  
  
                                 [1XA HAP tutorial[101X
  
  
   [1X(See also an older tutorial ([7X../www/SideLinks/About/aboutContents.html[107X) or
                  mini-course notes ([7Xcomp.pdf[107X) or related book
  ([7Xhttps://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980[107X))
                 The [12XHAP[112X home page is here ([7X../www/index.html[107X)[101X
  
  
                                  Graham Ellis
  
  
  
  
  -------------------------------------------------------
  
  
  [1XContents (HAP commands)[101X
  
  1 [33X[0;0YSimplicial complexes & CW complexes[133X
    1.1 [33X[0;0YThe Klein bottle as a simplicial complex[133X
    1.2 [33X[0;0YOther simplicial surfaces[133X
    1.3 [33X[0;0YThe Quillen complex[133X
    1.4 [33X[0;0YThe Quillen complex as a reduced CW-complex[133X
    1.5 [33X[0;0YSimple homotopy equivalences[133X
    1.6 [33X[0;0YCellular simplifications preserving homeomorphism type[133X
    1.7 [33X[0;0YConstructing a CW-structure on a knot complement[133X
    1.8 [33X[0;0YConstructing a regular CW-complex by attaching cells[133X
    1.9 [33X[0;0YConstructing a regular CW-complex from its face lattice[133X
    1.10 [33X[0;0YCup products[133X
    1.11 [33X[0;0YIntersection forms of [22X4[122X-manifolds[133X
    1.12 [33X[0;0YCohomology Rings[133X
    1.13 [33X[0;0YBockstein homomorphism[133X
    1.14 [33X[0;0YDiagonal maps on associahedra and other polytopes[133X
    1.15 [33X[0;0YCW maps and induced homomorphisms[133X
    1.16 [33X[0;0YConstructing a simplicial complex from a regular CW-complex[133X
    1.17 [33X[0;0YSome limitations to representing spaces as regular CW complexes[133X
    1.18 [33X[0;0YEquivariant CW complexes[133X
    1.19 [33X[0;0YOrbifolds and classifying spaces[133X
  2 [33X[0;0YCubical complexes & permutahedral complexes[133X
    2.1 [33X[0;0YCubical complexes[133X
    2.2 [33X[0;0YPermutahedral complexes[133X
    2.3 [33X[0;0YConstructing pure cubical and permutahedral complexes[133X
    2.4 [33X[0;0YComputations in dynamical systems[133X
  3 [33X[0;0YCovering spaces[133X
    3.1 [33X[0;0YCellular chains on the universal cover[133X
    3.2 [33X[0;0YSpun knots and the Satoh tube map[133X
    3.3 [33X[0;0YCohomology with local coefficients[133X
    3.4 [33X[0;0YDistinguishing between two non-homeomorphic homotopy equivalent spaces[133X
    3.5 [33X[0;0YSecond homotopy groups of spaces with finite fundamental group[133X
    3.6 [33X[0;0YThird homotopy groups of simply connected spaces[133X
      3.6-1 [33X[0;0YFirst example: Whitehead's certain exact sequence[133X
      3.6-2 [33X[0;0YSecond example: the Hopf invariant[133X
    3.7 [33X[0;0YComputing the second homotopy group of a space with infinite
    fundamental group[133X
  4 [33X[0;0YThree Manifolds[133X
    4.1 [33X[0;0YDehn Surgery[133X
    4.2 [33X[0;0YConnected Sums[133X
    4.3 [33X[0;0YDijkgraaf-Witten Invariant[133X
    4.4 [33X[0;0YCohomology rings[133X
    4.5 [33X[0;0YLinking Form[133X
    4.6 [33X[0;0YDetermining the homeomorphism type of a lens space[133X
    4.7 [33X[0;0YSurgeries on distinct knots can yield homeomorphic manifolds[133X
    4.8 [33X[0;0YFinite fundamental groups of [22X3[122X-manifolds[133X
    4.9 [33X[0;0YPoincare's cube manifolds[133X
    4.10 [33X[0;0YThere are at least 25 distinct cube manifolds[133X
      4.10-1 [33X[0;0YFace pairings for 25 distinct cube manifolds[133X
