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#(C) Graham Ellis, 2005-2006
#####################################################################
InstallGlobalFunction(ResolutionGraphOfGroups,
function(arg)
local
D,N,L,
PositionName,
Vertices, Edges, EdgeGroups,
ColouredVertices, ComplementaryEdges,
resVertices, resEdges, resEdgeGroups,
FpHoms, F,Frels,PhomF, FundGroup,
inc,
Dimension,
Elts,
PseudoBoundary,
FillPseudoBoundary,
Boundary,
hom, quohom,
pos,e,f,g,x,P,R,S,T;
####################################READ INPUT DATA##################
D:=arg[1];
N:=arg[2];
if Length(arg)>2 then L:=arg[3];
else L:=[];
fi;
########################################DATA READ####################
if not GraphOfGroupsTest(D) then
Print("The list D does not represent a Graph of Groups \n");
return fail;
fi;
#####################################################################
PositionName:=function(L,x)
return PositionProperty(L,n->Name(n)=Name(x));
end;
#####################################################################
###########EXTRACT VERTICES ABD EDGES OF GRAPH OF GROUPS ############
Vertices:=[];
Edges:=[];
for x in D do
if IsGroup(x) then Append(Vertices,[x]); fi;
if IsList(x) then Append(Edges,[x]); fi;
od;
EdgeGroups:=List(Edges,e->Source(e[1]));
##########EXTRACTED##################################################
####################FIND MAXIMAL TREE IN GRAPH OF GROUPS#############
ColouredVertices:=[];
ComplementaryEdges:=[];
for e in Edges do
if Name(Range(e[1])) in ColouredVertices
and Name(Range(e[2])) in ColouredVertices
or Name(Range(e[1]))=Name(Range(e[2]))
then
Append(ComplementaryEdges,[[Name(Range(e[1])),Name(Range(e[2]))]]);
fi;
Append(ColouredVertices,[Name(Range(e[1])),Name(Range(e[2]))]);
od;
#########################TREE FOUND##################################
##########CREATE GRAPH OF RESOLUTIONS################################
resVertices:=[];
for x in Vertices do
pos:=Position(List(L,r->r!.group),x);
if pos=fail then
if Order(x)<121 then #This suggests I should improve
R:=ResolutionFiniteGroup(x,N); #ResolutionGenericGroup() !!!
else
R:=ResolutionGenericGroup(x,N);
fi;
else
R:=L[pos];
fi;
SetName(R!.group,Name(x));
Append(resVertices,[R]);
od;
resEdgeGroups:=[];
for x in EdgeGroups do
pos:=Position(List(L,r->r!.group),x);
if pos=fail then
#R:= ResolutionFiniteGroup(x,N-1);
R:= ResolutionGenericGroup(x,N-1);
else
R:=L[pos];
fi;
SetName(R!.group,Name(x));
Append(resEdgeGroups,[R]);
od;
resEdges:=[];
for x in Edges do
R:=resVertices[PositionName(Vertices,Range(x[1]))];
S:=resEdgeGroups[PositionName(EdgeGroups,Source(x[1]))];
T:=resVertices[PositionName(Vertices,Range(x[2]))];
e:=[EquivariantChainMap(S,R,x[1]),EquivariantChainMap(S,T,x[2])];
Append(resEdges,[e]);
od;
##########CREATED####################################################
##########CREATE HOMOMORPHISM TO FP GROUP############################
F:=FreeGroup(Length(ComplementaryEdges)+
Sum(List(resVertices,R->R!.dimension(1))));
FpHoms:=[];
Frels:=[];
x:=0;
for R in resVertices do
P:=PresentationOfResolution(R);
PhomF:=GroupHomomorphismByImagesNC(P.freeGroup,F,
GeneratorsOfGroup(P.freeGroup),
GeneratorsOfGroup(F){[x+1..x+R!.dimension(1)]});
Append(Frels, List(P.relators,r->Image(PhomF,r)));
Append(FpHoms,
[GroupHomomorphismByImagesNC(R!.group,F,
List([1..R!.dimension(1)],i->R!.elts[R!.boundary(1,i)[2][2]]), #CHANGED March 2021
GeneratorsOfGroup(F){[x+1..x+R!.dimension(1)]}
)]);
x:=x+R!.dimension(1);
od;
pos:=Sum(List(resVertices,R->R!.dimension(1)));
for e in Edges do
if not [Name(Range(e[1])),Name(Range(e[2]))] in ComplementaryEdges
and
not [Name(Range(e[2])),Name(Range(e[1]))] in ComplementaryEdges
then
for x in GeneratorsOfGroup(Source(e[1])) do
Append(Frels,[
Image(FpHoms[PositionName(Vertices,Range(e[1]))],
Image(e[1],x))*
Image(FpHoms[PositionName(Vertices,Range(e[2]))],
Image(e[2],x))^-1
]);
od;
else
pos:=pos+1;
for x in GeneratorsOfGroup(Source(e[1])) do
Append(Frels,[
Image(FpHoms[PositionName(Vertices,Range(e[1]))],
Image(e[1],x))*GeneratorsOfGroup(F)[pos]*
Image(FpHoms[PositionName(Vertices,Range(e[2]))],
