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<h1>A HAP tutorial</h1>


<h2>(See also an <span class="URL"><a href="../www/SideLinks/About/aboutContents.html">older tutorial</a></span> or <span class="URL"><a href="comp.pdf">mini-course notes</a></span> or related <span class="URL"><a href="https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980">book</a></span>) <span class="URL"><a href="../www/index.html">The <strong class="button">HAP</strong> home page is here</a></span></h2>

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<p><b>Graham Ellis</b>
</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#X7E5EA9587D4BCFB4">1 <span class="Heading">Simplicial complexes & CW complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X85691C6980034524">1.1 <span class="Heading">The Klein bottle as a simplicial complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7B8F88487B1B766C">1.2 <span class="Heading">Other simplicial surfaces</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80A72C347D99A58E">1.3 <span class="Heading">The Quillen complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7C4A2B8B79950232">1.4 <span class="Heading">The Quillen complex as a reduced CW-complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X782AAB84799E3C44">1.5 <span class="Heading">Simple homotopy equivalences</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80474C7885AC1578">1.6 <span class="Heading">Cellular simplifications preserving homeomorphism type</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A15484C7E680AC9">1.7 <span class="Heading">Constructing a CW-structure on a knot complement</span></a>
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</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X829793717FB6DDCE">1.8 <span class="Heading">Constructing a regular CW-complex by attaching cells</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7B7354E68025FC92">1.9 <span class="Heading">Constructing a regular CW-complex from its face lattice</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X823FA6A9828FF473">1.10 <span class="Heading">Cup products</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7F9B01CF7EE1D2FC">1.11 <span class="Heading">Intersection forms of <span class="SimpleMath">4</span>-manifolds</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80B6849C835B7F19">1.12 <span class="Heading">Cohomology Rings</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X83035DEC7C9659C6">1.13 <span class="Heading">Bockstein homomorphism</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X87135D067B6CDEEC">1.14 <span class="Heading">Diagonal maps on associahedra and other polytopes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8771FF2885105154">1.15 <span class="Heading">CW maps and induced homomorphisms</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X853D6B247D0E18DB">1.16 <span class="Heading">Constructing a simplicial complex from a regular CW-complex</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7900FD197F175551">1.17 <span class="Heading">Some limitations to representing spaces as regular CW complexes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X85A579217DCB6CC8">1.18 <span class="Heading">Equivariant CW complexes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X86881717878ADCD6">1.19 <span class="Heading">Orbifolds and classifying spaces</span></a>
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<div class="ContChap"><a href="chap2.html#X7F8376F37AF80AAC">2 <span class="Heading">Cubical complexes &  permutahedral complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7D67D5F3820637AD">2.1 <span class="Heading">Cubical complexes</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X85D8195379F2A8CA">2.2 <span class="Heading">Permutahedral complexes</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X78D3037283B506E0">2.3 <span class="Heading">Constructing pure cubical and permutahedral complexes</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8462CF66850CC3A8">2.4 <span class="Heading">Computations in dynamical systems</span></a>
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<div class="ContChap"><a href="chap3.html#X87472058788D76C0">3 <span class="Heading">Covering spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85FB4CA987BC92CC">3.1 <span class="Heading">Cellular chains on the universal cover</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E5CC04E7E3CCDAD">3.2 <span class="Heading">Spun knots and the Satoh tube map</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C304A1C7EF0BA60">3.3 <span class="Heading">Cohomology with local coefficients</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7A4F34B780FA2CD5">3.4 <span class="Heading">Distinguishing between two non-homeomorphic homotopy equivalent spaces</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X869FD75B84AAC7AD">3.5 <span class="Heading"> Second homotopy groups of spaces with finite fundamental group</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X87F8F6C3812A7E73">3.6 <span class="Heading">Third homotopy groups of simply connected spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B506CF27DE54DBE">3.6-1 <span class="Heading">First example: Whitehead's certain exact sequence
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X828F0FAB86AA60E9">3.6-2 <span class="Heading">Second example: the Hopf invariant</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7EAF7E677FB9D53F">3.7 <span class="Heading">Computing the second homotopy group of a space with infinite fundamental group</span></a>
</span>
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</div>
