Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place.
Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place.
| Download
GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#(C) Graham Ellis
##################################################################
##################################################################
InstallGlobalFunction(SimplicialComplexToRegularCWComplex,
function(arg)
local
K,DM,NrCells,Boundaries,tmp,TMP,Coboundaries,Properties,
Orientation,
cnt,b,bb,k,n,s,x,i,j,dim ;
K:=arg[1];
if Length(arg)>1 then dim:=arg[2]; else dim:=Dimension(K); fi;
####################
NrCells:=function(n);
if n>dim then return 0; fi;
return Length(Filtered(Boundaries[n+1],x->not x[1]=0));
end;
####################
#dim:=Dimension(K);
Properties:=[["dimension",dim]];
#############################
Orientation:=[];
Orientation[1]:=ListWithIdenticalEntries(K!.nrSimplices(0),[1]);
for n in [1..dim] do
tmp:=[];
for i in [1..n+1] do
Add(tmp,(-1)^(i+1));
od;
Orientation[n+1]:=ListWithIdenticalEntries(K!.nrSimplices(n),tmp);
od;
#############################
### BOUNDARIES BEGIN ######################
Boundaries:=[]; #Boundaries[n+1] contains the info on n-cells
Boundaries[1]:=List([1..K!.nrSimplices(0)],x->[1,0]);
##We denote by 0 the unique vertex in dimension -1.
for n in [1..dim] do
Boundaries[n+1]:=[];
tmp:=List(Boundaries[1],x->[]);
TMP:=List(Boundaries[1],x->[]);
cnt:=0;
for s in K!.simplicesLst[n] do
cnt:=cnt+1;
Add(tmp[s[1]],s);
Add(TMP[s[1]],cnt);
od;
for k in [1..K!.nrSimplices(n)] do
bb:=K!.simplices(n,k);
bb:=SSortedList(bb);
b:=List(bb,x-> Difference(bb,[x]) );
Apply(b,x-> TMP[x[1]][Position(tmp[x[1]],x)] );
Boundaries[n+1][k]:=Concatenation([Length(b)],b);
od;
od;
Boundaries[dim+2]:=[];
### BOUNDARIES END ###############################
### COBOUNDARIES BEGIN ######################
Coboundaries:=[];; #Coboundaries[n+1] contains the info on n-cells.
for n in [0..dim] do
k:=n+3;
Coboundaries[n+1]:=List(Boundaries[n+1],i->[0]);
for j in [1..Length(Boundaries[n+2])] do
b:=Boundaries[n+2][j];
for i in b{[2..k]} do
Coboundaries[n+1][i][1]:=Coboundaries[n+1][i][1]+1;
Add(Coboundaries[n+1][i],j);
od;
od;
#for b in Coboundaries[n+1] do
#Append(b,List([1..Length(b)-1],a->1));
#od;
od;
Coboundaries[dim+1]:=List(Boundaries[dim+1],a->[0]);
### COBOUNDARIES END ###############################
return Objectify(HapRegularCWComplex,
rec(
nrCells:=NrCells,
boundaries:=Boundaries,
coboundaries:=Coboundaries,
orientation:=Orientation,
vectorField:=fail,
inverseVectorField:=fail,
criticalCells:=fail,
properties:=Properties));
end);
##################################################################
##################################################################
#############################################
#############################################
InstallOtherMethod(Dimension,
"Dimension of regular CW-complex",
[IsHapRegularCWComplex],
function(f) return EvaluateProperty(f,"dimension");
return EvaluateProperty(f,"dimension");
end);
#############################################
#############################################
#############################################
#############################################
InstallGlobalFunction(HAPContractRegularCWComplex,
function(Y)
local
Contract, nn, dim, bool, BOOL, FREE;
#############################################
##### The work-horse function.###############
Contract:=function(n)
local
b, C, i, j, t, cob, pos, bool,
Free, UBoundaries, UCoboundaries,
MBoundaries, MCoboundaries, LCoboundaries, U;
#This function removes pairs of n- and (n+1)-cells if possible.
