Quelle chainComplexOfSimplicialGroup.gi
Sprache: unbekannt
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rahmenlose Ansicht.gi DruckansichtUnknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
#############################################################################
#0
#F ChainComplexOfSimplicialGroup
## Input: A simplicial group or a morphism of simplicial groups or
## a sequence of morphisms of simplicial groups
## Output: The image of the input under the functor
## C: (simplicial groups)->(chain complexes)
##
InstallGlobalFunction(ChainComplexOfSimplicialGroup, function(X)
local
AddElement,
ChainComplexOf_Objpre,
ChainComplexOf_Obj,
ChainComplexOf_Morpre,
ChainComplexOf_Mor,
ChainComplexOf_Seq;
######################################################################
#1
#F AddElement
## Input: A list L=[[m_1,h_1,g_11,...,g_1n],...,[m_k,h_k,g_k1,...,g_kn]]
## and an element x=[m',h',g'_1,...,g'_n]
## Output: Add the element x into the list L
##
AddElement:=function(L,x)
local sx,nx,nL,flag,i,j;
sx:=StructuralCopy(x);
nx:=Length(sx);
nL:=Length(L);
for i in [1..nL] do
flag:=0;
for j in [2..nx] do
if L[i][j]<>sx[j] then
flag:=1;
break;
fi;
od;
if flag=0 then
L[i][1]:=L[i][1]+sx[1];
if L[i][1]=0 then
Remove(L,i);
fi;
return;
fi;
od;
Add(L,sx);
end;
##
#################### end of AddElement ##############################
######################################################################
#1
#F ChainComplexOf_Objpre
##
ChainComplexOf_Objpre:=function(N,Maps,Bar,Hap)
local
HapBoundary,Phi,Psi,Equiv,HapDimension,BarMap,
Dim,BoundChain,ImageBasis,
i,j,k,n,tmp,t,
M,m,ii,jj,
SearchPosition,Dimension,BelowDim,Boundary,
d0,dm;
#############################################################
#2
HapBoundary:= function(i,j,k)
return Hap[i+1]!.boundary(j,k);
end;
##
#############################################################
#2
Psi:=function(i,j,w)
return Bar[i+1]!.psi(j,w);
end;
##
#############################################################
#2
Phi:=function(i,j,w)
return Bar[i+1]!.phi(j,w);
end;#
##
#############################################################
#2
Equiv:=function(i,j,w)
return Bar[i+1]!.equiv(j,w);
end;
##
#############################################################
#2
HapDimension:=function(i,j)
return Hap[i+1]!.dimension(j);
end;
##
#############################################################
#############################################################
#2
#F BarMap
## Input: A position (i,j) and a word w
## Output: The image of w under del_n: B_{i}^j->B_{i-1}^j
##
BarMap:=function(i,j,w)
local Rew,sign,ii,jj,x,d,tmp;
if j mod 2 = 0 then
sign:=1;
else
sign:=-1;
fi;
Rew:=[];
for ii in [0..i] do
d:=Maps(i,ii);
for x in w do
tmp:=[sign*x[1]];
for jj in [1..j] do
Add(tmp,Image(d,x[jj+1]));
od;
AddElement(Rew,tmp);
od;
sign:=-sign;
od;
return Rew;
end;
##
########## end of BarMap ####################################
########## Compute the dimendion of K_n #####################
Dim:=[];
for i in [0..N] do
k:=0;
for j in [0..i] do;
k:=k+HapDimension(j,i-j);
od;
Dim[i+1]:=k;
od;
#############################################################
#2
Dimension:=function(n)
return Dim[n+1];
end;
##
#############################################################
########### Compute the sum of dimensions under the position(i,j)
BelowDim:=List([0..N],n->[]); ###i+1,j+1
for i in [0..N] do
for j in [0..N-i] do
n:=i+j;
tmp:=0;
for k in [0..j-1] do
tmp:=tmp+HapDimension(n-k,k);
od;
BelowDim[i+1][j+1]:=tmp;
od;
od;
