\BOOKMARK [0][-]{chapter.1}{Simplicial complexes \046 CW complexes}{}% 1
\BOOKMARK [1][-]{section.1.1}{The Klein bottle as a simplicial complex}{chapter.1}% 2
\BOOKMARK [1][-]{section.1.2}{Other simplicial surfaces}{chapter.1}% 3
\BOOKMARK [1][-]{section.1.3}{The Quillen complex}{chapter.1}% 4
\BOOKMARK [1][-]{section.1.4}{The Quillen complex as a reduced CW45complex}{chapter.1}% 5
\BOOKMARK [1][-]{section.1.5}{Simple homotopy equivalences}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.6}{Cellular simplifications preserving homeomorphism type}{chapter.1}% 7
\BOOKMARK [1][-]{section.1.7}{Constructing a CW45structure on a knot complement}{chapter.1}% 8
\BOOKMARK [1][-]{section.1.8}{Constructing a regular CW45complex by attaching cells}{chapter.1}% 9
\BOOKMARK [1][-]{section.1.9}{Constructing a regular CW45complex from its face lattice}{chapter.1}% 10
\BOOKMARK [1][-]{section.1.10}{Cup products}{chapter.1}% 11
\BOOKMARK [1][-]{section.1.11}{Intersection forms of 445manifolds}{chapter.1}% 12
\BOOKMARK [1][-]{section.1.12}{Cohomology Rings}{chapter.1}% 13
\BOOKMARK [1][-]{section.1.13}{Bockstein homomorphism}{chapter.1}% 14
\BOOKMARK [1][-]{section.1.14}{Diagonal maps on associahedra and other polytopes}{chapter.1}% 15
\BOOKMARK [1][-]{section.1.15}{CW maps and induced homomorphisms}{chapter.1}% 16
\BOOKMARK [1][-]{section.1.16}{Constructing a simplicial complex from a regular CW45complex}{chapter.1}% 17
\BOOKMARK [1][-]{section.1.17}{Some limitations to representing spaces as regular CW complexes}{chapter.1}% 18
\BOOKMARK [1][-]{section.1.18}{Equivariant CW complexes}{chapter.1}% 19
\BOOKMARK [1][-]{section.1.19}{Orbifolds and classifying spaces}{chapter.1}% 20
\BOOKMARK [0][-]{chapter.2}{Cubical complexes \046 permutahedral complexes}{}% 21
\BOOKMARK [1][-]{section.2.1}{Cubical complexes}{chapter.2}% 22
\BOOKMARK [1][-]{section.2.2}{Permutahedral complexes}{chapter.2}% 23
\BOOKMARK [1][-]{section.2.3}{Constructing pure cubical and permutahedral complexes}{chapter.2}% 24
\BOOKMARK [1][-]{section.2.4}{Computations in dynamical systems}{chapter.2}% 25
\BOOKMARK [0][-]{chapter.3}{Covering spaces}{}% 26
\BOOKMARK [1][-]{section.3.1}{Cellular chains on the universal cover}{chapter.3}% 27
\BOOKMARK [1][-]{section.3.2}{Spun knots and the Satoh tube map}{chapter.3}% 28
\BOOKMARK [1][-]{section.3.3}{Cohomology with local coefficients}{chapter.3}% 29
\BOOKMARK [1][-]{section.3.4}{Distinguishing between two non45homeomorphic homotopy equivalent spaces}{chapter.3}% 30
\BOOKMARK [1][-]{section.3.5}{ Second homotopy groups of spaces with finite fundamental group}{chapter.3}% 31
\BOOKMARK [1][-]{section.3.6}{Third homotopy groups of simply connected spaces}{chapter.3}% 32
\BOOKMARK [1][-]{section.3.7}{Computing the second homotopy group of a space with infinite fundamental group}{chapter.3}% 33
\BOOKMARK [0][-]{chapter.4}{Three Manifolds}{}% 34
\BOOKMARK [1][-]{section.4.1}{Dehn Surgery}{chapter.4}% 35
\BOOKMARK [1][-]{section.4.2}{Connected Sums}{chapter.4}% 36
\BOOKMARK [1][-]{section.4.3}{Dijkgraaf45Witten Invariant}{chapter.4}% 37
