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[MathJax off]
A short HAP tutorial
(
A more comprehensive tutorial is available here
and
A related book is available here
and
The
HAP
home page is here
)
Graham Ellis
Contents
1
Simplicial complexes & CW complexes
1.1
The Klein bottle as a simplicial complex
1.2
The Quillen complex
1.3
The Quillen complex as a reduced CW-complex
1.4
Constructing a regular CW-complex from its face lattice
1.5
Cup products
1.6
CW maps and induced homomorphisms
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
3.6-2
Second example
3.7
Computing the second homotopy group of a space with infinite fundamental group
4
Topological data analysis
4.1
Persistent homology
4.1-1
Background to the data
4.2
Mapper clustering
4.2-1
Background to the data
4.3
Digital image analysis
4.3-1
Background to the data
5
Group theoretic computations
5.1
Third homotopy group of a supsension of an Eilenberg-MacLane space
5.2
Representations of knot quandles
5.3
Aspherical
\(2\)
-complexes
5.4
Bogomolov multiplier
6
Cohomology of groups
6.1
Finite groups
6.2
Nilpotent groups
6.3
Crystallographic groups
6.4
Arithmetic groups
6.5
Artin groups
6.6
Graphs of groups
7
Cohomology operations
7.1
Steenrod operations on the classifying space of a finite
\(2\)
-group
7.2
Steenrod operations on the classifying space of a finite
\(p\)
-group
8
Bredon homology
8.1
Davis complex
8.2
Arithmetic groups
8.3
Crystallographic groups
9
Simplicial groups
9.1
Crossed modules
9.2
Eilenberg-MacLane spaces as simplicial groups (not recommended)
9.3
Eilenberg-MacLane spaces as simplicial free abelian groups (recommended)
9.4
Elementary theoretical information on
\(H^\ast(K(\pi,n),\mathbb Z)\)
9.5
The first three non-trivial homotopy groups of spheres
9.6
The first two non-trivial homotopy groups of the suspension and double suspension of a
\(K(G,1)\)
9.7
Postnikov towers and
\(\pi_5(S^3)\)
10
Congruence Subgroups, Cuspidal Cohomology and Hecke Operators
10.1
Eichler-Shimura isomorphism
10.2
Generators for
\(SL_2(\mathbb Z)\)
and the cubic tree
10.3
One-dimensional fundamental domains and generators for congruence subgroups
10.4
Cohomology of congruence subgroups
10.5
Cuspidal cohomology
10.6
Hecke operators
10.7
Reconstructing modular forms from cohomology computations
10.8
The Picard group
10.9
Bianchi groups
10.10
Some other infinite matrix groups
10.11
Ideals and finite quotient groups
10.12
Congruence subgroups for ideals
10.13
First homology
11
Parallel computation
11.1
An embarassingly parallel computation
11.2
An non-embarassingly parallel computation
References
Index
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