############################################################
#2
#F SearchPosition
## Input: Numbers n and t
## Output: The position (i,j) and the order of e_t
SearchPosition:=function(n,t)
local count,j,k;
count:=0;
for j in [0..n] do
k:=t-count;
count:=count+HapDimension(n-j,j);
if t <= count then
return [n-j,j,k];
break;
fi;
od;
end;
##
########## end of SearchPosition ############################
#############################################################
#2
#F d0
## Input: A position (i,j) and a basis element e_k
## Output: The image of e_k under the map d_0
##
d0:=function(i,j,k)
local n,t,beg,Bound,Rew;
n:=i+j;
if n=0 then
return [0];
fi;
Rew:=List([1..Dimension(n-1)],x->0);
if j=0 then
return Rew;
fi;
beg:=BelowDim[i+1][j]; ##below(i,j-1)
Bound:=HapBoundary(i,j,k);
for t in [1..HapDimension(i,j-1)] do
Rew[beg+t]:=Bound[t];
od;
return Rew;
end;
##
########## end of d0 ########################################
########## Compute d_m(e_k) at the postion (i,j) ############
ImageBasis:=[];
for i in [1..N] do
ImageBasis[i]:=[];
for j in [0..N-i]do
ImageBasis[i][j+1]:=[];
for m in [1..i] do
ImageBasis[i][j+1][m]:=[];
od;
od;
od;
######## Compute ImageBasis[i][j+1][1][k] ################
for i in [1..N] do
for j in [0..N-i] do
for k in [1..HapDimension(i,j)] do
ImageBasis[i][j+1][1][k]:=BarMap(i,j,Psi(i,j,[[1,k]]));
od;
od;
od;
######## Compute for m>1 ###############################
for i in [2..N]do
for j in [0..N-i]do
for m in [2..i] do
for k in [1..HapDimension(i,j)] do
tmp:=StructuralCopy(ImageBasis[i][j+1][m-1][k]);
ImageBasis[i][j+1][m][k]:=BarMap(i-m+1,j+m-1,
Equiv(i-m+1,j+m-2,tmp));
od;
od;
od;
od;
#############################################################
#2
#F dm
## Input: A position (i,j) and a basis element e_k
## Output: The image of e_k under the map d_m
##
dm:=function(i,j,m,k)
local n,t,beg,Rew,Phiw;
n:=i+j;
if n=0 then
return [0];
fi;
Rew:=List([1..Dimension(n-1)],x->0);
if m>i then
return Rew;
fi;
Phiw:= Phi(i-m,j+(m-1),ImageBasis[i][j+1][m][k]);
beg:=BelowDim[(i-m)+1][j+(m-1)+1];
for t in [1..HapDimension(i-m,j+(m-1))] do
Rew[beg+t]:=Phiw[t];
od;
return Rew;
end;
##
########## end of dm ########################################
BoundChain:=List([0..N],x->[]);
BoundChain[1][1]:=[0];
for n in [1..N] do
for t in [1..Dimension(n)] do
M:=SearchPosition(n,t);
i:=M[1];
j:=M[2];
k:=M[3];
tmp:=d0(i,j,k);
for m in [1..i] do
tmp:=tmp+dm(i,j,m,k);
od;
BoundChain[n+1][t]:=tmp;
od;
od;
#############################################################
#2
Boundary:=function(n,k)
return BoundChain[n+1][k];
end;
##
#############################################################
return Objectify( HapChainComplex, rec(
boundary:=Boundary,
dimension:=Dimension,
properties:= [ [ "length",N],
[ "type", "chainComplex" ],
[ "characteristic",0 ] ]
) );
end;
##
############### end of ChainComplexOf_Objpre ##########
######################################################################
#1
#F ChainComplexOf_Obj
## Input: A simplicial group G
## Output: A chain complex of G
##
ChainComplexOf_Obj:=function(G)
local N,Maps,Grps,Bar,Hap,Res;
N:=EvaluateProperty(G,"length");
Maps:=G!.boundariesList;
Grps:=G!.groupsList;
Res:=List([0..N],i->ResolutionGenericGroup(Grps(i),N-i));
Bar:=List([0..N],i->BarComplexEquivalence(Res[i+1]));
Hap:=List([0..N],i->TensorWithIntegers(Res[i+1]));
return ChainComplexOf_Objpre(N,Maps,Bar,Hap);
end;
##
############### end of ChainComplexOf_Obj ############################
######################################################################