<div class="ContChap"><a href="chap4.html#X7BFA4D1587D8DF49">4 <span class="Heading">Three Manifolds</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82D1348C79238C2D">4.1 <span class="Heading">Dehn Surgery</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X848EDEE882B36F6C">4.2 <span class="Heading">Connected Sums</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X78AE684C7DBD7C70">4.3 <span class="Heading">Dijkgraaf-Witten Invariant</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X80B6849C835B7F19">4.4 <span class="Heading">Cohomology rings</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F56BB4C801AB894">4.5 <span class="Heading">Linking Form</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X850C76697A6A1654">4.6 <span class="Heading">Determining the homeomorphism type of a lens space</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7EC6B008878CC77E">4.7 <span class="Heading">Surgeries on distinct knots can yield homeomorphic manifolds</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B425A3280A2AF07">4.8 <span class="Heading">Finite fundamental groups of <span class="SimpleMath">3</span>-manifolds</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X78912D227D753167">4.9 <span class="Heading">Poincare's cube manifolds
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8761051F84C6CEC2">4.10 <span class="Heading">There are at least 25 distinct cube manifolds</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D50795883E534A3">4.10-1 <span class="Heading">Face pairings for 25 distinct cube manifolds</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X837811BB8181666E">4.10-2 <span class="Heading">Platonic cube manifolds</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8084A36082B26D86">4.11 <span class="Heading">There are at most 41 distinct cube manifolds</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B63C22C80E53758">4.12 <span class="Heading">There are precisely 18 orientable cube manifolds, of which   9 are spherical and 5 are euclidean</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X796BF3817BD7F57D">4.13 <span class="Heading">Cube manifolds with boundary</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7EC4359B7DF208B0">4.14 <span class="Heading">Octahedral manifolds</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85FFF9B97B7AD818">4.15 <span class="Heading">Dodecahedral manifolds</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X78B75E2E79FBCC54">4.16 <span class="Heading">Prism manifolds</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F31DFDA846E8E75">4.17 <span class="Heading">Bipyramid manifolds</span></a>
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<div class="ContChap"><a href="chap5.html#X7B7E077887694A9F">5 <span class="Heading">Topological data analysis</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80A70B20873378E0">5.1 <span class="Heading">Persistent homology  </span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D512DA37F789B4C">5.1-1 <span class="Heading">Background to the data</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X849556107A23FF7B">5.2 <span class="Heading">Mapper clustering</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D512DA37F789B4C">5.2-1 <span class="Heading">Background to the data</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7BBDE0567DB8C5DA">5.3 <span class="Heading">Some tools for handling pure complexes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X79616D12822FDB9A">5.4 <span class="Heading">Digital image analysis and persistent homology</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8066F9B17B78418E">5.4-1 <span class="Heading">Naive example of image segmentation by automatic thresholding</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E6436E0856761F2">5.4-2 <span class="Heading">Refining the filtration</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D512DA37F789B4C">5.4-3 <span class="Heading">Background to the data</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7A8224DA7B00E0D9">5.5 <span class="Heading">A second example of digital image segmentation</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8290E7D287F69B98">5.6 <span class="Heading">A third example of digital image segmentation</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7957F329835373E9">5.7 <span class="Heading">Naive example of digital image contour extraction</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7D2CC9CB85DF1BAF">5.8 <span class="Heading">Alternative approaches to computing persistent homology</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86FD0A867EC9E64F">5.8-1 <span class="Heading">Non-trivial cup product</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X783EF0F17B629C46">5.8-2 <span class="Heading">Explicit homology generators</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80D0D8EB7BCD05E9">5.9 <span class="Heading">Knotted proteins</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X87AF06677F05C624">5.10 <span class="Heading">Random simplicial complexes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X875EE92F7DBA1E27">5.11 <span class="Heading">Computing homology of a clique complex (Vietoris-Rips complex) </span></a>
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<div class="ContChap"><a href="chap6.html#X7C07F4BD8466991A">6 <span class="Heading">Group theoretic computations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X86D7FBBD7E5287C9">6.1 <span class="Heading">Third homotopy group of a supsension of an Eilenberg-MacLane space </span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X803FDFFE78A08446">6.2 <span class="Heading">Representations of knot quandles</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7E4EFB987DA22017">6.3 <span class="Heading">Identifying knots</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X8664E986873195E6">6.4 <span class="Heading">Aspherical <span class="SimpleMath">2</span>-complexes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X84C0CB8B7C21E179">6.5 <span class="Heading">Group presentations and homotopical syzygies</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7F719758856A443D">6.6 <span class="Heading">Bogomolov multiplier</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X8333413B838D787D">6.7 <span class="Heading">Second group cohomology and group