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############################################################
## ##
## equispecseq.gi ##
## HAP subpackage for GAP (Groups Algorithms Programming) ##
## under the GNU GPL license (v. 3), 2012 ##
## by Alexander D. Rahm & Bui Anh Tuan ##
############################################################
InstallGlobalFunction("EquivariantSpectralSequencePage", function( C, n)
#########################################################################
#########################################################################
local reducedTorsionCells,celldata,j,N,l,w,P,Q,RP,RQ,g,Pt,k,pos,E1PageRec,p,q,
stabgrp,cell,i, E1page, T, EnPage, Differential, CohomologyOfGroup, stabres,stabcohom, inclusionMaps, groupname,name,sb,se,
maps,map,eqmap,tmp,BI,SGN,LstEl,s,r,t,multiple,E2page,CH1,Mat1, MatRank
;
if IsString(C) then
groupname:=Filtered(C,x->not(x='(' or x=')' or x=',' or x='[' or x=']'));
Read(Concatenation( DirectoriesPackageLibrary("HAP")[1]![1],
"Perturbations/Gcomplexes/",groupname));
celldata := StructuralCopy(HAP_GCOMPLEX_LIST);
name:=StructuralCopy(groupname);
RemoveCharacters(name,"torsion");
sb:=Position(name,'_');
se:=Length(name);
l:=Int(name{[sb+1..se]});
reducedTorsionCells:=[];
for i in [1..Size(celldata)] do
reducedTorsionCells[i]:=[];
for j in [1..Size(celldata[i])] do
reducedTorsionCells[i][j]:=[i-1,j];
od;
od;
N:=Size(reducedTorsionCells);
# if N>2 then return fail;fi;
else
#if not IsHapTorsionSubcomplex(C) then
# return fail;
#else
N:=Size(C!.reducedTorsionCells);
## We only consider the case when length of the subcomplex is
## less than 2 in order to get rid of the differential d2
# if N>2 then return fail;fi;
reducedTorsionCells:=C!.torsionCells;
celldata:=C!.celldata;
l:=C!.torsion;
groupname:=C!.groupname;
fi;
############### LIST THE STABILIZERS#######################
stabgrp:=[];
#stabres:=[];
#stabcohom:=[];
for j in [1..N] do
stabgrp[j]:=[];
for i in [1..Size(reducedTorsionCells[j])] do
cell:=reducedTorsionCells[j][i];
Add(stabgrp[j],celldata[cell[1]+1][cell[2]]!.TheMatrixStab);
od;
od;
###################### Rank of a matrix ###################
MatRank:=function(g)
if not (IsBound(g[1]) and IsBound(g[1][1])) then return 0;
else return RankMat(g);fi;
end;
###################### E_1 Page of Cohomology##############
E1page:=function(p,q)
local w,i;
if p=0 then return Differential(1,0,q)[1];
else return Differential(1,p-1,q)[2];
fi;
end;
######################End of E_1 Page######################
E2page:=function(p,q)
local t,M,lnth,N;
lnth:=Length(reducedTorsionCells);
if (p<0) or (p>lnth-1) then return 0;fi;
if p=0 then
M:=Differential(1,0,q)[3];
if IsEmpty(M) then return 0;
else return Size(M[1])-RankMat(M);
fi;
fi;
if p=lnth-1 then
M:=Differential(1,p-1,q)[3];
if IsEmpty(M) then return E1page(p,q);
else return E1page(p,q)-RankMat(M);
fi;
fi;
M:=Differential(1,p-1,q)[3];