#1
#F ChainComplexOf_Morpre
##
ChainComplexOf_Morpre:=function(N,map,MapsH,BarH,HapH,MapsG,BarG,HapG)
local
PsiH,PhiH,EquivH,HapDimensionH,DimensionH,BarMapH,DimH,
SearchPosH,ZeroVectorH,ImageBasisH,
PsiG,PhiG,EquivG,HapDimensionG,DimensionG,BarMapG,DimG,
SearchPosG,ZeroVectorG,ImgG,Imgs,
i,j,k,m,n,t,tmp,Tmp,itmp,w,LM,FF,LowDimG,Mapping,
BarMapHG;
############### All functions for H #########################
#############################################################
#2
PsiH:=function(i,j,w)
return BarH[i+1]!.psi(j,w);
end;
##
#############################################################
#2
PhiH:=function(i,j,w)
return BarH[i+1]!.phi(j,w);
end;
##
#############################################################
#2
EquivH:=function(i,j,w)
return BarH[i+1]!.equiv(j,w);
end;
##
#############################################################
#2
HapDimensionH:=function(i,j)
return HapH[i+1]!.dimension(j);
end;
##
#############################################################
#############################################################
#2
#F BarMapH
## Input: A position (i,j) and a word w
## Output: The image of w under del_n: BH_{i}^j->BH_{i-1}^j
##
BarMapH:=function(i,j,w)
local Rew,sign,ii,jj,x,d,tmp;
if j mod 2 = 0 then
sign:=1;
else
sign:=-1;
fi;
Rew:=[];
for ii in [0..i] do
d:=MapsH(i,ii);
for x in w do
tmp:=[sign*x[1]];
for jj in [1..j] do
Add(tmp,Image(d,x[jj+1]));
od;
AddElement(Rew,tmp);
od;
sign:=-sign;
od;
return Rew;
end;
##
########## end of BarMapH ###################################
########## Compute the dimendion of KH_n ####################
DimH:=[];
for i in [0..N] do
k:=0;
for j in [0..i] do;
k:=k+HapDimensionH(j,i-j);
od;
DimH[i+1]:=k;
od;
#############################################################
#2
DimensionH:=function(n)
return DimH[n+1];
end;
##
#############################################################
#############################################################
#2
#F SearchPosH
## Input: Numbers n and t
## Output: The position (i,j) and the order of e_t
SearchPosH:=function(n,t)
local count,j,k;
if t>DimensionH(n) then
return fail;
fi;
count:=0;
for j in [0..n] do
k:=t-count;
count:=count+HapDimensionH(n-j,j);
if t <= count then
return [n-j,j,k];
break;
fi;
od;
end;
##
########## end of SearchPosition ############################
############### All functions for G #########################
#############################################################
#2
PsiG:=function(i,j,w)
return BarG[i+1]!.psi(j,w);
end;
##
#############################################################
#2
PhiG:=function(i,j,w)
return BarG[i+1]!.phi(j,w);
end;
##
#############################################################
#2
EquivG:=function(i,j,w)
return BarG[i+1]!.equiv(j,w);
end;
##
#############################################################
#2
HapDimensionG:=function(i,j)
return HapG[i+1]!.dimension(j);
end;
##
#############################################################
#############################################################
#2
#F BarMapG
## Input: A position (i,j) and a word w
## Output: The image of w under del_n: BG_{i}^j->BG_{i-1}^j
##
BarMapG:=function(i,j,w)
local Rew,sign,ii,jj,x,d,tmp;
if j mod 2 = 0 then
sign:=1;
else
sign:=-1;
fi;
Rew:=[];
for ii in [0..i] do
d:=MapsG(i,ii);
for x in w do
tmp:=[sign*x[1]];
for jj in [1..j] do
Add(tmp,Image(d,x[jj+1]));
od;
AddElement(Rew,tmp);
od;
sign:=-sign;
od;
return Rew;
end;
##
########## end of BarMapG ###################################
