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##########################################################################
#0
#F ResolutionGTree
##
## This function use C.T.C Wall’s method to build a free ZG-resolution
## from a G-equivariant CW-complex given resolutions for all stabilisers
##
## Input: A non free ZG-resolution with resolutions for all stabilizers,
## a positive integer n
## Output: The first n+1 terms of a free ZG-reoslution of Z.
##
##
InstallGlobalFunction(ResolutionGTree,
function(arg)
local
R,n,G,i,L,
StabRes,StabGrps,Triple2Pair,Quad2One,GrpsRes,Pair2Quad,Quad2Pair,
Dimension,Hmap,Gmult,Gmultrec,Mult,AlgRed,CorrectList,Boundary,
Homotopy,FinalHomotopy,
StRes,hmap,Action,PseudoBoundary,PseudoHomotopy, ZeroDimensionHmap,
ZeroDimensionHtpy,
HmapRec,p,q,r,s,HtpyRec,k,g, ZeroDimensionHmapRec, Pair2QuadRec,
QuadToPairRec,DimensionRec;
R:=arg[1];
n:=arg[2];
G:=R!.group;
###################################################################
#1
#F Action
##
## Input: a triple of integers (p,r,q)
## Output: 1 or -1
##
Action:=function(p,r,g)
if not IsBound(R!.action) then
return 1;
else
return R!.action(p,r,g);
fi;
end;
###################################################################
AlgRed:=AlgebraicReduction;
Gmultrec:=[];
###################################################################
#1
#F Gmult
##
## The product of 2 elements in Elts
##
## Input: a pair of positive integers (i,j)
## Output: the position of the product in the list Elts
##
Gmult:=function(i,j)
local posit,g;
if not IsBound(Gmultrec[i]) then
Gmultrec[i]:=[];
fi;
if not IsBound(Gmultrec[i][j]) then
g:=R!.elts[i]*R!.elts[j];
posit:=Position(R!.elts,g);
if posit=fail then
Add(R!.elts,g);
posit:= Length(R!.elts);
fi;
Gmultrec[i][j]:=posit;
fi;
return Gmultrec[i][j];
end;
###################################################################
###################################################################
#1
#F Mult(g,w)
##
## Multiply gth-element with a list of words w
##
## Input: an integer g and a list of words w
## Output: the product of q and w
##
Mult:=function(g,w) # Multiply gth-element with a word
local
l,x;
l:=StructuralCopy(w);
if R!.elts[g]=[] then
return [];
fi;
Apply(l,y->[y[1],Gmult(g,y[2])]);
return l;
end;
###################################################################
###################################################################
#1
#F GrpsRes
##
## Return a free resolution for a given group
##
## Input: A group G and a positive integer n
## Output: The first n+1 term of a free ZG-resolution of Z
##
GrpsRes:=function(G,n) # Resolutions of Group
local
iso,Q,res,x;
if IsBound(R!.resolutions) and HasName(G) then
x:=Position(R!.resolutions[2], Name(G));
if not x=fail then
return R!.resolutions[1][x];
fi;
fi;
iso:=RegularActionHomomorphism(G);
Q:=Image(iso);
res:=ResolutionFiniteGroup(Q,n);
res!.group:=G;
res!.elts:=List(res!.elts,x->PreImagesRepresentative(iso,x));
return res;
end;
###################################################################
#Create list of stabilizer subgroups and their resolutions
StabGrps:= List([0..Length(R)],n->
List([1..R!.dimension(n)], k->R!.stabilizer(n,k)));
StabRes:=[];
for L in StabGrps do
Add(StabRes,List(L,g->ExtendScalars(GrpsRes(g,n),G,R!.elts)) );
od;
###################################################################
CorrectList:=function(list)
local
l,i;
if list=[] then return [];fi;
l:=StructuralCopy(list[1]);
for i in [2..Length(list)] do
Append(l,StructuralCopy(list[i]));
od;
return l;
end;
###################################################################
