<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (HAP commands) - Contents</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap0" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chap10_mj.html" >10</a> <a href="chap11_mj.html" >11</a> <a href="chap12_mj.html" >12</a> <a href="chap13_mj.html" >13</a> <a href="chap14_mj.html" >14</a> <a href="chap15_mj.html" >15</a> <a href="chap16_mj.html" >16</a> <a href="chap17_mj.html" >17</a> <a href="chap18_mj.html" >18</a> <a href="chap19_mj.html" >19</a> <a href="chap20_mj.html" >20</a> <a href="chap21_mj.html" >21</a> <a href="chap22_mj.html" >22</a> <a href="chap23_mj.html" >23</a> <a href="chap24_mj.html" >24</a> <a href="chap25_mj.html" >25</a> <a href="chap26_mj.html" >26</a> <a href="chap27_mj.html" >27</a> <a href="chap28_mj.html" >28</a> <a href="chap29_mj.html" >29</a> <a href="chap30_mj.html" >30</a> <a href="chap31_mj.html" >31</a> <a href="chap32_mj.html" >32</a> <a href="chap33_mj.html" >33</a> <a href="chap34_mj.html" >34</a> <a href="chap35_mj.html" >35</a> <a href="chap36_mj.html" >36</a> <a href="chap37_mj.html" >37</a> <a href="chap38_mj.html" >38</a> <a href="chap39_mj.html" >39</a> <a href="chap40_mj.html" >40</a> <a href="chapInd_mj.html" >Ind</a> </div div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0.html" >[MathJax off]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >Homological Algebra Programming</h1 >div >br />Email: <span class="URL" ><a href="mailto:Graham.Ellis@nuigalway.ie" >Graham.Ellis@nuigalway.ie</a></span >Address :</b><br />br /> National University of Irelnd, Galway <br /> Galway<br /> (Ireland)</p>"X7AA6C5737B711C89" name="X7AA6C5737B711C89" ></a></p>strong class="button" >HAP</strong > is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.70 was released on 19 July 2025 . <br /> The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis. <br /> This document describes the functions available in <strong class="button" >HAP</strong >. Examples illustrating these functions are available in the <span class="URL" ><a href="../tutorial/chap0.html" ><strong class="button" >HAP</strong > tutorial</a></span >.</p>"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>d postdoctoral researchers at NUI Galway, the Irish Research Council, Science Foundation Ireland, and the EU Marie Curie programme.</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"contents" name="contents" ></a></h3>div class="ContChap" ><a href="chap1_mj.html#X85BEB9F48106583E" >1 <span class="Heading" >Basic functionality for cellular complexes, fundamental groups and homology</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7F06418383E098EB" >1.1 <span class="Heading" > Data <span class="SimpleMath" >\(\longrightarrow\)</span > Cellular Complexes </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X85C818B87D9AC922" >1.1-1 RegularCWPolytope</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7910F39B7AB79096" >1.1-2 CubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X78A3981C878C7FB5" >1.1-3 PureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X869065F77C4761EC" >1.1-4 PureCubicalKnot</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7B432A6184CBAC75" >1.1-5 PurePermutahedralKnot</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X824625A27FF6DE6F" >1.1-6 PurePermutahedralComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X80CAD0357AF44E48" >1.1-7 CayleyGraphOfGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8187F6507BA14D5C" >1.1-8 EquivariantEuclideanSpace</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7FE0522B8134DF7C" >1.1-9 EquivariantOrbitPolytope</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X81E8E97278B1AE92" >1.1-10 EquivariantTwoComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7F8D4C4C7ED15A31" >1.1-11 QuillenComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X854B96757AF38A41" >1.1-12 RestrictedEquivariantCWComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7A3B6B647C8CF90B" >1.1-13 RandomSimplicialGraph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8394037487D3C17E" >1.1-14 RandomSimplicialTwoComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X83DB403087D02CC8" >1.1-15 ReadCSVfileAsPureCubicalKnot</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7BE9892784AA4990" >1.1-16 ReadImageAsPureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X84D89B96873308B7" >1.1-17 ReadImageAsFilteredPureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X80E8B89F7E95D101" >1.1-18 ReadImageAsWeightFunction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7D8681B079E019C0" >1.1-19 ReadPDBfileAsPureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7E278788808A9EE4" >1.1-20 ReadPDBfileAsPurepermutahedralComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X85C818B87D9AC922" >1.1-21 RegularCWPolytope</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X818F2E887FE5F7BE" >1.1-22 SimplicialComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X79CA51F27C07435C" >1.1-23 SymmetricMatrixToFilteredGraph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8227636B7E878448" >1.1-24 SymmetricMatrixToGraph</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7C0C080487641830" >1.2 <span class="Heading" > Metric Spaces</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7F8113757F7DD2F4" >1.2-1 CayleyMetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7A4560307BA911F5" >1.2-2 EuclideanMetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X789AE7CE8445A67C" >1.2-3 EuclideanSquaredMetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X79DA33CB7D46CAB4" >1.2-4 HammingMetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7BD62D75829F8701" >1.2-5 KendallMetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8763D1167EF519A1" >1.2-6 ManhattanMetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7C86B58A7CEA5513" >1.2-7 VectorsToSymmetricMatrix</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X80A49CAC84313990" >1.3 <span class="Heading" > Cellular Complexes <span class="SimpleMath" >\(\longrightarrow\)</span > Cellular Complexes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7AF313D387F6BA22" >1.3-1 BoundaryMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X848ED6C378A1C5C0" >1.3-2 CliqueComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X85FAD5E086DBD429" >1.3-3 ConcentricFiltration</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X861BA02C7902A4F4" >1.3-4 DirectProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7DB4D3B57E0DA723" >1.3-5 FiltrationTerm</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7B335342839E5146" >1.3-6 Graph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7966519E78BC6C18" >1.3-7 HomotopyGraph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X84560FF678621AE1" >1.3-8 Nerve</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7C2BEF7C871E54D7" >1.3-9 RegularCWComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X79967AC2859A9631" >1.3-10 RegularCWMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X82843E747FE622AF" >1.3-11 ThickeningFiltration</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7FD50DF6782F00A0" >1.4 <span class="Heading" > Cellular Complexes <span class="SimpleMath" >\(\longrightarrow\)</span > Cellular Complexes (Preserving Data Types)</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X840576107A2907B8" >1.4-1 ContractedComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7A46614B84FF25BE" >1.4-2 ContractibleSubcomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X86164F4481ACC485" >1.4-3 KnotReflection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7D86D13C822D59A9" >1.4-4 KnotSum</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X855537287E9C4E72" >1.4-5 OrientRegularCWComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7A266B5A7BE88E89" >1.4-6 PathComponent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7FF34B9E86E901DC" >1.4-7 PureComplexBoundary</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7D0C9B27845F0739" >1.4-8 PureComplexComplement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7FB5BE6C78D5C7C8" >1.4-9 PureComplexDifference</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8091C9BA819C2332" >1.4-10 PureComplexInterstection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X84A7E7A47F7BA09D" >1.4-11 PureComplexThickened</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X78014E027F28C2C8" >1.4-12 PureComplexUnion</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7E7AC0E77E25C45B" >1.4-13 SimplifiedComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X844174D37E70B9B4" >1.4-14 ZigZagContractedComplex</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7E25932F7DD535E8" >1.5 <span class="Heading" > Cellular Complexes <span class="SimpleMath" >\(\longrightarrow\)</span > Homotopy Invariants</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7DC474EE7A909563" >1.5-1 AlexanderPolynomial</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X83EF7B888014C363" >1.5-2 BettiNumber</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8307F8DB85F145AE" >1.5-3 EulerCharacteristic</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X78813B9A851B922A" >1.5-4 EulerIntegral</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7EAE7E4181546C17" >1.5-5 FundamentalGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X808733FF7EF6278E" >1.5-6 FundamentalGroupOfQuotient</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X78F2C5ED80D1C8DD" >1.5-7 IsAspherical</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X797F8D4A848DD9BC" >1.5-8 KnotGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X825539B57FBDDE86" >1.5-9 PiZero</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7EE96E8B7C1643BD" >1.5-10 PersistentBettiNumbers</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7C17A7897DDAE22C" >1.6 <span class="Heading" > Data <span class="SimpleMath" >\(\longrightarrow\)</span > Homotopy Invariants</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7F5B6CAD7CB2E985" >1.6-1 DendrogramMat</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X859286BF7F6047B7" >1.7 <span class="Heading" > Cellular Complexes <span class="SimpleMath" >\(\longrightarrow\)</span > Non Homotopy Invariants</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7A1C427578108B7E" >1.7-1 ChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7D4AF2E8785DA457" >1.7-2 ChainComplexEquivalence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7D77D18679E941D3" >1.7-3 ChainComplexOfQuotient</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7BCD94877DF261C4" >1.7-4 ChainMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7B8741FB7A3263EC" >1.7-5 CochainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8489A39F870FF08B" >1.7-6 CriticalCells</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7A4AD52D82627ABC" >1.7-7 DiagonalApproximation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X858ADA3B7A684421" >1.7-8 Size</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7B6F366F7A2D8FEE" >1.8 <span class="Heading" > (Co)chain Complexes <span class="SimpleMath" >\(\longrightarrow \)</span > (Co)chain Complexes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X829DD3868410FE2E" >1.8-1 FilteredTensorWithIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7BC291C47FEAC5B8" >1.8-2 FilteredTensorWithIntegersModP</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X788F3B5E7810E309" >1.8-3 HomToIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X8122D25786C83565" >1.8-4 TensorWithIntegersModP</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7BB8DC9783A4AF81" >1.9 <span class="Heading" > (Co)chain Complexes <span class="SimpleMath" >\(\longrightarrow \)</span > Homotopy Invariants</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X84CFC57B7E9CCCF7" >1.9-1 Cohomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X877825E57D79839C" >1.9-2 CupProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X85A9D5CB8605329C" >1.9-3 Homology</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X867BE1388467C939" >1.10 <span class="Heading" > Visualization</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X806A81EF79CE0DEF" >1.10-1 BarCodeDisplay</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X83D60A6682EBB6F1" >1.10-2 BarCodeCompactDisplay</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X80CAD0357AF44E48" >1.10-3 CayleyGraphOfGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X83A5C59278E13248" >1.10-4 Display</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7B98A3C4831D5B0D" >1.10-5 DisplayArcPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X861690C27BADC326" >1.10-6 DisplayCSVKnotFile</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7F4AA01E7C0A5C16" >1.10-7 DisplayDendrogram</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7E5A38F081B401BE" >1.10-8 DisplayDendrogramMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X822F54F385D7EF8A" >1.10-9 DisplayPDBfile</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X80EC50C27EFF2E12" >1.10-10 OrbitPolytope</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X7DF49EAD7C0B0E84" >1.10-11 ScatterPlot</a></span >div ></div >div >div class="ContChap" ><a href="chap2_mj.html#X84CA5C9B81900889" >2 <span class="Heading" >Basic functionality for <span class="SimpleMath" >\(\mathbb ZG\)</span >-resolutions and group cohomology</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7C0B125E7D5415B4" >2.1 <span class="Heading" > Resolutions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X868E2A04832619C5" >2.1-1 EquivariantChainMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X79EA11238403019D" >2.1-2 FreeGResolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7CA87AA478007468" >2.1-3 ResolutionBieberbachGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X81A5CEFC82A1897D" >2.1-4 ResolutionCubicalCrystGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X789B3E7C7CBB3751" >2.1-5 ResolutionFiniteGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7CBE6BDA7DB5AD7D" >2.1-6 ResolutionNilpotentGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8574D76D7C891A04" >2.1-7 ResolutionNormalSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X86934BE9858F7199" >2.1-8 ResolutionPrimePowerGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7E4556B078B209CE" >2.1-9 ResolutionSL2Z</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8518446086A3F7EA" >2.1-10 ResolutionSmallGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X79A0221B7E96B642" >2.1-11 ResolutionSubgroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X85EC9D8E7A15A570" >2.2 <span class="Heading" > Algebras <span class="SimpleMath" >\(\longrightarrow \)</span > (Co)chain Complexes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7D5DD19D7BA9D816" >2.2-1 LeibnizComplex</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7F9E1F1781479F7B" >2.3 <span class="Heading" > Resolutions <span class="SimpleMath" >\(\longrightarrow \)</span > (Co)chain Complexes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X788F3B5E7810E309" >2.3-1 HomToIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X81FED0E9858E413A" >2.3-2 HomToIntegralModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X83BA99787CBE2B7D" >2.3-3 TensorWithIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8122D25786C83565" >2.3-4 TensorWithIntegersModP</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X80B6849C835B7F19" >2.4 <span class="Heading" > Cohomology rings</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X79C31EED8406A3E9" >2.4-1 AreIsomorphicGradedAlgebras</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X83DC2F1A805BA7A3" >2.4-2 HAPDerivation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7B93B7D082A50E61" >2.4-3 HilbertPoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X803D9B5E7A26F749" >2.4-4 HomologyOfDerivation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X855D2D747B6C54E1" >2.4-5 IntegralCohomologyGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7F5D00C97A46D686" >2.4-6 LHSSpectralSequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X828D20AC8735152B" >2.4-7 LHSSpectralSequenceLastSheet</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7DEFADD17CAA6308" >2.4-8 ModPCohomologyGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X796632C585D47245" >2.4-9 ModPCohomologyRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X831034A284F3906F" >2.4-10 Mod2CohomologyRingPresentation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7BCF8D907D237A03" >2.5 <span class="Heading" > Group Invariants</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7D1658EF810022E5" >2.5-1 GroupCohomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7F0A19E97980FD57" >2.5-2 GroupHomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7A30C1CC7FB6B2E9" >2.5-3 PrimePartDerivedFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X828B81D9829328F8" >2.5-4 PoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X828B81D9829328F8" >2.5-5 PoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7EFE814686C4EEF5" >2.5-6 RankHomologyPGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X86CDD4B77CBE3087" >2.6 <span class="Heading" > <span class="SimpleMath" >\(\mathbb F_p\)</span >-modules</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X85758F95832207D2" >2.6-1 GroupAlgebraAsFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X84B5182E831D0928" >2.6-2 Radical</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7929281B848A9FBE" >2.6-3 RadicalSeries</a></span >div ></div >div >div class="ContChap" ><a href="chap3_mj.html#X7F52C4747A402789" >3 <span class="Heading" >Basic functionality for homological group theory</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X85A9B66278AF63D9" >3.1 <span class="Heading" > Cocycles</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8343D6CA811C1E50" >3.1-1 CcGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7C4C64EE864B04D5" >3.1-2 CocycleCondition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A69F5007F07F478" >3.1-3 StandardCocycle</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7D02CE0A83211FB7" >3.2 <span class="Heading" > G-Outer Groups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X787C8FD6879771D9" >3.2-1 ActedGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8115386782214B38" >3.2-2 ActingGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X847ABE6F781C7FE8" >3.2-3 Centre</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X842035BD7E0B81EF" >3.2-4 GOuterGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7C2A5A4F84DC70CB" >3.3 <span class="Heading" > <span class="SimpleMath" >\(G\)</span >-cocomplexes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7D5E7FB97BF38DF1" >3.3-1 CohomologyModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7CF7B8A3842D498B" >3.3-2 HomToGModule</a></span >div ></div >div >div class="ContChap" ><a href="chap4_mj.html#X79D98558806BCE54" >4 <span class="Heading" >Basic functionality for parallel computation</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X835EF64B87BD48C7" >4.1 <span class="Heading" > Six Core Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X780C7FD2866D6C2B" >4.1-1 ChildCreate</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X780C7FD2866D6C2B" >4.1-2 ChildCreate</a></span >div ></div >div >div class="ContChap" ><a href="chap5_mj.html#X8735FC5E7BB5CE3A" >5 <span class="Heading" >Resolutions of the ground ring</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7CFDEEC07F15CF82" >5.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7D8875C87BC9C379" >5.1-1 TietzeReducedResolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X808535C3851CA4D4" >5.1-2 ResolutionArithmeticGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79EA11238403019D" >5.1-3 FreeGResolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8108E1047C31A058" >5.1-4 ResolutionGTree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7E4556B078B209CE" >5.1-5 ResolutionSL2Z</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79CB82D77A1FAE9D" >5.1-6 ResolutionAbelianGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79107B5F857DC27B" >5.1-7 ResolutionAlmostCrystalGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X839D1B3B78B672BB" >5.1-8 ResolutionAlmostCrystalQuotient</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X82A0D2B986724BB1" >5.1-9 ResolutionArtinGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X87DABAF98575DC13" >5.1-10 ResolutionAsphericalPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7CA87AA478007468" >5.1-11 ResolutionBieberbachGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7A20180E7D45038F" >5.1-12 ResolutionCoxeterGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7FCE801781AD83E1" >5.1-13 ResolutionDirectProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79C83C4881B6A656" >5.1-14 ResolutionExtension</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X87C346747F3B7C8C" >5.1-15 ResolutionFiniteDirectProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79EDB1D584C2776F" >5.1-16 ResolutionFiniteExtension</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X789B3E7C7CBB3751" >5.1-17 ResolutionFiniteGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7AF48FA77F677E75" >5.1-18 ResolutionFiniteSubgroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X78E504557FD75664" >5.1-19 ResolutionGraphOfGroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7CBE6BDA7DB5AD7D" >5.1-20 ResolutionNilpotentGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8574D76D7C891A04" >5.1-21 ResolutionNormalSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X86934BE9858F7199" >5.1-22 ResolutionPrimePowerGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8521359A87D46462" >5.1-23 ResolutionSmallFpGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79A0221B7E96B642" >5.1-24 ResolutionSubgroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X80CE971A7C2C538B" >5.1-25 ResolutionSubnormalSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7A8E03D57895D04A" >5.1-26 TwistedTensorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X864044D679AE4E25" >5.1-27 ConjugatedResolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X82646B64875E5560" >5.1-28 RecalculateIncidenceNumbers</a></span >div ></div >div >div class="ContChap" ><a href="chap6_mj.html#X841673BA782D0D1D" >6 <span class="Heading" > Resolutions of modules</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7CFDEEC07F15CF82" >6.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X795E37107DE0D0BD" >6.1-1 ResolutionFpGModule</a></span >div ></div >div >div class="ContChap" ><a href="chap7_mj.html#X7E91068780486C3A" >7 <span class="Heading" > Induced equivariant chain maps</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7CFDEEC07F15CF82" >7.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X868E2A04832619C5" >7.1-1 EquivariantChainMap</a></span >div ></div >div >div class="ContChap" ><a href="chap8_mj.html#X78D1062D78BE08C1" >8 <span class="Heading" > Functors</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7CFDEEC07F15CF82" >8.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X81BA486D7E532469" >8.1-1 ExtendScalars</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X788F3B5E7810E309" >8.1-2 HomToIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7E0216028756963B" >8.1-3 HomToIntegersModP</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X81FED0E9858E413A" >8.1-4 HomToIntegralModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7F5BAB35811AB0D1" >8.1-5 TensorWithIntegralModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7CF7B8A3842D498B" >8.1-6 HomToGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7D686D5D78FEF5C9" >8.1-7 InduceScalars</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X8456E06D7E76707B" >8.1-8 LowerCentralSeriesLieAlgebra</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X83BA99787CBE2B7D" >8.1-9 TensorWithIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X829DD3868410FE2E" >8.1-10 FilteredTensorWithIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7A0B33D085067A38" >8.1-11 TensorWithTwistedIntegers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X8122D25786C83565" >8.1-12 TensorWithIntegersModP</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X873096CB823BFD1B" >8.1-13 TensorWithTwistedIntegersModP</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X809BA8A87F61EEDA" >8.1-14 TensorWithRationals</a></span >div ></div >div >div class="ContChap" ><a href="chap9_mj.html#X7A06103979B92808" >9 <span class="Heading" > Chain complexes</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X7CFDEEC07F15CF82" >9.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7A1C427578108B7E" >9.1-1 ChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X838AF689838BA681" >9.1-2 ChainComplexOfPair</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7D84631C7B16C703" >9.1-3 ChevalleyEilenbergComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7D5DD19D7BA9D816" >9.1-4 LeibnizComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X86EC96CC7EB5957E" >9.1-5 SuspendedChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X83340F8C868BDE60" >9.1-6 ReducedSuspendedChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X82F1E19E7B11095A" >9.1-7 CoreducedChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7ADC193D813C82F7" >9.1-8 TensorProductOfChainComplexes</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7992AE7B7C8201F9" >9.1-9 LefschetzNumber</a></span >div ></div >div >div class="ContChap" ><a href="chap10_mj.html#X856F202D823280F8" >10 <span class="Heading" > Sparse Chain complexes</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X7CFDEEC07F15CF82" >10.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X81D5E16D81934320" >10.1-1 SparseMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7C85DF92798C625A" >10.1-2 TransposeOfSparseMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8011873278AB2827" >10.1-3 ReverseSparseMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8551FB1D81A85362" >10.1-4 SparseRowMult</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X824CD2E6862435EB" >10.1-5 SparseRowInterchange</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X84E8FA687CABA3CD" >10.1-6 SparseRowAdd</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7BE58D2983505606" >10.1-7 SparseSemiEchelon</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7A7B47D380A14F28" >10.1-8 RankMatDestructive</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7B21AE7987D4FB31" >10.1-9 RankMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7F10BD65823B1632" >10.1-10 SparseChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8156FB007C49C020" >10.1-11 SparseChainComplexOfRegularCWComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7BA17DDC81AA855D" >10.1-12 SparseBoundaryMatrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7C3327917BE532FD" >10.1-13 Bettinumbers</a></span >div ></div >div >div class="ContChap" ><a href="chap11_mj.html#X782177107A5D6D19" >11 <span class="Heading" > Homology and cohomology groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X7CFDEEC07F15CF82" >11.