#U=Upper, M=Middle and L=Lower dimensional cells.
####################
####################
if Y!.vectorField=fail then
Y!.vectorField:=List([1..Dimension(Y)],i->[]);
Y!.inverseVectorField:=List([1..Dimension(Y)],i->[]);
Y!.bnd:=StructuralCopy(Y!.boundaries);
Y!.cobnd:=StructuralCopy(Y!.coboundaries);
fi;
####################
####################
MCoboundaries:=Y!.cobnd[n+1];
MBoundaries:=Y!.bnd[n+1];
UCoboundaries:=Y!.cobnd[n+2];
UBoundaries:=Y!.bnd[n+2];
if n>0 then
LCoboundaries:=Y!.cobnd[n];
fi;
C:=Length(MCoboundaries);
#######################
#######################THIS TAKES ALL THE TIME
if not IsBound(FREE) then FREE:=[1..C]; fi;
Free:=[];
for i in FREE do
if MCoboundaries[i][1]=1 then Add(Free,i);fi;
od;
#Print([Length(FREE),Length(Free)]," ");
if Length(Free)=0 then Unbind(FREE); return false;fi;
#######################
#######################
for i in Free do
if MCoboundaries[i][1]=1 then
Y!.vectorField[n+1][MCoboundaries[i][2]]:=i;
Y!.inverseVectorField[n+1][i]:=MCoboundaries[i][2];
###
if n>0 then
b:=MBoundaries[i];
for j in StructuralCopy(b{[2..1+b[1]]}) do
t:=LCoboundaries[j][1];
LCoboundaries[j][1]:=LCoboundaries[j][1]-1;
cob:=LCoboundaries[j];
pos:=Position(cob{[2..t+1]},i);
LCoboundaries[j]:=Concatenation(cob{[1..pos]},cob{[2+pos..Length(cob)]});
od;
fi;
###
U:=MCoboundaries[i][2];
b:=UBoundaries[U];
for j in StructuralCopy(b{[2..1+b[1]]}) do
t:=MCoboundaries[j][1];
MCoboundaries[j][1]:=MCoboundaries[j][1]-1;
if t=2 then Add(Free,j);fi;############################ADDED
cob:=MCoboundaries[j];
pos:=Position(cob{[2..t+1]},U);
MCoboundaries[j]:=Concatenation(cob{[1..pos]},cob{[2+pos..Length(cob)]});
od;
###
MBoundaries[i]:=[0];
UBoundaries[U]:=[0];
UCoboundaries[U]:=[0];
MCoboundaries[i]:=[0];
fi;
od;
Y!.bnd[n+2]:=UBoundaries;
Y!.cobnd[n+2]:=UCoboundaries;
Y!.bnd[n+1]:=MBoundaries;
Y!.cobnd[n+1]:=MCoboundaries;
if n>0 then
Y!.cobnd[n]:=LCoboundaries;
fi;
Y!.nrCells:=function(k);
if k>EvaluateProperty(Y,"dimension") then return 0; fi;
return Length(Filtered(Y!.bnd[k+1],x->not x[1]=0));
end;
if Length(Free)>0 then FREE:=Free; return true;
else Unbind(FREE); return false; fi;
end;
####End of work-horse function.#############
############################################
dim:=EvaluateProperty(Y,"dimension");
bool:=true;
BOOL:=true;
nn:=dim-1;
while BOOL or nn>0 do
BOOL:=false;
for nn in Reversed([0..dim-1]) do
while bool do
bool:=Contract(nn);
if bool=true then BOOL:=true; fi;
od;
bool:=true;
od;
od;
end);
############################################
############################################
#####################################################################
