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|
#############################################################################
##
#W CWcomplexThings_GroupRingRep.gi HAPcryst package Marc Roeder
##
##
##
##
#Y Copyright (C) 2006 Marc Roeder
#Y
#Y This program is free software; you can redistribute it and/or
#Y modify it under the terms of the GNU General Public License
#Y as published by the Free Software Foundation; either version 2
#Y of the License, or (at your option) any later version.
#Y
#Y This program is distributed in the hope that it will be useful,
#Y but WITHOUT ANY WARRANTY; without even the implied warranty of
#Y MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
#Y GNU General Public License for more details.
#Y
#Y You should have received a copy of the GNU General Public License
#Y along with this program; if not, write to the Free Software
#Y Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA
##
#############################################################################
##
##
InstallMethod(UndirectedWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList],
function(resolution,word)
if not IsFreeZGWordNoTermCheck_LargeGroupRep(resolution,word)
then
Error("<word> is not a valid word");
fi;
return UndirectedWordNC_LargeGroupRep(resolution,word);
end);
#############################################################################
##
InstallMethod(UndirectedWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList],
function(resolution,word)
local fam, returnword, term, coeffs, i;
fam:=FamilyObj(Zero(GroupRingOfResolution(resolution)));
returnword:=[];
for term in [1..Size(word)]
do
coeffs:=CoefficientsAndMagmaElementsAsLists(word[term]);
for i in [1..Size(coeffs[1])]
do
if coeffs[1][i]<>0
then
coeffs[1][i]:=1;
fi;
od;
returnword[term]:=ElementOfMagmaRing(fam,0,coeffs[1],coeffs[2]);
od;
return returnword;
end);
#############################################################################
##
InstallMethod(IsUndirectedWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList],
function(resolution,word)
local zero, term;
if not IsFreeZGWordNoTermCheck_LargeGroupRep(resolution,word)
then
return false;
fi;
zero:=Zero(GroupRingOfResolution(resolution));
for term in word
do
if term<>zero
then
if not Set(CoefficientsAndMagmaElementsAsLists(term)[1])=[1]
then
return false;
fi;
fi;
od;
return true;
end);
#############################################################################
##
InstallMethod(IsUndirectedWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,term,word)
local zero;
if not IsFreeZGWord_LargeGroupRep(resolution,term,word)
then
return false;
fi;
zero:=Zero(GroupRingOfResolution(resolution));
for term in word
do
if term<>zero
then
if not Set(CoefficientsAndMagmaElementsAsLists(term)[1])=[1]
then
return false;
fi;
fi;
od;
return true;
end);
#############################################################################
##
#O OneCoefficientPartOfWord
##
## just the parts parts of the ZG elements with coefficient=1.
## This can be used to calculate "intersections" of undirected words.
##
InstallMethod(OneCoefficientPartOfWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsVector],
function(resolution,word)
if not IsFreeZGWordNoTermCheck_LargeGroupRep(resolution,word)
then
Error("<word> is not valid");
fi;
return OneCoefficientPartOfWordNC_LargeGroupRep(resolution,word);
end);
############################################################
InstallMethod(OneCoefficientPartOfWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsVector],
function(resolution,word)
local returnword, zero, one, fam, term, coeffs, i;
returnword:=ShallowCopy(word);
zero:=Zero(GroupRingOfResolution(resolution));
one:=ZeroCoefficient(zero)^0;
fam:=FamilyObj(zero);
for term in [1..Size(word)]
do
if word[term]<>zero
then
coeffs:=CoefficientsAndMagmaElementsAsLists(word[term]);
for i in [1..Size(coeffs[1])]
do
if coeffs[1][i]<>one
then
Unbind(coeffs[1][i]);
Unbind(coeffs[2][i]);
fi;
od;
returnword[term]:=ElementOfMagmaRing(fam,0,Compacted(coeffs[1]),Compacted(coeffs[2]));
fi;
od;
return returnword;
end);