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X84CFC57B7E9CCCF7" >11.1-1 Cohomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7D5E7FB97BF38DF1" >11.1-2 CohomologyModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X86F3E9F17BF08BC0" >11.1-3 CohomologyPrimePart</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7D1658EF810022E5" >11.1-4 GroupCohomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7F0A19E97980FD57" >11.1-5 GroupHomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7F1A5C7D8288480F" >11.1-6 PersistentHomologyOfQuotientGroupSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X82FFCD8F8567BC95" >11.1-7 PersistentCohomologyOfQuotientGroupSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7998017B7B6C93B8" >11.1-8 UniversalBarCode</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X81598A7F7D0B1A07" >11.1-9 PersistentHomologyOfSubGroupSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X85542FBF7C1AEE55" >11.1-10 PersistentHomologyOfFilteredChainComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7E687DBD787A68BD" >11.1-11 PersistentHomologyOfCommutativeDiagramOfPGroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7A5DF30985E2738C" >11.1-12 PersistentHomologyOfFilteredPureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X80F604A579165F5C" >11.1-13 PersistentHomologyOfPureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7B4743ED799C2A16" >11.1-14 ZZPersistentHomologyOfPureCubicalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X86D0AEEC79FD104A" >11.1-15 RipsHomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7A12329E85BD4842" >11.1-16 BarCode</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X806A81EF79CE0DEF" >11.1-17 BarCodeDisplay</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X85A9D5CB8605329C" >11.1-18 Homology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X877EC6437EA89C45" >11.1-19 HomologyPb</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7CA212D0806A89FA" >11.1-20 HomologyVectorSpace</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7AE7B4857D0348AC" >11.1-21 HomologyPrimePart</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X841C3E3E86529CBF" >11.1-22 LeibnizAlgebraHomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X79FC84787D45273D" >11.1-23 LieAlgebraHomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7A30C1CC7FB6B2E9" >11.1-24 PrimePartDerivedFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7EFE814686C4EEF5" >11.1-25 RankHomologyPGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X81875BCD7A7A217A" >11.1-26 RankPrimeHomology</a></span >div ></div >div >div class="ContChap" ><a href="chap12_mj.html#X850CDAFE801E2B2A" >12 <span class="Heading" > Poincare series</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X7CFDEEC07F15CF82" >12.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X84117EA684724D53" >12.1-1 EfficientNormalSubgroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7EBC620581DCB4D6" >12.1-2 ExpansionOfRationalFunction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X828B81D9829328F8" >12.1-3 PoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X876B3DFB7B64688C" >12.1-4 PoincareSeriesPrimePart</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7E1A4C8781A02CD0" >12.1-5 PoincareSeriesLHS</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X82CBD11D84D50CBD" >12.1-6 Prank</a></span >div ></div >div >div class="ContChap" ><a href="chap13_mj.html#X7A9561E47A4994F5" >13 <span class="Heading" > Cohomology ring structure</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X7CFDEEC07F15CF82" >13.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X8152396B78D7F28C" >13.1-1 IntegralCupProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7F42615F7C10EEA0" >13.1-2 IntegralRingGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7DEFADD17CAA6308" >13.1-3 ModPCohomologyGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X796632C585D47245" >13.1-4 ModPCohomologyRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X8225CCD787C16ECB" >13.1-5 ModPRingGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X831034A284F3906F" >13.1-6 Mod2CohomologyRingPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap14_mj.html#X78C726EB7F6CDAC0" >14 <span class="Heading" > Cohomology rings of <span class="SimpleMath" >\(p\)</span >-groups (mainly <span class="SimpleMath" >\(p=2)\)</span ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X7CFDEEC07F15CF82" >14.