  
  
                                    [1m[4m[31m[1msimpcomp[1m[4m[31m[0m
  
  
                     [1m[4m[31mA [1mGAP[1m[4m[31m toolbox for simplicial complexes[0m
  
  
                                 Version 1.1.21
  
  
                                  January 2010
  
  
                               Felix Effenberger
  
                                Jonathan Spreer
  
  
  
  Felix Effenberger
      Email:    [34mmailto:effenberger@mathematik.uni-stuttgart.de[0m
  Jonathan Spreer
      Email:    [34mmailto:spreer@mathematik.uni-stuttgart.de[0m
  
  
  Address: University of Stuttgart
           Department of Mathematics
           Institute of Geometry and Topology
           D-70550 Stuttgart, Germany
  
  
  -------------------------------------------------------
  [1m[4m[31mAbstract[0m
  [1msimpcomp[0m  is  a [1mGAP[0m package for working with simplicial complexes. It allows
  the  computation of many properties of simplicial complexes (such as the f-,
  g-  and  h-vectors,  the face lattice, the automorphism group, (co-)homology
  with  explicit  basis computation, intersection form, etc.) and provides the
  user  with  functions  to  compute new complexes from old (simplex links and
  stars,  connected  sums,  cartesian  products,  handle  additions, bistellar
  flips,  etc.).  Furthermore,  it  comes  with  an extensive library of known
  triangulations  of  manifolds  and provides the user with the possibility to
  create own complex libraries.
  [1msimpcomp[0m  caches  computed properties of a simplicial complex, thus avoiding
  unnecessary  computations,  internally  handles  the  vertex labeling of the
  complexes and insures the consistency of a simplicial complex throughout all
  operations.
  [1msimpcomp[0m  relies  on  the  [1mGAP[0m  package  [1mhomology[0m  [DH+04]  for its homology
  computation,  but also provides the user with an own (co-)homology algorithm
  in  case  the  packacke  [1mhomology[0m  is  not available. For automorphism group
  computation  the  [1mGAP[0m  package  [1mGRAPE[0m  [S06] is used, which in turn uses the
  program  [22m[32mnauty[0m  by  Brendan  McKay  [M84].  An  internal  automorphism group
  calculation  algorithm  in  used  as  fallback  if  the [1mGRAPE[0m package is not
  available.
  
  
  -------------------------------------------------------
  [1m[4m[31mCopyright[0m
  (C) 2009 by Felix Effenberger and Jonathan Spreer.
  
  Permission  is granted to copy, distribute and/or modify this document under
  the  terms  of  the GNU Free Documentation License, Version 1.2 or any later
  version     published    by    the    Free    Software    Foundation,    see
  [34mhttp://www.fsf.org/licensing/licenses/fdl.html[0m  for a copy. [1msimpcomp[0m is free
  software.  The code of [1msimpcomp[0m is released under the GPL version 2 or later
  (at  your  preference).  For the text of the GPL see the file [22m[32mCOPYING[0m in the
  [1msimpcomp[0m directory or [34mhttp://www.gnu.org/licenses/[0m.
  
  
  -------------------------------------------------------
  [1m[4m[31mAcknowledgements[0m
  A  few  functions  of  [1msimpcomp[0m  are  based  on code from other authors. The
  bistellar  flips implementation is based upon work of Frank Lutz [L03]. Some
  functions  were  carried  over  from  the  [1mHomology[0m  package by Dumas et al.
  [DH+04]  --  these  functions are marked in the documentation and the source
  code. The internal (co)homology algorithms were implemented by Armin Weiss.
  
  Most  of  the complexes in the simplicial complex library are taken from the
  "Manifold Page" by Frank Lutz [ManifoldPage].
  
  The first author acknowledges support by the Deutsche Forschungsgemeinschaft
  (DFG). This work was carried out as part of the DFG project "Ku 1203/5-2".
  