#############################################################################
##
#O IntersectingUndirectedWords_LargeGroupRep
##
## returns true if <word1> and <word2> have a non-trivial intersection.
## The words have to be undirected. Otherwise the intersection is not defined.
##
InstallMethod(IntersectingUndirectedWords_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList,IsDenseList],
function(resolution,word1,word2)
if not Size(word1)=Size(word2)
then
Error("words of different length");
elif not (IsUndirectedWord_LargeGroupRep(resolution,word1)
and IsUndirectedWord_LargeGroupRep(resolution,word2)
)
then
Error("<word1> and <word2> must be valid words");
fi;
return IntersectingUndirectedWordsNC_LargeGroupRep(resolution,word1,word2);
end);
#############################################################################
##
InstallMethod(IntersectingUndirectedWordsNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList,IsDenseList],
function(resolution,word1,word2)
local zero, term, gElts1, gElts2;
zero:=Zero(GroupRingOfResolution(resolution));
for term in [1..Size(word1)]
do
if word1[term]<>zero and word2[term]<>zero
then
gElts1:=CoefficientsAndMagmaElementsAsLists(word1[term])[2];
gElts2:=Set(CoefficientsAndMagmaElementsAsLists(word2[term])[2]);
if ForAny(gElts1,i->i in gElts2)
then
return true;
fi;
fi;
od;
return false;
end);
#############################################################################
##
InstallMethod(IsUndirectedSubWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList,IsDenseList],
function(resolution,word1,word2)
if IsUndirectedWord_LargeGroupRep(resolution,word1)
and IsUndirectedWord_LargeGroupRep(resolution,word2)
then
return IsUndirectedSubWordNC_LargeGroupRep(resolution,word1,word2);
else
Error("words must be undirected words of <resolution>");
fi;
end);
#############################################################################
##
InstallMethod(IsUndirectedSubWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList,IsDenseList],
function(resolution,word1,word2)
local term, groupelts1, groupelts2;
if Size(word1)<>Size(word2)
then
return false;
else
for term in [1..Size(word1)]
do
groupelts1:=CoefficientsAndMagmaElementsAsLists(word1[term])[2];
groupelts2:=CoefficientsAndMagmaElementsAsLists(word2[term])[2];
if not IsSubset(groupelts1,groupelts2)
then
return false;
fi;
od;
fi;
return true;
end);
#############################################################################
##
## undirectedBoundary calculates just the cells occuring in the boundary.
## signs and multiplicities are ignored.
##
InstallMethod(UndirectedBoundaryOfFreeZGLetter_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,cell)
if not IsFreeZGLetter_LargeGroupRep(resolution,dim,cell)
then
Error("invalid letter");
fi;
return UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,cell);
end);
#############################################################################
##
InstallMethod(UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,cell)
local zero, fam, boundary, term, coeffsAndGroupElts,
coeffs;
zero:=Zero(GroupRingOfResolution(resolution));
fam:=FamilyObj(zero);
boundary:=BoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,cell);
return UndirectedWord_LargeGroupRep(resolution,boundary);
# for term in [1..Size(boundary)]
# do
# if term<>zero
# then
# coeffsAndGroupElts:=CoefficientsAndMagmaElementsAsLists(boundary[term]);
# coeffs:=coeffsAndGroupElts[1];
# Apply(coeffs,function(i) if i<>0 then return 1; else return 0; fi; end);
# boundary[term]:=ElementOfMagmaRing(fam,0,coeffs,coeffsAndGroupElts[2]);
# fi;
# od;
# return boundary;
end);
