chap0.txt

  
  
                                    simpcomp
  
  
                     A GAP toolbox for simplicial complexes
  
  
                                 Version 2.1.7
  
  
                                 September 2017
  
  
                               Felix Effenberger
  
                                Jonathan Spreer
  
  
  
  Felix Effenberger
      Email:    mailto:[email protected]
      Address:  Max Planck Institute for Mathematics in the Sciences
                Inselstr. 22
                D-04103 Leipzig, Germany
  
  
  Jonathan Spreer
      Email:    mailto:[email protected]
      Address:  University of Queensland
                School of Mathematics and Physics
                Brisbane QLD 4072 Australia
  
  
  
  -------------------------------------------------------
  Abstract
  simpcomp  is  an  extension  (a  so  called package) to GAP for working with
  simplicial  complexes  in the context of combinatorial topology. The package
  enables  the  user  to  compute numerous properties of (abstract) simplicial
  complexes  (such  as  the  f-,  g-  and  h-vectors,  the  face  lattice, the
  fundamental group, the automorphism group, (co-)homology with explicit basis
  computation,  etc.).  It provides functions to generate simplicial complexes
  from  facet  lists,  orbit representatives or difference cycles. Moreover, a
  variety  of  infinite  series of combinatorial manifolds and pseudomanifolds
  (such  as  the  simplex,  the  cross  polytope, transitive handle bodies and
  sphere  bundles,  etc.)  is given and it is possible to create new complexes
  from  existing  ones  (links and stars, connected sums, simplicial cartesian
  products,  handle  additions, bistellar flips, etc.). simpcomp ships with an
  extensive  library  of known triangulations of manifolds and a census of all
  combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices.
  Furthermore, it provides the user with the possibility to create own complex
  libraries.  In  addition, functions related to slicings and polyhedral Morse
  theory  as  well  as  a  combinatorial  version of algebraic blowups and the
  possibility   to   resolve   isolated   singularities   of  4-manifolds  are
  implemented.
  simpcomp  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.
  If possible, simpcomp makes use of the GAP package homology [DHSW11] for its
  homology  computation  but  also  provides  the  user with own (co-)homology
  algorithms. For automorphism group computation the GAP package GRAPE [Soi12]
  is  used,  which  in turn uses the program nauty by Brendan McKay [MP14]. An
  internal automorphism group calculation algorithm is used as fallback if the
  GRAPE package is not available.
  
  
  -------------------------------------------------------
  Copyright
  © 2017 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 http://www.fsf.org/licensing/licenses/fdl.html
  for a copy.
  
  simpcomp  is  free  software. The code of simpcomp is released under the GPL
  version  2  or  later  (at your preference). For the text of the GPL see the
  file COPYING in the simpcomp directory or http://www.gnu.org/licenses/.
  
  
  -------------------------------------------------------
  Acknowledgements
  A  few  functions  of  simpcomp  are  based  on code from other authors. The
  bistellar flips implementation, the algorithm to collapse bounded simplicial
  complexes   as   well   as   the  classification  algorithm  for  transitive
  triangulations  is  based  upon  work of Frank Lutz (see [Lut03] and the GAP
  programs BISTELLAR and MANIFOLD_VT from [ManifoldPage]). Some functions were
  carried  over  from  the  homology package by Dumas et al. [DHSW11] -- 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  authors  acknowledge  support  by  the  Deutsche Forschungsgemeinschaft
  (DFG):  simpcomp  has been developed within the DFG projects Ku 1203/5-2 and
  Ku 1203/5-3.
  
  
  -------------------------------------------------------
  
  
  Contents (simpcomp)
  
