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##
## simpcomp / blowups.gi
##
## Functions for bistellar moves
##
## $Id: bistellar.gi 68 2010-04-23 14:08:15Z jonathan $
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### computing possible flips in complex ###
### computing possible flips in boundary ###
### initial parameters ###
### strategy for selecting options ###
### computing possible flips in boundary ###
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### test, if isomorphic ###
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##<#GAPDoc Label="SCBlowup">
## <ManSection>
## <Prop Name="SCBlowup" Arg="pseudomanifold,singularity[,mappingCyl]"/>
## <Returns> simplicial complex of type <C>SCSimplicialComplex</C> upon
## success, <K>fail</K> otherwise.</Returns>
## <Description>
## If <C>singularity</C> is an ordinary double point of a combinatorial
## <M>4</M>-pseudomanifold <Arg>pseudomanifold</Arg>
## (lk(<C>singularity</C><M>) = \mathbb{R}P^3</M>) the blowup of
## <C>pseudomanifold</C> at <C>singularity</C> is computed. If it is a
## singularity of type <M>S^2 \times S^1</M>, <M>S^2 \dtimes S^1</M> or
## <M>L(k,1)</M>, <M>k \leq 4</M>, the canonical resolution of
## <C>singularity</C> is computed using the bounded complexes provided in
## the source code below. <P/>
##
## If the optional argument <C>mappingCyl</C> of type
## <C>SCIsSimplicialComplex</C> is given, this complex will be used to to
## resolve the singularity <C>singularity</C>.<P/>
##
## Note that bistellar moves do not necessarily preserve any orientation.
## Thus, the orientation of the blowup has to be checked in order to verify
## which type of blowup was performed. Normally, repeated computation results
## in both versions.
## <Example><![CDATA[
## gap> SCLib.SearchByName("Kummer variety");
## [ [ 519, "4-dimensional Kummer variety (VT)" ] ]
## gap> c:=SCLib.Load(last[1][1]);;
## gap> d:= SCBlowup(c,1);
## #I SCBlowup: checking if singularity is a combinatorial manifold...
## #I SCBlowup: ...true
## #I SCBlowup: checking type of singularity...
## #I SCReduceComplexEx: complexes are bistellarly equivalent.
## #I SCBlowup: ...ordinary double point (supported type).
## #I SCBlowup: starting blowup...
## #I SCBlowup: map boundaries...
## #I SCBlowup: boundaries not isomorphic, initializing bistellar moves...
## #I SCBlowup: found complex with smaller boundary: f = [ 15, 74, 118, 59 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 14, 70, 112, 56 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 14, 69, 110, 55 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 14, 68, 108, 54 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 13, 65, 104, 52 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 13, 64, 102, 51 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 13, 63, 100, 50 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 13, 62, 98, 49 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 13, 61, 96, 48 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 12, 57, 90, 45 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 12, 56, 88, 44 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 12, 55, 86, 43 ].
## #I SCBlowup: found complex with smaller boundary: f = [ 11, 51, 80, 40 ].
## #I SCBlowup: found complex with isomorphic boundaries.
## #I SCBlowup: ...boundaries mapped succesfully.
## #I SCBlowup: build complex...
## #I SCBlowup: ...done.
## #I SCBlowup: ...blowup completed.
## #I SCBlowup: You may now want to reduce the complex via 'SCReduceComplex'.
## <SimplicialComplex: unnamed complex 2735 \ star([ 1 ]) in unnamed complex 2735\
## cup unnamed complex 2739 cup unnamed complex 2737 | dim = 4 | n = 39>
## ]]></Example>
## <Example><![CDATA[
## gap> # resolving the singularities of a 4 dimensional Kummer variety
## gap> SCLib.SearchByName("Kummer variety");
## [ [ 519, "4-dimensional Kummer variety (VT)" ] ]
## gap> c:=SCLib.Load(last[1][1]);;
## gap> for i in [1..16] do
## for j in SCLabels(c) do
## lk:=SCLink(c,j);
## if lk.Homology = [[0],[0],[0],[1]] then continue; fi;
## singularity := j; break;
## od;
## c:=SCBlowup(c,singularity);
## od;
## gap> d.IsManifold;
## true
## gap> d.Homology;
## [ [ 0, [ ] ], [ 0, [ ] ], [ 22, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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##<#GAPDoc Label="SCMappingCylinder">
## <ManSection>
## <Func Name="SCMappingCylinder" Arg="k"/>
## <Returns> simplicial complex of type <C>SCSimplicialComplex</C> upon
## success, <K>fail</K> otherwise.</Returns>
## <Description>
## Generates a bounded version of <M>\mathbb{C}P^2</M> (a so-called mapping
## cylinder for a simplicial blowup, compare
## <Cite Key="Spreer09CombPorpsOfK3"/>) with boundary
## <M>L(</M><C>k</C><M>,1)</M>.
## <Example><![CDATA[
## gap> mapCyl:=SCMappingCylinder(3);;
## gap> mapCyl.Homology;
## [ [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ], [ 0, [ ] ], [ 0, [ ] ] ]
## gap> l31:=SCBoundary(mapCyl);;
## gap> l31.Homology;
## [ [ 0, [ ] ], [ 0, [ 3 ] ], [ 0, [ ] ], [ 1, [ ] ] ]
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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