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<h1><strong class="pkg">simpcomp</strong></h1>
<h2>A <strong class="pkg">GAP</strong> toolbox for simplicial complexes</h2>
<p>Version 2.1.14</p>
<p>15/03/2022
</p>
</div>
<p><b>
Felix Effenberger
</b>
<br />Email: <span class="URL"><a href="mailto:exilef@gmail.com">exilef@gmail.com</a></span>
</p><p><b>
Jonathan Spreer
</b>
<br />Email: <span class="URL"><a href="mailto:jonathan.spreer@sydney.edu.au">jonathan.spreer@sydney.edu.au</a></span>
<br />Address: <br />School of Mathematics and Statistics F07<br /> The University of Sydney<br /> NSW 2006 Australia
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p><strong class="pkg">simpcomp</strong> is an extension (a so called package) to <strong class="pkg">GAP</strong> 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 <span class="SimpleMath">f</span>-, <span class="SimpleMath">g</span>- and <span class="SimpleMath">h</span>-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.). <strong class="pkg">simpcomp</strong> ships with an extensive library of known triangulations of manifolds and a census of all combinatorial <span class="SimpleMath">3</span>-manifolds with transitive cyclic symmetry up to <span class="SimpleMath">22</span> 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 <span class="SimpleMath">4</span>-manifolds are implemented.<br /> <strong class="pkg">simpcomp</strong> 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.<br /> If possible, <strong class="pkg">simpcomp</strong> makes use of the <strong class="pkg">GAP</strong> package <strong class="pkg">homology</strong> <a href="chapBib.html#biBDumas04Homology">[DHSW11]</a> for its homology computation but also provides the user with own (co-)homology algorithms. For automorphism group computation the <strong class="pkg">GAP</strong> package <strong class="pkg">GRAPE</strong> <a href="chapBib.html#biBSoicher06GRAPE">[Soi12]</a> is used, which in turn uses the program <code class="code">nauty</code> by Brendan McKay <a href="chapBib.html#biBMcKay84Nauty">[MP14]</a>. An internal automorphism group calculation algorithm is used as fallback if the <strong class="pkg">GRAPE</strong> package is not available.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2021 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 <span class="URL"><a href="http://www.fsf.org/licensing/licenses/fdl.html">http://www.fsf.org/licensing/licenses/fdl.html</a></span> for a copy.</p>
<p><strong class="pkg">simpcomp</strong> is free software. The code of <strong class="pkg">simpcomp</strong> is released under the GPL version 2 or later (at your preference). For the text of the GPL see the file <code class="code">COPYING</code> in the <strong class="pkg">simpcomp</strong> directory or <span class="URL"><a href="https://www.gnu.org/licenses/">https://www.gnu.org/licenses/</a></span>.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>A few functions of <strong class="pkg">simpcomp</strong> 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 <a href="chapBib.html#biBLutz03TrigMnfFewVertVertTrans">[Lut03]</a> and the <strong class="pkg">GAP</strong> programs BISTELLAR and MANIFOLD_VT from <a href="chapBib.html#biBLutz08ManifoldPage">[Lut]</a>). Some functions were carried over from the <strong class="pkg">homology</strong> package by Dumas et al. <a href="chapBib.html#biBDumas04Homology">[DHSW11]</a> -- these functions are marked in the documentation and the source code. The internal (co-)homology algorithms were implemented by Armin Weiss.</p>
<p>Most of the complexes in the simplicial complex library are taken from the "Manifold Page" by Frank Lutz <a href="chapBib.html#biBLutz08ManifoldPage">[Lut]</a>.</p>
<p>The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG): <strong class="pkg">simpcomp</strong> has been developed within the DFG projects Ku 1203/5-2 and Ku 1203/5-3.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A1272437CD1F7AD">1.1 <span class="Heading">What is new</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7AA5F8168379C5E8">1.2 <span class="Heading"><strong class="pkg">simpcomp</strong> benefits</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X78BBA61C83276770">1.3 <span class="Heading">How to save time reading this document</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8336AF737AFAA468">1.4 <span class="Heading">Organization of this document</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X86F613867AA3A382">1.5 <span class="Heading">How to assure <strong class="pkg">simpcomp</strong> works correctly</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X85A6577886459361">1.6 <span class="Heading">Controlling <strong class="pkg">simpcomp</strong> log messages</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80002B357CE9B674">1.7 <span class="Heading">How to cite <strong class="pkg">simpcomp</strong></span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7E15BCD07F132C67">2 <span class="Heading">Theoretical foundations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7D819D6180502D18">2.1 <span class="Heading">Polytopes and polytopal complexes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7AA9180F7CF65B41">2.2 <span class="Heading">Simplices and simplicial complexes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84B178117BE7DD1C">2.3 <span class="Heading">From geometry to combinatorics</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BE7221B7C38B27D">2.4 <span class="Heading">Discrete Normal surfaces</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86275D5979B4B531">2.5 <span class="Heading">Polyhedral Morse theory and slicings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84451D2C879613E6">2.6 <span class="Heading">Discrete Morse theory</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X842354467CE73393">2.7 <span class="Heading">Tightness and tight triangulations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7E1BE673859191F0">2.8 <span class="Heading">Simplicial blowups</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X83DEEDF8835622AD">3 <span class="Heading">The new GAP object types of <strong class="pkg">simpcomp</strong></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X82FAB51B7C625240">3.1 <span class="Heading">Accessing properties of a <code class="code">SCPolyhedralComplex</code>
object</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X840691C285AB3AAD">4 <span class="Heading">Functions and operations for the GAP object type <code class="code">SCPolyhedralComplex</code></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7D35F510803714A8">4.1 <span class="Heading">Computing properties of objects of type
<code class="code">SCPolyhedralComplex</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BDD568184E3419D">4.1-1 SCFacets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87DC942881235E25">4.1-2 SCFacetsEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X849B0CD0796298EA">4.1-3 SCVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83D02BCD7D19FC6F">4.1-4 SCVerticesEx</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B69B61E867E748E">4.2 <span class="Heading">Vertex labelings and label operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C02CB9E8422EEE8">4.2-1 SCLabelMax</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7FC2186B7D346BD3">4.2-2 SCLabelMin</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X826E9B4482AF2671">4.2-3 SCLabels</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81C8A7EC84F5DEE9">4.2-4 SCName</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A812340843BCED3">4.2-5 SCReference</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B6011907B74EDDA">4.2-6 SCRelabel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78E22E3B787DDE90">4.2-7 SCRelabelStandard</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83276FCB844628EC">4.2-8 SCRelabelTransposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84AA5E217F1EAE23">4.2-9 SCRename</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87184048876229F3">4.2-10 SCSetReference</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80A64720826EA264">4.2-11 SCUnlabelFace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8188E7BE7E9BEF7E">4.3 <span class="Heading">Operations on objects of type <code class="code">SCPolyhedralComplex</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84646E6786FB7993">4.3-1 SCAntiStar</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X874059937F4C709B">4.3-2 SCLink</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EFA115B7B26F9AD">4.3-3 SCLinks</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78B725AD7E747A63">4.3-4 SCStar</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83F3E7487A1EF355">4.3-5 SCStars</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X78B454D3799549A9">5 <span class="Heading">The GAP object types <code class="code">SCSimplicialComplex</code> and <code class="code">SCNormalSurface</code></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7E7034FC82152AE6">5.1 <span class="Heading">The object type <code class="code">SCSimplicialComplex</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X82E3D6D97951997D">5.1-1 SCIsSimplicialComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B20035E791441C5">5.1-2 SCDetails</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X80F13BB484B3E9B2">5.1-3 SCCopy</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7AAEB857865D9B4A">5.1-4 ShallowCopy (SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8155B18D7EEF06DE">5.1-5 SCPropertiesDropped</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X790E779979A985FF">5.2 <span class="Heading">Overloaded operators of <code class="code">SCSimplicialComplex</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7D932F3C7F853443">5.2-1 Operation + (SCSimplicialComplex, Integer)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X84853235810E68E7">5.2-2 Operation - (SCSimplicialComplex, Integer)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X78329E8F7999F2F3">5.2-3 Operation mod (SCSimplicialComplex, Integer)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7CBC2E7A87726BA8">5.2-4 Operation ^ (SCSimplicialComplex, Integer)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7E49B2337FCCD890">5.2-5 Operation + (SCSimplicialComplex, SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F9794A381478434">5.2-6 Operation - (SCSimplicialComplex, SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X80C8CB3983D4356C">5.2-7 Operation * (SCSimplicialComplex, SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7A3887227BA8E878">5.2-8 