  
  -------------------------------------------------------
  
  
  [1m[4m[31mContent (SimpComp)[0m
  
  1. Introduction
    1.1 Why [1msimpcomp[0m
    1.2 How to save time reading this document
    1.3 Organization of this document
    1.4 How to assure [1msimpcomp[0m works correctly
    1.5 How to cite [1msimpcomp[0m
  2. Theoretical foundations
    2.1 Polytopes and polytopal complexes
    2.2 Simplices and simplicial complexes
    2.3 From geometry to combinatorics
    2.4 Normal surfaces and combinatorial slicings
  3. The GAP object types SCSimplicicalComplex and SCNormalSurface
    3.1 The object types [22m[32mSCSimplicialComplex[0m and [22m[32mSCPropertyObject[0m
      3.1-1 SCIsSimplicialComplex
      3.1-2 SCCopy
      3.1-3 ShallowCopy (SCSimplicialComplex)
      3.1-4 SCForceRecalc
    3.2 Accessing properties of a [22m[32mSCSimplicialComplex[0m object
    3.3 Overloaded operators of [22m[32mSCSimplicialComplex[0m
      3.3-1 Operation + (SCSimplicialComplex, Integer)
      3.3-2 Operation - (SCSimplicialComplex, Integer)
      3.3-3 Operation mod (SCSimplicialComplex, Integer)
      3.3-4 Operation ^ (SCSimplicialComplex, Integer)
      3.3-5 Operation + (SCSimplicialComplex, SCSimplicialComplex)
      3.3-6 Operation - (SCSimplicialComplex, SCSimplicialComplex)
      3.3-7 Operation * (SCSimplicialComplex, SCSimplicialComplex)
      3.3-8 Operation = (SCSimplicialComplex, SCSimplicialComplex)
      3.3-9 Operation Union (SCSimplicialComplex, SCSimplicialComplex)
      3.3-10 Operation Difference (SCSimplicialComplex, SCSimplicialComplex)
      3.3-11 Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)
    3.4 [22m[32mSCSimplicialComplex[0m as a subtype of [22m[32mSet[0m
      3.4-1 Size (SCSimplicialComplex)
      3.4-2 Length (SCSimplicialComplex)
      3.4-3 Operation [] (SCSimplicialComplex)
      3.4-4 IsBound (SCSimplicialComplex)
      3.4-5 Iterator (SCSimplicialComplex)
    3.5 The object types [22m[32mSCNormalSurface[0m as asubtyope of [22m[32mSCPropertyObject[0m
    3.6 Overloaded operators of [22m[32mSCNormalSurface[0m
      3.6-1 Operation + (SCNormalSurface, Integer)
      3.6-2 Operation - (SCNormalSurface, Integer)
      3.6-3 Operation mod (SCNormalSurface, Integer)
      3.6-4 Operation = (SCNormalSurface, SCNormalSurface)
      3.6-5 Operation Union (SCNormalSurface, SCNormalSurface)
    3.7 [22m[32mSCNormalSurface[0m as a subtype of [22m[32mSet[0m
  4. Functions for simplicial complexes
    4.1 Creating an [22m[32mSCSimplicialComplex[0m object from a facet list
      4.1-1 SCFromFacets
      4.1-2 SC
      4.1-3 SCFromDifferenceCycles
      4.1-4 SCFromGenerators
    4.2 Generating some standard triangulations
      4.2-1 SCBdCrossPolytope
      4.2-2 SCBdSimplex
      4.2-3 SCEmpty
      4.2-4 SCSimplex
    4.3 Generating new complexes from old
      4.3-1 SCCartesianPower
      4.3-2 SCCartesianProduct
      4.3-3 SCConnectedComponents
      4.3-4 SCConnectedProduct
      4.3-5 SCConnectedSum