  1 Introduction
    1.1 What is new
    1.2 simpcomp benefits
    1.3 How to save time reading this document
    1.4 Organization of this document
    1.5 How to assure simpcomp works correctly
    1.6 Controlling simpcomp log messages
    1.7 How to cite simpcomp
  2 Theoretical foundations
    2.1 Polytopes and polytopal complexes
    2.2 Simplices and simplicial complexes
    2.3 From geometry to combinatorics
    2.4 Discrete Normal surfaces
    2.5 Polyhedral Morse theory and slicings
    2.6 Discrete Morse theory
    2.7 Tightness and tight triangulations
    2.8 Simplicial blowups
  3 The new GAP object types of simpcomp
    3.1 Accessing properties of a SCPolyhedralComplex object
  4 Functions and operations for the GAP object type SCPolyhedralComplex
    4.1 Computing properties of objects of type SCPolyhedralComplex
      4.1-1 SCFacets
      4.1-2 SCFacetsEx
      4.1-3 SCVertices
      4.1-4 SCVerticesEx
    4.2 Vertex labelings and label operations
      4.2-1 SCLabelMax
      4.2-2 SCLabelMin
      4.2-3 SCLabels
      4.2-4 SCName
      4.2-5 SCReference
      4.2-6 SCRelabel
      4.2-7 SCRelabelStandard
      4.2-8 SCRelabelTransposition
      4.2-9 SCRename
      4.2-10 SCSetReference
      4.2-11 SCUnlabelFace
    4.3 Operations on objects of type SCPolyhedralComplex
      4.3-1 SCAntiStar
      4.3-2 SCLink
      4.3-3 SCLinks
      4.3-4 SCStar
      4.3-5 SCStars
  5 The GAP object types SCSimplicialComplex and SCNormalSurface
    5.1 The object type SCSimplicialComplex
      5.1-1 SCIsSimplicialComplex
      5.1-2 SCCopy
      5.1-3 ShallowCopy (SCSimplicialComplex)
      5.1-4 SCPropertiesDropped
    5.2 Overloaded operators of SCSimplicialComplex
      5.2-1 Operation + (SCSimplicialComplex, Integer)
      5.2-2 Operation - (SCSimplicialComplex, Integer)
      5.2-3 Operation mod (SCSimplicialComplex, Integer)
      5.2-4 Operation ^ (SCSimplicialComplex, Integer)
      5.2-5 Operation + (SCSimplicialComplex, SCSimplicialComplex)
      5.2-6 Operation - (SCSimplicialComplex, SCSimplicialComplex)
      5.2-7 Operation * (SCSimplicialComplex, SCSimplicialComplex)
      5.2-8 Operation = (SCSimplicialComplex, SCSimplicialComplex)
    5.3 SCSimplicialComplex as a subtype of Set
      5.3-1 Operation Union (SCSimplicialComplex, SCSimplicialComplex)
      5.3-2 Operation Difference (SCSimplicialComplex, SCSimplicialComplex)
      5.3-3 Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)
      5.3-4 Size (SCSimplicialComplex)
      5.3-5 Length (SCSimplicialComplex)
      5.3-6 Operation [] (SCSimplicialComplex)
      5.3-7 Iterator (SCSimplicialComplex)
    5.4 The object type SCNormalSurface
    5.5 Overloaded operators of SCNormalSurface
      5.5-1 Operation + (SCNormalSurface, Integer)
      5.5-2 Operation - (SCNormalSurface, Integer)
      5.5-3 Operation mod (SCNormalSurface, Integer)
    5.6 SCNormalSurface as a subtype of Set
      5.6-1 Operation Union (SCNormalSurface, SCNormalSurface)
  6 Functions and operations for SCSimplicialComplex
    6.1 Creating an SCSimplicialComplex object from a facet list
      6.1-1 SCFromFacets
      6.1-2 SC
      6.1-3 SCFromDifferenceCycles
      6.1-4 SCFromGenerators
    6.2 Isomorphism signatures
      6.2-1 SCExportToString