\BOOKMARK [1][-]{section.4.4}{Cohomology rings}{chapter.4}% 38
\BOOKMARK [1][-]{section.4.5}{Linking Form}{chapter.4}% 39
\BOOKMARK [1][-]{section.4.6}{Determining the homeomorphism type of a lens space}{chapter.4}% 40
\BOOKMARK [1][-]{section.4.7}{Surgeries on distinct knots can yield homeomorphic manifolds}{chapter.4}% 41
\BOOKMARK [1][-]{section.4.8}{Finite fundamental groups of 345manifolds}{chapter.4}% 42
\BOOKMARK [1][-]{section.4.9}{Poincare's cube manifolds}{chapter.4}% 43
\BOOKMARK [1][-]{section.4.10}{There are at least 25 distinct cube manifolds}{chapter.4}% 44
\BOOKMARK [1][-]{section.4.11}{There are at most 41 distinct cube manifolds}{chapter.4}% 45
\BOOKMARK [1][-]{section.4.12}{There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean}{chapter.4}% 46
\BOOKMARK [1][-]{section.4.13}{Cube manifolds with boundary}{chapter.4}% 47
\BOOKMARK [1][-]{section.4.14}{Octahedral manifolds}{chapter.4}% 48
\BOOKMARK [1][-]{section.4.15}{Dodecahedral manifolds}{chapter.4}% 49
\BOOKMARK [1][-]{section.4.16}{Prism manifolds}{chapter.4}% 50
\BOOKMARK [1][-]{section.4.17}{Bipyramid manifolds}{chapter.4}% 51
\BOOKMARK [0][-]{chapter.5}{Topological data analysis}{}% 52
\BOOKMARK [1][-]{section.5.1}{Persistent homology }{chapter.5}% 53
\BOOKMARK [1][-]{section.5.2}{Mapper clustering}{chapter.5}% 54
\BOOKMARK [1][-]{section.5.3}{Some tools for handling pure complexes}{chapter.5}% 55
\BOOKMARK [1][-]{section.5.4}{Digital image analysis and persistent homology}{chapter.5}% 56
\BOOKMARK [1][-]{section.5.5}{A second example of digital image segmentation}{chapter.5}% 57
\BOOKMARK [1][-]{section.5.6}{A third example of digital image segmentation}{chapter.5}% 58
\BOOKMARK [1][-]{section.5.7}{Naive example of digital image contour extraction}{chapter.5}% 59
\BOOKMARK [1][-]{section.5.8}{Alternative approaches to computing persistent homology}{chapter.5}% 60
\BOOKMARK [1][-]{section.5.9}{Knotted proteins}{chapter.5}% 61
\BOOKMARK [1][-]{section.5.10}{Random simplicial complexes}{chapter.5}% 62
\BOOKMARK [1][-]{section.5.11}{Computing homology of a clique complex \(Vietoris45Rips complex\) }{chapter.5}% 63
\BOOKMARK [0][-]{chapter.6}{Group theoretic computations}{}% 64
\BOOKMARK [1][-]{section.6.1}{Third homotopy group of a supsension of an Eilenberg45MacLane space }{chapter.6}% 65
\BOOKMARK [1][-]{section.6.2}{Representations of knot quandles}{chapter.6}% 66
\BOOKMARK [1][-]{section.6.3}{Identifying knots}{chapter.6}% 67
\BOOKMARK [1][-]{section.6.4}{Aspherical 245complexes}{chapter.6}% 68
\BOOKMARK [1][-]{section.6.5}{Group presentations and homotopical syzygies}{chapter.6}% 69
\BOOKMARK [1][-]{section.6.6}{Bogomolov multiplier}{chapter.6}% 70
\BOOKMARK [1][-]{section.6.7}{Second group cohomology and group extensions}{chapter.6}% 71
\BOOKMARK [1][-]{section.6.8}{Cocyclic groups: a convenient way of representing certain groups}{chapter.6}% 72
\BOOKMARK [1][-]{section.6.9}{Effective group presentations}{chapter.6}% 73
\BOOKMARK [1][-]{section.6.10}{Second group cohomology and cocyclic Hadamard matrices}{chapter.6}% 74
\BOOKMARK [1][-]{section.6.11}{Third group cohomology and homotopy 245types}{chapter.6}% 75
