resFiniteDirectProd.gi

#(C) Graham Ellis, 2005-2006
#RT:=0;
#####################################################################
InstallGlobalFunction(ResolutionFiniteDirectProduct,
function(arg)
local
	R,S,
	G,H,E,K,GhomE,HhomE,EhomG,EhomH,EltsE,eltse,elts2intrec,
        ghome, hhome, ehomg, ehomh, 
 	DimensionR,BoundaryR,HomotopyR,
        DimensionS,BoundaryS,HomotopyS,
	Lngth,Dimension,Boundary,Homotopy,
	PseudoBoundary,
	DimPQ,
	Int2Pair, Pair2Int,
	Charact,
	AddWrds,
	Int2Vector, Int2Vectorrec,
	Vector2Int, Vector2IntRec,
	Elts2Int,
	HomotopyGradedGen,
	HomotopyRec,
	HomotopyOfWord,
	FinalHomotopy,
	HorizontalBoundaryGen,
	HorizontalBoundaryWord,
	F,FhomE,
	gensE, gensE1, gensE2, 
	Boole,HGrec, DimPQrec,
	i,j,k,p,q,r,s,b,g,h,fn;

R:=arg[1];
S:=arg[2];

G:=R!.group; 
H:=S!.group; 


####################### DIRECT PRODUCT OF GROUPS ###########
if Length(arg)=2 then

if (not IsFinite(G)) or (not IsFinite(H)) then
return ResolutionDirectProduct(R,S); fi;

E:=DirectProduct(G,H);
if Size(G)=infinity or Size(H)=infinity then SetSize(E,infinity);fi;
GhomE:=Embedding(E,1);
HhomE:=Embedding(E,2);
EhomG:=Projection(E,1);
EhomH:=Projection(E,2);

else  	#if G and H both lie in a group K, and if they commute and have
	#have trivial intersection then we create their direct product as
	#a subgroup of K. We treat pcp groups as a seperate case.

#####PCP CASE #######################
if IsPcpGroup(G) then
K:=PcpGroupByCollector(Collector(Identity(G)));

gensE:=Igs(Concatenation(GeneratorsOfGroup(G),GeneratorsOfGroup(H)));
E:=Group(gensE);
       
       fn:=function(x,S)
       local v,w,y;
       v:=GenExpList(x);
       v:=List([1..Length(v)/2],i->[Igs(K)[v[2*i-1]],v[2*i]]);
       w:=Identity(K);
       for y in v do
       if y[1] in S then w:=w*y[1]^y[2]; fi;
       od;
       return w;
       end;
       
GhomE:=GroupHomomorphismByFunction(G,E,x->x);
HhomE:=GroupHomomorphismByFunction(H,E,x->x);
EhomG:=GroupHomomorphismByFunction(E,G,x->fn(x,G));
EhomH:=GroupHomomorphismByFunction(E,H,x->fn(x,H));
fi;
############PCP CASE DONE###########

############OTHER CASE##############
if not IsPcpGroup(G) then
gensE:=Concatenation(GeneratorsOfGroup(G),GeneratorsOfGroup(H));
E:=Group(gensE);

gensE1:=Concatenation(GeneratorsOfGroup(G),
 List([1..Length(GeneratorsOfGroup(H))],x->Identity(G)));
 gensE2:=Concatenation(List([1..Length(GeneratorsOfGroup(G))],x->Identity(H)),
              GeneratorsOfGroup(H));

GhomE:=GroupHomomorphismByFunction(G,E,x->x);
HhomE:=GroupHomomorphismByFunction(H,E,x->x);
EhomG:=GroupHomomorphismByImagesNC(E,G,gensE,gensE1);
EhomH:=GroupHomomorphismByImagesNC(E,H,gensE,gensE2);
fi;
###########OTHER CASE DONE#########

fi;
################ DIRECT PRODUCT OF GROUPS CONSTRUCTED #########



EltsE:=[Identity(E)];
for g in R!.elts do					
for h in S!.elts do					
AddSet(EltsE,Image(GhomE,g)*Image(HhomE,h));		
AddSet(EltsE,Image(HhomE,h)*Image(GhomE,g));		
od;							
od;						
i:=Position(EltsE,Identity(E));			
EltsE[i]:=EltsE[1];				
EltsE[1]:=Identity(E);				




PseudoBoundary:=[];
DimensionR:=R!.dimension; 
DimensionS:=S!.dimension; 
BoundaryS:= S!.boundary;
	   
BoundaryR:=R!.boundary;  
HomotopyR:=R!.homotopy;
HomotopyS:=S!.homotopy;  