#####################################################################
InstallOtherMethod(Size,
"Volume of a regular CW-complex",
[IsHapRegularCWComplex],
function(Y) return Sum(List( [1..Length(Y!.boundaries)],i->Y!.nrCells(i-1)));
end);
#####################################################################
#####################################################################
#####################################################################
#####################################################################
InstallGlobalFunction(HAPRemoveCellFromRegularCWComplex,
function(Y,dim,n)
local bnd, x,tmp, cobnd;
#Remove the n-th cell in dimension dim
####################
####################
if Y!.vectorField=fail then
Y!.vectorField:=List([1..Dimension(Y)],i->[]);
Y!.inverseVectorField:=List([1..Dimension(Y)],i->[]);
Y!.bnd:=StructuralCopy(Y!.boundaries);
Y!.cobnd:=StructuralCopy(Y!.coboundaries);
fi;
####################
####################
dim:=dim+1;
bnd:=Y!.bnd[dim][n];
bnd:=bnd{[2..Length(bnd)]};
Y!.bnd[dim][n]:=[0];
cobnd:=Y!.cobnd[dim][n]; ####Added this loop July 2012
cobnd:=cobnd{[2..Length(cobnd)]}; #
for x in cobnd do #
tmp:=Y!.bnd[dim+1][x]; #
tmp[1]:=tmp[1]-1; #
tmp[Position(tmp{[2..Length(tmp)]},n)+1]:=-42; #
tmp:=Filtered(tmp,i->not i = -42); #
Y!.bnd[dim+1][x]:=tmp; #
od; ##############
if dim=1 then return [dim-1,n]; fi;
for x in bnd do
tmp:=Y!.cobnd[dim-1][x];
tmp[1]:=tmp[1]-1;
tmp[Position(tmp{[2..Length(tmp)]},n)+1]:=-42;
tmp:=Filtered(tmp,i->not i = -42);
Y!.cobnd[dim-1][x]:=tmp;
if IsBound(Y!.free) then
if IsBound(Y!.free[dim-1]) then
if tmp[1]=1 then AddSet(Y!.free[dim-1],x); fi;
fi;
fi;
od;
return [dim-1,n];
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallGlobalFunction(CriticalCellsOfRegularCWComplex,
function(arg)
local Y,ContractSpace,cells,dim,c,pos,ppos, b,x, bbooll;
Y:=arg[1];
if not Y!.criticalCells=fail then
return Y!.criticalCells;
fi;
##############################
if Length(arg)>1 then
cells:=CocriticalCellsOfRegularCWComplex(Y,arg[2]);
if arg[2]<EvaluateProperty(Y,"dimension") then
Y!.criticalCells:=fail; fi;
return cells;
fi;
##############################
ContractSpace:=HAPContractRegularCWComplex;
#######
dim:=0;
while true do
if Y!.nrCells(dim)=0 then break; fi;
dim:=dim+1;
od;
dim:=dim-1;
#######
cells:=[];
ContractSpace(Y);
while true do
if
Sum(List( [1..Length(Y!.bnd)],i->Y!.nrCells(i-1)))=0
then
Y!.criticalCells:=cells;
Y!.nrCells:=function(k);
if k>EvaluateProperty(Y,"dimension") then return 0; fi;
return Length(Filtered(Y!.boundaries[k+1],x->not x[1]=0));
end;
Unbind(Y!.bnd);
Unbind(Y!.cobnd);
return cells; fi;
pos:=0;
while true do