Operation = (SCSimplicialComplex, SCSimplicialComplex)</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7CB571C97C426BE2">5.3 <span class="Heading"><code class="code">SCSimplicialComplex</code> as a subtype of <code class="code">Set</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X82C94EE47E339DD8">5.3-1 Operation Union (SCSimplicialComplex, SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X80CFABE083100541">5.3-2 Operation Difference (SCSimplicialComplex, SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X851CE49F7F7437C3">5.3-3 Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B3E2F12853D4303">5.3-4 Size (SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86F0D20F8529E0DB">5.3-5 Length (SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85B5C18F7EFB94A4">5.3-6 Operation [] (SCSimplicialComplex)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F8511457D591474">5.3-7 Iterator (SCSimplicialComplex)</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80F19FA07C71EDDC">5.4 <span class="Heading">The object type <code class="code">SCNormalSurface</code></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X874D22B47BCD48D4">5.5 <span class="Heading">Overloaded operators of <code class="code">SCNormalSurface</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8295612F7BF5B611">5.5-1 Operation + (SCNormalSurface, Integer)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8428D7E5857EEAB5">5.5-2 Operation - (SCNormalSurface, Integer)</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8596775A7EBAAD7D">5.5-3 Operation mod (SCNormalSurface, Integer)</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X84C01E097C009E76">5.6 <span class="Heading"><code class="code">SCNormalSurface</code> as a subtype of <code class="code">Set</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8462E960847F8B83">5.6-1 Operation Union (SCNormalSurface, SCNormalSurface)</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6.html#X82BDBFFC81D080D1">6 <span class="Heading">Functions and operations for <code class="code">SCSimplicialComplex</code></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7A93E4B08536E2C8">6.1 <span class="Heading">Creating an <code class="code">SCSimplicialComplex</code> object from a facet list</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B5A874584FF34A7">6.1-1 SCFromFacets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B5470FD7E2320DE">6.1-2 SC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X827D29DD79A82CFA">6.1-3 SCFromDifferenceCycles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X804A0B1F85B333C2">6.1-4 SCFromGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X867E1FF580230E20">6.2 <span class="Heading">Isomorphism signatures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E95E36680C188C4">6.2-1 SCExportToString</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X80098C5F7B80A621">6.2-2 SCExportIsoSig</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E915DA7821DD513">6.2-3 SCFromIsoSig</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X79072405786FEA0B">6.3 <span class="Heading">Generating some standard triangulations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E04DD807AF33B78">6.3-1 SCBdCyclicPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X839F3BD37DBA3F3C">6.3-2 SCBdSimplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X856E48967BBFCF0E">6.3-3 SCEmpty</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A23532F7A8A3988">6.3-4 SCSimplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8664A90879248282">6.3-5 SCSeriesTorus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87C67A0087F645C1">6.3-6 SCSurface</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X831315CD80BA3654">6.3-7 SCFVectorBdCrossPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C010361858F0214">6.3-8 SCFVectorBdCyclicPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C3E0F7687AC966E">6.3-9 SCFVectorBdSimplex</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X814FE0267D7C54A9">6.4 <span class="Heading">Generating infinite series of transitive triangulations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EA6421A8156EBDF">6.4-1 SCSeriesAGL</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85E6FD6D84FF762B">6.4-2 SCSeriesBrehmKuehnelTorus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X786AD599875BD006">6.4-3 SCSeriesBdHandleBody</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8787A3A4788E950C">6.4-4 SCSeriesBid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C0223DF83CC961B">6.4-5 SCSeriesC2n</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E2927DA7F60D957">6.4-6 SCSeriesConnectedSum</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7D1CEF9F86D3AE66">6.4-7 SCSeriesCSTSurface</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C56D2B7858A80C7">6.4-8 SCSeriesD2n</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7CCBF8F487036415">6.4-9 SCSeriesHandleBody</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8519C1B678C101BF">6.4-10 SCSeriesHomologySphere</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78DA125479E1D77F">6.4-11 SCSeriesK</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B8300428516DAD8">6.4-12 SCSeriesKu</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X813C5B0E7FA7C1A3">6.4-13 