#################DETERMINE VARIOUS PROPERTIES########################
Lngth:=Minimum(EvaluateProperty(R,"length"),EvaluateProperty(S,"length"));

if EvaluateProperty(R,"characteristic")=0
and EvaluateProperty(S,"characteristic")=0
then Charact:=EvaluateProperty(R,"characteristic");
fi;

if EvaluateProperty(R,"characteristic")=0
and EvaluateProperty(S,"characteristic")>0
then Charact:=EvaluateProperty(S,"characteristic");
fi;

if EvaluateProperty(R,"characteristic")>0
and EvaluateProperty(S,"characteristic")=0
then Charact:=EvaluateProperty(R,"characteristic");
fi;

if EvaluateProperty(R,"characteristic")>0
and EvaluateProperty(S,"characteristic")>0
then Charact:=Product(Intersection([
DivisorsInt(EvaluateProperty(R,"characteristic")),
DivisorsInt(EvaluateProperty(S,"characteristic"))
]));
fi;

if Charact=0 then AddWrds:=AddFreeWords; else
        AddWrds:=function(v,w);
        return AddFreeWordsModP(v,w,Charact);
        end;
fi;
####################PROPERTIES DETERMINED############################


#####################################################################
Dimension:=function(i)
local D,j;
if i<0 then return 0; fi;
if i=0 then return 1; else
D:=0;

for j in [0..i] do
D:=D+DimensionR(j)*DimensionS(i-j);
od;

return D; fi;
end;
#####################################################################

for i in [1..Lngth] do
PseudoBoundary[i]:=[1..Dimension(i)];
od;

DimPQrec:=[];
for i in [1..Lngth+1] do
DimPQrec[i]:=[];
od;

#####################################################################
DimPQ:=function(p,q)
local D,j;

if (p<0) or (q<0) then return 0; fi;
if not IsBound(DimPQrec[p+1][q+1]) then 
D:=0;
for j in [0..q] do
D:=D+DimensionR(p+q-j)*DimensionS(j);
od;
DimPQrec[p+1][q+1]:=D;
fi;

return DimPQrec[p+1][q+1];
end;
#####################################################################

#####################################################################
Int2Pair:=function(i,p,q)       #Assume that x<=DimR(p)*DimS(q).
local s,r,x;
                                #The idea is that the generator f_i in F
				#corresponds to a tensor (e_r x e_s)
x:=AbsoluteValue(i)-DimPQ(p+1,q-1);     #with e_r in R_p, e_s in S_q. If we
s:= x mod DimensionS(q);                #input i we get output [r,s].
r:=(x-s)/DimensionS(q);

if s=0 then return [SignInt(i)*r,DimensionS(q)];
else return [SignInt(i)*(r+1),s]; fi;

end;
#####################################################################

#####################################################################
Pair2Int:=function(x,p,q)
local y;                        #Pair2Int is the inverse of Int2Pair.
y:=[AbsoluteValue(x[1]),AbsoluteValue(x[2])];
return SignInt(x[1])*SignInt(x[2])*((y[1]-1)*DimensionS(q)+y[2]+DimPQ(p+1,q-1));
end;
#####################################################################

Int2Vectorrec:=[];
for i in [1..Lngth+1] do
Int2Vectorrec[i]:=[];
od;

#####################################################################
Int2Vector:=function(k,j)
local tmp,p,q;

if not IsBound(Int2Vectorrec[k+1][j]) then
p:=k;q:=0;
while j>=DimPQ(p,q)+1 do
p:=p-1;q:=q+1;
od;				#p,q are now computed from k,j

tmp:=Int2Pair(j,p,q);
Int2Vectorrec[k+1][j]:=[p,q,tmp[1],tmp[2]];
fi;
return Int2Vectorrec[k+1][j];
end;
#####################################################################

Vector2IntRec:=[];
for p in [1..Lngth+1] do
Vector2IntRec[p]:=[];
for q in [1..Lngth+1] do
Vector2IntRec[p][q]:=[];
for r in [1..R!.dimension(p-1)] do
Vector2IntRec[p][q][r]:=[];
od;od;od;
#####################################################################
Vector2Int:=function(p,q,r,s)
local rr, ss;
rr:=AbsInt(r); ss:=AbsInt(s);
if not IsBound(Vector2IntRec[p+1][q+1][rr][ss]) then
Vector2IntRec[p+1][q+1][rr][ss]:= Pair2Int([rr,ss],p,q);
fi;
return SignInt(r)*SignInt(s)*Vector2IntRec[p+1][q+1][rr][ss];
end;
#####################################################################