pos:=pos+1;
ppos:=PositionProperty(Y!.bnd[dim+1]{[pos..Length(Y!.bnd[dim+1])]}, x->x[1]>0); if ppos=fail then dim:=dim-1; break; fi;
pos:=pos+ppos-1;
#######
#######
if dim=0 then bbooll:=true; else
bbooll:=false;
for b in Y!.bnd[dim+1][pos]{[2..Length(Y!.bnd[dim+1][pos])]} do
if bbooll then break; fi;
if Y!.cobnd[dim][b][1]=2 then bbooll:=true; break; fi;
od;
fi;
#######
#######
c:=HAPRemoveCellFromRegularCWComplex(Y,dim,pos);
Add(cells,c);
if bbooll then
ContractSpace(Y);
fi;
od;
od;
Y!.criticalCells:=cells;
Y!.nrCells:=function(k);
if k>EvaluateProperty(Y,"dimension") then return 0; fi;
return Length(Filtered(Y!.boundaries[k+1],x->not x[1]=0));
end;
Unbind(Y!.bnd);
Unbind(Y!.cobnd);
return cells;
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallGlobalFunction(CubicalComplexToRegularCWComplex,
function(arg)
local M,dim,C, Properties, Boundaries, Coboundaries, BinLst,
LstBin, bnd, Boundary, ArrayValueDim, Orientation,
Dimension, n, i, j, k, b, v;
M:=arg[1];
if Length(arg)>1 then dim:=arg[2];
else
dim:=EvaluateProperty(M,"dimension");
fi;
ArrayValueDim:=ArrayValueFunctions(EvaluateProperty(M,"dimension"));
C:=ChainComplex(M);
BinLst:=C!.coordinateToPosition;
LstBin:=C!.positionToCoordinate;
Properties:=[["dimension",dim]];
if Length(arg)=1 then
Dimension:=C!.dimension;
else
###
Dimension:=function(n);
if n<dim then return C!.dimension(n);
else return 0; fi;
end;
#######
fi;
#######################################
Boundary:=function(n,j)
local x,poscells,negcells,nn,a,b,cnt;
poscells:=[];
negcells:=[];
cnt:=0;
nn:=LstBin[n+1][j];
for x in [1..Length(nn)] do
if IsEvenInt(nn[x]) then
cnt:=cnt+1;
a:=StructuralCopy(nn);
a[x]:=a[x]+1;
b:=StructuralCopy(nn);
b[x]:=b[x]-1;
if IsOddInt(cnt) then
Add(poscells,a);
Add(negcells,b);
else
Add(poscells,b);
Add(negcells,a);
fi;
fi;
od;
Apply(poscells,x->ArrayValueDim(BinLst,x));
Apply(negcells,x->ArrayValueDim(BinLst,x));
return [poscells,negcells];
end;
################################
##############################
Boundaries:=[];
Boundaries[1]:=List([1..C!.dimension(0)],x->[1,0]);
Orientation:=[];
Orientation[1]:=List([1..C!.dimension(0)],x->[1]);
for n in [1..dim] do
Boundaries[n+1]:=[];
Orientation[n+1]:=[];
for i in [1..C!.dimension(n)] do
v:=StructuralCopy(Boundary(n,i));
bnd:=Concatenation([Length(v[1])+Length(v[2])],Flat(v));
Add(Boundaries[n+1],bnd);
Add(Orientation[n+1],
Concatenation(List([1..Length(v[1])],a->1),List([1..Length(v[2])],a->-1)));
od;
od;
Boundaries[dim+2]:=[];
##############################
### COBOUNDARIES BEGIN ######################
Coboundaries:=[];; #Coboundaries[n+1] contains the info on n-cells.
