|
#############################################################################
#0
#F BarResolutionEquivalence
## Input: A HAP resolution
## Output: An equivariant chain homotopy between the bar & the HAP resolution
##
InstallGlobalFunction(BarResolutionEquivalence,function(R)
local
nElts,Elts,e,nR,n,k,
SearchPosition,AddElement,HapHomotopy,BarHomotopy,BarBoundary,
PsiBasis,BoundBasis,TmpPsi,tmp1,tmp2,sign,base,g,Stmp,
Phi,Psi,Equiv;
Elts:=R!.elts;
nElts:=Length(Elts);
e:=Identity(R!.group);
nR:=EvaluateProperty(R,"length");
######################################################################
#1
## SearchPosition
## Input: An element g of G
## Output: The position of g in Elts
##
SearchPosition:=function(g)
local n,i,pos;
pos:=Position(Elts,g);
if not pos=fail then return pos;
else Add(Elts,g); return Length(Elts);
fi;
#Adjusted 27/03/2021
#n:=Length(Elts);
#for i in [1..n] do
# if Elts[i]=g then
# return i;
# fi;
#od;
#Add(Elts,g);
#return n+1;
end;
##
############### end of SearchPosition ################################
######################################################################
#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 BarBoundary
## Input: A word w =[[m_1,h_1,g_11,...,g_1n],...,[m_k,h_k,g_k1,...,g_kn]]
## and number n
## Output: The image of w under the boudary d_n:B_n->B_{n-1}
##
BarBoundary:=function(n,w)
local i,j,tmp,x,Rew;
Rew:=[];
for x in w do
############### Compute 0 #####################
tmp:=[x[1],x[2]*x[3]];
for j in [2..n] do
Add(tmp,x[j+2]);
od;
AddElement(Rew,tmp);
############### Compute 1 -> n-1 ##############
for i in [1..n-1] do
tmp:=[(-1)^i*x[1],x[2]];
for j in [1..i-1] do
Add(tmp,x[j+2]);
od;
Add(tmp,x[i+2]*x[i+3]);
for j in [i+2..n]do
Add(tmp,x[j+2]);
od;
AddElement(Rew,tmp);
od;
############### Compute n #####################
tmp:=[(-1)^n*x[1],x[2]];
for j in [1..n-1] do
Add(tmp,x[j+2]);
od;
AddElement(Rew,tmp);
od;
return Rew;
end;
##
################### end of BarBoundary ###############################
######################################################################
#1
#F HapHomotopy
## Input: A word w:=[[m_1,e_1,pos_1],...,[m_k,e_k,pos_k]] and number n
## Output: The image of w under the boundary h_n: R_n->R_{n+1}
##
HapHomotopy:=function(n,w)
local Rew, x, Hw,iHw,m;
Rew:=[];
for x in w do
m:=x[1];
Hw:=R!.homotopy(n,[x[2],x[3]]);
for iHw in Hw do
if iHw[1]>0 then
AddElement(Rew,[m,iHw[1],iHw[2]]);
else
AddElement(Rew,[-m,-iHw[1],iHw[2]]);
fi;
od;
od;
return Rew;
end;
##
############### end of HapHomotopy ###################################
######################################################################
#1
#F BarHomotopy
## Input: A word w =[[m_1,h_1,g_11,...,g_1n],...,[m_k,h_k,g_k1,...,g_kn]]
## and number n
## Output: The image of w under the boundary h_n: B_n->B_{n+1}
##
BarHomotopy:=function(n,w)
local i,x,Rew,tmp;
Rew:=[];
for x in w do
tmp:=[x[1],e,x[2]];
for i in [1..n] do
Add(tmp,x[i+2]);
od;
AddElement(Rew,tmp);
od;
return Rew;
end;
##
############### end of HapHomotopy ###################################
########## Compute the image of basis of R under the map psi:R->B ####
PsiBasis:=List([0..nR],x->[]);
PsiBasis[1][1]:=[[1,e]]; ##[0+1][1]
for n in [1..nR] do
for k in [1..R!.dimension(n)] do
TmpPsi:=[];
BoundBasis:=R!.boundary(n,k); ##ex:[[2,3],[-3,5]]
for tmp1 in BoundBasis do
if tmp1[1]<0 then
sign:=-1;
base:=-tmp1[1];
else
sign:=1;
base:=tmp1[1];
fi;
g:=Elts[tmp1[2]];
Stmp:=StructuralCopy(PsiBasis[n][base]); ##n = (n-1) +1
for tmp2 in Stmp do
tmp2[1]:=sign*tmp2[1];
tmp2[2]:=g*tmp2[2];
od;
Append(TmpPsi,Stmp);
od;
PsiBasis[n+1][k]:=BarHomotopy(n-1,TmpPsi);
od;
od;
######################################################################
#1
#F Psi
## Input: A word w:=[[m1,e1,pos1],...,[mk,ek,posk]] and number n
## Output: The image of w under the map psi_n: R_n->B_n
##
Psi:= function(n,w)
local Rew,m,h,x,u,Psix;
Rew:=[];
for x in w do
m:=x[1];
h:=Elts[x[3]];
Psix:=StructuralCopy(PsiBasis[n+1][x[2]]);
for u in Psix do
u[1]:=m*u[1];
u[2]:=h*u[2];
AddElement(Rew,u);
od;
od;
return Rew;
end;
##
################ end of Psi ##########################################
######################################################################
#1
#F Phi
## Input: A word w =[[m1,h1,g11,g12,g13,..,g1n],...,[mk,hk,gk1,...gkn]]
## Output: The image of w under the map phi_n: B_n->R_n
##
Phi:=function(n,w)
local x,Rew,Rex,h,u,cw;
cw:=StructuralCopy(w);
Rew:=[];
if n=0 then
for x in cw do
AddElement(Rew,[x[1],1,SearchPosition(x[2])]);
od;
return Rew;
fi;
for x in cw do
h:=x[2];
x[2]:=e;
Rex:=HapHomotopy(n-1,Phi(n-1,
BarBoundary(n,[x])));
for u in Rex do
u[3]:=SearchPosition(h*Elts[u[3]]);
AddElement(Rew,u);
od;
od;
return Rew;
end;
##
################ end of Phi ##########################################
######################################################################
#1
#F Equiv
## Input: A word w =[[m1,h1,g11,g12,g13,..,g1n],...,[mk,hk,gk1,...gkn]]
## Output: The image of w under the homotopy map H_n: B_n->B_{n+1}
##
Equiv:=function(n,w)
local
cw,h,x,
PsiPhix,HBx,HLx,
tmp,Rex,Rew,u;
cw:=StructuralCopy(w);
if n = 0 then
return [];
fi;
Rew:=[];
for x in cw do
h:=x[2];
x[2]:=e;
HBx:=Equiv(n-1,BarBoundary(n,[x]));
AddElement(HBx,x);
for tmp in HBx do
tmp[1]:=-tmp[1];
od;
PsiPhix:= Psi(n,Phi(n,[x]));
HLx:= Concatenation(PsiPhix,HBx);
Rex:=BarHomotopy(n,HLx);
for u in Rex do
u[2]:=h*u[2];
AddElement(Rew,u);
od;
od;
return Rew;
end;
##
################ end of Equiv ########################################
return rec(
phi:=Phi,
psi:=Psi,
equiv:=Equiv,
barBoundary:=BarBoundary,
barHomotopy:=BarHomotopy
);
end);
##
################### end of BarResolutionEquivalence #########################
[ Dauer der Verarbeitung: 0.35 Sekunden
(vorverarbeitet)
]
|