#####################################################################
Elts2Int:=function(x)
local pos;

pos:=Position(EltsE,x);
if IsPosInt(pos) then return pos;
else
	Append(EltsE,[x]);
	return Length(EltsE); 
fi;
end;
#####################################################################
eltse:=Elements(E);
elts2intrec:=List([1..Length(eltse)],i->Elts2Int(eltse[i]));
#####################################################################
Elts2Int:=function(x);
return elts2intrec[PositionSorted(eltse,x)];
end;
#####################################################################

###############################################
ghome:=List([1..Order(G)],i->Elts2Int(Image(GhomE,R!.elts[i])));
hhome:=List([1..Order(H)],i->Elts2Int(Image(HhomE,S!.elts[i])));
ehomg:=List([1..Order(E)],i->Position(R!.elts,Image(EhomG,EltsE[i])));
ehomh:=List([1..Order(E)],i->Position(S!.elts,Image(EhomH,EltsE[i])));
###############################################

#####################################################################
Boundary:=function(k,jj)
local j, p,q,r,s,tmp, horizontal, vertical;

if k<1 then return []; fi;
j:=AbsoluteValue(jj);
	#################IF BOUNDARY NOT ALREADY COMPUTED############
if IsInt(PseudoBoundary[k][j]) then
tmp:=Int2Vector(k,j);
p:=tmp[1]; q:=tmp[2]; r:=tmp[3]; s:=tmp[4];

horizontal:=ShallowCopy(BoundaryR(p,r));
#Apply(horizontal,x->[x[1],Elts2Int(   Image(GhomE,R!.elts[x[2]])   )  ]);
Apply(horizontal,x->[x[1],1*ghome[x[2]]]);
Apply(horizontal,x->[Vector2Int(p-1,q,x[1],s),x[2]]);

vertical:=ShallowCopy(BoundaryS(q,s));
#Apply(vertical,x->[x[1],Elts2Int(   Image(HhomE,S!.elts[x[2]])  )    ]);
Apply(vertical,x->[x[1],1*hhome[x[2]] ]);
Apply(vertical,x->[Vector2Int(p,q-1,r,x[1]),x[2]]);
if IsOddInt(p) then
vertical:=NegateWord(vertical);
fi;


PseudoBoundary[k][j]:= Concatenation(horizontal, vertical);
fi;
	################IF ENDS######################################

if SignInt(jj)=1 then
return PseudoBoundary[k][j];
else
return NegateWord(PseudoBoundary[k][j]);
fi;
end;
#####################################################################

#####################################################################
HorizontalBoundaryGen:=function(n,y)
local a,i, p,q,r,s, tmp,horizontal;

a:=AbsoluteValue(y[1]);
tmp:=Int2Vector(n,a);

p:=tmp[1]; q:=tmp[2]; r:=tmp[3]; s:=tmp[4];

horizontal:=StructuralCopy(BoundaryR(p,r));

#Apply(horizontal,x->[x[1],Elts2Int( EltsE[y[2]]*Image(GhomE,R!.elts[x[2]]) )]);
Apply(horizontal,x->[x[1],Elts2Int( EltsE[y[2]]*EltsE[ghome[x[2]]])   ]);


Apply(horizontal,x->[Vector2Int(p-1,q,x[1],s),x[2]]);

return horizontal;

end;
#####################################################################

#####################################################################
HorizontalBoundaryWord:=function(n,w)
local x, bnd;

bnd:=[];
for x in w do
Append(bnd,HorizontalBoundaryGen(n,x));
od;
return bnd;

end;
#####################################################################

HGrec:=[];
for p in [1..Lngth+1] do
HGrec[p]:=[];
for q in [1..Lngth-p+1] do
HGrec[p][q]:=[];
for r in [1..R!.dimension(p)+2] do  #why +2?
HGrec[p][q][r]:=[];
for s in [1..S!.dimension(q)+2] do  #why +2?
HGrec[p][q][r][s]:=[];
for b in [1,2] do
HGrec[p][q][r][s][b]:=[];
od;od;od;od;od;

#####################################################################
HomotopyGradedGen:=function(g,p,q,r,s,bool)    #Assume EltsE[g] exists!
local aa,hty, hty1, Eg, Eg1, Eg2, g1, g2,b;	#bool=true for vertical homotopy

if bool=true then b:=1; else b:=2; fi;
if IsBound(HGrec[p+1][q+1][r+1][s+1][b][g]) then 
return 1*HGrec[p+1][q+1][r+1][s+1][b][g]; fi;