for n in [0..dim] do
k:=2*(n+1)+1;#k:=1+2^(n+1);
Coboundaries[n+1]:=List(Boundaries[n+1],i->[0]);
for j in [1..Length(Boundaries[n+2])] do
b:=Boundaries[n+2][j];
#k:=Length(b);
for i in b{[2..k]} do
Coboundaries[n+1][i][1]:=Coboundaries[n+1][i][1]+1;
Add(Coboundaries[n+1][i],j);
od;
od;
# for b in Coboundaries[n+1] do
# Append(b,List([1..Length(b)-1],a->1));
# od;
od;
Coboundaries[dim+1]:=List(Boundaries[dim+1],a->[0]);
### COBOUNDARIES END ###############################
return Objectify(HapRegularCWComplex,
rec(
nrCells:=Dimension,
boundaries:=Boundaries,
coboundaries:=Coboundaries,
vectorField:=fail,
inverseVectorField:=fail,
criticalCells:=fail,
orientation:=Orientation,
properties:=Properties));
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallGlobalFunction(ChainComplexOfRegularCWComplex,
function(Y)
local
C, Dimension, Boundary, one, zero, n, dim, characteristic;
#Dimension:=Y!.nrCells;
##########################
Dimension:=function(n);
if n<0 then return 0; fi;
return Length(Y!.boundaries[n+1]);
end;
##########################
dim:=EvaluateProperty(Y,"dimension");
zero:=[];
for n in [1..dim+1] do
zero[n]:=List([1..Dimension(n-1)],i->0);
od;
if not IsBound(Y!.orientation) then
characteristic:=2;
one:=One(GF(2));
######################
Boundary:=function(n,k)
local b,i,j,B;
b:=StructuralCopy(zero[n]);
B:=Y!.boundaries[n+1][k];
for i in [2..Length(B)] do
b[B[i]]:=1;
od;
return one*b;
end;
######################
else
characteristic:=0;
######################
Boundary:=function(n,k)
local b,i,j,B,sn;
b:=StructuralCopy(zero[n]);
B:=Y!.boundaries[n+1][k];
sn:=Y!.orientation[n+1][k];
for i in [2..Length(B)] do
b[B[i]]:=sn[i-1];
od;
return b;
end;
######################
fi;
return
Objectify(HapChainComplex,
rec(
dimension:=Dimension,
boundary:=Boundary,
properties:=[
["length",dim],
["type","chainComplex"],
["characteristic",characteristic]]
));
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallGlobalFunction(ChainComplexOfRegularCWComplexWithVectorField,
function(arg)
local
Y, basis, bool, bij,Dimension, Boundary, one, zero, b, n, dim, characteristic, DeformCell,DeformCellSgn, DCSrec, AlgRed;
Y:=arg[1];
if Length(arg)=2 then bool:=arg[2]; else bool:=false; fi;
#############################################
AlgRed:=function(w)
local cnt, tog;
cnt:=1;
tog:=true;
while tog do
tog:=false;
while cnt<Length(w) do
if w[cnt]=-w[cnt+1] then
w[cnt]:=0; w[cnt+1]:=0;
cnt:=cnt+2; tog:=true;
else cnt:=cnt+1;
fi;
od;
w:=Filtered(w,i->not i=0);
od;
return w;
end;
#############################################
if not bool then AlgRed:=AlgebraicReduction;fi;
dim:=EvaluateProperty(Y,"dimension");
basis:=[];
bij:=[];
for n in [0..dim] do
basis[n+1]:=Filtered(CriticalCellsOfRegularCWComplex(Y),x->x[1]=n);
Apply(basis[n+1],x->x[2]);
bij[n+1]:=[];
for b in [1..Length(basis[n+1])] do
bij[n+1][basis[n+1][b]]:=b;
od;
od;
###############################
Dimension:=function(n);
return Length(basis[n+1]);
end;
###############################
zero:=[];
for n in [1..dim+1] do
zero[n]:=List([1..Dimension(n-1)],i->0);
od;
###############################
###############################
DeformCell:=function(n,k)
local x,f,bnd,def; #This will return a list of n-cells
#into which the k-th n-cell is deformed.
if [n,k] in Y!.criticalCells then
return [k];
fi;
if n>0 then
if IsBound(Y!.vectorField[n][k]) then return []; fi;
fi;
f:=Y!.inverseVectorField[n+1][k];
bnd:=Y!.boundaries[n+2][f];
def:=[];
for x in [2..Length(bnd)] do
if not bnd[x]=k then
Append(def,DeformCell(n,bnd[x]));
fi;
od;
return def;
end;
###############################
###############################
DCSrec:=List([1..dim+1],i->[]);;
###############################
###############################
DeformCellSgn:=function(n,kk)
local sgnn,x,f,k,sgnk,cnt,bnd,def,sn,tog,def1,def2;
#This will return a list of signed n-cells
#into which the k-th n-cell is deformed.
