A GAP package for working with simplicial complexes
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                       The GAP 4 package 'simpcomp'


simpcomp is a GAP package for working with simplicial complexes. It allows the computation of many properties of simplicial complexes (such as the f-, g- and h-vecors, 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. 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. simpcomp relies on the GAP package homology for its homology computation, but also provides the user with an own (co-)homology algorithm in case the packacke homology is not available. For automorphism group computation the GAP package GRAPE is used, which in turn uses the program nauty by Brendan McKay. An internal automorphism group calculation algorithm in used as fallback if the GRAPE package is not available.


simpcomp is free software. The code 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/.


  • GAP (>=4.4)
  • Depending on GAP packages: GAPDoc (>= 0.9999), part of the core GAP system.
  • Strongly suggested other packages: Homology (>= 1.4.2), GRAPE (>= 4.2)

Download and installation

** Under Linux, run "rsync --port=8731 -azv rsync.gap-system.org::gap4r5/gapsync/gapsync ." followed by "./gapsync" (see [http://www.math.rwth-aachen.de/~Frank.Luebeck/gap/rsync/index.html] for details). You may need admin rights to perform this step.

** Under Windows: download the Windows installer under [http://www.gap-system.org/pub/gap/windowsinstaller/gap4r7p5_2014_05_24-20_02.exe] and follow the instructions given under [http://www.gap-system.org/ukrgap/wininst/].

** Under Mac OS X (or Linux): follow the instructions given under [http://gap-system.github.io/bob/]

  • (Optional) Download the current [https://github.com/simpcomp-team/simpcomp simpcomp] Version (This step is only required if you want to use features of simpcomp which are not yet included in the official release). Copy folder GAPROOT/pkg/simpcomp to GAPROOT/pkg/simpcomp-old and unpack the archive to the directory GAPROOT/pkg/ (default for GAPROOT is /usr/local/lib/gap4r5).

  • (Optional) Run "chmod +x configure; ./configure; make; make install; make clean" (This step is only required if you want to use SCReduceComplexFast()). You may need admin rights to perform this step.

  • From within GAP, load simpcomp using the following command which should return true. Based on which packages are already loaded, you will see something like the following:

  gap> SCLoadPackage("simpcomp");
  Loading  AutoDoc 2014.08.21 (Generate documentation from GAP source code)
  by Sebastian Gutsche (http://wwwb.math.rwth-aachen.de/~gutsche/) and
     Max Horn (http://www.quendi.de/math).
  Homepage: http://wwwb.math.rwth-aachen.de/~gutsche/gap_packages/AutoDoc
  Loading  Gauss 2013.06.26 (Extended Gauss functionality for GAP)
  by Simon Goertzen (http://wwwb.math.rwth-aachen.de/goertzen).
  Homepage: http://homalg.math.rwth-aachen.de/index.php/extensions/gaussgaussforhomalg
  Loading  Homology 1.4.5 (Simplicial Homology)
  by Jean-Guillaume Dumas (http://ljk.imag.fr/membres/Jean-Guillaume.Dumas),
     Frank Heckenbach (http://www.mi.uni-erlangen.de/~heckenb),
     B. David Saunders (http://www.cis.udel.edu/~saunders), and
     Volkmar Welker (http://www.mathematik.uni-marburg.de/~welker).
  Homepage: http://www.cis.udel.edu/~dumas/Homology/Homology
  Loading  ToolsForHomalg 2014.04.10 (Provides special methods and knowledge propagation tools)
  by Mohamed Barakat (http://www.mathematik.uni-kl.de/~barakat/),
     Sebastian Gutsche (http://wwwb.math.rwth-aachen.de/~gutsche/), and
     Markus Lange-Hegermann (http://wwwb.math.rwth-aachen.de/~markus/).
  Homepage: http://wwwb.math.rwth-aachen.de/~gutsche/gap_packages/ToolsForHomalg
  Loading  homalg 2014.09.14 (A homological algebra meta-package for computable Abelian categories)
  by Mohamed Barakat (http://www.mathematik.uni-kl.de/~barakat/),
     Sebastian Gutsche (http://wwwb.math.rwth-aachen.de/~gutsche/), and
     Markus Lange-Hegermann (http://wwwb.math.rwth-aachen.de/~markus/).
  Homepage: http://homalg.math.rwth-aachen.de/index.php/core-packages/homalg-package
  Loading  MatricesForHomalg 2014.08.26 (Lazy evaluated matrices with clever operations for the homalg project)
  by Mohamed Barakat (http://www.mathematik.uni-kl.de/~barakat/),
     Markus Lange-Hegermann (http://wwwb.math.rwth-aachen.de/~markus/),
     Martin Leuner (http://wwwb.math.rwth-aachen.de/Mitarbeiter/leuner.php), and
     Vinay Wagh (http://www.iitg.ernet.in/vinay.wagh/).
  Homepage: http://homalg.math.rwth-aachen.de/index.php/core-packages/matricesforhomalg
  Loading  GaussForHomalg 2013.09.20 (Gauss functionality for the homalg project)
  by Simon Goertzen (http://wwwb.math.rwth-aachen.de/goertzen).
  Homepage: http://homalg.math.rwth-aachen.de/index.php/extensions/gaussgaussforhomalg
  Loading  Modules 2014.07.02 (A homalg based package for the Abelian category of finitely presented modules over computable rings)
  by Thomas Bächler (http://wwwb.math.rwth-aachen.de/~thomas/),
     Mohamed Barakat (http://www.mathematik.uni-kl.de/~barakat/),
     Florian Diebold (),
     Sebastian Gutsche (http://wwwb.math.rwth-aachen.de/~gutsche/), and
     Markus Lange-Hegermann (http://wwwb.math.rwth-aachen.de/~markus/).
  Homepage: http://homalg.math.rwth-aachen.de/index.php/core-packages/modules
  Loading  GRAPE 4.6.1 (GRaph Algorithms using PErmutation groups)
  by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~leonard/).
  Homepage: http://www.maths.qmul.ac.uk/~leonard/grape/
  Loading simpcomp 2.1.8
  by F.Effenberger and J.Spreer
  • Run "SCRunTest();" for a quick self test of simpcomp that assures the package works correctly. The output of the test function should look like this (the number printed after GAP4stones is a performance measure of your system and thus may differ from computer to computer):
  gap> SCRunTest();
  + simpcomp package test
  + GAP4stones: 28579
  • If you want simpcomp to automatically load upon GAP startup, either set the variable Autoload:=true in GAPROOT/pkg/simpcomp/PackageInfo.g (in line 106) or add the command LoadPackage("simpcomp"); to your*/.gaprc*.