#############################################################################
##
## undirectedBoundary for words.
## This does NOT calculate the boundary and then kills all the multiplicities.
## It calculates all dim-1 faces which touch the word <word>.
##
InstallMethod(UndirectedBoundaryOfFreeZGWord,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
local boundary_large;
if IsFreeZGWord_LargeGroupRep(resolution,dim,word)
then
return UndirectedBoundaryOfFreeZGWordNC_LargeGroupRep(resolution,dim,word);
elif IsFreeZGWord(resolution,dim,word)
then
boundary_large:=UndirectedBoundaryOfFreeZGWordNC_LargeGroupRep(resolution,dim,ConvertStandardWord(resolution,dim,word));
return ConvertWordToStandardRep(resolution,dim-1,boundary_large);
else
Error("invalid word");
fi;
end);
#############################################################################
##
InstallMethod(UndirectedBoundaryOfFreeZGWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
if not IsFreeZGWord_LargeGroupRep(resolution,dim,word)
then
Error("invalid word");
fi;
return UndirectedBoundaryOfFreeZGWordNC_LargeGroupRep(resolution,dim,word);
end);
#############################################################################
##
InstallMethod(UndirectedBoundaryOfFreeZGWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
local zero, fam, boundary, term, thistermbound;
zero:=Zero(GroupRingOfResolution(resolution));
fam:=FamilyObj(zero);
boundary:=[];
for term in [1..Size(word)]
do
if word[term]<>zero
then
thistermbound:=BoundaryOfGenerator_LargeGroupRep(resolution,dim,term);
boundary:=boundary+UndirectedWord_LargeGroupRep(resolution,thistermbound)*word[term];
fi;
od;
return UndirectedWord_LargeGroupRep(resolution,boundary);
end);
#############################################################################
##
InstallMethod(LowerSpaceFromWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList],
function(resolution,term,word,lowerterms)
local lowerspaceparts, zero, one, zerovec, thisterm, i,
lowerspace, position, returnspaces, lsp, returnspace,
thispart;
if not (IsSet(lowerterms) and ForAll(lowerterms,i->IsInt(i) and i>=0))
then
Error("<lowerterms> must be a set of non-negative integers");
fi;
if term<Maximum(lowerterms)
then
Error("term<Maximum(lowerterm)");
fi;
if not IsFreeZGWord_LargeGroupRep(resolution,term,word)
then
Error("word not valid");
fi;
if lowerterms=[term]
then
return [word];
fi;
lowerspaceparts:=List(lowerterms,term->List([1..Size(word)],i->[]));
zero:=Zero(GroupRingOfResolution(resolution));
one:=zero^0;
zerovec:=ListWithIdenticalEntries(Size(word),zero);
thisterm:=term;
for i in [1..Size(word)]
do
lowerspace:=ShallowCopy(zerovec);
lowerspace[i]:=one;
while thisterm>Minimum(lowerterms)
do
lowerspace:=UndirectedBoundaryOfFreeZGWord_LargeGroupRep(resolution,thisterm,lowerspace);
thisterm:=thisterm-1;
if thisterm in lowerterms
then
position:=Position(lowerterms,thisterm);
lowerspaceparts[position][i]:=lowerspace;
fi;
od;
od;
Info(InfoHAPcryst,2,"template done");
returnspaces:=[];
for lsp in [1..Size(lowerterms)]
do
Info(InfoHAPcryst,2,"dimension ",lowerterms[lsp]);
if lowerterms[lsp]=term
then
returnspace:=word;
else
returnspace:=ListWithIdenticalEntries(Size(lowerspaceparts[lsp]),zero);
for i in [1..Size(word)]
do
thispart:=lowerspaceparts[lsp][i];
returnspace:=returnspace+List(thispart,x->x*word[i]);
od;
fi;
returnspace:=UndirectedWord_LargeGroupRep(resolution,returnspace);
Add(returnspaces,returnspace);
od;
return returnspaces;
end);
#############################################################################
##
##
InstallMethod(SubspaceListFromWord,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
local spaces_large;
if IsFreeZGWord_LargeGroupRep(resolution,dim,word)
then
spaces_large:=SubspaceListFromWordNC_LargeGroupRep(resolution,dim,word);
return spaces_large;
elif IsFreeZGWord(resolution,dim,word)
then
spaces_large:=SubspaceListFromWordNC_LargeGroupRep(resolution,
dim,
ConvertStandardWord(resolution,dim,word)
);
return List([0..Size(spaces_large)-1],
i->ConvertWordToStandardRep(resolution,i,spaces_large[i+1]));
else
Error("<word> is not a valid word");
fi;
end);
#############################################################################
##
InstallMethod(SubspaceListFromWord_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
if IsFreeZGWord_LargeGroupRep(resolution,dim,word)
then
return SubspaceListFromWordNC_LargeGroupRep(resolution,dim,word);
else
Error("<word> is not a valid word in large group representation");
fi;
end);
#############################################################################
##
InstallMethod(SubspaceListFromWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
local subspaces, i;
subspaces:=List([0..dim],i->[]);
subspaces[dim+1]:=UndirectedWord_LargeGroupRep(resolution,word);
for i in [dim-1,dim-2..0]
do
subspaces[i+1]:=UndirectedBoundaryOfFreeZGWordNC_LargeGroupRep(resolution,i+1,subspaces[i+2]);
od;
return subspaces;
end);
