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simpcomp

A GAP toolbox for simplicial complexes

Version 2.1.6

February 2016

Felix Effenberger
Email: felix.effenberger@mis.mpg.de
Address:
Max Planck Institute for Mathematics in the Sciences
Inselstr. 22
D-04103 Leipzig, Germany

Jonathan Spreer
Email: j.spreer@uq.edu.au
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

© 2016 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

1 Introduction
2 Theoretical foundations
3 The new GAP object types of simpcomp
4 Functions and operations for the GAP object type SCPolyhedralComplex
5 The GAP object types SCSimplicialComplex and SCNormalSurface
6 Functions and operations for SCSimplicialComplex
 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
7 Functions and operations for SCNormalSurface
8 (Co-)Homology of simplicial complexes
9 Bistellar flips
10 Simplicial blowups
11 Polyhedral Morse theory
12 Forman's discrete Morse theory
13 Library and I/O
14 Interfaces to other software packages
15 Miscellaneous functions
16 Property handlers
17 A demo session with simpcomp
18 simpcomp internals
References
Index

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