###################################################################
#1
#F Quad2One
##
## return nth-generator of F_(p,q) from (r,s)th-generator of
## stabilizer
##
## Input: A quadruple of integers (p,q,r,s)
## Output: n-th generator of F_{p,q}
##
Quad2One:=function(p,q,r,s)
local
n,d,i,j;
n:=0;
i:=SignInt(s);
s:=AbsInt(s);
d:=List([1..R!.dimension(p)],x->StabRes[p+1][x]!.dimension(q));
for j in [1..r-1] do
n:=n+d[j];
od;
n:=n+s;
if q=0 and n>R!.dimension(p) then
n:=R!.dimension(p);
fi;
return i*n;
end;
###################################################################
###################################################################
#1
#F Triple2Pair
##
## Input: A triple of integers (p,q,n)
## Output: a pair of integers (r,s)
##
Triple2Pair:=function(p,q,n)
local
r,s,d,i;
r:=0;
d:=List([1..R!.dimension(p)],x->StabRes[p+1][x]!.dimension(q));
i:=SignInt(n);
n:=AbsInt(n);
while n>0 do
r:=r+1;
s:=n;
n:=n-d[r];
od;
return [r,i*s];
end;
###################################################################
# Create a record for horizontal map: d_1
HmapRec:=[];
for p in [1..2] do
HmapRec[p]:=[];
for q in [1..n+1] do
HmapRec[p][q]:=[];
for r in [1..R!.dimension(p-1)] do
HmapRec[p][q][r]:=[];
od;
od;
od;
ZeroDimensionHmapRec:=[];
###################################################################
#1
#F ZeroDimensionHmap
##
## Input: An integer k
## Output: The map d_1 at degree 0
##
ZeroDimensionHmap:=function(k)
local i,j,pk;
pk:=AbsInt(k);
if not IsBound(ZeroDimensionHmapRec[pk]) then
j:=0;
for i in [1..pk-1] do
j:=j+StabRes[1][i]!.dimension(0);
od;
j:=j+1;
ZeroDimensionHmapRec[pk]:=j;
fi;
if k>0 then
return ZeroDimensionHmapRec[pk];
else return -ZeroDimensionHmapRec[pk];fi;
end;
###################################################################
###################################################################
#1
#F ZeroDimensionHmap
##
## Horiziontal map d_1:A(p,q)->A(p-1,q), acts on the
## (r,s) th-generator of A(p,q)
##
## Input: An integer k
## Output: The map d_1 at degree 0
##
Hmap:=function(p,q,r,s)
local
i,l,d0,m,bdr,ps,d1d0,w;
ps:=AbsInt(s);
if p<>1 then
return [];
else
if not IsBound(HmapRec[p+1][q+1][r][ps]) then
if q=0 then bdr:=StructuralCopy(R!.boundary(1,1));
Apply(bdr,w->[ZeroDimensionHmap(w[1]),w[2]]);
HmapRec[p+1][q+1][r][ps]:=bdr;
else
l:=[];m:=[];
d0:=StructuralCopy(List(StabRes[p+1][r]!.boundary(q,ps),
x->[Action(p,r,x[2])*x[1],x[2]]));
for w in d0 do
Append(m,Mult(w[2],Hmap(p,q-1,r,w[1])));
od;
Apply(m,x->[Triple2Pair(p-1,q-1,x[1]),x[2]]);
for w in m do
Append(l,List(StabRes[p][w[1][1]]!.homotopy
(q-1,[w[1][2],w[2]]),y->
[Quad2One(p-1,q,w[1][1],y[1]),y[2]]));
od;
HmapRec[p+1][q+1][r][ps]:=AlgRed(l);
fi;
fi;
fi;
if SignInt(s)=1 then
return HmapRec[p+1][q+1][r][ps];
else
return NegateWord(HmapRec[p+1][q+1][r][ps]);
fi;
end;
###################################################################
Pair2QuadRec:=[];
###################################################################
#1
#F Pair2Quad
##
## Input: A pair of integers (k,n)
## Output: A triple of integers
##
Pair2Quad:=function(k,nn)
local
x,n,nnn, p,q,r,s,i,temp,j1,j2;
i:=SignInt(nn);
n:=AbsInt(nn);
nnn:=n;
if not IsBound(Pair2QuadRec[k+1]) then
Pair2QuadRec[k+1]:=[];
fi;
if not IsBound(Pair2QuadRec[k+1][n]) then
temp:=0;
for j1 in [0..k] do
for j2 in [1..R!.dimension(j1)] do
temp:=temp+StabRes[j1+1][j2]!.dimension(k-j1);
od;
od;
p:=-1;
while n>0 do;
p:=p+1;
r:=0;
while (n>0 and r<R!.dimension(p)) do
r:=r+1;
s:=n;
n:=n-StabRes[p+1][r]!.dimension(k-p);
od;
od;
q:=k-p;
Pair2QuadRec[k+1][nnn]:=[p,q,r,s];
fi;
x:=Pair2QuadRec[k+1][nnn];
return [x[1],x[2],x[3],i*x[4]];
end;
###################################################################
QuadToPairRec:=[];
###################################################################
#1
#F Pair2Quad
##
## Input: A pair of integers (k,n)
## Output: A triple of integers
##
Quad2Pair:=function(p,q,r,s)
local
k,n,i,j,p1,q1,r1,s1;
p1:=p+1;q1:=q+1;r1:=r+1;s1:=AbsInt(s)+1;
if not IsBound(QuadToPairRec[p1]) then
QuadToPairRec[p1]:=[];
fi;
if not IsBound(QuadToPairRec[p1][q1]) then
QuadToPairRec[p1][q1]:=[];
fi;
if not IsBound(QuadToPairRec[p1][q1][r1]) then
QuadToPairRec[p1][q1][r1]:=[];
fi;
if not IsBound(QuadToPairRec[p1][q1][r1][s1]) then
k:=p+q;
n:=0;
for i in [0..p-1] do
for j in [1..R!.dimension(i)] do
n:=n+StabRes[i+1][j]!.dimension(k-i);
od;
od;
for i in [1..r-1] do
n:=n+StabRes[p+1][i]!.dimension(k-p);
od;
n:=n+AbsInt(s);
QuadToPairRec[p+1][q+1][r+1][AbsInt(s)+1]:=[k,n];
fi;
k:=QuadToPairRec[p1][q1][r1][s1];
return [k[1],SignInt(s)*k[2]];
end;
###################################################################
# Create an empty list for the pseudo boundary
PseudoBoundary:=[];
for k in [1..n+1] do
PseudoBoundary[k]:=[];
od;
###################################################################
#1
#F Boundary
##
## Input: A pair of integers (k,n)
## Output: the boundary d(k,n)
##
Boundary:=function(k,n)
local
d,l,p,q,r,s,w,pn;
pn:=AbsInt(n);
if not IsBound(PseudoBoundary[k+1][pn]) then
w:=Pair2Quad(k,pn);
p:=w[1];q:=w[2];r:=w[3];s:=w[4];
d:=[];
if q<>0 then
l:=StructuralCopy(List(StabRes[p+1][r]!.boundary(q,s),
x->[Quad2Pair(p,q-1,r,Action(p,r,x[2])*x[1])[2],x[2]]));
Append(d,StructuralCopy(l));
fi;
if IsEvenInt(q) then
Append(d,StructuralCopy(Hmap(p,q,r,s)));
else
Append(d,StructuralCopy(NegateWord(Hmap(p,q,r,s))));
fi;
PseudoBoundary[k+1][pn]:=AlgRed(d);
fi;
if SignInt(n)=1 then
return PseudoBoundary[k+1][pn];
else
return NegateWord(PseudoBoundary[k+1][pn]);
fi;
end;
###################################################################
DimensionRec:=[];
###################################################################
#1
#F Boundary
##
## Input: A pair of integers (k,n)
## Output: A triple of integers
##
Dimension:=function(n)
local
dim,p,i;
if not IsBound(DimensionRec[n+1]) then
dim:=0;
for p in [0..n] do
for i in [1..R!.dimension(p)] do
dim:=dim+StabRes[p+1][i]!.dimension(n-p);
od;
od;
DimensionRec[n+1]:= dim;
fi;
return DimensionRec[n+1];
end;
###################################################################
# Create a record for the homotopy map
HtpyRec:=[];
for k in [1..n] do
HtpyRec[k]:=[];
for s in [1..Dimension(k-1)] do
HtpyRec[k][s]:=[];
od;
od;
###################################################################
#1
#F ZeroDimensionHtpy
##
## Input: An integer k
## Output: the homotopy h(0,[1,k]) of the chain complex
##
ZeroDimensionHtpy:=function(k)
local i,j,r;
i:=0;
while k>0 do
i:=i+1;
k:=k-StabRes[1][i]!.dimension(0);
r:=i;
od;
return r;
end;
###################################################################
###################################################################
#1
#F Homotopy
##
## Input: An integer n and a word w=[f,g]
## Output: the homotopy h(n,w)
##
Homotopy:=function(n,w)
local
t,g,h0,h11,e,h,dh,
p,q,r,s,v,m,pt,ppt,
h1,d1h1,x,k,y,ps;
t:=w[1];
g:=w[2];
e:=[];
h:=[];
dh:=[];
pt:=AbsInt(t);
v:=Pair2Quad(n,pt);#Print(v);
p:=v[1];q:=v[2];r:=v[3];s:=v[4];
if not IsBound(HtpyRec[n+1][pt][g]) then
if n=0 then
ppt:=ZeroDimensionHtpy(pt);
h1:=StructuralCopy(R!.homotopy(n,[ppt,g]));
d1h1:=StructuralCopy(AlgRed(CorrectList(List(h1,x->
Mult(x[2],Hmap(p+1,q,1,x[1]))))));
for x in d1h1 do
k:=Pair2Quad(n,x[1]);
y:=StructuralCopy(StabRes[k[1]+1][k[3]]!.
