A short HAP tutorial
  
  
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  (../www/SideLinks/About/aboutContents.html) and A related book is available
                                      here
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                                  Graham Ellis
  
  
  
  
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  Contents (HAP commands)
  
  1 Simplicial complexes & CW complexes
    1.1 The Klein bottle as a simplicial complex
    1.2 Other simplicial surfaces
    1.3 The Quillen complex
    1.4 The Quillen complex as a reduced CW-complex
    1.5 Simple homotopy equivalences
    1.6 Cellular simplifications preserving homeomorphism type
    1.7 Constructing a CW-structure on a knot complement
    1.8 Constructing a regular CW-complex by attaching cells
    1.9 Constructing a regular CW-complex from its face lattice
    1.10 Cup products
    1.11 Cohomology Rings
    1.12 Intersection forms of 4-manifolds
    1.13 CW maps and induced homomorphisms
    1.14 Constructing a simplicial complex from a regular CW-complex
    1.15 Equivariant CW complexes
    1.16 Orbifolds and classifying spaces
  2 Cubical complexes & permutahedral complexes
    2.1 Cubical complexes
    2.2 Permutahedral complexes
    2.3 Constructing pure cubical and permutahedral complexes
    2.4 Computations in dynamical systems
  3 Covering spaces
    3.1 Cellular chains on the universal cover
    3.2 Spun knots and the Satoh tube map
    3.3 Cohomology with local coefficients
    3.4 Distinguishing between two non-homeomorphic homotopy equivalent spaces
    3.5 Second homotopy groups of spaces with finite fundamental group
    3.6 Third homotopy groups of simply connected spaces
      3.6-1 First example: Whitehead's certain exact sequence
      3.6-2 Second example: the Hopf invariant
    3.7 Computing the second homotopy group of a space with infinite
    fundamental group
  4 Three Manifolds
    4.1 Dehn Surgery
    4.2 Connected Sums
    4.3 Dijkgraaf-Witten Invariant
    4.4 Cohomology rings
    4.5 Linking Form
    4.6 Determining the homeomorphism type of a lens space
    4.7 Surgeries on distinct knots can yield homeomorphic manifolds
    4.8 Finite fundamental groups of 3-manifolds
  5 Topological data analysis
    5.1 Persistent homology
      5.1-1 Background to the data
    5.2 Mapper clustering
      5.2-1 Background to the data
    5.3 Digital image analysis
      5.3-1 Background to the data
    5.4 Random simplicial complexes
  6 Group theoretic computations
    6.1 Third homotopy group of a supsension of an Eilenberg-MacLane space
    6.2 Representations of knot quandles
    6.3 Identifying knots
    6.4 Aspherical 2-complexes
    6.5 Bogomolov multiplier
    6.6 Second group cohomology and group extensions
    6.7 Second group cohomology and cocyclic Hadamard matrices
    6.8 Third group cohomology and homotopy 2-types
  7 Cohomology of groups
    7.1 Finite groups
    7.2 Nilpotent groups
    7.3 Crystallographic and Almost Crystallographic groups
    7.4 Arithmetic groups
    7.5 Artin groups
    7.6 Graphs of groups
    7.7 Cohomology with coefficients in a module
    7.8 Cohomology as a functor of the first variable
    7.9 Cohomology as a functor of the second variable and the long exact
    coefficient sequence
    7.10 Transfer Homomorphism
    7.11 Cohomology rings of finite fundamental groups of 3-manifolds
    7.12 Explicit cocycles
  8 Cohomology operations
    8.1 Steenrod operations on the classifying space of a finite 2-group
    8.2 Steenrod operations on the classifying space of a finite p-group
  9 Bredon homology
    9.1 Davis complex
    9.2 Arithmetic groups
    9.3 Crystallographic groups
  10 Chain Complexes
    10.1 Chain complex of a simplicial complex and simplicial pair
    10.2 Chain complex of a cubical complex and cubical pair
    10.3 Chain complex of a regular CW-complex
    10.4 Chain Maps of simplicial and regular CW maps
    10.5 Constructions for chain complexes
    10.6 Filtered chain complexes
    10.7 Sparse chain complexes
  11 Resolutions
    11.1 Resolutions for small finite groups
    11.2 Resolutions for very small finite groups
    11.3 Resolutions for finite groups acting on orbit polytopes
    11.4 Minimal resolutions for finite p-groups over F_p
    11.5 Resolutions for abelian groups
    11.6 Resolutions for nilpotent groups
    11.7 Resolutions for groups with subnormal series
    11.8 Resolutions for groups with normal series
    11.9 Resolutions for polycyclic (almost) crystallographic groups
    11.10 Resolutions for Bieberbach groups
    11.11 Resolutions for arbitrary crystallographic groups
    11.12 Resolutions for crystallographic groups admitting cubical
    fundamental domain
    11.13 Resolutions for Coxeter groups
    11.14 Resolutions for Artin groups
    11.15 Resolutions for G=SL_2( Z[1/m])
    11.16 Resolutions for selected groups G=SL_2( mathcal O( Q(sqrtd) )
    11.17 Resolutions for selected groups G=PSL_2( mathcal O( Q(sqrtd) )
    11.18 Resolutions for a few higher-dimensional arithmetic groups
    11.19 Resolutions for finite-index subgroups
    11.20 Simplifying resolutions
    11.21 Resolutions for graphs of groups and for groups with aspherical
    presentations
    11.22 Resolutions for FG-modules
  12 Simplicial groups
    12.1 Crossed modules
    12.2 Eilenberg-MacLane spaces as simplicial groups (not recommended)
    12.3 Eilenberg-MacLane spaces as simplicial free abelian groups
    (recommended)
    12.4 Elementary theoretical information on H^∗(K(π,n), Z)
    12.5 The first three non-trivial homotopy groups of spheres
    12.6 The first two non-trivial homotopy groups of the suspension and
    double suspension of a K(G,1)
    12.7 Postnikov towers and π_5(S^3)
    12.8 Towards π_4(Σ K(G,1))
    12.9 Enumerating homotopy 2-types
    12.10 Identifying cat^1-groups of low order
    12.11 Identifying crossed modules of low order
  13 Congruence Subgroups, Cuspidal Cohomology and Hecke Operators
    13.1 Eichler-Shimura isomorphism
    13.2 Generators for SL_2( Z) and the cubic tree
    13.3 One-dimensional fundamental domains and generators for congruence
    subgroups
    13.4 Cohomology of congruence subgroups
      13.4-1 Cohomology with rational coefficients
    13.5 Cuspidal cohomology
    13.6 Hecke operators on forms of weight 2
    13.7 Hecke operators on forms of weight ge 2
    13.8 Reconstructing modular forms from cohomology computations
    13.9 The Picard group
    13.10 Bianchi groups
    13.11 Some other infinite matrix groups
    13.12 Ideals and finite quotient groups
    13.13 Congruence subgroups for ideals
    13.14 First homology
  14 Parallel computation
    14.1 An embarassingly parallel computation
    14.2 An non-embarassingly parallel computation
  15 Regular CW-structure on knots
    15.1 Knot complements in the 3-ball
    15.2 Tubular neighbourhoods
    15.3 Knotted surface complements in the 4-ball