      6.2-2 SCExportIsoSig
      6.2-3 SCFromIsoSig
    6.3 Generating some standard triangulations
      6.3-1 SCBdCyclicPolytope
      6.3-2 SCBdSimplex
      6.3-3 SCEmpty
      6.3-4 SCSimplex
      6.3-5 SCSeriesTorus
      6.3-6 SCSurface
      6.3-7 SCFVectorBdCrossPolytope
      6.3-8 SCFVectorBdCyclicPolytope
      6.3-9 SCFVectorBdSimplex
    6.4 Generating infinite series of transitive triangulations
      6.4-1 SCSeriesAGL
      6.4-2 SCSeriesBrehmKuehnelTorus
      6.4-3 SCSeriesBdHandleBody
      6.4-4 SCSeriesBid
      6.4-5 SCSeriesC2n
      6.4-6 SCSeriesConnectedSum
      6.4-7 SCSeriesCSTSurface
      6.4-8 SCSeriesD2n
      6.4-9 SCSeriesHandleBody
      6.4-10 SCSeriesHomologySphere
      6.4-11 SCSeriesK
      6.4-12 SCSeriesKu
      6.4-13 SCSeriesL
      6.4-14 SCSeriesLe
      6.4-15 SCSeriesLensSpace
      6.4-16 SCSeriesPrimeTorus
      6.4-17 SCSeriesSeifertFibredSpace
      6.4-18 SCSeriesS2xS2
    6.5 A census of regular and chiral maps
      6.5-1 SCChiralMap
      6.5-2 SCChiralMaps
      6.5-3 SCChiralTori
      6.5-4 SCNrChiralTori
      6.5-5 SCNrRegularTorus
      6.5-6 SCRegularMap
      6.5-7 SCRegularMaps
      6.5-8 SCRegularTorus
      6.5-9 SCSeriesSymmetricTorus
    6.6 Generating new complexes from old
      6.6-1 SCCartesianPower
      6.6-2 SCCartesianProduct
      6.6-3 SCConnectedComponents
      6.6-4 SCConnectedProduct
      6.6-5 SCConnectedSum
      6.6-6 SCConnectedSumMinus
      6.6-7 SCDifferenceCycleCompress
      6.6-8 SCDifferenceCycleExpand
      6.6-9 SCStronglyConnectedComponents
    6.7 Simplicial complexes from transitive permutation groups
      6.7-1 SCsFromGroupExt
      6.7-2 SCsFromGroupByTransitivity
    6.8 The classification of cyclic combinatorial 3-manifolds
      6.8-1 SCNrCyclic3Mflds
      6.8-2 SCCyclic3MfldTopTypes
      6.8-3 SCCyclic3Mfld
      6.8-4 SCCyclic3MfldByType
      6.8-5 SCCyclic3MfldListOfGivenType
    6.9 Computing properties of simplicial complexes
      6.9-1 SCAltshulerSteinberg
      6.9-2 SCAutomorphismGroup
      6.9-3 SCAutomorphismGroupInternal
      6.9-4 SCAutomorphismGroupSize
      6.9-5 SCAutomorphismGroupStructure
      6.9-6 SCAutomorphismGroupTransitivity
      6.9-7 SCBoundary
      6.9-8 SCDehnSommervilleCheck
      6.9-9 SCDehnSommervilleMatrix
      6.9-10 SCDifferenceCycles
      6.9-11 SCDim
      6.9-12 SCDualGraph
      6.9-13 SCEulerCharacteristic
      6.9-14 SCFVector
      6.9-15 SCFaceLattice
      6.9-16 SCFaceLatticeEx
      6.9-17 SCFaces
      6.9-18 SCFacesEx
      6.9-19 SCFacets
      6.9-20 SCFacetsEx
      6.9-21 SCFpBettiNumbers
      6.9-22 SCFundamentalGroup
      6.9-23 SCGVector
      6.9-24 SCGenerators
      6.9-25 SCGeneratorsEx
      6.9-26 SCHVector
      6.9-27 SCHasBoundary
      6.9-28 SCHasInterior
      6.9-29 SCHeegaardSplittingSmallGenus
      6.9-30 SCHeegaardSplitting
      6.9-31 SCHomologyClassic
      6.9-32 SCIncidences
      6.9-33 SCIncidencesEx
      6.9-34 SCInterior
      6.9-35 SCIsCentrallySymmetric
      6.9-36 SCIsConnected
      6.9-37 SCIsEmpty
      6.9-38 SCIsEulerianManifold
      6.9-39 SCIsFlag
      6.9-40 SCIsHeegaardSplitting
      6.9-41 SCIsHomologySphere
      6.9-42 SCIsInKd
      6.9-43 SCIsKNeighborly
      6.9-44 SCIsOrientable