      4.3-6 SCConnectedSumMinus
      4.3-7 SCDifferenceCycleCompress
      4.3-8 SCDifferenceCycleExpand
      4.3-9 SCFVectorBdCrossPolytope
      4.3-10 SCFVectorBdSimplex
      4.3-11 SCStronglyConnectedComponents
    4.4 Vertex labelings and label operations
      4.4-1 SCLabelMax
      4.4-2 SCLabelMin
      4.4-3 SCLabels
      4.4-4 SCRelabel
      4.4-5 SCRelabelStandard
      4.4-6 SCRelabelTransposition
      4.4-7 SCRename
      4.4-8 SCSortComplex
      4.4-9 SCUnlabelFace
    4.5 Computing properties of simplicial complexes
      4.5-1 SCAltshulerSteinberg
      4.5-2 SCAutomorphismGroup
      4.5-3 SCAutomorphismGroupInternal
      4.5-4 SCBoundary
      4.5-5 SCDim
      4.5-6 SCDualGraph
      4.5-7 SCEulerCharacteristic
      4.5-8 SCFVector
      4.5-9 SCFaceLattice
      4.5-10 SCFaceLatticeEx
      4.5-11 SCFaces
      4.5-12 SCFacesEx
      4.5-13 SCFacets
      4.5-14 SCFacetsEx
      4.5-15 SCFpBettiNumbers
      4.5-16 SCFundamentalGroup
      4.5-17 SCGVector
      4.5-18 SCGenerators
      4.5-19 SCGeneratorsEx
      4.5-20 SCHVector
      4.5-21 SCHasBoundary
      4.5-22 SCHasInterior
      4.5-23 SCHomology
      4.5-24 SCIncidences
      4.5-25 SCIncidencesEx
      4.5-26 SCInterior
      4.5-27 SCIsCentrallySymmetric
      4.5-28 SCIsConnected
      4.5-29 SCIsEmpty
      4.5-30 SCIsEulerianManifold
      4.5-31 SCIsHomologySphere
      4.5-32 SCIsInKd
      4.5-33 SCIsKNeighborly
      4.5-34 SCIsOrientable
      4.5-35 SCIsPseudoManifold
      4.5-36 SCIsPure
      4.5-37 SCIsShellable
      4.5-38 SCIsStronglyConnected
      4.5-39 SCMinimalNonFaces
      4.5-40 SCMinimalNonFacesEx
      4.5-41 SCName
      4.5-42 SCNeighborliness
      4.5-43 SCNumFaces
      4.5-44 SCOrientation
      4.5-45 SCSkel
      4.5-46 SCSkelEx
      4.5-47 SCSpanningTree
      4.5-48 SCVertices
      4.5-49 SCVerticesEx
    4.6 Operations on simplicial complexes
      4.6-1 SCAlexanderDual
      4.6-2 SCCollapseGreedy
      4.6-3 SCCone
      4.6-4 SCDeletedJoin
      4.6-5 SCDifference
      4.6-6 SCHandleAddition
      4.6-7 SCIntersection
      4.6-8 SCIsIsomorphic
      4.6-9 SCIsSubcomplex
      4.6-10 SCIsomorphism
      4.6-11 SCIsomorphismEx
      4.6-12 SCJoin
      4.6-13 SCLink
      4.6-14 SCLinks
      4.6-15 SCNeighbors
      4.6-16 SCNeighborsEx
      4.6-17 SCShelling
      4.6-18 SCShellingExt
      4.6-19 SCShellings
      4.6-20 SCSpan
      4.6-21 SCStar
      4.6-22 SCStars
      4.6-23 SCSuspension
      4.6-24 SCUnion
      4.6-25 SCVertexIdentification
      4.6-26 SCWedge
  5. (Co-)Homology of simplicial complexes
    5.1 Homology computation
      5.1-1 SCBoundaryOperatorMatrix
      5.1-2 SCBoundarySimplex
      5.1-3 SCHomologyBasis
      5.1-4 SCHomologyBasisAsSimplices
      5.1-5 SCHomologyInternal
    5.2 Cohomology computation
      5.2-1 SCCoboundaryOperatorMatrix
      5.2-2 SCCohomology
      5.2-3 SCCohomologyBasis
      5.2-4 SCCohomologyBasisAsSimplices
      5.2-5 SCCupProduct
      5.2-6 SCIntersectionForm