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X831034A284F3906F" >14.1-1 Mod2CohomologyRingPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7E1A4C8781A02CD0" >14.1-2 PoincareSeriesLHS</a></span >div ></div >div >div class="ContChap" ><a href="chap15_mj.html#X86DE968B7B20BD48" >15 <span class="Heading" > Commutator and nonabelian tensor computations</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X7CFDEEC07F15CF82" >15.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7DFB0FC4834DF183" >15.1-1 BaerInvariant</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7D1614F87AAF5B97" >15.1-2 BogomolovMultiplier</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X87A1E2E279597B8D" >15.1-3 Bogomology</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7C374F188523A659" >15.1-4 Coclass</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X819E8AEC835F8CD1" >15.1-5 EpiCentre</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7EAF8A2A79C86181" >15.1-6 NonabelianExteriorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X8155A3747A66FE81" >15.1-7 NonabelianSymmetricKernel</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7856BC54856452DE" >15.1-8 NonabelianSymmetricSquare</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X829476FA82D5759B" >15.1-9 NonabelianTensorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7C0DF7C97F78C666" >15.1-10 NonabelianTensorSquare</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7EF8F3FF7FB900F8" >15.1-11 RelativeSchurMultiplier</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X854F2DB382723504" >15.1-12 TensorCentre</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X79D037908746C65C" >15.1-13 ThirdHomotopyGroupOfSuspensionB</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7967760B7E0F1E5F" >15.1-14 UpperEpicentralSeries</a></span >div ></div >div >div class="ContChap" ><a href="chap16_mj.html#X7A3DC9327EE1BE6C" >16 <span class="Heading" > Lie commutators and nonabelian Lie tensors</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap16_mj.html#X7CFDEEC07F15CF82" >16.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X80BBA6247ED4DCCF" >16.1-1 LieCoveringHomomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7BEEE3D380CF22F1" >16.1-2 LeibnizQuasiCoveringHomomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7B384F9486A7C92B" >16.1-3 LieEpiCentre</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X849324D680C0EE5E" >16.1-4 LieExteriorSquare</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X809B166C835516EB" >16.1-5 LieTensorSquare</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X802F2E417D872042" >16.1-6 LieTensorCentre</a></span >div ></div >div >div class="ContChap" ><a href="chap17_mj.html#X7A2144518112F830" >17 <span class="Heading" > Generators and relators of groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap17_mj.html#X7CFDEEC07F15CF82" >17.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7F49A86A82EB2420" >17.1-1 CayleyGraphOfGroupDisplay</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X82C8B87287602BFA" >17.1-2 IdentityAmongRelatorsDisplay</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X78F2C5ED80D1C8DD" >17.1-3 IsAspherical</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X878938C3835871D7" >17.1-4 PresentationOfResolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7F71698178AF48DD" >17.1-5 TorsionGeneratorsAbelianGroup</a></span >div ></div >div >div class="ContChap" ><a href="chap18_mj.html#X7CD67FEA7A1B6345" >18 <span class="Heading" > Orbit polytopes and fundamental domains</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X7CFDEEC07F15CF82" >18.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X802CD86983E8384B" >18.1-1 CoxeterComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7B60301179A6E7D2" >18.1-2 ContractibleGcomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X82B7AF6A876AA021" >18.1-3 QuotientOfContractibleGcomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X8092E70D8425CEBB" >18.1-4 TruncatedGComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X79F4F7938116201E" >18.1-5 FundamentalDomainStandardSpaceGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X80EC50C27EFF2E12" >18.1-6 OrbitPolytope</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7A4234867B232E34" >18.1-7 PolytopalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7E03550580383C01" >18.1-8 PolytopalGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X797A90E17C18FC89" >18.1-9 VectorStabilizer</a></span >div ></div >div >div class="ContChap" ><a href="chap19_mj.html#X85A9B66278AF63D9" >19 <span class="Heading" > Cocycles</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap19_mj.html#X7CFDEEC07F15CF82" >19.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X8343D6CA811C1E50" >19.1-1 CcGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7C4C64EE864B04D5" >19.1-2 CocycleCondition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7A69F5007F07F478" >19.1-3 StandardCocycle</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X8157A77284F56BAD" >19.1-4 Syzygy</a></span >div ></div >div >div class="ContChap" ><a href="chap20_mj.html#X8276B4377D092A80" >20 <span class="Heading" > Words in free <span class="SimpleMath" >\(ZG\)</span >-modules </span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap20_mj.html#X7CFDEEC07F15CF82" >20.