\BOOKMARK [0][-]{chapter.7}{Cohomology of groups \(and Lie Algebras\)}{}% 76
\BOOKMARK [1][-]{section.7.1}{Finite groups }{chapter.7}% 77
\BOOKMARK [1][-]{section.7.2}{Nilpotent groups}{chapter.7}% 78
\BOOKMARK [1][-]{section.7.3}{Crystallographic and Almost Crystallographic groups}{chapter.7}% 79
\BOOKMARK [1][-]{section.7.4}{Arithmetic groups}{chapter.7}% 80
\BOOKMARK [1][-]{section.7.5}{Artin groups}{chapter.7}% 81
\BOOKMARK [1][-]{section.7.6}{Graphs of groups}{chapter.7}% 82
\BOOKMARK [1][-]{section.7.7}{Lie algebra homology and free nilpotent groups}{chapter.7}% 83
\BOOKMARK [1][-]{section.7.8}{Cohomology with coefficients in a module}{chapter.7}% 84
\BOOKMARK [1][-]{section.7.9}{Cohomology as a functor of the first variable}{chapter.7}% 85
\BOOKMARK [1][-]{section.7.10}{Cohomology as a functor of the second variable and the long exact coefficient sequence}{chapter.7}% 86
\BOOKMARK [1][-]{section.7.11}{Transfer Homomorphism}{chapter.7}% 87
\BOOKMARK [1][-]{section.7.12}{Cohomology rings of finite fundamental groups of 345manifolds }{chapter.7}% 88
\BOOKMARK [1][-]{section.7.13}{Explicit cocycles }{chapter.7}% 89
\BOOKMARK [1][-]{section.7.14}{Quillen's complex and the p45part of homology }{chapter.7}% 90
\BOOKMARK [1][-]{section.7.15}{Homology of a Lie algebra}{chapter.7}% 91
\BOOKMARK [1][-]{section.7.16}{Covers of Lie algebras}{chapter.7}% 92
\BOOKMARK [0][-]{chapter.8}{Cohomology rings and Steenrod operations for groups}{}% 93
\BOOKMARK [1][-]{section.8.1}{Mod45p cohomology rings of finite groups}{chapter.8}% 94
\BOOKMARK [1][-]{section.8.2}{Poincare Series for Mod45p cohomology}{chapter.8}% 95
\BOOKMARK [1][-]{section.8.3}{Functorial ring homomorphisms in Mod45p cohomology}{chapter.8}% 96
\BOOKMARK [1][-]{section.8.4}{Steenrod operations for finite 245groups}{chapter.8}% 97
\BOOKMARK [1][-]{section.8.5}{Steenrod operations on the classifying space of a finite p45group}{chapter.8}% 98
\BOOKMARK [1][-]{section.8.6}{Mod45p cohomology rings of crystallographic groups}{chapter.8}% 99
\BOOKMARK [0][-]{chapter.9}{Bredon homology}{}% 100
\BOOKMARK [1][-]{section.9.1}{Davis complex}{chapter.9}% 101
\BOOKMARK [1][-]{section.9.2}{Arithmetic groups}{chapter.9}% 102
\BOOKMARK [1][-]{section.9.3}{Crystallographic groups}{chapter.9}% 103
\BOOKMARK [0][-]{chapter.10}{Chain Complexes}{}% 104
\BOOKMARK [1][-]{section.10.1}{Chain complex of a simplicial complex and simplicial pair}{chapter.10}% 105
\BOOKMARK [1][-]{section.10.2}{Chain complex of a cubical complex and cubical pair}{chapter.10}% 106
\BOOKMARK [1][-]{section.10.3}{Chain complex of a regular CW45complex}{chapter.10}% 107
\BOOKMARK [1][-]{section.10.4}{Chain Maps of simplicial and regular CW maps}{chapter.10}% 108
\BOOKMARK [1][-]{section.10.5}{Constructions for chain complexes}{chapter.10}% 109
\BOOKMARK [1][-]{section.10.6}{Filtered chain complexes}{chapter.10}% 110
\BOOKMARK [1][-]{section.10.7}{Sparse chain complexes}{chapter.10}% 111
\BOOKMARK [0][-]{chapter.11}{Resolutions}{}% 112
\BOOKMARK [1][-]{section.11.1}{Resolutions for small finite groups}{chapter.11}% 113