      5.2-7 SCIntersectionFormParity
      5.2-8 SCIntersectionFormDimensionality
      5.2-9 SCIntersectionFormSignature
  6. Bistellar flips
    6.1 Theory
    6.2 Functions for bistellar flips
      6.2-1 SCBistellarOptions
      6.2-2 SCEquivalent
      6.2-3 SCExamineComplexBistellar
      6.2-4 SCIntFunc.SCChooseMove
      6.2-5 SCIsKStackedSphere
      6.2-6 SCIsManifold
      6.2-7 SCIsMovableComplex
      6.2-8 SCMove
      6.2-9 SCMoves
      6.2-10 SCRMoves
      6.2-11 SCReduceAsSubcomplex
      6.2-12 SCReduceComplex
      6.2-13 SCReduceComplexEx
  7. Functions for normal surfaces
    7.1 Creating an [22m[32mSCNormalSurface[0m object
      7.1-1 SCNSEmpty
      7.1-2 SCNSFromFacets
      7.1-3 SCNS
      7.1-4 SCNSSlicing
    7.2 Generating new objects from normal surfaces
      7.2-1 SCNSCopy
      7.2-2 SCNSSubdivision
    7.3 Properties of [22m[32mSCNormalSurface[0m objects
      7.3-1 SCNSDim
      7.3-2 SCNSEulerCharacteristic
      7.3-3 SCNSFVector
      7.3-4 SCNSFaceLattice
      7.3-5 SCNSFaceLatticeEx
      7.3-6 SCNSFpBettiNumbers
      7.3-7 SCNSGenus
      7.3-8 SCNSHomology
      7.3-9 SCNSSkel
      7.3-10 SCNSSkelEx
      7.3-11 SCNSTopologicalType
  8. Library and I/O
    8.1 Simplicial complex library
      8.1-1 SCIsLibRepository
      8.1-2 SCLib
      8.1-3 SCLibAdd
      8.1-4 SCLibAllComplexes
      8.1-5 SCLibDelete
      8.1-6 SCLibDetermineTopologicalType
      8.1-7 SCLibFlush
      8.1-8 SCLibInit
      8.1-9 SCLibIsLoaded
      8.1-10 SCLibSearchByAttribute
      8.1-11 SCLibSearchByName
      8.1-12 SCLibSize
      8.1-13 SCLibUpdate
      8.1-14 SCLibStatus
    8.2 [1msimpcomp[0m input / output functions
      8.2-1 SCLoad
      8.2-2 SCSave
      8.2-3 SCExportPolymake
      8.2-4 SCImportPolymake
      8.2-5 SCExportLatexTable
      8.2-6 SCExportJavaView
  9. Miscellaneous functions
    9.1 [1msimpcomp[0m error handling
      9.1-1 SCErrorBreak
      9.1-2 SCErrorMail
    9.2 Email notification system
      9.2-1 SCMailClearPending
      9.2-2 SCMailIsEnabled
      9.2-3 SCMailIsPending
      9.2-4 SCMailSend
      9.2-5 SCMailSendPending
      9.2-6 SCMailSetAddress
      9.2-7 SCMailSetEnabled
      9.2-8 SCMailSetMinInterval
    9.3 Testing the functionality of [1msimpcomp[0m
      9.3-1 SCRunTest
  10. Property handlers
    10.1 Property handlers of a [22m[32mSCSimplicialComplex[0m
    10.2 Property handlers of a [22m[32mSCNormalSurface[0m
  11. A demo session with SimpComp
    11.1 Creating a [22m[32mSCSimplicialComplex[0m object
    11.2 Working with a [22m[32mSCSimplicialComplex[0m object
    11.3 Calculating properties of a [22m[32mSCSimplicialComplex[0m object
    11.4 Creating new complexes from a [22m[32mSCSimplicialComplex[0m object
    11.5 Homology related calculations
    11.6 Bistellar flips
  12. [1msimpcomp[0m internals
    12.1 Example of a common property handler
    12.2 Writing a property handler
  
  
  -------------------------------------------------------