k:=AbsInt(kk);
sgnk:=SignInt(kk);
if [n,k] in Y!.criticalCells then
return [kk];
fi;
if n>0 then
if IsBound(Y!.vectorField[n][k]) then return []; fi;
fi;
if IsBound(DCSrec[n+1][k]) then
if sgnk=1 then return DCSrec[n+1][k];
else
return -Reversed(DCSrec[n+1][k]);
fi;
fi;
f:=Y!.inverseVectorField[n+1][k];
bnd:=Y!.boundaries[n+2][f];
sn:=Y!.orientation[n+2][f];
def:=[]; def1:=[];def2:=[];
for x in [2..Length(bnd)] do
if not bnd[x]=k then
Add(def1,sn[x-1]*bnd[x]);
else
sgnn:=sn[x-1];
break;
fi;
od;
cnt:=x+1;
for x in [cnt..Length(bnd)] do
Add(def2,sn[x-1]*bnd[x]);
od;
if sgnn=1 then
def:=-Concatenation(Reversed(def1),Reversed(def2));
else
def:=Concatenation(def2,def1);
fi;
Apply(def,x->DeformCellSgn(n,x));
def:=Flat(def);
Apply(def,x->[x,0]);
def:=AlgRed(def);
#def:=AlgebraicReduction(def); #ONLY VALID FOR HOMOLOGY, NOT HOMOTOPY
Apply(def,x->x[1]);
DCSrec[n+1][k]:=def;
if sgnk=1 then return def;
else
return -Reversed(def);
fi;
end;
###############################
###############################
if not IsBound(Y!.orientation) then
characteristic:=2;
one:=One(GF(2));
######################
Boundary:=function(n,k)
local b,i,j,B;
b:=StructuralCopy(zero[n]);
B:=Y!.boundaries[n+1][basis[n+1][k]];
B:=B{[2..Length(B)]};
Apply(B,x->DeformCell(n-1,x));
B:=Concatenation(B);
Apply(B,i->bij[n][i]);
for i in B do
b[i]:=b[i]+1;
od;
return one*b;
end;
######################
else
characteristic:=0;
######################
Boundary:=function(n,k)
local b,i,j,B,sn;
b:=StructuralCopy(zero[n]);
B:=Y!.boundaries[n+1][basis[n+1][k]];
B:=B{[2..Length(B)]};
sn:=Y!.orientation[n+1][basis[n+1][k]];
B:=List([1..Length(B)],i->sn[i]*B[i]);
Apply(B,x->DeformCellSgn(n-1,x));
B:=Concatenation(B);
Apply(B,i->SignInt(i)*bij[n][AbsInt(i)]);
for i in B do
#b[AbsInt(i)]:=b[AbsInt(i)]+SignInt(i)*sn[AbsInt(i)];
b[AbsInt(i)]:=b[AbsInt(i)]+SignInt(i);
od;
return b;
end;
######################
fi;
if IsBound(Y!.orientation) then DeformCell:=DeformCellSgn; fi;
return
Objectify(HapChainComplex,
rec(
dimension:=Dimension,
boundary:=Boundary,
deform:=DeformCell,
basis:=basis,
bij:=bij,
properties:=[
["length",dim],
["type","chainComplex"],
["characteristic",characteristic]]
));
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallMethod( ChainComplex,
"for regular CW spaces, using discrete vector fields",
[IsHapRegularCWComplex],
function(arg) local Y,bool;
Y:=arg[1]; if Length(arg)=2 then bool:=arg[2]; else bool:=false; fi;
CriticalCellsOfRegularCWComplex(Y);
return ChainComplexOfRegularCWComplexWithVectorField(Y,bool);
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallMethod( SparseChainComplex,
"for regular CW spaces, using discrete vector fields",
[IsHapRegularCWComplex],
function(Y)
CriticalCellsOfRegularCWComplex(Y);
return SparseChainComplexOfRegularCWComplexWithVectorField(Y);