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X82D71B3F85D5BE77" >20.1-1 AddFreeWords</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7CCBD61A7EEBD996" >20.1-2 AddFreeWordsModP</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7E3F96C27E64EEF9" >20.1-3 AlgebraicReduction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X844F2C187CC85C47" >20.1-4 MultiplyWord</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7CA8A7CB81820EBB" >20.1-5 Negate</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X818F12EC81BA4788" >20.1-6 NegateWord</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7FFAEBBD7B35F84C" >20.1-7 PrintZGword</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7D769FB3873B9527" >20.1-8 TietzeReduction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7D043C4282B27B69" >20.1-9 ResolutionBoundaryOfWord</a></span >div ></div >div >div class="ContChap" ><a href="chap21_mj.html#X81A2A3C97C09685E" >21 <span class="Heading" > <span "SimpleMath" >\(FpG\)</span >-modules</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7CFDEEC07F15CF82" >21.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78A33AD27F99EF01" >21.1-1 CompositionSeriesOfFpGModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7BECAE4182051B42" >21.1-2 DirectSumOfFpGModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7BADE250862DDD5F" >21.1-3 FpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X834924DF873A86B3" >21.1-4 FpGModuleDualBasis</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X851DB60D862610A5" >21.1-5 FpGModuleHomomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8380187081CE1237" >21.1-6 DesuspensionFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X87A9EA448236280E" >21.1-7 RadicalOfFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X87B150217D5AB909" >21.1-8 RadicalSeriesOfFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7CEC4AFB8181D098" >21.1-9 GeneratorsOfFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7AC4426279F2C004" >21.1-10 ImageOfFpGModuleHomomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X85758F95832207D2" >21.1-11 GroupAlgebraAsFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X83D91CE68582A650" >21.1-12 IntersectionOfFpGModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7F1A1FA280D8C5D0" >21.1-13 IsFpGModuleHomomorphismData</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78F8BC20784828A7" >21.1-14 MaximalSubmoduleOfFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X85C8FD707F10C22F" >21.1-15 MaximalSubmodulesOfFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C94F977801C1B97" >21.1-16 MultipleOfFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X804F7A1B79BEC521" >21.1-17 ProjectedFpGModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X822272997C4A8352" >21.1-18 RandomHomomorphismOfFpGModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X827146F37E2AA841" >21.1-19 Rank</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7A6C598E82D5A7F8" >21.1-20 SumOfFpGModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X86CAE52783D8E343" >21.1-21 SumOp</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8290448D87105F0B" >21.1-22 VectorsToFpGModuleWords</a></span >div ></div >div >div class="ContChap" ><a href="chap22_mj.html#X85B05BBA78ED7BE2" >22 <span class="Heading" > Meataxe modules</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap22_mj.html#X7CFDEEC07F15CF82" >22.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7AF88FB07F26E4F1" >22.1-1 DesuspensionMtxModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X78679D82835AEC25" >22.1-2 FpG_to_MtxModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X8594DD3B7F69265E" >22.1-3 GeneratorsOfMtxModule</a></span >div ></div >div >div class="ContChap" ><a href="chap23_mj.html#X7D02CE0A83211FB7" >23 <span class="Heading" > G-Outer Groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X7CFDEEC07F15CF82" >23.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X842035BD7E0B81EF" >23.1-1 GOuterGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7F681DB67F556FDF" >23.1-2 GOuterGroupHomomorphismNC</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7B4CE3397CAED0EC" >23.1-3 GOuterHomomorphismTester</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X847ABE6F781C7FE8" >23.1-4 Centre</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7F5C49A38455A64C" >23.1-5 DirectProductGog</a></span >div ></div >div >div class="ContChap" ><a href="chap24_mj.html#X7B54B8CA841C517B" >24 <span class="Heading" > Cat-1-groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X7CFDEEC07F15CF82" >24.1 <span class="Heading" > </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X79EB568C79A9EF01" >24.1-1 AutomorphismGroupAsCatOneGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7F2E058F7AF17E82" >24.1-2 HomotopyGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X866044CB7F43E1D2" >24.1-3 HomotopyModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7FD749E97F92A32C" >24.1-4 QuasiIsomorph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7DC403BA7DBA3ECE" >24.1-5 ModuleAsCatOneGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X80CF81187D3A7C0A" >24.1-6 MooreComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X857D6511876DEC0E" >24.1-7 NormalSubgroupAsCatOneGroup</a></span >quality 100%
¤ Dauer der Verarbeitung: 0.25 Sekunden
(vorverarbeitet)
¤ *© Formatika GbR, Deutschland