extensions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7F04FA5E81FFA848">6.8 <span class="Heading">Cocyclic groups: a convenient way of representing  certain groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X863080FE8270468D">6.9 <span class="Heading">Effective group presentations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7C60E2B578074532">6.10 <span class="Heading">Second group cohomology and cocyclic Hadamard matrices</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X78040D8580D35D53">6.11 <span class="Heading">Third group cohomology and homotopy <span class="SimpleMath">2</span>-types</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap7.html#X787E37187B7308C9">7 <span class="Heading">Cohomology of groups (and Lie Algebras)</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X807B265978F90E01">7.1 <span class="Heading">Finite groups </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80A721AC7A8D30A3">7.1-1 <span class="Heading">Naive homology computation for a very small group</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X838CEA3F850DFC82">7.1-2 <span class="Heading">A more efficient homology computation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X842E93467AD09EC1">7.1-3 <span class="Heading">Computation of an induced homology homomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8754D2937E6FD7CE">7.1-4 <span class="Heading">Some other finite group homology computations</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X8463EF6A821FFB69">7.2 <span class="Heading">Nilpotent groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X82E8FAC67BC16C01">7.3 <span class="Heading">Crystallographic and Almost Crystallographic groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7AFFB32587D047FE">7.4 <span class="Heading">Arithmetic groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X800CB6257DC8FB3A">7.5 <span class="Heading">Artin groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7BAFCA3680E478AE">7.6 <span class="Heading">Graphs of groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7CE849E58706796C">7.7 <span class="Heading">Lie algebra homology and free nilpotent groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7C3DEDD57BB4D537">7.8 <span class="Heading">Cohomology with coefficients in a module</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7E573EA582CCEF2E">7.9 <span class="Heading">Cohomology as a functor of the first variable</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X796731727A7EBE59">7.10 <span class="Heading">Cohomology as a functor of the second variable and the long exact coefficient sequence</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X80F6FD3E7C7E4E8D">7.11 <span class="Heading">Transfer Homomorphism</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X79B1406C803FF178">7.12 <span class="Heading">Cohomology rings of finite fundamental groups of 3-manifolds
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X833A19F0791C3B06">7.13 <span class="Heading">Explicit cocycles </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7C5233E27D2D603E">7.14 <span class="Heading">Quillen's complex and the p-part of homology
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X865CC8E0794C0E61">7.15 <span class="Heading">Homology of a Lie algebra</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X86B4EE4783A244F7">7.16 <span class="Heading">Covers of Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7DFF32A67FF39C82">7.16-1 <span class="Heading">Computing a cover</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap8.html#X7ED29A58858AAAF2">8 <span class="Heading">Cohomology rings and Steenrod operations for groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X877CAF8B7E64DE04">8.1 <span class="Heading">Mod-<span class="SimpleMath">p</span> cohomology rings of finite groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X870E0299782638AF">8.1-1 <span class="Heading">Ring presentations (for the commutative <span class="SimpleMath">p=2</span> case)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X862538218748627F">8.2 <span class="Heading">Poincare Series for Mod-<span class="SimpleMath">p</span> cohomology</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X780DF87680C3F52B">8.3 <span class="Heading">Functorial ring homomorphisms in Mod-<span class="SimpleMath">p</span> cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X834CED9D7A104695">8.3-1 <span class="Heading">Testing homomorphism properties</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7A0D505D844F0CD4">8.3-2 <span class="Heading">Testing functorial properties</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X855764877FA44225">8.3-3 <span class="Heading">Computing with larger groups</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X80114B0483EF9A67">8.4 <span class="Heading">Steenrod operations for finite <span class="SimpleMath">2</span>-groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7D5ACA56870A40E9">8.5 <span class="Heading">Steenrod operations on the classifying space of a finite <span class="SimpleMath">p</span>-group</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7D2D26C0784A0E14">8.6 <span class="Heading">Mod-<span class="SimpleMath">p</span> cohomology rings of crystallographic groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X81C107C07CF02F0E">8.6-1 <span class="Heading">Poincare series for crystallographic groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7F5C242F7BC938A5">8.6-2 <span class="Heading">Mod <span class="SimpleMath">2</span> cohomology rings of <span class="SimpleMath">3</span>-dimensional crystallographic groups</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap9.html#X786DB80A8693779E">9 <span class="Heading">Bredon homology</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7B0212F97F3D442A">9.1 <span class="Heading">Davis complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7AFFB32587D047FE">9.2 <span class="Heading">Arithmetic groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7DEBF2BB7D1FB144">9.3 <span class="Heading">Crystallographic groups</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap10.html#X7A06103979B92808">10 <span class="Heading">Chain Complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X782DE78884DD6992">10.1 <span class="Heading">Chain complex of a simplicial complex and simplicial pair</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X79E7A13E7DE9C412">10.2 <span class="Heading">Chain complex of a cubical complex and cubical pair</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X86C38E87817F2EAD">10.3 <span class="Heading">Chain complex of a regular CW-complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7F9662EF83A1FA76">10.4 <span class="Heading">Chain Maps of simplicial and regular CW maps</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X8127E17383F45359">10.5 <span class="Heading">Constructions for chain complexes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7AAAB26682CD8AC4">10.6 <span class="Heading">Filtered chain complexes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X856F202D823280F8">10.7 <span class="Heading">Sparse chain complexes</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap11.html#X7C0B125E7D5415B4">11 <span class="Heading">Resolutions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X83E8F9DA7CDC0DA7">11.1 <span class="Heading">Resolutions for small finite groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7EEA738385CC3AEA">11.2 <span class="Heading">Resolutions for very small finite groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X86C0983E81F706F5">11.3 <span class="Heading">Resolutions for finite groups acting on orbit polytopes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X85374EA47E3D97CF">11.4 <span class="Heading">Minimal resolutions for finite <span class="SimpleMath">p</span>-groups over <span class="SimpleMath">F_p</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X866C8D91871D1170">11.5 <span class="Heading">Resolutions for abelian groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7B332CBE85120B38">11.6 <span class="Heading">Resolutions for nilpotent groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7B03997084E00509">11.7 <span class="Heading">Resolutions for groups with subnormal series</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X814FFCE080B3A826">11.8 <span class="Heading">Resolutions for groups with normal series</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X81227BF185C417AF">11.9 <span class="Heading">Resolutions for polycyclic (almost) crystallographic groups </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X814BCDD6837BB9C5">11.10 <span class="Heading">Resolutions for Bieberbach groups </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X87ADCB7D7FC0B4D3">11.11 <span class="Heading">Resolutions for arbitrary crystallographic groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7B9B3AF487338A9B">11.12 <span class="Heading">Resolutions for crystallographic groups  admitting cubical fundamental domain</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X78DD8D068349065A">11.13 <span class="Heading">Resolutions for Coxeter groups </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7C69E7227F919CC9">11.14 <span class="Heading">Resolutions for Artin groups </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X8032647F8734F4EB">11.15 <span class="Heading">Resolutions for <span class="SimpleMath">G=SL_2( Z[1/m])</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7BE4DE82801CD38E">11.16 <span class="Heading">Resolutions for selected groups 
<span class="SimpleMath">G=SL_2( mathcal O( Q(sqrtd) )</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7D9CCB2C7DAA2310">11.17 <span class="Heading">Resolutions for selected groups
<span class="SimpleMath">G=PSL_2( mathcal O( Q(sqrtd) )</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7F699587845E6DB1">11.18 <span class="Heading">Resolutions for a few higher-dimensional arithmetic groups
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X7812EB3F7AC45F87">11.19 <span class="Heading">Resolutions for finite-index subgroups
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X84CAAA697FAC8E0D">11.20 <span class="Heading">Simplifying resolutions
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X780C3F038148A1C7">11.21 <span class="Heading">Resolutions for graphs of groups and for groups with aspherical presentations
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X85AB973F8566690A">11.22 <span class="Heading">Resolutions for <span class="SimpleMath">FG</span>-modules
</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap12.html#X7D818E5F80F4CF63">12 <span class="Heading">Simplicial groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X808C6B357F8BADC1">12.1 <span class="Heading">Crossed modules</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X795E339978B42775">12.2 <span class="Heading">Eilenberg-MacLane spaces as simplicial groups (not recommended)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7D91E64D7DD7F10F">12.3 <span class="Heading">Eilenberg-MacLane spaces as simplicial free abelian groups (recommended)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X84ABCA497C577132">12.4 <span class="Heading">Elementary theoretical information on  