      4.10-2 [33X[0;0YPlatonic cube manifolds[133X
    4.11 [33X[0;0YThere are at most 41 distinct cube manifolds[133X
    4.12 [33X[0;0YThere are precisely 18 orientable cube manifolds, of which 9 are
    spherical and 5 are euclidean[133X
    4.13 [33X[0;0YCube manifolds with boundary[133X
    4.14 [33X[0;0YOctahedral manifolds[133X
    4.15 [33X[0;0YDodecahedral manifolds[133X
    4.16 [33X[0;0YPrism manifolds[133X
    4.17 [33X[0;0YBipyramid manifolds[133X
  5 [33X[0;0YTopological data analysis[133X
    5.1 [33X[0;0YPersistent homology[133X
      5.1-1 [33X[0;0YBackground to the data[133X
    5.2 [33X[0;0YMapper clustering[133X
      5.2-1 [33X[0;0YBackground to the data[133X
    5.3 [33X[0;0YSome tools for handling pure complexes[133X
    5.4 [33X[0;0YDigital image analysis and persistent homology[133X
      5.4-1 [33X[0;0YNaive example of image segmentation by automatic thresholding[133X
      5.4-2 [33X[0;0YRefining the filtration[133X
      5.4-3 [33X[0;0YBackground to the data[133X
    5.5 [33X[0;0YA second example of digital image segmentation[133X
    5.6 [33X[0;0YA third example of digital image segmentation[133X
    5.7 [33X[0;0YNaive example of digital image contour extraction[133X
    5.8 [33X[0;0YAlternative approaches to computing persistent homology[133X
      5.8-1 [33X[0;0YNon-trivial cup product[133X
      5.8-2 [33X[0;0YExplicit homology generators[133X
    5.9 [33X[0;0YKnotted proteins[133X
    5.10 [33X[0;0YRandom simplicial complexes[133X
    5.11 [33X[0;0YComputing homology of a clique complex (Vietoris-Rips complex)[133X
  6 [33X[0;0YGroup theoretic computations[133X
    6.1 [33X[0;0YThird homotopy group of a supsension of an Eilenberg-MacLane space[133X
    6.2 [33X[0;0YRepresentations of knot quandles[133X
    6.3 [33X[0;0YIdentifying knots[133X
    6.4 [33X[0;0YAspherical [22X2[122X-complexes[133X
    6.5 [33X[0;0YGroup presentations and homotopical syzygies[133X
    6.6 [33X[0;0YBogomolov multiplier[133X
    6.7 [33X[0;0YSecond group cohomology and group extensions[133X
    6.8 [33X[0;0YCocyclic groups: a convenient way of representing certain groups[133X
    6.9 [33X[0;0YEffective group presentations[133X
    6.10 [33X[0;0YSecond group cohomology and cocyclic Hadamard matrices[133X
    6.11 [33X[0;0YThird group cohomology and homotopy [22X2[122X-types[133X
  7 [33X[0;0YCohomology of groups (and Lie Algebras)[133X
    7.1 [33X[0;0YFinite groups[133X
      7.1-1 [33X[0;0YNaive homology computation for a very small group[133X
      7.1-2 [33X[0;0YA more efficient homology computation[133X
      7.1-3 [33X[0;0YComputation of an induced homology homomorphism[133X
      7.1-4 [33X[0;0YSome other finite group homology computations[133X
    7.2 [33X[0;0YNilpotent groups[133X
    7.3 [33X[0;0YCrystallographic and Almost Crystallographic groups[133X
    7.4 [33X[0;0YArithmetic groups[133X
    7.5 [33X[0;0YArtin groups[133X
    7.6 [33X[0;0YGraphs of groups[133X
    7.7 [33X[0;0YLie algebra homology and free nilpotent groups[133X
    7.8 [33X[0;0YCohomology with coefficients in a module[133X
    7.9 [33X[0;0YCohomology as a functor of the first variable[133X
    7.10 [33X[0;0YCohomology as a functor of the second variable and the long exact
    coefficient sequence[133X
    7.11 [33X[0;0YTransfer Homomorphism[133X
    7.12 [33X[0;0YCohomology rings of finite fundamental groups of 3-manifolds[133X
    7.13 [33X[0;0YExplicit cocycles[133X