N:=Differential(1,p,q)[3];
return E1page(p,q)-MatRank(M)-MatRank(N);
end;
##############End of E_2 Page######################
###################### Cohomology of Group ################
CohomologyOfGroup:=function(k)
local p,w,lnth;
Print("\n Users please self-aware that this attribute only works correctly in those cases whose d2 differentials are trivial \n");
w:=0;
lnth:=Length(reducedTorsionCells);
for p in [0..Minimum(k,lnth-1)] do
w:=w+E2page(p,k-p);
od;
return w;
return fail;
end;
######################End of Cohomology####################
inclusionMaps:=[];
maps:=[];
multiple:=[];
# N:=2;
for k in [1..N-1] do
maps[k]:=[];
inclusionMaps[k]:=[];
multiple[k]:=[];
for i in [1..Size(reducedTorsionCells[k+1])] do
cell:= reducedTorsionCells[k+1][i];
maps[k][i]:=[];
inclusionMaps[k][i]:=[];
multiple[k][i]:=[];
tmp:=celldata[k+1][cell[2]].BoundaryImage;
BI:=tmp.ListIFace;
SGN:=tmp.ListSign;
LstEl:=tmp.ListElt;
P:=StructuralCopy(stabgrp[k+1][i]);
if IsPNormal(P,l) then
P:=Normalizer(P,Center(SylowSubgroup(P,l)));
fi;
RP:=ResolutionFiniteGroup(P,n);
for r in [1..Size(reducedTorsionCells[k])] do
s:=reducedTorsionCells[k][r][2];
pos:=Positions(BI,s);
multiple[k][i][r]:=Sum(SGN{pos}) mod l;
# Print([k,i,r,s],BI,pos,multiple[k][i][r],"\n");
if not multiple[k][i][r]=0 then
Q:=StructuralCopy(stabgrp[k][r]);
if IsPNormal(Q,l) then
Q:=Normalizer(Q,Center(SylowSubgroup(Q,l)));
fi;
RQ:=ResolutionFiniteGroup(Q,n);
t:=Position(BI,s);
Pt:=ConjugateGroup(P,LstEl[t]);
for g in stabgrp[k][r] do
if IsSubgroup(Q,ConjugateGroup(Pt,g)) then
break;
fi;
od;
map:=GroupHomomorphismByFunction(P,
Q,x->(LstEl[t]*g)^-1*x*(LstEl[t]*g));
inclusionMaps[k][i][r]:=LstEl[t];
eqmap:=EquivariantChainMap(RP,RQ,map);
T:=HomToIntegersModP(eqmap,l);
maps[k][i][r]:=T;
else
maps[k][i][r]:=0;
fi;
od;
od;
od;
stabres:=[];
for k in [1..N-1] do
stabres[k]:=[];
od;
for k in [1..N-1] do
for j in [1..Size(maps[k][1])] do
if not IsBound(stabres[k][j]) then
i:=1;
while i<=Size(maps[k]) do
if maps[k][i][j]=0 then
i:=i+1;
else break;
fi;
od;
if i>Size(maps[k]) then
P:=StructuralCopy(stabgrp[k][j]);
if IsPNormal(P,l) then
fi;
RP:=ResolutionFiniteGroup(P,n);
stabres[k][j]:=HomToIntegersModP(RP,l);
fi;
fi;
od;
for i in [1..Size(maps[k])] do
if not IsBound(stabres[k][i]) then
j:=1;
while j<=Size(maps[k][i]) do
if maps[k][i][j]=0 then
j:=j+1;
else break;
fi;
od;
if j>Size(maps[k][i]) then
P:=StructuralCopy(stabgrp[k][i]);
if IsPNormal(P,l) then
fi;
RP:=ResolutionFiniteGroup(P,n);
stabres[k+1][i]:=HomToIntegersModP(RP,l);
fi;
fi;
od;
od;
###################### d1 differential#### ################
Differential:=function(k,p,q)
local w,i,j,A,B,CH,temp,x,M,Mat,BMat,t;
if k=1 then
N:=Length(reducedTorsionCells);
if (p < 0) or (p > N-2) then return [];fi;
CH:=[];
Mat:=[];
for i in [1..Size(reducedTorsionCells[p+2])] do
CH[i]:=[];
Mat[i]:=[];
for j in [1..Size(reducedTorsionCells[p+1])] do