#############################################################################
##
## Tests if a word represents a connected supspace.
##
InstallMethod(IsConnectedWord,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
local converted_word;
if IsFreeZGWord_LargeGroupRep(resolution,dim,word)
then
return IsConnectedWordNC_LargeGroupRep(resolution,dim,word);
elif IsFreeZGWord(resolution,dim,word)
then
converted_word:=ConvertStandardWord(resolution,dim,word);
return IsConnectedWordNC_LargeGroupRep(resolution,dim,converted_word);
else
Error("invalid input");
fi;
end);
#############################################################################
##
InstallMethod(IsConnectedWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,word)
local zero, fam, zerovec, lettersAndBound, term, groupels,
g, letter, startblob, blobbound, addToBlob,
addToBlobBound;
zero:=Zero(GroupRingOfResolution(resolution));
fam:=FamilyObj(zero);
zerovec:=List([1..Size(word)],i->zero);
lettersAndBound:=[];
for term in [1..Size(word)]
do
if word[term]<>zero
then
groupels:=CoefficientsAndMagmaElementsAsLists(word[term])[2];
for g in groupels
do
letter:=ShallowCopy(zerovec);
letter[term]:=ElementOfMagmaRing(fam,0,[1],[g]);
Add(lettersAndBound,[letter,
UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,letter)]
);
od;
fi;
od;
lettersAndBound:=Set(lettersAndBound);
# a "blob" is just that. A connected part of <word>.
# We don't generate the blob. As we are just interesed in it's size.
startblob:=Remove(lettersAndBound);
blobbound:=startblob[2];
repeat
addToBlob:=Filtered(lettersAndBound,i->
IntersectingUndirectedWordsNC_LargeGroupRep(resolution,i[2],blobbound)
);
if addToBlob<>[]
then
SubtractSet(lettersAndBound,addToBlob);
addToBlobBound:=Sum(List(addToBlob,i->i[2]));
blobbound:=UndirectedWord_LargeGroupRep(resolution,blobbound+addToBlobBound);
fi;
until lettersAndBound=[] or addToBlob=[];
if lettersAndBound=[]
then
return true;
elif addToBlob=[]
then
return false;
fi;
end);
#############################################################################
##
## connect a cell <cell> to the subspace <cellblob>.
##
InstallMethod(ConnectingPath,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList,IsDenseList],
function(resolution,dim,area,cellblob,cell)
local cell_large, cellblob_large, area_large, path_large;
if IsFreeZGLetter(resolution,dim,cell)
then
cell_large:=ConvertStandardLetter(resolution,dim,cell);
cellblob_large:=ConvertStandardWord(resolution,dim,cellblob);
area_large:=ConvertStandardWord(resolution,dim,area);
path_large:=ConnectingPathNC_LargeGroupRep(resolution,
dim,
area_large,
cellblob_large,
cell_large
);
if path_large=fail
then
return fail;
else
return ConvertWordToStandardRep(resolution,dim,path_large);
fi;
elif IsFreeZGLetter_LargeGroupRep(resolution,dim,cell)
then
path_large:=ConnectingPath_LargeGroupRep(resolution,
dim,
area,
cellblob,
cell
);
return path_large;
else
TryNextMethod();
fi;
end);
#############################################################################
##
InstallMethod(ConnectingPath_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList,IsDenseList],
function(resolution,dim,area,cellblob,cell)
if not (IsUndirectedWord_LargeGroupRep(resolution,area)
and IsFreeZGWord_LargeGroupRep(resolution,dim,cellblob)
)
then
Error("<area> and <cellblob> must be undirected words");
elif not (IsFreeZGLetter_LargeGroupRep(resolution,dim,cell)
and IsUndirectedWord_LargeGroupRep(resolution,cell)
)
then
Error("<cell> is not a valid undirected letter");
elif not IsUndirectedSubWordNC_LargeGroupRep(resolution,area,cellblob) and cell in area
then
Error("<area> does not contain <cellblob> and <cell>");
fi;
return ConnectingPathNC_LargeGroupRep(resolution,dim,area,cellblob,cell);
end);
#############################################################################
##
InstallMethod(ConnectingPathNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList,IsDenseList],
function(resolution,dim,area,cellblob,cell)
local pathfinder, zero, fam, zerovec, sphereAndBounds,
term, groupels, g, letter, path;