homotopy(q,[k[4],x[2]]));
Apply(y,w->[Quad2Pair(k[1],k[2]+1,k[3],w[1])[2],w[2]]);
Append(e,y);
od;
h0:=StructuralCopy(StabRes[p+1][r]!.homotopy(0,[s,g]));
Apply(h0,w->[Quad2Pair(p,q+1,r,w[1])[2],w[2]]);
h11:=List(h1,x->[Quad2Pair(p+1,q,Triple2Pair(p+1,q,x[1])[1],
Triple2Pair(p+1,q,x[1])[2])[2],x[2]]);
Append(h,NegateWord(e));
Append(h,h0);
Append(h,h11);
HtpyRec[n+1][pt][g]:=AlgRed(h);
else
if p=0 then
h0:=StructuralCopy(StabRes[p+1][r]!.homotopy(q,[s,g]));
Apply(h0,w->[Quad2Pair(p,q+1,r,w[1])[2],w[2]]);
Append(h,h0);
else
ps:=Action(1,1,g)*s;
m:=StructuralCopy(StabRes[p+1][r]!.homotopy(q,[ps,g]));
Apply(m,x->[Action(1,1,x[2])*x[1],x[2]]);
h0:=List(m,x->[Quad2Pair(p,q+1,r,x[1])[2],x[2]]);
Append(h,h0);
dh:=AlgRed(CorrectList(List(m,x->
Mult(x[2],Hmap(p,q+1,1,x[1])))));
for x in dh do
k:=Pair2Quad(n,x[1]);
y:=StructuralCopy(StabRes[k[1]+1][k[3]]!.
homotopy(k[2],[k[4],x[2]]));
Apply(y,w->[Quad2Pair(k[1],k[2]+1,k[3],w[1])[2],w[2]]);
Append(e,y);
od;
if IsEvenInt(q) then
Append(h,e);
else
Append(h,NegateWord(e));
fi;
fi;
HtpyRec[n+1][pt][g]:=AlgRed(h);
fi;
fi;
if SignInt(t)=1 then
return HtpyRec[n+1][pt][g];
else
return NegateWord(HtpyRec[n+1][pt][g]);
fi;
end;
###################################################################
###################################################################
#1
#F FinalHomotopy
##
## Input: An integer n and a word w=[f,g]
## Output: the homotopy h(n,w)
##
FinalHomotopy:=function(n,g)
if R!.homotopy=fail then
return fail;
else
return Homotopy(n,g);
fi;
end;
####ADDED MAY 2012################################################
StRes:=function(n,k)
return StabRes[n+1][k];
end;
######################################################################
return Objectify(HapResolution,
rec(
dimension:=Dimension,
boundary:=Boundary,
homotopy:=FinalHomotopy,
elts:=R!.elts,
group:=R!.group,
stabres:=StRes,
properties:=
[["length",n],
["initial_inclusion",true],
["type","resolution"],
["characteristic",EvaluateProperty(
R,"characteristic")] ] ));
end);
##
################### end of ResolutionGTree ###########################
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