05/02/2018 - Version 2.1.8: MINOR UPDATE

  • Minor changes in documentation/manual
  • Adjusted output of test file for GAP 4.9

29/09/2017 - Version 2.1.7: Minor update

  • Adjusted SCRunTest (simpcomp testing facility)
  • Minor changes in documentation/manual
  • Added characterisation of tight 3-manifolds to SCIsTight (see http://arxiv.org/abs/1601.00065)
  • Fixed bug in SCIsFlag

01/02/2016 - Version 2.1.6: Minor update

  • Minor changes in documentation/manual

22/01/2016 - Version 2.1.5: Minor update

23/12/2015 - Version 2.1.4: Minor update

  • Fixed minor bug in SCsFromGroupByTransitivity();

04/06/2015 - Version 2.1.1: Minor update

  • Minor changes in documentation/manual

20/05/2015 - Version 2.1.0: Major update, support for discrete Morse theory added

  • Fixed bug in SCFVector, non-manifolds where treated like the simplex or the boundary of the simplex
  • Improved collapsing sequence (it is now using a Hasse diagram, SCIntFunc.SCHasseDiagram())
  • Improved high dimensional bistellar moves flip strategy (both for SCReduceComplex() and SCReduceComplexFast())
  • Updated references
  • Added chapter about discrete Morse theory in documentation
  • Added support for discrete Morse theory (SCHasseDiagram, SCMorseEngstrom, SCMorseRandom, SCMorseUST, SCMorseSpec, SCMorseRandomLex, SCMorseRandomRevLex)
  • Added Wilson's algorithm for uniform spanning trees SCSpanningTreeRandom
  • Added new functionality based on discrete Morse theory: SCHomologyEx, SCHomology, SCIsSphere, SCIsManifold
  • Old homology and manifold verification code is now available under SCHomologyClassic and SCBistellarIsManifold
  • Fixed bug in SCSkelEx and improved performance
  • Fixed bug in SCCyclic3Mfld(22,X), X = 1437, 2087, 2660, 2827, 2844, 3079
  • Fixed bug in SCMove for d-moves
  • Fixed bug in SCStronglyConnectedComponents and improved performance
  • Fixed bug in SCNumFaces
  • Improved functions SCIsPseudoManifold, SCHasBoundary, SCBoundarEx and SCBoundary

12/12/2013 - Version 2.0.0: FIRST RELEASE AS GAP SHARED PACKAGE

  • Added functions SCExportIsoSig, SCExportToString, SCFromIsoSig for isomorphism signature support and compressing
  • Added infinite family of transitive triangulations (S2xS2)^k (function SCSeriesS2xS2(k))
  • Added function SCExportRecognizer to export 3-manifolds to Matveev's 3-manifold recognizer

31/01/2013 - Version 1.6.1: Minor update.