##################################################
##
## The recursive function "pathfinder" assumes that connectTo is not empty.
## It calculates a path from a "disk" that connects a given starting part
## with the space of known homotopies.
##
pathfinder:=function(resolution, connectTo, sphereAndBounds, startingBit,startingbitboundary)
local thingsThatCouldBeAdded, endpoint,
newSphereAndBounds, addface, newstartingbitboundary,
newstartingBit, returnpath;
thingsThatCouldBeAdded:=Filtered(sphereAndBounds,i->
IntersectingUndirectedWords_LargeGroupRep(resolution,i[2],startingbitboundary)
);
endpoint:=First(thingsThatCouldBeAdded,i->
IntersectingUndirectedWords_LargeGroupRep(resolution,i[2],connectTo)
);
if endpoint<>fail
then
return UndirectedWord_LargeGroupRep(resolution,startingBit+endpoint[1]);
else
newSphereAndBounds:=Difference(sphereAndBounds,thingsThatCouldBeAdded);
repeat
if thingsThatCouldBeAdded=[]
then
return [];
fi;
addface:=Remove(thingsThatCouldBeAdded);
newstartingbitboundary:=UndirectedWord_LargeGroupRep(resolution,startingbitboundary+addface[2]);
newstartingBit:=UndirectedWord_LargeGroupRep(resolution,startingBit+addface[1]);
returnpath:=pathfinder(resolution,
connectTo,
newSphereAndBounds,
newstartingBit,
newstartingbitboundary
);
until returnpath<>[];
return UndirectedWord_LargeGroupRep(resolution,returnpath);
fi;
end;
zero:=Zero(GroupRingOfResolution(resolution));
fam:=FamilyObj(zero);
zerovec:=List([1..Size(area)],i->zero);
sphereAndBounds:=[];
for term in [1..Size(area)]
do
if area[term]<>zero
then
groupels:=CoefficientsAndMagmaElementsAsLists(area[term])[2];
for g in groupels
do
letter:=ShallowCopy(zerovec);
letter[term]:=ElementOfMagmaRing(fam,0,[1],[g]);
Add(sphereAndBounds,[letter,
UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,letter)]
);
od;
fi;
od;
if cell in cellblob
then
return List(cell,i->zero);
fi;
path:=pathfinder(resolution,
UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,cell),
sphereAndBounds,
cell,
UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,cell)
);
if path=[]
then
return fail;
else
return path;
fi;
end);
#############################################################################
##
## given a word in the <dim>th term of <resolution>, this returns true
## if and only if this word represents a contractible subspace.
##
## connectedness is not tested.
## Is this right, anyway?
##
InstallMethod(IsContractibleWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,subspace)
local chaincomplex, i;
chaincomplex:=ChainComplexFromWordNC_LargeGroupRep(resolution,dim,subspace);
return HomologyPb(chaincomplex,dim)=[];
end);
#############################################################################
##
InstallMethod(IsContractibleWord,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,subspace)
local converted_subspace;
if IsFreeZGWord_LargeGroupRep(resolution,dim,subspace)
then
return IsContractibleWordNC_LargeGroupRep(resolution,dim,subspace);
elif IsFreeZGWord(resolution,dim,subspace)
then
converted_subspace:=ConvertWordToStandardRep(resolution,dim,subspace);
return IsContractibleWordNC_LargeGroupRep(resolution,dim,converted_subspace);
else
Error("invalid input");
fi;
end);
#############################################################################
##
##
InstallMethod(IsContractiblePartialSpace,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,spacelist)
local spacelist_large;
if ForAll([1..Size(spacelist)],subspace->
IsFreeZGWord_LargeGroupRep(resolution,subspace-1,spacelist[subspace]))
then
return IsContractiblePartialSpaceNC_LargeGroupRep(resolution,dim,spacelist);
elif ForAll([1..Size(spacelist)],subspace->
IsFreeZGWord(resolution,subspace-1,spacelist[subspace]))
then
spacelist_large:=List(spacelist,space->ConvertStandardWord(resolution,dim,space));
return IsContractiblePartialSpaceNC_LargeGroupRep(resolution,dim,spacelist_large);
else
Error("subspacelist does not consist of valid words");
fi;
end);
#############################################################################
##
InstallMethod(IsContractiblePartialSpaceNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,spacelist)
local chaincomplex;
chaincomplex:=ChainComplexFromPartialSpaceNC_LargeGroupRep(resolution,spacelist{[1..dim+1]});
return Homology(chaincomplex,dim)=[];
end);