########## Compute the dimendion of KG_n ####################
DimG:=[];
for i in [0..N] do
k:=0;
for j in [0..i] do;
k:=k+HapDimensionG(j,i-j);
od;
DimG[i+1]:=k;
od;
#############################################################
#2
DimensionG:=function(n)
return DimG[n+1];
end;
##
#############################################################
#############################################################
#2
#F SearchPosG
## Input: Numbers n and t
## Output: The position (i,j) and the order of e_t
SearchPosG:=function(n,t)
local count,j,k;
if t>DimensionG(n) then
return fail;
fi;
count:=0;
for j in [0..n] do
k:=t-count;
count:=count+HapDimensionG(n-j,j);
if t <= count then
return [n-j,j,k];
break;
fi;
od;
end;
##
########## end of SearchPosition ############################
#############################################################
#2
#F BarMapHG
## Input: A position (i,j) and a word w
## Output: The image of w under the map_i: BH->BG
##
BarMapHG:=function(i,j,w)
local x,f,jj,tmp,Rew;
f:=map(i);
Rew:=[];
for x in w do
tmp:=[x[1]];
for jj in [2..j+1] do
Add(tmp,Image(f,x[jj]));
od;
AddElement(Rew,tmp);
od;
return Rew;
end;
##
########## end of BarMapHG ##################################
########## Compute d_m(e_k) at the postion (i,j) in BH ######
ImageBasisH:=[];
for i in [0..N] do
ImageBasisH[i+1]:=[];
for j in [0..N-i]do
ImageBasisH[i+1][j+1]:=[];
for m in [0..i] do
ImageBasisH[i+1][j+1][m+1]:=[];
od;
od;
od;
######### Compute for m =1 ##################################
for i in [0..N] do
for j in [0..N-i] do
for m in [0..i] do
for k in [1..HapDimensionH(i,j)] do
ImageBasisH[i+1][j+1][m+1][k]:=PsiH(i,j,[[1,k]]);
od;
od;
od;
od;
######### Compute for m >1 ##################################
for i in [1..N]do
for j in [0..N-i]do
for m in [1..i] do
for k in [1..HapDimensionH(i,j)] do
tmp:=StructuralCopy(ImageBasisH[i+1][j+1][m][k]);
ImageBasisH[i+1][j+1][m+1][k]:=EquivH(i-m,j+(m-1),
(BarMapH(i-(m-1),j+(m-1),tmp)));
od;
od;
od;
od;
########## Compute d_m(e_k) at the postion (i,j) from BH ->BG
ImgG:=[];
for i in [0..N] do
ImgG[i+1]:=[];
for j in [0..N-i]do
ImgG[i+1][j+1]:=[];
for m in [0..i] do
ImgG[i+1][j+1][m+1]:=[];
od;
od;
od;
for i in [0..N]do
for j in [0..N-i]do
for m in [0..i] do
for k in [1..HapDimensionH(i,j)] do
ImgG[i+1][j+1][m+1][k]:=BarMapHG(i-m,j+m,
ImageBasisH[i+1][j+1][m+1][k]);
od;
od;
od;
od;
Imgs:=[];
for i in [0..N] do
Imgs[i+1]:=[];
for j in [0..N-i]do
Imgs[i+1][j+1]:=[];
for m in [0..i] do
Imgs[i+1][j+1][m+1]:=[];
od;
od;
od;
for i in [0..N]do
for j in [0..N-i]do
for k in [1..HapDimensionH(i,j)] do
Tmp:=List([0..i],m->[]);
for m in [0..i] do ##Create Tmp(m,0)
Tmp[m+1][0+1]:=ImgG[i+1][j+1][m+1][k];
od;
for m in [0..i-1] do
for n in [1..i-m] do
w:=StructuralCopy(Tmp[m+1][n]);
Tmp[m+1][n+1]:=BarMapG(i-(m+(n-1)),j+(m+n),
EquivG(i-(m+(n-1)),j+(m+(n-1)),w));
od;
od;
for m in [0..i] do
itmp:=[];
for n in [0..m] do
Append(itmp,Tmp[n+1][m-n+1]);
od;
Imgs[i+1][j+1][m+1][k]:=PhiG(i-m,j+m,itmp);
od;
od;
od;
od;
########### Compute the sum of dimensions under position(i,j)
LowDimG:=List([0..N],n->[]);
for i in [0..N] do
for j in [0..N-i] do
n:=i+j;
tmp:=0;
for k in [0..j-1] do
tmp:=tmp+HapDimensionG(n-k,k);
od;
LowDimG[i+1][j+1]:=tmp;
od;
od;
####################################
FF:=List([0..N],n->[]);
for n in [0..N] do
for t in [1..DimensionH(n)] do
LM:=SearchPosH(n,t);
i:=LM[1];
j:=LM[2];
k:=LM[3];
Tmp:=List([1..LowDimG[i+1][j+1]],m->0);
for m in [0..i] do
Append(Tmp,Imgs[i+1][j+1][m+1][k]);
od;
FF[n+1][t]:=Tmp;
od;