  • Fixed minor bug in SCChiralMap and SCRegularMap.
  • Updated deployment system.

22/01/2013 - Version 1.6.0: Major update.

  • Added functions SCHeegaardSplitting, SCHeegaardSplittingSmallGenus and SCIsHeegaardSplitting to handle Heegaard splittings.
  • Added infinte series of d-tori SCSeriesTorus.
  • Added infinte series of lens spaces SCSeriesLensSpace.
  • Fixed bug in SCLibDelete.
  • Recalculated examples of documentation.
  • Fixed inconsistency in SCsFromGroupExt (non-strongly connected 2-pseudomanifolds are now all computed).
  • Added external code for bistellar flips.
  • Fixed bug in SCOrientation.
  • Added function SCSeriesBrehmKuehnelTorus.
  • Fixed minor bug in SCCartesianProduct (typo in description of topological type).
  • Added classification of cyclic combinatorial 3-mflds up to 22 vertices.
  • Added K3-surface K3_17 with standard PL-structure to the library.
  • Fixed bug in SCIntersectionFormSignature.
  • Revised documentation.
  • Added SCSurface, generates a surface of prescribed orientability and genus.
  • Added SCDifferenceCycles (analogue of SCGenerators for cyclic combinatorial manifolds).
  • Added census of highly symmetric surfaces. This includes the functions: SCNrRegularTorus, SCNrChiralTori, SCRegularTorus, SCChiralTori, SCSeriesSymmetricTorus, SCRegularMap, SCChiralMap, SCRegularMaps and SCChiralMaps.
  • Fixed dependencies on external UNIX programs to avoid problems with simpcomp running under Windows.
  • Added further combinatorial 3-manifolds with transitive cyclic symmetry. In particular, added functions SCSeriesHomologySphere, SCSeriesConnectedSum, SCSeriesSeifertFibredSpace and extended SCSeriesLensSpace.
  • New deployment system for full support of Google Code Git repository.

23/11/2011 - Version 1.5.4: GAP 4.5 release: required GAP-version is now 4.5

  • Updated references to other packages

28/10/2011 - Version 1.5.3: Tiny update

  • Replaced test file in PackageInfo.g

27/10/2011 - Version 1.5.2: Minor update

  • Fixed bug in function SCNumFacesOp
  • Fixed bug in function SCsFromGroupExt
  • Removed unnecessary .git subdirectory from package archive

27/10/2011 - Version 1.5.2: Minor update

  • Fixed bug in function SCNumFacesOp
  • Fixed bug in function SCsFromGroupExt
  • Removed unnecessary .git subdirectory from package archive

16/08/2011 - Version 1.5.1: Minor update

  • Added the GAP package ''io'' to the list of ''NeededOtherPackages''.
  • Fixed bug in function SCMove (thanks to Paul VanKoughnett for reporting!), bistellar move was performed on argument, too.
  • Fixed bug in shelling code (thanks to Samuel Kolins for reporting!), shelling intersection was not checked for pureness.
  • Extended SCExportJavaView to work with higher-dimensional complexes. For complexes of dim > 2, the 2-skeleton is exported now.
  • Added new series of triangulations of certain Hamiltonian subcomplexes of the cross polytope in arbitrary dimension: SCSeriesBid.

01/06/2011 - Version 1.5.0: Major update. Reworked most parts of package.

  • Completely rewritten type system, simpcomp is now compliant to the standard GAP object format. Introduced new object type SCPolyhedralComplex, from which both SCSimplicialComplex and SCNormalSurface are derived.
  • Changed naming of .-operator for simpcomp objects -- now there exist short and long versions of method names.
  • SCFaces and operator [] for polyhedral complex are now synonyms for SCSkel.
  • Vertex labels must now be elements with a total ordering.
  • Changed behaviour of operator = for SCSimplicialComplex: complexes are now only equal if and only if they have the same facet lists in standard labelling 1..n (used to include combinatorial isomorphy).
  • Fixed bug in SCRelabel... functions. As a side-effect, function SCSortComplex was deprecated and is now gone.
  • Fixed serveral small bugs in the classification functions for transitive manifolds with the prefix SCs... -- now all manifolds should be found.
  • Heavily reworked SCsFromGroup...-functions, should be much faster now.
  • Added and changed a lot of new functionality reagarding infinite series of triangulations, function prefix SCSeries...
  • Changed file format to binary one. Note that SCSave/SCLoad now use the binary format. If you want to continue using XML, use SCSaveXML/SCLoadXML. The package falls back to old format where neccessary.
  • Fixed bugs in SCDetermineTopologicalType and SCSpanningTree reported by Jack Schmidt (thank you!). In some cases, the spanning tree was not computed correctly. SCDetermineTopological did not work correctly with user libraries.
  • SCIsInKd now only checks one link if the complex has a transitive automorphism group.
  • Fixed polymake export/import SCExportPolymake/SCImportPolymake.
  • Fixed small bug in SCAutmorphismGroupInternal.
  • Changed the way the documentation is generated (using GAPDoc) -- the examples should now always be up to date and in sync with the code.
  • Changed package tests. Got rid of testshort.gap and testall.gap, the only function you need from now on is SCRunTest().
  • Added interface to GAP package "homalg" by Mohamed Barakat -- the package can now be used for homology-related computations. In particular, simpcomp can now calculate homology groups with arbitrary coefficients using homalg.