    7.14 [33X[0;0YQuillen's complex and the [22Xp[122X-part of homology[133X
    7.15 [33X[0;0YHomology of a Lie algebra[133X
    7.16 [33X[0;0YCovers of Lie algebras[133X
      7.16-1 [33X[0;0YComputing a cover[133X
  8 [33X[0;0YCohomology rings and Steenrod operations for groups[133X
    8.1 [33X[0;0YMod-[22Xp[122X cohomology rings of finite groups[133X
      8.1-1 [33X[0;0YRing presentations (for the commutative [22Xp=2[122X case)[133X
    8.2 [33X[0;0YPoincare Series for Mod-[22Xp[122X cohomology[133X
    8.3 [33X[0;0YFunctorial ring homomorphisms in Mod-[22Xp[122X cohomology[133X
      8.3-1 [33X[0;0YTesting homomorphism properties[133X
      8.3-2 [33X[0;0YTesting functorial properties[133X
      8.3-3 [33X[0;0YComputing with larger groups[133X
    8.4 [33X[0;0YSteenrod operations for finite [22X2[122X-groups[133X
    8.5 [33X[0;0YSteenrod operations on the classifying space of a finite [22Xp[122X-group[133X
    8.6 [33X[0;0YMod-[22Xp[122X cohomology rings of crystallographic groups[133X
      8.6-1 [33X[0;0YPoincare series for crystallographic groups[133X
      8.6-2 [33X[0;0YMod [22X2[122X cohomology rings of [22X3[122X-dimensional crystallographic groups[133X
  9 [33X[0;0YBredon homology[133X
    9.1 [33X[0;0YDavis complex[133X
    9.2 [33X[0;0YArithmetic groups[133X
    9.3 [33X[0;0YCrystallographic groups[133X
  10 [33X[0;0YChain Complexes[133X
    10.1 [33X[0;0YChain complex of a simplicial complex and simplicial pair[133X
    10.2 [33X[0;0YChain complex of a cubical complex and cubical pair[133X
    10.3 [33X[0;0YChain complex of a regular CW-complex[133X
    10.4 [33X[0;0YChain Maps of simplicial and regular CW maps[133X
    10.5 [33X[0;0YConstructions for chain complexes[133X
    10.6 [33X[0;0YFiltered chain complexes[133X
    10.7 [33X[0;0YSparse chain complexes[133X
  11 [33X[0;0YResolutions[133X
    11.1 [33X[0;0YResolutions for small finite groups[133X
    11.2 [33X[0;0YResolutions for very small finite groups[133X
    11.3 [33X[0;0YResolutions for finite groups acting on orbit polytopes[133X
    11.4 [33X[0;0YMinimal resolutions for finite [22Xp[122X-groups over [22XF_p[122X[133X
    11.5 [33X[0;0YResolutions for abelian groups[133X
    11.6 [33X[0;0YResolutions for nilpotent groups[133X
    11.7 [33X[0;0YResolutions for groups with subnormal series[133X
    11.8 [33X[0;0YResolutions for groups with normal series[133X
    11.9 [33X[0;0YResolutions for polycyclic (almost) crystallographic groups[133X
    11.10 [33X[0;0YResolutions for Bieberbach groups[133X
    11.11 [33X[0;0YResolutions for arbitrary crystallographic groups[133X
    11.12 [33X[0;0YResolutions for crystallographic groups admitting cubical
    fundamental domain[133X
    11.13 [33X[0;0YResolutions for Coxeter groups[133X
    11.14 [33X[0;0YResolutions for Artin groups[133X
    11.15 [33X[0;0YResolutions for [22XG=SL_2( Z[1/m])[122X[133X
    11.16 [33X[0;0YResolutions for selected groups [22XG=SL_2( mathcal O( Q(sqrtd) )[122X[133X
    11.17 [33X[0;0YResolutions for selected groups [22XG=PSL_2( mathcal O( Q(sqrtd) )[122X[133X
    11.18 [33X[0;0YResolutions for a few higher-dimensional arithmetic groups[133X
    11.19 [33X[0;0YResolutions for finite-index subgroups[133X
    11.20 [33X[0;0YSimplifying resolutions[133X
    11.21 [33X[0;0YResolutions for graphs of groups and for groups with aspherical
    presentations[133X
    11.22 [33X[0;0YResolutions for [22XFG[122X-modules[133X