SCSeriesL</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EAC6828812A241A">6.4-14 SCSeriesLe</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8280ED8280FF9218">6.4-15 SCSeriesLensSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7DDC1B127F21CFA4">6.4-16 SCSeriesPrimeTorus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7CC3944D7E2F6458">6.4-17 SCSeriesSeifertFibredSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B71BC8B7D74AFD5">6.4-18 SCSeriesS2xS2</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7899878881EA47F8">6.5 <span class="Heading">A census of regular and chiral maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X867D2AFC79B11405">6.5-1 SCChiralMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85BB97CB8240E59B">6.5-2 SCChiralMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B362D25784E7217">6.5-3 SCChiralTori</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7AB15983833FCA6B">6.5-4 SCNrChiralTori</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87180AD07F799C5A">6.5-5 SCNrRegularTorus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X83D0946E7E2C4163">6.5-6 SCRegularMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F75F5E183CC097E">6.5-7 SCRegularMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X79B6F47187668CDF">6.5-8 SCRegularTorus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87CE08247BE77E44">6.5-9 SCSeriesSymmetricTorus</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7F4308DB7C3699D1">6.6 <span class="Heading">Generating new complexes from old</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8255A2F97A7432F9">6.6-1 SCCartesianPower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X859DA29B83BDE35E">6.6-2 SCCartesianProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X82C9F57780C0B7F8">6.6-3 SCConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C63CDF28162C755">6.6-4 SCConnectedProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X81338CE18195607C">6.6-5 SCConnectedSum</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78B843417D63B408">6.6-6 SCConnectedSumMinus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X84F3487182AB102A">6.6-7 SCDifferenceCycleCompress</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8510B6CF85070A28">6.6-8 SCDifferenceCycleExpand</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8278E1157A318C32">6.6-9 SCStronglyConnectedComponents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X87C1C49987E75A9C">6.7 <span class="Heading">Simplicial complexes from transitive permutation groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C1592677A76A3E5">6.7-1 SCsFromGroupExt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A04D77085D9BE4E">6.7-2 SCsFromGroupByTransitivity</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X81FDA1407B1E96C9">6.8 <span class="Heading">The classification of cyclic combinatorial 3-manifolds</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X805BBDF58568614F">6.8-1 SCNrCyclic3Mflds</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X84477B9E7CAEED7B">6.8-2 SCCyclic3MfldTopTypes</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X781256E37DA1B69F">6.8-3 SCCyclic3Mfld</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8490744E81DA45BF">6.8-4 SCCyclic3MfldByType</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X86F8AED6843FCD65">6.8-5 SCCyclic3MfldListOfGivenType</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X81CE90127800B91A">6.9 <span class="Heading">Computing properties of simplicial complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B69B327809F67A0">6.9-1 SCAltshulerSteinberg</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B88925386E197AC">6.9-2 SCAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A33B8177A7ACD3A">6.9-3 SCAutomorphismGroupInternal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78F6EF808047772C">6.9-4 SCAutomorphismGroupSize</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EAC3A5D7A3339BB">6.9-5 SCAutomorphismGroupStructure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E9D5C257F88E5E0">6.9-6 SCAutomorphismGroupTransitivity</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X836DC73380EA7414">6.9-7 SCBoundary</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X812AE7397B4FC88E">6.9-8 SCDehnSommervilleCheck</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X859C3981831B4B81">6.9-9 SCDehnSommervilleMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X80B33CAF7B5476C0">6.9-10 SCDifferenceCycles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X82351AAE793DCB68">6.9-11 SCDim</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X798175C58050DDBD">6.9-12 SCDualGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X788BAE187D584103">6.9-13 SCEulerCharacteristic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X81F8071385FD9C1D">6.9-14 SCFVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F8B561C823DDDBA">6.9-15 SCFaceLattice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7AE0029985BD0775">6.9-16 SCFaceLatticeEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F6FE9B27B8D6922">6.9-17 SCFaces</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B40DFE780A47109">6.9-18 SCFacesEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7BDD568184E3419D">6.9-19 