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# Called as CohomologyGenerators(C,n), the objective
# of this function is to lift ALL of the generators
# n-1 times, discovering the generators of degree > 1
# in the process. In particular, this means that the
# degree 1 generators will be lifted to maps P_n -> P_{n-1}.
InstallMethod(CohomologyGenerators,
"Computes cohomology generators to degree n",
[IsCObject,IsPosInt],
function(C,n)
local G,K,R,L,B,D,P,BB,H,h,j,k,l,m,d,N;
ProjectiveResolution(C,n);
G:=C!.G; K:=C!.K; R:=C!.R; L:=C!.L; B:=C!.B;
if not IsBound(C!.D) then C!.D:=List([1..B[2]],x->1);fi;
if not IsBound(C!.P) then C!.P:=[];fi;
D:=C!.D; P:=C!.P;
# Set up cochains in degree 1.
if not IsBound(C!.H) then
BB:=IdentityMat(B[2],K);
C!.H:=List(BB,x->FirstLift(C,x));
fi;
H:=C!.H;
for m in [Size(H[1])..n-1] do
# Size(H[1]) is the number of lifts already computed.
h:=Size(H);
# h is the number of generators there were at the
# beginning of round m. We need this information later.
for j in [1..h] do
d:=D[j];
LiftChainMap(C,H[j],d,m);
# Digging out the first column of each block column of
# the matrix we just computed amounts to multiplying
# H[j] by a degree m map, producing a degree d+m map.
# We stick them all into P[d+m] for now.
if not IsBound(P[d+m]) then P[d+m]:=[];fi;
for k in [1..B[m+1]] do
Append(P[d+m],[ExtractColumn(H[j][m+1],(k-1)*Size(G)+1)]);
od;
od;
# At this point, all products of generators
# of degree < m+1 which produce maps of degree m+1
# have been computed put in P[m+1].
N:=BaseSteinitzVectors(
IdentityMat(B[m+2],K),BaseMat(P[m+1])).factorspace;
# Voila! The new generators in degree m+1. It only
# remains to repeat all of the above computations
# m times for each of the |N| new generators so that
# all the generators will have the same number m
# of lifts not including the first map P_d -> P_0.
for j in [1..Size(N)] do
D[h+j]:=m+1;
H[h+j]:=FirstLift(C,N[j]);
ProjectiveResolution(C,2*m+1);
for k in [1..m] do
LiftChainMap(C,H[h+j],m+1,k);
if not IsBound(P[m+k+1]) then P[m+k+1]:=[];fi;
for l in [1..B[k+1]] do
Append(P[m+k+1],[ExtractColumn(H[h+j][k+1],(l-1)*Size(G)+1)]);
od;
od;
od;
od;
return D;
end
);
[ Dauer der Verarbeitung: 0.20 Sekunden
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