      6.9-45 SCIsPseudoManifold
      6.9-46 SCIsPure
      6.9-47 SCIsShellable
      6.9-48 SCIsStronglyConnected
      6.9-49 SCMinimalNonFaces
      6.9-50 SCMinimalNonFacesEx
      6.9-51 SCNeighborliness
      6.9-52 SCNumFaces
      6.9-53 SCOrientation
      6.9-54 SCSkel
      6.9-55 SCSkelEx
      6.9-56 SCSpanningTree
    6.10 Operations on simplicial complexes
      6.10-1 SCAlexanderDual
      6.10-2 SCClose
      6.10-3 SCCone
      6.10-4 SCDeletedJoin
      6.10-5 SCDifference
      6.10-6 SCFillSphere
      6.10-7 SCHandleAddition
      6.10-8 SCIntersection
      6.10-9 SCIsIsomorphic
      6.10-10 SCIsSubcomplex
      6.10-11 SCIsomorphism
      6.10-12 SCIsomorphismEx
      6.10-13 SCJoin
      6.10-14 SCNeighbors
      6.10-15 SCNeighborsEx
      6.10-16 SCShelling
      6.10-17 SCShellingExt
      6.10-18 SCShellings
      6.10-19 SCSpan
      6.10-20 SCSuspension
      6.10-21 SCUnion
      6.10-22 SCVertexIdentification
      6.10-23 SCWedge
  7 Functions and operations for SCNormalSurface
    7.1 Creating an SCNormalSurface object
      7.1-1 SCNSEmpty
      7.1-2 SCNSFromFacets
      7.1-3 SCNS
      7.1-4 SCNSSlicing
    7.2 Generating new objects from discrete normal surfaces
      7.2-1 SCCopy
      7.2-2 SCNSTriangulation
    7.3 Properties of SCNormalSurface objects
      7.3-1 SCConnectedComponents
      7.3-2 SCDim
      7.3-3 SCEulerCharacteristic
      7.3-4 SCFVector
      7.3-5 SCFaceLattice
      7.3-6 SCFaceLatticeEx
      7.3-7 SCFpBettiNumbers
      7.3-8 SCGenus
      7.3-9 SCHomology
      7.3-10 SCIsConnected
      7.3-11 SCIsEmpty
      7.3-12 SCIsOrientable
      7.3-13 SCSkel
      7.3-14 SCSkelEx
      7.3-15 SCTopologicalType
      7.3-16 SCUnion
  8 (Co-)Homology of simplicial complexes
    8.1 Homology computation
      8.1-1 SCBoundaryOperatorMatrix
      8.1-2 SCBoundarySimplex
      8.1-3 SCHomologyBasis
      8.1-4 SCHomologyBasisAsSimplices
      8.1-5 SCHomologyInternal
    8.2 Cohomology computation
      8.2-1 SCCoboundaryOperatorMatrix
      8.2-2 SCCohomology
      8.2-3 SCCohomologyBasis
      8.2-4 SCCohomologyBasisAsSimplices
      8.2-5 SCCupProduct
      8.2-6 SCIntersectionForm
      8.2-7 SCIntersectionFormParity
      8.2-8 SCIntersectionFormDimensionality
      8.2-9 SCIntersectionFormSignature
  9 Bistellar flips
    9.1 Theory
    9.2 Functions for bistellar flips
      9.2-1 SCBistellarOptions
      9.2-2 SCEquivalent
      9.2-3 SCExamineComplexBistellar
      9.2-4 SCIntFunc.SCChooseMove
      9.2-5 SCIsKStackedSphere
      9.2-6 SCBistellarIsManifold
      9.2-7 SCIsMovableComplex
      9.2-8 SCMove
      9.2-9 SCMoves
      9.2-10 SCRMoves
      9.2-11 SCRandomize
      9.2-12 SCReduceAsSubcomplex
      9.2-13 SCReduceComplex
      9.2-14 SCReduceComplexEx
      9.2-15 SCReduceComplexFast
  10 Simplicial blowups
    10.1 Theory
    10.2 Functions related to simplicial blowups
      10.2-1 SCBlowup
      10.2-2 SCMappingCylinder
  11 Polyhedral Morse theory
    11.1 Polyhedral Morse theory related functions
      11.1-1 SCIsTight
      11.1-2 SCMorseIsPerfect
      11.1-3 SCSlicing
      11.1-4 SCMorseMultiplicityVector
      11.1-5 SCMorseNumberOfCriticalPoints
  12 Forman's discrete Morse theory
    12.1 Functions using discrete Morse theory