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallOtherMethod( Homology,
"Homology of a regular CW spaces, using discrete vector fields",
[IsHapRegularCWComplex,IsInt],
function(Y,n) local C, H, m, bool;
if not IsBound(Y!.orientation) then
Print("Can only compute the mod 2 homology as no orientation is available.\n");
fi;
m:=Minimum(n+1,Dimension(Y));
bool:=Y!.vectorField=fail or Y!.criticalCells=fail;
if bool then
if m=Dimension(Y) then CriticalCellsOfRegularCWComplex(Y);
else
CocriticalCellsOfRegularCWComplex(Y,m); fi;
fi;
C:=ChainComplex(Y);
H:=Homology(C,n);
if m<Dimension(Y) and bool then
Y!.vectorField:=fail;
Y!.criticalCells:=fail;
Y!.properties:=Filtered(Y!.properties,x->not x[1]="codim");
fi;
return H;
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallGlobalFunction(SparseChainComplexOfRegularCWComplex,
function(Y)
local
C, Dimension, Boundary, one, n, dim, characteristic;
#Dimension:=Y!.nrCells;
##########################
Dimension:=function(n);
if n<0 then return 0; fi;
return Length(Y!.boundaries[n+1]);
end;
##########################
dim:=EvaluateProperty(Y,"dimension");
if not IsBound(Y!.orientation) then
characteristic:=2;
one:=One(GF(2));
######################
Boundary:=function(n,k)
local b,i,j,B;
b:=[];
B:=Y!.boundaries[n+1][k];
for i in [2..Length(B)] do
Add(b,[B[i],one]);
od;
return b;
end;
######################
else
characteristic:=0;
######################
Boundary:=function(n,k)
local b,i,j,B,sn;
b:=[];
B:=Y!.boundaries[n+1][k];
sn:=Y!.orientation[n+1][k];
for i in [2..Length(B)] do
Add(b,[B[i],sn[i-1]]);
od;
return b;
end;
######################
fi;
return
Objectify(HapSparseChainComplex,
rec(
dimension:=Dimension,
boundary:=Boundary,
properties:=[
["length",dim],
["type","chainComplex"],
["characteristic",characteristic]]
));
end);
##########################################################
##########################################################
##########################################################
##########################################################
InstallGlobalFunction(SparseChainComplexOfRegularCWComplexWithVectorField,
function(Y)
local
basis, bij,Dimension, Boundary, one, zero, b, n, dim, characteristic, DeformCell,DeformCellSgn;
dim:=EvaluateProperty(Y,"dimension");
basis:=[];
bij:=[];
for n in [0..dim] do
basis[n+1]:=Filtered(CriticalCellsOfRegularCWComplex(Y),x->x[1]=n);
Apply(basis[n+1],x->x[2]);
bij[n+1]:=[];
for b in [1..Length(basis[n+1])] do
bij[n+1][basis[n+1][b]]:=b;
od;
od;
###############################
Dimension:=function(n);
return Length(basis[n+1]);
end;
###############################
zero:=[];
for n in [1..dim+1] do
zero[n]:=List([1..Dimension(n-1)],i->0);
od;
###############################
###############################
DeformCell:=function(n,k)
local x,f,bnd,def; #This will return a list of n-cells
#into which the k-th n-cell is deformed.