if maps[p+1][i][j]=0 then CH[i][j]:=0;
else
CH[i][j]:=Cohomology(maps[p+1][i][j],q);
fi;
od;
od;
A:=[];
for j in [1..Size(CH[1])] do
i:=1;
while i<=Size(CH) do
if CH[i][j]=0 then
i:=i+1;
else
A[j]:=Size(AbelianInvariants(Source(CH[i][j]))); break;
fi;
od;
if i>Size(CH) then
A[j]:=Cohomology(stabres[p+1][j],q);
fi;
od;
B:=[];
for i in [1..Size(CH)] do
j:=1;
while j<=Size(CH[1]) do
if CH[i][j]=0 then j:=j+1;
else B[i]:=Size(AbelianInvariants(Target(CH[i][j]))); break;
fi;
od;
if j>Size(CH[1]) then
B[i]:=Cohomology(stabres[p+2][i],q);
fi;
od;
for i in [1..Size(reducedTorsionCells[p+2])] do
Mat[i]:=[];
for j in [1..Size(reducedTorsionCells[p+1])] do
if maps[p+1][i][j]=0 then
if A[j]*B[i]=0 then Mat[i][j]:=[];
else Mat[i][j]:=0*RandomMat(B[i],A[j]);fi;
else
Mat[i][j]:=multiple[p+1][i][j]*GroupHomomorphismToMatrix(CH[i][j],l);
fi;
od;
od;
BMat:=[];
for i in [1..Size(Mat)] do
M:=[];
for t in [1..Size(Mat[1])] do
t:=Size(Mat[i][1]);
if not t=0 then break;fi;
od;
for r in [1..t] do
M[r]:=[];
for j in [1..Size(Mat[1])] do
if not IsEmpty(Mat[i][j]) then
Append(M[r],Mat[i][j][r]);
fi;
od;
od;
Append(BMat,M);
od;
#Print(1/0);
return [Sum(A),Sum(B),BMat];
else
return fail;
fi;
end;
######################End of d1 differential###############
######################### E_n Page ########################
EnPage:=function(k,p,q)
if k=1 then return E1page(p,q);
elif k=2 then return E2page(p,q);
else return fail;
fi;
end;
######################End of d1 differential###############
###########################################################
return Objectify(HapEquivariantSpectralSequencePage,
rec(
page:=EnPage,
differential:=Differential,
groupname:=groupname,
torsion:=l,
cohomology:=CohomologyOfGroup,
maps:=maps,
multiple:=multiple,
inclusionMaps:= inclusionMaps
));
end);
###########################################################
InstallGlobalFunction("GroupHomomorphismToMatrix", function( phi,p)
local i,x,vectors,fgensA,fgensB,A,B,M,FreeGenerators, ElementToWord;
ElementToWord:=function( G,L, g)
local i,x,vectors,a;
vectors:=CombinationDisjointSets(List([1..Size(L)],w->p));
for x in vectors do
a:=One(G);
for i in [1..Size(x)] do
a:=a*L[i]^x[i];
od;
if g=a then return x;fi;
od;
return false;
end;
A:=Source(phi);
B:=Target(phi);
FreeGenerators:=function(G)
local gens,i,fgens;
gens:=GeneratorsOfGroup(G);
fgens:=[];
for x in gens do
if not Order(x)=1 then
if ElementToWord(G,fgens,x)=false then
Add(fgens,x);
fi;
fi;
od;
return fgens;
end;
fgensA:=FreeGenerators(A);
fgensB:=FreeGenerators(B);
M:=[];
for i in [1..Size(fgensA)] do
Add(M,ElementToWord(B,fgensB,Image(phi,fgensA[i])));
od;
return TransposedMat(M);
end);
######################################################
DeclareGlobalFunction("MatrixSize");
InstallGlobalFunction(MatrixSize,
function(M)
return [Length(M),Length(M[1])];
end);
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