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# SPDX-License-Identifier: GPL-2.0-or-later
# SCO: SCO - Simplicial Cohomology of Orbifolds
#
# Implementations
#
## Implementation stuff for matrix creation.
## <#GAPDoc Label="BoundaryOperator">
## <ManSection>
## <Func Arg="i, L, mu" Name="BoundaryOperator"/>
## <Returns>List B</Returns>
## <Description>
## This returns the <A>i</A>th boundary of <A>L</A>, which has to be an
## element of a simplicial set. <A>mu</A> is the function <M>\mu</M> that
## has to be taken into account when computing orbifold boundaries. This
## function is used for matrix creation, there should not be much reason
## for calling it independently.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( BoundaryOperator,
function( i, L, mu)
local n, tau, rho, j, boundary;
boundary := ShallowCopy( L );
n := ( Length( L ) - 1 ) / 2;
if i < 0 or i > n then
Error( "BoundaryOperator index i is out of bounds ([0..", n, "])! i = ", i );
fi;
tau := Intersection( Filtered( L, x->IsList( x ) ) );
if i = 0 then
boundary := L{[3..Length( L )]};
rho := Intersection( Filtered( boundary, x->IsList( x ) ) );
elif i = n then
boundary := L{[1..Length( L ) - 2]};
rho := Intersection( Filtered( boundary, x->IsList( x ) ) );
fi;
if i = 0 or i = n then
for j in [2,4..Length( boundary ) - 1] do
boundary[j] := mu( [ tau, rho, boundary[j-1], boundary[j+1] ] )( boundary[j] );
od;
return boundary;
fi;
boundary := boundary{Difference( [1..Length( boundary )],[2*i+1,2*i+2] )};
rho := Intersection( Filtered( boundary, x->IsList( x ) ) );
for j in [2,4..Length( boundary ) - 1] do
if j = 2*i then
boundary[j] := mu( [ tau, rho, L[j-1], L[j+3] ] )( L[j] * L[j+2] );
else
boundary[j] := mu( [ tau, rho, boundary[j-1], boundary[j+1] ] )( boundary[j] );
fi;
od;
return boundary;
end
);
## <#GAPDoc Label="CreateCoboundaryMatrices">
## <ManSection>
## <Meth Arg="S[, d], R" Name="CreateCoboundaryMatrices"/>
## <Returns>List <A>M</A></Returns>
## <Description>
## This returns the list <A>M</A> of homalg matrices over the homalg ring
## <A>R</A> up to dimension <A>d</A>, corresponding to the coboundary matrices
## induced by the simplicial set <A>S</A>. If <A>d</A> is not given, the
## current dimension of <A>S</A> is used.
## <Example><![CDATA[
## gap> S := SimplicialSet( Teardrop );
## <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
## imension 0 with Length vector [ 4 ]>
## gap> M := CreateCoboundaryMatrices( S, 4, HomalgRingOfIntegers() );;
## gap> S;
## <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
## imension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( CreateCoboundaryMatrices, "for internal and external rings",
[ IsSimplicialSet, IsInt, IsHomalgRing ],
function( ss, d, R )
local S, x, matrices, one, minusone, k, p, i, ind, pos, res;
S := x -> SimplicialSet( ss, x );
matrices := [];
one := One( R );
minusone := MinusOne( R );
for k in [ 1 .. d + 1 ] do
if Length( S(k) ) = 0 then
matrices[k] := HomalgZeroMatrix( Length( S(k-1) ), 0, R );
else
matrices[k] := HomalgInitialMatrix( Length( S(k-1) ), Length( S(k) ), R );
for p in [ 1 .. Length( S(k) ) ] do #column iterator
for i in [ 0 .. k ] do #row iterator
ind := PositionSet( S(k-1), BoundaryOperator( i, S(k)[p], ss!.orbifold_triangulation!.mu ) );
if not ind = fail then
if IsEvenInt( i ) then
AddToMatElm( matrices[k], ind, p, one );
else
AddToMatElm( matrices[k], ind, p, minusone );
fi;
fi;
od;
od;
ResetFilterObj( matrices[k], IsInitialMatrix );
fi;
od;
return matrices;
end
);
InstallMethod( CreateCoboundaryMatrices,
[ IsSimplicialSet, IsHomalgRing ],
function( ss, R )
return CreateCoboundaryMatrices( ss, ss!.dimension - 1, R );
end
);
## <#GAPDoc Label="CreateBoundaryMatrices">
## <ManSection>
## <Meth Arg="S, d, R" Name="CreateBoundaryMatrices"/>
## <Returns>List <A>M</A></Returns>
## <Description>
## This returns the list <A>M</A> of homalg matrices over the homalg ring
## <A>R</A> up to dimension <A>d</A>, corresponding to the boundary matrices
## induced by the simplicial set <A>S</A>. If <A>d</A> is not given, the
## current dimension of <A>S</A> is used.
## <Example><![CDATA[
## gap> S := SimplicialSet( Teardrop );
## <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
## imension 0 with Length vector [ 4 ]>
## gap> M := CreateBoundaryMatrices( S, 4, HomalgRingOfIntegers() );;
## gap> S;
## <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
## imension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( CreateBoundaryMatrices, "for internal and external rings",
[ IsSimplicialSet, IsInt, IsHomalgRing ],
function( ss, d, R )
local S, x, matrices, one, minusone, k, p, i, ind, pos, res;
S := x -> SimplicialSet( ss, x );
matrices := [];
one := One( R );
minusone := MinusOne( R );
for k in [ 1 .. d + 1 ] do
if Length( S(k) ) = 0 then
matrices[k] := HomalgZeroMatrix( 0, Length( S(k-1) ), R );
else
matrices[k] := HomalgInitialMatrix( Length( S(k) ), Length( S(k-1) ), R );
for p in [ 1 .. Length( S(k) ) ] do #column iterator
for i in [ 0 .. k ] do #row iterator
ind := PositionSet( S(k-1), BoundaryOperator( i, S(k)[p], ss!.orbifold_triangulation!.mu ) );
if not ind = fail then
if IsEvenInt( i ) then
AddToMatElm( matrices[k], p, ind, one );
else
AddToMatElm( matrices[k], p, ind, minusone );
fi;
fi;
od;
od;
ResetFilterObj( matrices[k], IsInitialMatrix );
fi;
od;
return matrices;
end
);
InstallMethod( CreateBoundaryMatrices,
[ IsSimplicialSet, IsHomalgRing ],
function( ss, R )
return CreateBoundaryMatrices( ss, ss!.dimension - 1, R );
end
);
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