#############################################################################
##
## find the sphere that contains <cell>.
## The list of cells <space> must induce a chain complex with <dim>th
## homology [0].
##
#############################################################################
##
## check the input and delegate...
##
InstallMethod(SphereContainingCell,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList],
function(resolution,dim,space,cell)
local complex;
if not IsFreeZGLetter_LargeGroupRep(resolution,dim,cell)
then
Error("<cell> is not a valid letter");
elif not IsFreeZGWord_LargeGroupRep(resolution,dim,space)
then
Error("<space> is not a valid word");
elif not IsUndirectedSubWord_LargeGroupRep(resolution,space,cell)
then
Error("<cell> not in <space>");
fi;
complex:=ChainComplexFromWordNC_LargeGroupRep(resolution,dim,space);
if not Homology(complex,dim)=[0]
then
Error("<space> does not contain a unique sphere");
fi;
return SphereContainingCellNC_LargeGroupRep(resolution,dim,space,cell);
end);
#############################################################################
##
InstallMethod(SphereContainingCell_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList],
function(resolution,dim,space,cell)
local complex;
if not IsFreeZGLetter_LargeGroupRep(resolution,dim,cell)
then
Error("<cell> is not a valid letter");
elif not IsFreeZGWord_LargeGroupRep(resolution,dim,space)
then
Error("<space> is not a valid word");
elif not IsUndirectedSubWord_LargeGroupRep(resolution,space,cell)
then
Error("<cell> not in <space>");
fi;
complex:=ChainComplexFromWordNC_LargeGroupRep(resolution,dim,space);
if not Homology(complex,dim)=[0]
then
Error("<space> does not contain a unique sphere");
fi;
return SphereContainingCellNC_LargeGroupRep(resolution,dim,space,cell);
end);
#############################################################################
##
##
InstallMethod(SphereContainingCellNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList,IsDenseList],
function(resolution,dim,space,cell)
local zero, fam, zerovec, space_and_bounds, term, groupels,
g, letter, sphere, spherebound, sphere_done,
subspacelist, complex, newcells, new_subspaces, i;
zero:=Zero(GroupRingOfResolution(resolution));
fam:=FamilyObj(zero);
zerovec:=List([1..Size(space)],i->zero);
space_and_bounds:=[];
for term in [1..Size(space)]
do
if space[term]<>zero
then
groupels:=CoefficientsAndMagmaElementsAsLists(space[term])[2];
for g in groupels
do
letter:=ShallowCopy(zerovec);
letter[term]:=ElementOfMagmaRing(fam,0,[1],[g]);
Add(space_and_bounds,[letter,
UndirectedBoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,dim,letter)]
);
od;
fi;
od;
Sort(space_and_bounds);
sphere:=cell;
sphere_done:=false;
subspacelist:=SubspaceListFromWord_LargeGroupRep(resolution,dim,sphere);
spherebound:=subspacelist[dim];
complex:=ChainComplexFromPartialSpaceNC_LargeGroupRep(resolution,
subspacelist
);
while Homology(complex,dim)<>[0]
do
newcells:=Filtered(space_and_bounds,c->
IntersectingUndirectedWordsNC_LargeGroupRep(resolution,c[2],spherebound)
);
SubtractSet(space_and_bounds,newcells);
sphere:=UndirectedWordNC_LargeGroupRep(resolution,sphere+Sum(List(newcells,i->i[1])));
new_subspaces:=SubspaceListFromWordNC_LargeGroupRep(resolution,
dim,
Sum(List(newcells,i->i[1]))
);
for i in [1..Size(new_subspaces)]
do
subspacelist[i]:=UndirectedWordNC_LargeGroupRep(resolution,subspacelist[i]+new_subspaces[i]);
od;
spherebound:=subspacelist[dim];
complex:=ChainComplexFromPartialSpaceNC_LargeGroupRep(resolution,subspacelist);
od;
return sphere;
end);
#############################################################################
##
## Generate a chain complex from a word
##
InstallMethod(ChainComplexFromWord,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,subspace)
local converted_subspace;
if IsFreeZGWord_LargeGroupRep(resolution,dim,subspace)