  12 [33X[0;0YSimplicial groups[133X
    12.1 [33X[0;0YCrossed modules[133X
    12.2 [33X[0;0YEilenberg-MacLane spaces as simplicial groups (not recommended)[133X
    12.3 [33X[0;0YEilenberg-MacLane spaces as simplicial free abelian groups
    (recommended)[133X
    12.4 [33X[0;0YElementary theoretical information on [22XH^∗(K(π,n), Z)[122X[133X
    12.5 [33X[0;0YThe first three non-trivial homotopy groups of spheres[133X
    12.6 [33X[0;0YThe first two non-trivial homotopy groups of the suspension and
    double suspension of a [22XK(G,1)[122X[133X
    12.7 [33X[0;0YPostnikov towers and [22Xπ_5(S^3)[122X[133X
    12.8 [33X[0;0YTowards [22Xπ_4(Σ K(G,1))[122X[133X
    12.9 [33X[0;0YEnumerating homotopy 2-types[133X
    12.10 [33X[0;0YIdentifying cat[22X^1[122X-groups of low order[133X
    12.11 [33X[0;0YIdentifying crossed modules of low order[133X
  13 [33X[0;0YCongruence Subgroups, Cuspidal Cohomology and Hecke Operators[133X
    13.1 [33X[0;0YEichler-Shimura isomorphism[133X
    13.2 [33X[0;0YGenerators for [22XSL_2( Z)[122X and the cubic tree[133X
    13.3 [33X[0;0YOne-dimensional fundamental domains and generators for congruence
    subgroups[133X
    13.4 [33X[0;0YCohomology of congruence subgroups[133X
      13.4-1 [33X[0;0YCohomology with rational coefficients[133X
    13.5 [33X[0;0YCuspidal cohomology[133X
    13.6 [33X[0;0YHecke operators on forms of weight 2[133X
    13.7 [33X[0;0YHecke operators on forms of weight [22X≥ 2[122X[133X
    13.8 [33X[0;0YReconstructing modular forms from cohomology computations[133X
    13.9 [33X[0;0YThe Picard group[133X
    13.10 [33X[0;0YBianchi groups[133X
    13.11 [33X[0;0Y(Co)homology of Bianchi groups and [22XSL_2(cal O_-d)[122X[133X
    13.12 [33X[0;0YSome other infinite matrix groups[133X
    13.13 [33X[0;0YIdeals and finite quotient groups[133X
    13.14 [33X[0;0YCongruence subgroups for ideals[133X
    13.15 [33X[0;0YFirst homology[133X
  14 [33X[0;0YFundamental domains for Bianchi groups[133X
    14.1 [33X[0;0YBianchi groups[133X
    14.2 [33X[0;0YSwan's description of a fundamental domain[133X
    14.3 [33X[0;0YComputing a fundamental domain[133X
    14.4 [33X[0;0YExamples[133X
    14.5 [33X[0;0YEstablishing correctness of a fundamental domain[133X
    14.6 [33X[0;0YComputing quotient [22X2[122X-complexes[133X
    14.7 [33X[0;0YComputing a free resolution for [22XSL_2(mathcal O_-d)[122X[133X
    14.8 [33X[0;0YSome sanity checks[133X
      14.8-1 [33X[0;0YEquivariant Euler characteristic[133X
      14.8-2 [33X[0;0YBoundary squares to zero[133X
      14.8-3 [33X[0;0YCompare different algorithms or implementations[133X
      14.8-4 [33X[0;0YCompare geometry to algebra[133X
    14.9 [33X[0;0YGroup presentations[133X
      14.9-1 [33X[0;0YSwan's generators[133X
    14.10 [33X[0;0YFinite index subgroups[133X
    14.11 [33X[0;0YTotally real quadratic fields / Hilbert modular group[133X
    14.12 [33X[0;0YCalling Magma's Voronoi algorithm directly[133X
  15 [33X[0;0YParallel computation[133X
    15.1 [33X[0;0YAn embarassingly parallel computation[133X
    15.2 [33X[0;0YA non-embarassingly parallel computation[133X
    15.3 [33X[0;0YParallel persistent homology[133X
  16 [33X[0;0YRegular CW-structure on knots (written by Kelvin Killeen)[133X
    16.1 [33X[0;0YKnot complements in the 3-ball[133X
    16.2 [33X[0;0YTubular neighbourhoods[133X
    16.3 [33X[0;0YKnotted surface complements in the 4-ball[133X
  
  
  [32X