\BOOKMARK [1][-]{section.11.2}{Resolutions for very small finite groups}{chapter.11}% 114
\BOOKMARK [1][-]{section.11.3}{Resolutions for finite groups acting on orbit polytopes}{chapter.11}% 115
\BOOKMARK [1][-]{section.11.4}{Minimal resolutions for finite p45groups over Fp}{chapter.11}% 116
\BOOKMARK [1][-]{section.11.5}{Resolutions for abelian groups}{chapter.11}% 117
\BOOKMARK [1][-]{section.11.6}{Resolutions for nilpotent groups}{chapter.11}% 118
\BOOKMARK [1][-]{section.11.7}{Resolutions for groups with subnormal series}{chapter.11}% 119
\BOOKMARK [1][-]{section.11.8}{Resolutions for groups with normal series}{chapter.11}% 120
\BOOKMARK [1][-]{section.11.9}{Resolutions for polycyclic \(almost\) crystallographic groups }{chapter.11}% 121
\BOOKMARK [1][-]{section.11.10}{Resolutions for Bieberbach groups }{chapter.11}% 122
\BOOKMARK [1][-]{section.11.11}{Resolutions for arbitrary crystallographic groups}{chapter.11}% 123
\BOOKMARK [1][-]{section.11.12}{Resolutions for crystallographic groups admitting cubical fundamental domain}{chapter.11}% 124
\BOOKMARK [1][-]{section.11.13}{Resolutions for Coxeter groups }{chapter.11}% 125
\BOOKMARK [1][-]{section.11.14}{Resolutions for Artin groups }{chapter.11}% 126
\BOOKMARK [1][-]{section.11.15}{Resolutions for G=SL2\(Z[1/m]\)}{chapter.11}% 127
\BOOKMARK [1][-]{section.11.16}{Resolutions for selected groups G=SL2\( O\(Q\(d\) \)}{chapter.11}% 128
\BOOKMARK [1][-]{section.11.17}{Resolutions for selected groups G=PSL2\( O\(Q\(d\) \)}{chapter.11}% 129
\BOOKMARK [1][-]{section.11.18}{Resolutions for a few higher45dimensional arithmetic groups }{chapter.11}% 130
\BOOKMARK [1][-]{section.11.19}{Resolutions for finite45index subgroups }{chapter.11}% 131
\BOOKMARK [1][-]{section.11.20}{Simplifying resolutions }{chapter.11}% 132
\BOOKMARK [1][-]{section.11.21}{Resolutions for graphs of groups and for groups with aspherical presentations }{chapter.11}% 133
\BOOKMARK [1][-]{section.11.22}{Resolutions for FG45modules }{chapter.11}% 134
\BOOKMARK [0][-]{chapter.12}{Simplicial groups}{}% 135
\BOOKMARK [1][-]{section.12.1}{Crossed modules}{chapter.12}% 136
\BOOKMARK [1][-]{section.12.2}{Eilenberg45MacLane spaces as simplicial groups \(not recommended\)}{chapter.12}% 137
\BOOKMARK [1][-]{section.12.3}{Eilenberg45MacLane spaces as simplicial free abelian groups \(recommended\)}{chapter.12}% 138
\BOOKMARK [1][-]{section.12.4}{Elementary theoretical information on H\(K\(,n\),Z\)}{chapter.12}% 139
\BOOKMARK [1][-]{section.12.5}{The first three non45trivial homotopy groups of spheres}{chapter.12}% 140
\BOOKMARK [1][-]{section.12.6}{The first two non45trivial homotopy groups of the suspension and double suspension of a K\(G,1\)}{chapter.12}% 141
\BOOKMARK [1][-]{section.12.7}{Postnikov towers and 5\(S3\) }{chapter.12}% 142
\BOOKMARK [1][-]{section.12.8}{Towards 4\(K\(G,1\)\) }{chapter.12}% 143
\BOOKMARK [1][-]{section.12.9}{Enumerating homotopy 245types}{chapter.12}% 144
\BOOKMARK [1][-]{section.12.10}{Identifying cat145groups of low order}{chapter.12}% 145