then
return ChainComplexFromWordNC_LargeGroupRep(resolution,dim,subspace);
elif IsFreeZGWord(resolution,dim,subspace)
then
converted_subspace:=ConvertStandardWord(resolution,dim,subspace);
return ChainComplexFromWordNC_LargeGroupRep(resolution,dim,subspace);
else
Error("invalid input");
fi;
end);
InstallMethod(ChainComplexFromWordNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsInt,IsDenseList],
function(resolution,dim,subspace)
local spaces;
spaces:=SubspaceListFromWordNC_LargeGroupRep(resolution,dim,subspace);
return ChainComplexFromPartialSpaceNC_LargeGroupRep(resolution,spaces);
end);
#############################################################################
##
## Generate a chain complex from a list of words
##
InstallMethod(ChainComplexFromPartialSpace_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList],
function(resolution,subspaces)
if not ForAll([1..Size(subspaces)],dim->
IsFreeZGWord_LargeGroupRep(resolution,dim-1,subspaces[dim])
)
then
Error("subspace list contains invalid words");
fi;
return ChainComplexFromPartialSpaceNC_LargeGroupRep(resolution,subspaces);
end);
InstallMethod(ChainComplexFromPartialSpaceNC_LargeGroupRep,
[IsHapLargeGroupResolutionRep,IsDenseList],
function(resolution,subspaces)
local word2vec, boundary, dimension, zero, zeroCoeff,
oneCoeff, fam, generatorGroupEltsList, ccGenerators,
term, thistermdimension, zerovec, dim, g, generator,
complex, properties;
##################################################
# /begin functions/
## assuming that <generators> is a list of length <word>
## where each entry is a list of group elements,
## we calculate the representation in the chain complex
## by just concatenating the coefficients.
word2vec:=function(generators,word)
local vec, term, coeffsAndGroupElts, termgenerators, g,
pos;
if Size(generators)<>Size(word)
then
Error("dimension mismatch");
fi;
vec:=List(generators,i->0*[1..Size(i)]);
for term in [1..Size(word)]
do
coeffsAndGroupElts:=CoefficientsAndMagmaElementsAsLists(word[term]);
termgenerators:=generators[term];
for g in [1..Size(coeffsAndGroupElts[2])]
do
pos:=Position(termgenerators,coeffsAndGroupElts[2][g]);
if pos<>fail
then
vec[term][pos]:=coeffsAndGroupElts[1][g];
else
Error("word-vector conversion error");
fi;
od;
od;
return Concatenation(vec);
end;
boundary:=function(k,j)
local letter, boundaryAsWord;
if k=Size(subspaces)+1
then
return [];
fi;
boundaryAsWord:=BoundaryOfFreeZGLetterNC_LargeGroupRep(resolution,k,ccGenerators[k+1][j]);
if boundaryAsWord=[]
then
return [];
else
return word2vec(generatorGroupEltsList[k],boundaryAsWord);
fi;
end;
dimension:=function(k)
if k<Size(subspaces)
then
return Size(ccGenerators[k+1]);
elif k=Size(subspaces)
then
return 0;
else
Error("chain complex too short");
fi;
end;
# / end functions/
#############################################
# / begin program/
zero:=Zero(GroupRingOfResolution(resolution));
zeroCoeff:=ZeroCoefficient(zero);
oneCoeff:=zeroCoeff^0;
fam:=FamilyObj(zero);
generatorGroupEltsList:=List(subspaces,s->
List(s,i->Set(CoefficientsAndMagmaElementsAsLists(i)[2]))
);;
ccGenerators:=List(generatorGroupEltsList,i->[]);
for term in [1..Size(generatorGroupEltsList)]
do
thistermdimension:=Size(generatorGroupEltsList[term]);
zerovec:=List([1..thistermdimension],i->zero);
for dim in [1..thistermdimension]
do
for g in [1..Size(generatorGroupEltsList[term][dim])]
do
generator:=ShallowCopy(zerovec);
generator[dim]:=ElementOfMagmaRing(fam,zeroCoeff,[oneCoeff],[generatorGroupEltsList[term][dim][g]]);
Add(ccGenerators[term],generator);
od;
od;
od;
complex:=Objectify(HapChainComplex,
rec(dimension:=dimension,
boundary:=boundary,
subspaces:=List(subspaces),
properties:=
[["length", Size(subspaces)-1],
["characteristic", 0],
["type", "chainComplex"]
])
);
return complex;
end);
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