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Graham Ellis
Contents
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
Intersection forms of
\(4\)
-manifolds
1.12
Cohomology Rings
1.13
Bockstein homomorphism
1.14
Diagonal maps on associahedra and other polytopes
1.15
CW maps and induced homomorphisms
1.16
Constructing a simplicial complex from a regular CW-complex
1.17
Some limitations to representing spaces as regular CW complexes
1.18
Equivariant CW complexes
1.19
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
4.9
Poincare's cube manifolds
4.10
There are at least 25 distinct cube manifolds
4.10-1
Face pairings for 25 distinct cube manifolds
4.10-2
Platonic cube manifolds
4.11
There are at most 41 distinct cube manifolds
4.12
There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean
4.13
Cube manifolds with boundary
4.14
Octahedral manifolds
4.15
Dodecahedral manifolds
4.16
Prism manifolds
4.17
Bipyramid 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
Some tools for handling pure complexes
5.4
Digital image analysis and persistent homology
5.4-1
Naive example of image segmentation by automatic thresholding
5.4-2
Refining the filtration
5.4-3
Background to the data
5.5
A second example of digital image segmentation
5.6
A third example of digital image segmentation
5.7
Naive example of digital image contour extraction
5.8
Alternative approaches to computing persistent homology
5.8-1
Non-trivial cup product
5.8-2
Explicit homology generators
5.9
Knotted proteins
5.10
Random simplicial complexes
5.11
Computing homology of a clique complex (Vietoris-Rips complex)
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
Group presentations and homotopical syzygies
6.6
Bogomolov multiplier
6.7
Second group cohomology and group extensions
6.8
Second group cohomology and cocyclic Hadamard matrices
6.9
Third group cohomology and homotopy
\(2\)
-types
7
Cohomology of groups (and Lie Algebras)
7.1
Finite groups
7.1-1
Naive homology computation for a very small group
7.1-2
A more efficient homology computation
7.1-3
Computation of an induced homology homomorphism
7.1-4
Some other finite group homology computations
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
Lie algebra homology and free nilpotent groups
7.8
Cohomology with coefficients in a module
7.9
Cohomology as a functor of the first variable
7.10
Cohomology as a functor of the second variable and the long exact coefficient sequence
7.11
Transfer Homomorphism
7.12
Cohomology rings of finite fundamental groups of 3-manifolds
7.13
Explicit cocycles
7.14
Quillen's complex and the
\(p\)
-part of homology
7.15
Homology of a Lie algebra
7.16
Covers of Lie algebras
7.16-1
Computing a cover
8
Cohomology rings and Steenrod operations for groups
8.1
Mod-
\(p\)
cohomology rings of finite groups
8.1-1
Ring presentations (for the commutative
\(p=2\)
case)
8.2
Functorial ring homomorphisms in Mod-
\(p\)
cohomology
8.2-1
Testing homomorphism properties
8.2-2
Testing functorial properties
8.2-3
Computing with larger groups
8.3
Cohomology rings of finite
\(2\)
-groups
8.4
Steenrod operations for finite
\(2\)
-groups
8.5
Steenrod operations on the classifying space of a finite
\(p\)
-group
8.6
Mod-
\(p\)
cohomology rings of crystallographic groups
8.6-1
Poincare series for crystallographic groups
8.6-2
Mod
\(2\)
cohomology rings of
\(3\)
-dimensional crystallographic groups
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
\(\mathbb F_p\)
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
\(G=SL_2(\mathbb Z[1/m])\)
11.16
Resolutions for selected groups
\(G=SL_2( {\mathcal O}(\mathbb Q(\sqrt{d}) )\)
11.17
Resolutions for selected groups
\(G=PSL_2( {\mathcal O}(\mathbb Q(\sqrt{d}) )\)
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
\(\mathbb FG\)
-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
\(H^\ast(K(\pi,n),\mathbb Z)\)
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
\(\pi_5(S^3)\)
12.8
Towards
\(\pi_4(\Sigma K(G,1))\)
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
\(SL_2(\mathbb Z)\)
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
Fundamental domains for Bianchi groups
14.1
Bianchi groups
14.2
Swan's description of a fundamental domain
14.3
Computing a fundamental domain
14.4
Examples
15
Parallel computation
15.1
An embarassingly parallel computation
15.2
A non-embarassingly parallel computation
15.3
Parallel persistent homology
16
Regular CW-structure on knots (written by Kelvin Killeen)
16.1
Knot complements in the 3-ball
16.2
Tubular neighbourhoods
16.3
Knotted surface complements in the 4-ball
References
Index
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