if [n,k] in Y!.criticalCells then
return [k];
fi;
if n>0 then
if IsBound(Y!.vectorField[n][k]) then return []; fi;
fi;
f:=Y!.inverseVectorField[n+1][k];
bnd:=Y!.boundaries[n+2][f];
def:=[];
for x in [2..Length(bnd)] do
if not bnd[x]=k then
Append(def,DeformCell(n,bnd[x]));
fi;
od;
return def;
end;
###############################
###############################
###############################
###############################
DeformCellSgn:=function(n,kk)
local sgnn,x,f,k,sgnk,cnt,bnd,def,sn,tog,def1,def2;
#This will return a list of signed n-cells
#into which the k-th n-cell is deformed.
k:=AbsInt(kk);
sgnk:=SignInt(kk);
if [n,k] in Y!.criticalCells then
return [kk];
fi;
if n>0 then
if IsBound(Y!.vectorField[n][k]) then return []; fi;
fi;
f:=Y!.inverseVectorField[n+1][k];
bnd:=Y!.boundaries[n+2][f];
sn:=Y!.orientation[n+2][f];
def:=[]; def1:=[];def2:=[];
for x in [2..Length(bnd)] do
if not bnd[x]=k then
Add(def1,sn[x-1]*bnd[x]);
else
sgnn:=sn[x-1];
break;
fi;
od;
cnt:=x+1;
for x in [cnt..Length(bnd)] do
Add(def2,sn[x-1]*bnd[x]);
od;
if sgnn=1 then
def:=-Concatenation(Reversed(def1),Reversed(def2));
else
def:=Concatenation(def2,def1);
fi;
Apply(def,x->DeformCellSgn(n,x));
if sgnk=1 then return Flat(def);
else
return -Reversed(Flat(def));
fi;
end;
###############################
###############################
if not IsBound(Y!.orientation) then
characteristic:=2;
one:=One(GF(2));
######################
Boundary:=function(n,k)
local b,i,j,B;
b:=[];
B:=Y!.boundaries[n+1][basis[n+1][k]];
B:=B{[2..Length(B)]};
Apply(B,x->DeformCell(n-1,x));
B:=Concatenation(B);
Apply(B,i->bij[n][i]);
for i in B do
Add(b,[i,one]);
#b[i]:=b[i]+1;
od;
b:=SortedList(b);
for i in [1..Length(b)-1] do
if b[i][1]=b[i+1][1] then
b[i+1][2]:=b[i+1][2]+b[i][2];
b[i][2]:=0;
fi;
od;
b:=Filtered(b,x->not IsZero(x[2]));
return b;
end;
######################
else
characteristic:=0;
######################
Boundary:=function(n,k)
local b,i,j,B,sn;
b:=[];
B:=Y!.boundaries[n+1][basis[n+1][k]];
B:=B{[2..Length(B)]};
sn:=Y!.orientation[n+1][basis[n+1][k]];
B:=List([1..Length(B)],i->sn[i]*B[i]);
Apply(B,x->DeformCellSgn(n-1,x));
B:=Concatenation(B);
Apply(B,i->SignInt(i)*bij[n][AbsInt(i)]);
for i in B do
#b[AbsInt(i)]:=b[AbsInt(i)]+SignInt(i);
Add(b,[AbsInt(i),SignInt(i)]);
od;
b:=SortedList(b);
for i in [1..Length(b)-1] do
if b[i][1]=b[i+1][1] then
b[i+1][2]:=b[i+1][2]+b[i][2];
b[i][2]:=0;
fi;
od;
b:=Filtered(b,x->not IsZero(x[2]));
return b;
end;
######################
fi;
if IsBound(Y!.orientation) then DeformCell:=DeformCellSgn; fi;
return
Objectify(HapSparseChainComplex,
rec(
dimension:=Dimension,
boundary:=Boundary,
deform:=DeformCell,
basis:=basis,
bij:=bij,
properties:=[
["length",dim],
["type","chainComplex"],
["characteristic",characteristic]]
));
end);
##########################################################
##########################################################