<span class="SimpleMath">H^∗(K(π,n), Z)</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7F828D8D8463CC20">12.5 <span class="Heading">The first three non-trivial homotopy groups of spheres</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X81E2F80384ADF8C2">12.6 <span class="Heading">The first two non-trivial homotopy groups of the suspension and double suspension of a <span class="SimpleMath">K(G,1)</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X83EAC40A8324571F">12.7 <span class="Heading">Postnikov towers and <span class="SimpleMath">π_5(S^3)</span> </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X8227000D83B9A17F">12.8 <span class="Heading">Towards <span class="SimpleMath">π_4(Σ K(G,1))</span> </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7F5E6C067B2AE17A">12.9 <span class="Heading">Enumerating homotopy 2-types</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7D99B7AA780D8209">12.10 <span class="Heading">Identifying cat<span class="SimpleMath">^1</span>-groups of low order</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7F386CF078CB9A20">12.11 <span class="Heading">Identifying crossed modules of low order</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap13.html#X86D5DB887ACB1661">13 <span class="Heading">Congruence Subgroups, Cuspidal Cohomology  and Hecke Operators</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X79A1974B7B4987DE">13.1 <span class="Heading">Eichler-Shimura isomorphism</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7BFA2C91868255D9">13.2 <span class="Heading">Generators for <span class="SimpleMath">SL_2( Z)</span> and the cubic tree</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7D1A56967A073A8B">13.3 <span class="Heading">One-dimensional fundamental domains and  
generators for congruence subgroups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X818BFA9A826C0DB3">13.4 <span class="Heading">Cohomology of congruence subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F55F8EA82FE9122">13.4-1 <span class="Heading">Cohomology with rational coefficients</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X84D30F1580CD42D1">13.5 <span class="Heading">Cuspidal cohomology</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X80861D3F87C29C43">13.6 <span class="Heading">Hecke operators on forms of weight 2</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X831BB0897B988DA3">13.7 <span class="Heading">Hecke operators on forms of weight <span class="SimpleMath">≥ 2</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X84CC51EE8525E0D9">13.8 <span class="Heading">Reconstructing modular forms from cohomology computations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X8180E53C834301EF">13.9 <span class="Heading">The Picard group</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X858B1B5D8506FE81">13.10 <span class="Heading">Bianchi groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X851390E07C3B3BB1">13.11 <span class="Heading">(Co)homology of Bianchi groups and <span class="SimpleMath">SL_2(cal O_-d)</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X86A6858884B9C05B">13.12 <span class="Heading">Some other infinite matrix groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7EF5D97281EB66DA">13.13 <span class="Heading">Ideals and finite quotient groups</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7D1F72287F14C5E1">13.14 <span class="Heading">Congruence subgroups for ideals</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X85E912617AFE03F4">13.15 <span class="Heading">First homology</span></a>
</span>
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</div>
<div class="ContChap"><a href="chap14.html#X805848868005D528">14 <span class="Heading">Fundamental domains for Bianchi groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X858B1B5D8506FE81">14.1 <span class="Heading">Bianchi groups</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X872D22507F797001">14.2 <span class="Heading">Swan's description of a fundamental domain
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7B9DE54F7ECB7E44">14.3 <span class="Heading">Computing a fundamental domain</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7A489A5D79DA9E5C">14.4 <span class="Heading">Examples</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X86CD59CB7A04EE5A">14.5 <span class="Heading">Establishing correctness of a fundamental domain</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X78476F127B73BBD1">14.6 <span class="Heading">Computing a free resolution for <span class="SimpleMath">SL_2(mathcal O_-d)</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X784B2156823AEB15">14.7 <span class="Heading">Some sanity checks</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7E5A36D47F9D4A47">14.7-1 <span class="Heading">Equivariant Euler characteristic</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X852CDAFF84C5DF01">14.7-2 <span class="Heading">Boundary squares to zero</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7E64819A7C058EDD">14.7-3 <span class="Heading">Compare different algorithms or implementations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8223864085412705">14.7-4 <span class="Heading">Compare geometry to algebra</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X78BC9D077956089A">14.8 <span class="Heading">Group presentations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X786CFAA17C0A6E7A">14.9 <span class="Heading">Finite index subgroups</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap15.html#X7F571E8F7BBC7514">15 <span class="Heading">Parallel computation</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7EAE286B837D27BA">15.1 <span class="Heading">An embarassingly parallel computation</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X80F359DD7C54D405">15.2 <span class="Heading">A non-embarassingly parallel computation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X8496786F7FCEC24A">15.3 <span class="Heading">Parallel persistent homology</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap16.html#X7C57D4AB8232983E">16 <span class="Heading">Regular CW-structure on knots (written by Kelvin Killeen)</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X86F56A85848347FF">16.1 <span class="Heading">Knot complements in the 3-ball</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X83EA2A38801E7A4C">16.2 <span class="Heading">Tubular neighbourhoods</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X78C28038837300BD">16.3 <span class="Heading">Knotted surface complements in the 4-ball</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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