od;
#############################################################
#2
#F Mapping
##
Mapping:=function(v,n)
local Rew,len,k;
Rew:=List([1..DimensionG(n)],x->0);
len:=Length(v);
for k in [1..len] do;
if v[k] <> 0 then
Rew:=Rew+v[k]*FF[n+1][k];
fi;
od;
return Rew;
end;
##
########## end of Mapping ###################################
return Mapping;
end;
##
############## end of ChainComplexOf_Morpre ##########################
######################################################################
#1
#F ChainComplexOf_Mor
## Input: A simplicial map Sf:G->G'
## Output: A chain complex of Sf
##
ChainComplexOf_Mor:=function(Sf)
local
map,N,
H,GrpsH,MapsH,RH,HapH,BarH,
G,GrpsG,MapsG,RG,HapG,BarG;
H:=Sf!.source;
G:=Sf!.target;
map:=Sf!.mapping;
N:=EvaluateProperty(Sf,"length");
GrpsH:=H!.groupsList;
MapsH:=H!.boundariesList;
RH:=List([0..N],i->ResolutionGenericGroup(GrpsH(i),(N+1)-i));
HapH:=List([0..N],i->TensorWithIntegers(RH[i+1]));
BarH:=List([0..N],i->BarComplexEquivalence(RH[i+1]));
GrpsG:=G!.groupsList;
MapsG:=G!.boundariesList;
RG:=List([0..N],i->ResolutionGenericGroup(GrpsG(i),(N+1)-i));
HapG:=List([0..N],i->TensorWithIntegers(RG[i+1]));
BarG:=List([0..N],i->BarComplexEquivalence(RG[i+1]));
return Objectify( HapChainMap, rec(
source:=ChainComplexOf_Objpre(N,MapsH,BarH,HapH),
target:=ChainComplexOf_Objpre(N,MapsG,BarG,HapG),
mapping:=ChainComplexOf_Morpre(N,map,MapsH,BarH,
HapH,MapsG,BarG,HapG),
properties:= [ [ "type", "chainMap" ],
[ "characteristic",0 ] ]
));
end;
##
############### end of ChainComplexOf_Mor ############################
######################################################################
#1
#F ChainComplexOf_Seq
## Input: A squence of simplicial maps L
## Output: A chain complex of L
##
ChainComplexOf_Seq:=function(L)
local
nL,k,
LSG,G,N,Grps,Maps,R,Hap,Bar,KG,RewL,map;
nL:=Length(L);
LSG:=[];
for k in [1..nL] do;
LSG[k]:=L[k]!.source;
od;
LSG[nL+1]:=L[nL]!.target;
Hap:=[];
Bar:=[];
Maps:=[];
KG:=[];
for k in [1..nL+1] do
G:=LSG[k];
N:=EvaluateProperty(G,"length");
Grps:=G!.groupsList;
Maps[k]:=G!.boundariesList;
R:=List([0..N],i->ResolutionGenericGroup(Grps(i),(N+1)-i));
Hap[k]:=List([0..N],i->TensorWithIntegers(R[i+1]));
Bar[k]:=List([0..N],i->BarComplexEquivalence(R[i+1]));
KG[k]:=ChainComplexOf_Objpre(N,Maps[k],Bar[k],Hap[k]);
od;
RewL:=[];
for k in [1..nL] do
N:=EvaluateProperty(L[k],"length");
map:=L[k]!.mapping;
RewL[k]:=Objectify( HapChainMap, rec(
source := KG[k],
target := KG[k+1],
mapping := ChainComplexOf_Morpre(N,map,Maps[k],
Bar[k],Hap[k],Maps[k+1],Bar[k+1],Hap[k+1]),
properties:= [ [ "type", "chainMap" ],
[ "characteristic",0 ] ]
) );
od;
return RewL;
end;
##
############## end of ChainComplexOf_Seq #############################
if IsHapSimplicialGroup(X) then
return ChainComplexOf_Obj(X);
fi;
if IsHapSimplicialGroupMorphism(X) then
return ChainComplexOf_Mor(X);
fi;
if IsList(X) then
return ChainComplexOf_Seq(X);
fi;
end);
##
################### end of ChainComplexOfSimplicialGroup ####################
##########################################################
##########################################################
InstallOtherMethod(ChainComplex,
"for simplicial groups",
[IsHapSimplicialGroup],
function(G)
return
ChainComplexOfSimplicialGroup(G);
end);
##########################################################
##########################################################
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2026-03-28
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