Image(e[2],x))^-1*GeneratorsOfGroup(F)[pos]^-1
]);
od;
fi;
od;
FundGroup:=F/Frels; ###Added Aug 2013
quohom:=
GroupHomomorphismByImages(F,FundGroup, ###
GeneratorsOfGroup(F), ###
GeneratorsOfGroup(FundGroup)); ###
FundGroup:=F; ###
x:=0;
FpHoms:=[];
Elts:=[];
for R in resVertices do
P:=PresentationOfResolution(R);
Append(FpHoms,
[GroupHomomorphismByImagesNC(R!.group,FundGroup,
List([1..R!.dimension(1)],i->R!.elts[R!.boundary(1,i)[2][2]]), ###CHANGED March 2021
GeneratorsOfGroup(FundGroup){[x+1..x+R!.dimension(1)]}
)]);
Append(Elts,Image(FpHoms[Length(FpHoms)],R!.elts));
x:=x+R!.dimension(1);
od;
##########DREATED####################################################
#####################################################################
Dimension:=function(k)
local dim,R;
if k<0 then return 0; fi;
dim:=0;
for R in resVertices do
dim:=dim+R!.dimension(k);
od;
for R in resEdgeGroups do
dim:=dim+R!.dimension(k-1);
od;
return dim;
end;
#####################################################################
#####################################################################
inc:=function(x,n) #x is the vertex or edge group/resolution and
local pos,i,increment; #n is the dimension in the TOTAL resolution
increment:=0;
pos:=PositionName(Vertices,x);
if not pos=fail then
for i in [1..pos-1] do
increment:=increment + resVertices[i]!.dimension(n);
od;
return increment;
else
for i in [1..Length(resVertices)] do
increment:=increment + resVertices[i]!.dimension(n);
od;
pos:=PositionName(EdgeGroups,x);
for i in [1..pos-1] do
increment:=increment + resEdgeGroups[i]!.dimension(n-1);
od;
return increment;
fi;
end;
#####################################################################
PseudoBoundary:=[];
#####################################################################
FillPseudoBoundary:=function()
local b,f,g,i,k,x,hom,hom1,hom2,R,dim,dim1,dim2,bnd;
for k in [1..N] do
PseudoBoundary[k]:=[];
for R in resVertices do
x:=PositionName(List(FpHoms,f->Source(f)),R!.group);
hom:=FpHoms[x];
dim:=inc(R!.group,k-1);
for i in [1..R!.dimension(k)] do
Append(PseudoBoundary[k],[
List(R!.boundary(k,i),a->
[
SignInt(a[1])*(AbsoluteValue(a[1])+dim),
Position(Elts,Image(hom,R!.elts[a[2]]))
])]);
od;
od;
#######################
for e in Edges do
hom:=e[1];
x:=PositionName(EdgeGroups,Source(e[1]));
R:=Source(resEdges[x][1]);
dim:=inc(R!.group,k-1);
S:=Target(resEdges[x][1]);
T:=Target(resEdges[x][2]);
f:=resEdges[x][1];
g:=resEdges[x][2];
dim1:=inc(S!.group,k-1);
x:=PositionName(List(FpHoms,f->Source(f)),S!.group);
hom1:=FpHoms[x];
dim2:=inc(T!.group,k-1);
x:=PositionName(List(FpHoms,f->Source(f)),T!.group);
hom2:=FpHoms[x];
for i in [1..R!.dimension(k-1)] do
bnd:=[];
b:=List( f!.mapping([[i,1]],k-1),a->
[
SignInt(a[1])*(AbsoluteValue(a[1])+dim1),
Position(Elts,Image(hom1,S!.elts[a[2]]))
]);
if IsOddInt(k) then
Append(bnd,b);
else
Append(bnd,NegateWord(b));
fi;
b:=List( g!.mapping([[i,1]],k-1),a->
[
SignInt(a[1])*(AbsoluteValue(a[1])+dim2),
# Position(Elts,Image(hom1,T!.elts[a[2]])) #Typo found Jan 2012
Position(Elts,Image(hom2,T!.elts[a[2]]))
]);
if IsOddInt(k) then
Append(bnd,NegateWord(b));
else
Append(bnd,b);
fi;
b:=List(R!.boundary(k-1,i),a->
[
SignInt(a[1])*(AbsoluteValue(a[1])+dim),
Position(Elts,Image(hom1,Image(hom,R!.elts[a[2]])))
]);
Append(bnd,b);
Append(PseudoBoundary[k],[bnd]);
od;
od;
od;
end;
#####################################################################
FillPseudoBoundary();
#####################################################################
Boundary:=function(k,i)
if k<1 then return 0; fi;
if i>0 then return PseudoBoundary[k][i];fi;
return NegateWord(PseudoBoundary[k][-i]);
end;
####################################################################
Apply(Elts,x->Image(quohom,x));
return Objectify(HapResolution,
rec(
dimension:=Dimension,
boundary:=Boundary,
homotopy:=fail,
elts:=Elts,
group:=Range(quohom),
properties:=
[["length",N],
["reduced",true],
["type","resolution"],
["characteristic",0] ]));
end);
#####################################################################
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