\BOOKMARK [1][-]{section.12.11}{Identifying crossed modules of low order}{chapter.12}% 146
\BOOKMARK [0][-]{chapter.13}{Congruence Subgroups, Cuspidal Cohomology and Hecke Operators}{}% 147
\BOOKMARK [1][-]{section.13.1}{Eichler45Shimura isomorphism}{chapter.13}% 148
\BOOKMARK [1][-]{section.13.2}{Generators for SL2\(Z\) and the cubic tree}{chapter.13}% 149
\BOOKMARK [1][-]{section.13.3}{One45dimensional fundamental domains and generators for congruence subgroups}{chapter.13}% 150
\BOOKMARK [1][-]{section.13.4}{Cohomology of congruence subgroups}{chapter.13}% 151
\BOOKMARK [1][-]{section.13.5}{Cuspidal cohomology}{chapter.13}% 152
\BOOKMARK [1][-]{section.13.6}{Hecke operators on forms of weight 2}{chapter.13}% 153
\BOOKMARK [1][-]{section.13.7}{Hecke operators on forms of weight \0402}{chapter.13}% 154
\BOOKMARK [1][-]{section.13.8}{Reconstructing modular forms from cohomology computations}{chapter.13}% 155
\BOOKMARK [1][-]{section.13.9}{The Picard group}{chapter.13}% 156
\BOOKMARK [1][-]{section.13.10}{Bianchi groups}{chapter.13}% 157
\BOOKMARK [1][-]{section.13.11}{\(Co\)homology of Bianchi groups and SL2\(O-d\)}{chapter.13}% 158
\BOOKMARK [1][-]{section.13.12}{Some other infinite matrix groups}{chapter.13}% 159
\BOOKMARK [1][-]{section.13.13}{Ideals and finite quotient groups}{chapter.13}% 160
\BOOKMARK [1][-]{section.13.14}{Congruence subgroups for ideals}{chapter.13}% 161
\BOOKMARK [1][-]{section.13.15}{First homology}{chapter.13}% 162
\BOOKMARK [0][-]{chapter.14}{Fundamental domains for Bianchi groups}{}% 163
\BOOKMARK [1][-]{section.14.1}{Bianchi groups}{chapter.14}% 164
\BOOKMARK [1][-]{section.14.2}{Swan's description of a fundamental domain}{chapter.14}% 165
\BOOKMARK [1][-]{section.14.3}{Computing a fundamental domain}{chapter.14}% 166
\BOOKMARK [1][-]{section.14.4}{Examples}{chapter.14}% 167
\BOOKMARK [1][-]{section.14.5}{Establishing correctness of a fundamental domain}{chapter.14}% 168
\BOOKMARK [1][-]{section.14.6}{Computing a free resolution for SL2\(O-d\)}{chapter.14}% 169
\BOOKMARK [1][-]{section.14.7}{Some sanity checks}{chapter.14}% 170
\BOOKMARK [1][-]{section.14.8}{Group presentations}{chapter.14}% 171
\BOOKMARK [1][-]{section.14.9}{Finite index subgroups}{chapter.14}% 172
\BOOKMARK [1][-]{section.14.10}{Totally real quadratic fields / Hilbert modular group}{chapter.14}% 173
\BOOKMARK [1][-]{section.14.11}{Calling Magma's Voronoi algorithm directly}{chapter.14}% 174
\BOOKMARK [0][-]{chapter.15}{Parallel computation}{}% 175
\BOOKMARK [1][-]{section.15.1}{An embarassingly parallel computation}{chapter.15}% 176
\BOOKMARK [1][-]{section.15.2}{A non45embarassingly parallel computation}{chapter.15}% 177
\BOOKMARK [1][-]{section.15.3}{Parallel persistent homology}{chapter.15}% 178
\BOOKMARK [0][-]{chapter.16}{Regular CW45structure on knots \(written by Kelvin Killeen\)}{}% 179
\BOOKMARK [1][-]{section.16.1}{Knot complements in the 345ball}{chapter.16}% 180
\BOOKMARK [1][-]{section.16.2}{Tubular neighbourhoods}{chapter.16}% 181
\BOOKMARK [1][-]{section.16.3}{Knotted surface complements in the 445ball}{chapter.16}% 182
\BOOKMARK [0][-]{chapter*.2}{References}{}% 183