#This function seems to work! But I should really check the maths again!!

g2:=1*ehomh[g];
g1:=1*ehomg[g];
Eg1:=EltsE[ghome[g1]];
Eg2:=EltsE[hhome[g2]];

hty:=HomotopyS(q,[s,g2]);
if Length(hty)>0 then
#Apply(hty,x->[ Vector2Int(p,q+1,r,x[1]), Image(HhomE,S!.elts[x[2]])]); 
Apply(hty,x->[ Vector2Int(p,q+1,r,x[1]), hhome[x[2]]]);
Apply(hty,x->[ x[1], Elts2Int(Eg1*EltsE[x[2]])]);
if IsOddInt(p) then
hty:=NegateWord(hty); fi;
fi;

if (p=0 and q>0) or bool then return hty; fi;

if p>0 then 
if Length(hty)>0 then
hty1:=HomotopyOfWord(p+q,1*HorizontalBoundaryWord(p+q+1,hty),false);
Append(hty, NegateWord(hty1));
fi;
fi;

if q>0 then  return hty; fi;


hty1:=HomotopyR(p,[r,g1]);
if Length(hty1)>0 then
#Apply(hty1,x->[ Vector2Int(p+1,q,x[1],s), Image(GhomE,R!.elts[x[2]])]);
Apply(hty1,x->[ Vector2Int(p+1,q,x[1],s), ghome[x[2]]]);
#Apply(hty1,x->[ x[1], Elts2Int(x[2])]); #Here

Append(hty,hty1);

hty1:=HomotopyOfWord(p+q,StructuralCopy(HorizontalBoundaryWord(p+q+1,hty1)),true);

Append(hty,NegateWord(hty1));

hty1:=HomotopyOfWord(p+q,StructuralCopy(HorizontalBoundaryWord(p+q+1,hty1)),false);


Append(hty,hty1); 	#I think this perturbation term is always zero and
			#thus not necessary.
fi;
HGrec[p+1][q+1][r+1][s+1][b][g]:=hty;

return 1*HGrec[p+1][q+1][r+1][s+1][b][g];

end;
#####################################################################

#####################################################################
Homotopy:=function(n,x,bool)
local vec,a;


a:=AbsoluteValue(x[1]);
vec:=Int2Vector(n,a);
if SignInt(x[1])=1 then 
return HomotopyGradedGen(x[2],vec[1],vec[2],vec[3],vec[4],bool);
else
return NegateWord(HomotopyGradedGen(x[2],vec[1],vec[2],vec[3],vec[4],bool));
fi;
end;
#####################################################################

#####################################################################
HomotopyOfWord:=function(n,w,bool)
local x, hty;

hty:=[];
for x in w do
Append(hty,Homotopy(n,x,bool));
od;

return hty;

end;
#####################################################################

HomotopyRec:=[];
for i in [1..Lngth] do
HomotopyRec[i]:=[];
for j in [1..Dimension(i-1)] do
HomotopyRec[i][j]:=[];
od;od;

#####################################################################
FinalHomotopy:=function(n,x)
local a;


a:=AbsInt(x[1]);
if not IsBound(HomotopyRec[n+1][a][x[2]]) then
HomotopyRec[n+1][a][x[2]]:=Homotopy(n,[a,x[2]],false);
fi;

if SignInt(x[1])=1 then
return StructuralCopy(HomotopyRec[n+1][a][x[2]]);
else
return NegateWord(StructuralCopy(HomotopyRec[n+1][a][x[2]]));
fi;
end;
#####################################################################


if HomotopyR=fail or HomotopyS=fail then
FinalHomotopy:=fail;
fi;

for i in [1..Lngth] do
for j in [1..Dimension(i)] do
g:=Boundary(i,j);
od;
od;



Boole:=false;
if EvaluateProperty(R,"reduced")=true
and  EvaluateProperty(S,"reduced")=true
then Boole:=true;
fi;


return Objectify(HapResolution,
	    rec(
            dimension:=Dimension,
	    boundary:=Boundary,
	    homotopy:=FinalHomotopy,
	    elts:=EltsE,
	    group:=E,
	    firstProjection:=EhomG,
	    secondProjection:=EhomH,
            Int2Vector:=Int2Vector,
            Vector2Int:=Vector2Int,
	    properties:=
	    [["type","resolution"],
	    ["length",Lngth],
	    ["reduced",Boole],
	    ["characteristic",Charact] ]));

end);
#####################################################################