17/08/2010 - Version 1.3.3: Minor bug-fix

  • Adjusted test files to accommodate some changes made in the versions 1.3.1 and 1.3.2

12/08/2010 - Version 1.3.2: Minor bug-fix

  • Fixed dependencies on other packages

05/08/2010 - Version 1.3.1: Minor bug-fix

  • Fixed minor issues with undeclared local variables
  • Fixed wrong output format of SCAutomorphismGroupInternal

26/05/2010 - Version 1.3.0: Major update. New functions, updated library and bug fixes.

  • simpcomp has learned to generate all triangulations with k-transitive automorphism groups given a prescribed dimension and vertex number. See the function prefix SCsFromGroup.
  • Expanded the library to also contain pseudomanifolds -- the library now contains ca. 650 manifolds and ca. 7000 pseudomanifolds, it indexes the properties IsManifold and IsPM and supports them in property-based searches.
  • SCExportLatexTable now uses SCFacets instead of SCFacetsEx and thus respects vertex labelings. Take care not to choose to wild labels though, these are not checked for latex compatibility during the export.
  • Changed naming scheme of automorphism groups -- StructureDescription is only called if the group cannot be found in GAPs group libraries out of speed considerations.
  • Fixed small error in SCIntFunc.ReadPermGroup resulting in an error when trying to load complexes with trivial automorphism groups

30/04/2010 - Version 1.2.73: Midsize update. simpcomp now has a mascot!

  • Added support for polyhedral Morse functions, prefix: SCMorse..., including a (naive) test for tightness of a triangulation (SCIsTight).
  • Minor updates of the complex library.

07/04/2010 - Version 1.1.43

  • IMPORTANT FIX: fixed incorrect calculation of g-vectors (last entry was missing) and updated library accordingly
  • Added SCDehnSommervilleCheck and SCDehnSommervilleMatrix
  • Added SCFVectorBdCyclicPolytope and SCBdCyclicPolytope
  • Removed cyclic polytopes from library (see above)
  • Added SCIsFlag to check whether a complex is flag
  • Added SCRandomize to support randomizing of complexes via bistellar moves
  • Again modified f-vector routine; should be faster now
  • SCFromGenerators now saves the automorphism group to the complex
  • Fixed small error in SCSortComplex

14/01/2010 - Version 1.1.21

  • Improved performance of f-vector calculation (does not need face lattice any more)

26/11/2009 - Version 1.1.9

  • Fixed loading errors of package when package `homology' is not loaded
  • Added support for normal surfaces, along with additional documentation
  • Complexes in library updated

16/11/2009 - Version 1.0.440

  • Updated file and directory attributes to comply with GAP standard.
  • Added function SCIntersectionFormSignature. Thanks to Michael Eisermann for helpful comments on calculating signatures of matrices without the need of calculating eigenvalues.

11/11/2009 - Version 1.0.435

  • Updated Documentation
  • Updated GRAPE availability test to reliably work with GRAPE 4.3

26/10/2009 - Version 1.0.403

  • Minor compatibility fixes in this release
  • Fixed some error messages on undefined global variables during loading of simpcomp when the packages homology' and/orGRAPE' are not present
  • Fallback to functions of `homology' package are now handled correctly
  • Added extended testing functionality and SCRunTest

13/10/2009 - Version 1.0.389

  • Initial version


The following features are planned to be supported in future versions of simpcomp:

  • Better simplification strategies
  • Coverings (finite index subgroups)
  • Presentation two-complex (from fp-group)


We have thorougly tested simpcomp on errors. If you none the less encouter errors or inconsistencies when working with simpcomp, please contact the authors.


Felix Effenberger, felix.effenberger@mis.mpg.de

Address: Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig GERMANY


Jonathan Spreer, j.spreer@uq.edu.au

Address: Computational Geometry and Topology Group School of Mathematics and Physics The University of Queensland St. Lucia, Queensland, 4072 AUSTRALIA