaryList[n]:=[];
StabilizerGroups[n]:=[];
RotSubGroups[n]:=[];
for k in [1..Dimension(n-1)] do
#if not name in InfGrps then
Append(Elts,Elements(data[n][k].TheMatrixStab));
#fi;
Add(StabilizerGroups[n],data[n][k].TheMatrixStab);
Add(RotSubGroups[n],data[n][k].TheRotSubgroup);
od;
od;
####
Elts:=SSortedList(Elts);
#######
for n in [1..lnth+1] do
boundaryList[n]:=[];
for k in [1..Dimension(n-1)] do
tmp:=data[n][k].BoundaryImage;
# Print(tmp,"\n");
BI:=tmp.ListIFace;
SGN:=tmp.ListSign;
LstEl:=StructuralCopy(tmp.ListElt);
Append(Elts,Difference(LstEl,Elts));
for s in [1..Length(BI)] do
BI[s]:=[SGN[s]*BI[s],Position(Elts,LstEl[s])];
od;
Add(boundaryList[n],BI);
od;
od;
####
G:=Group(Elts);
####################
Boundary:=function(n,k);
if k>0 then
return boundaryList[n+1][k];
else
return NegateWord(boundaryList[n+1][-k]);
fi;
end;
####################
####################
Stabilizer:=function(n,k);
return StabilizerGroups[n+1][k];
end;
####################
####################
Action:=function(n,k,g)
return 1;
end;
#############################
return Objectify(HapNonFreeResolution,
rec(
stabilizer:=Stabilizer,
elts:=Elts,
group:=G,
length:=lnth,
boundary:=Boundary,
homotopy:=fail,
action:=Action,
dimension:=Dimension,
torsion:=C!.torsion,
groupname:=C!.groupname,
torsionCells:=C!.torsionCells,
# torsionPosition:=torsionPosition,
# originalData:=celldata,
celldata:= data,
# numberOfTorsionCells:= numberOfTorsionCells,
# stabilizerCardinalities:= stabilizerCardinalities,
# warned:= warned,
properties:=
[["length",Maximum(1000,lnth)],
["characteristic",0],
["type","resolution"]] ));
end);
|