      12.1-1 SCCollapseGreedy
      12.1-2 SCCollapseLex
      12.1-3 SCCollapseRevLex
      12.1-4 SCHasseDiagram
      12.1-5 SCMorseEngstroem
      12.1-6 SCMorseRandom
      12.1-7 SCMorseRandomLex
      12.1-8 SCMorseRandomRevLex
      12.1-9 SCMorseSpec
      12.1-10 SCMorseUST
      12.1-11 SCSpanningTreeRandom
      12.1-12 SCHomology
      12.1-13 SCHomologyEx
      12.1-14 SCIsSimplyConnected
      12.1-15 SCIsSimplyConnectedEx
      12.1-16 SCIsSphere
      12.1-17 SCIsManifold
      12.1-18 SCIsManifoldEx
  13 Library and I/O
    13.1 Simplicial complex library
      13.1-1 SCIsLibRepository
      13.1-2 SCLib
      13.1-3 SCLibAdd
      13.1-4 SCLibAllComplexes
      13.1-5 SCLibDelete
      13.1-6 SCLibDetermineTopologicalType
      13.1-7 SCLibFlush
      13.1-8 SCLibInit
      13.1-9 SCLibIsLoaded
      13.1-10 SCLibSearchByAttribute
      13.1-11 SCLibSearchByName
      13.1-12 SCLibSize
      13.1-13 SCLibUpdate
      13.1-14 SCLibStatus
    13.2 simpcomp input / output functions
      13.2-1 SCLoad
      13.2-2 SCLoadXML
      13.2-3 SCSave
      13.2-4 SCSaveXML
      13.2-5 SCExportMacaulay2
      13.2-6 SCExportPolymake
      13.2-7 SCImportPolymake
      13.2-8 SCExportLatexTable
      13.2-9 SCExportJavaView
      13.2-10 SCExportPolymake
      13.2-11 SCExportSnapPy
  14 Interfaces to other software packages
    14.1 Interface to the GAP-package homalg
      14.1-1 SCHomalgBoundaryMatrices
      14.1-2 SCHomalgCoboundaryMatrices
      14.1-3 SCHomalgHomology
      14.1-4 SCHomalgHomologyBasis
      14.1-5 SCHomalgCohomology
      14.1-6 SCHomalgCohomologyBasis
  15 Miscellaneous functions
    15.1 simpcomp logging
      15.1-1 SCInfoLevel
    15.2 Email notification system
      15.2-1 SCMailClearPending
      15.2-2 SCMailIsEnabled
      15.2-3 SCMailIsPending
      15.2-4 SCMailSend
      15.2-5 SCMailSendPending
      15.2-6 SCMailSetAddress
      15.2-7 SCMailSetEnabled
      15.2-8 SCMailSetMinInterval
    15.3 Testing the functionality of simpcomp
      15.3-1 SCRunTest
  16 Property handlers
    16.1 Property handlers of SCPolyhedralComplex
    16.2 Property handlers of SCSimplicialComplex
    16.3 Property handlers of SCNormalSurface
    16.4 Property handlers of SCLibRepository
  17 A demo session with simpcomp
    17.1 Creating a SCSimplicialComplex object
    17.2 Working with a SCSimplicialComplex object
    17.3 Calculating properties of a SCSimplicialComplex object
    17.4 Creating new complexes from a SCSimplicialComplex object
    17.5 Homology related calculations
    17.6 Bistellar flips
    17.7 Simplicial blowups
    17.8 Discrete normal surfaces and slicings
  18 simpcomp internals
    18.1 The GAP object type SCPropertyObject
      18.1-1 SCProperties
      18.1-2 SCPropertiesFlush
      18.1-3 SCPropertiesManaged
      18.1-4 SCPropertiesNames
      18.1-5 SCPropertiesTmp
      18.1-6 SCPropertiesTmpNames
      18.1-7 SCPropertyByName
      18.1-8 SCPropertyDrop
      18.1-9 SCPropertyHandlersSet
      18.1-10 SCPropertySet
      18.1-11 SCPropertySetMutable
      18.1-12 SCPropertyTmpByName
      18.1-13 SCPropertyTmpDrop
      18.1-14 SCPropertyTmpSet
    18.2 Example of a common attribute
    18.3 Writing a method for an attribute