chap10.txt

  
  10 Simplicial blowups
  
  
  10.1 Theory
  
  In this chapter functions are provided to perform simplicial blowups as well
  as   the   resolution   of   isolated  singularities  of  certain  types  of
  combinatorial  4-manifolds.  As  of  today  singularities  where the link is
  homeomorphic  to  RP^3, S^2 × S^1, S^2 dtimes S^1 and the lens spaces L(k,1)
  are  supported.  In  addition,  the program provides the possibility to hand
  over  additional  types  of  mapping  cylinders  to  cover  other  types  of
  singularities.
  
  Please  note  that  the  program  is  based  on  a heuristic algorithm using
  bistellar  moves.  Hence,  the  search  for a suitable sequence of bistellar
  moves  to  perform the blowup does not always terminate. However, especially
  in the case of ordinary double points (singularities of type RP^3), a lot of
  blowups  have  already  been  successful.  For  a very short introduction to
  simplicial blowups see 2.8, for further information see [SK11].
  
  
  10.2 Functions related to simplicial blowups
  
  10.2-1 SCBlowup
  
  SCBlowup( pseudomanifold, singularity[, mappingCyl] )  property
  Returns:  simplicial  complex of type SCSimplicialComplex upon success, fail
            otherwise.
  
  If   singularity   is   an   ordinary   double   point  of  a  combinatorial
  4-pseudomanifold  pseudomanifold  (lk(singularity)  =  RP^3)  the  blowup of
  pseudomanifold  at  singularity  is computed. If it is a singularity of type
  S^2  ×  S^1,  S^2  dtimes  S^1 or L(k,1), k ≤ 4, the canonical resolution of
  singularity  is  computed using the bounded complexes provided in the source
  code below.
  
  If  the optional argument mappingCyl of type SCIsSimplicialComplex is given,
  this complex will be used to to resolve the singularity singularity.
  
  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  
     gap> SCLib.SearchByName("Kummer variety");
     [ [ 7493, "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
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 2735 \ star([ 1 ]) in unnamed complex 2735 cup unnamed \
     complex 2739 cup unnamed complex 2737"
      Dim=4
     
     /SimplicialComplex]
     
  
  
    Example  
     gap> # resolving the singularities of a 4 dimensional Kummer variety
     gap> SCLib.SearchByName("Kummer variety");
     [ [ 7488, "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, [ ] ] ]
     
  
  
  10.2-2 SCMappingCylinder
  
  SCMappingCylinder( k )  function
  Returns:  simplicial  complex of type SCSimplicialComplex upon success, fail
            otherwise.
  
  Generates  a  bounded  version  of  CP^2 (a so-called mapping cylinder for a
  simplicial blowup, compare [SK11]) with boundary L(k,1).
  
    Example  
     gap> mapCyl:=SCMappingCylinder(3);;
     gap> mapCyl.Homology;              
     [ [ 0, [  ] ], [ 0, [  ] ], [ 1, [  ] ], [ 0, [  ] ], [ 0, [  ] ] ]
     gap> l31:=SCBoundary(mapCyl);;
     gap> l31.Homology;
     [ [ 0, [  ] ], [ 0, [ 3 ] ], [ 0, [  ] ], [ 1, [  ] ] ]