SCFacets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87DC942881235E25">6.9-20 SCFacetsEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X79F60850875BB683">6.9-21 SCFpBettiNumbers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X875963367A7745FB">6.9-22 SCFundamentalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B9F77A885E1BABE">6.9-23 SCGVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X863CA73D7F66B295">6.9-24 SCGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X789F8FC77FC0E701">6.9-25 SCGeneratorsEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X84FBF0A685547ECD">6.9-26 SCHVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X796EBADE7803C622">6.9-27 SCHasBoundary</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C2A5B4D7E77E444">6.9-28 SCHasInterior</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A0547F67BBB6546">6.9-29 SCHeegaardSplittingSmallGenus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C7335667C162AFA">6.9-30 SCHeegaardSplitting</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X864978877E7D4DA0">6.9-31 SCHomologyClassic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B0C12F5780FDD9B">6.9-32 SCIncidences</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B4CA6FE78A9880F">6.9-33 SCIncidencesEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X862926A079F6DFC2">6.9-34 SCInterior</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8123A6E282CD0174">6.9-35 SCIsCentrallySymmetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X81AF20DC814B51A6">6.9-36 SCIsConnected</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X860375D980E9A801">6.9-37 SCIsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X83E01C957D2F2458">6.9-38 SCIsEulerianManifold</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X843C3E7F79D8093F">6.9-39 SCIsFlag</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X83F0246384A766F2">6.9-40 SCIsHeegaardSplitting</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E66FE0C83A3D371">6.9-41 SCIsHomologySphere</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87BC29AF878E7FD8">6.9-42 SCIsInKd</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F4BECCA7E67B1B2">6.9-43 SCIsKNeighborly</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78C860DC851167F7">6.9-44 SCIsOrientable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X79DFCA08808665B7">6.9-45 SCIsPseudoManifold</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7AAA757F842EA23A">6.9-46 SCIsPure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EA4F4DB78758652">6.9-47 SCIsShellable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A2BD5657BBE1CC7">6.9-48 SCIsStronglyConnected</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B935899849C8E40">6.9-49 SCMinimalNonFaces</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7DE069A0823BD56E">6.9-50 SCMinimalNonFacesEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X82A224DF787A97BE">6.9-51 SCNeighborliness</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7AC2427184B44C65">6.9-52 SCNumFaces</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78DACE3478340DB8">6.9-53 SCOrientation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8026B46F8236124D">6.9-54 SCSkel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X84ACF7D580FE8B76">6.9-55 SCSkelEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E770DE27938B140">6.9-56 SCSpanningTree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X8284003382F863A0">6.10 <span class="Heading">Operations on simplicial complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X818757A17DA5CFFC">6.10-1 SCAlexanderDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X82D22356858062D6">6.10-2 SCClose</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7CF53D8D7E0FA702">6.10-3 SCCone</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X800BDDD878DFCBDB">6.10-4 SCDeletedJoin</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7FB3D29178076EB4">6.10-5 SCDifference</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C8D11C684825ADC">6.10-6 SCFillSphere</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7DCB16857D49EC37">6.10-7 SCHandleAddition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B4BE2C783E6D0BF">6.10-8 SCIntersection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85659EC77DFF8183">6.10-9 SCIsIsomorphic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X81D9CC438313F589">6.10-10 SCIsSubcomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8623B92580E8B4E4">6.10-11 SCIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X86576B7287686E2B">6.10-12 SCIsomorphismEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X86AC8D81837CC677">6.10-13 SCJoin</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8482E1F67C927BB7">6.10-14 SCNeighbors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F8FBEF17ACD0D4F">6.10-15 SCNeighborsEx</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EDA334983025D3D">6.10-16 SCShelling</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F967A717D4E41C0">6.10-17 SCShellingExt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8373598C7FF5D28E">6.10-18 SCShellings</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7987AAE481C31F38">6.10-19 SCSpan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7AAA4669793C57DC">6.10-20 SCSuspension</a></ | |
