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Homological Algebra Programming

HAP

Version 1.24

05 December 2019

Graham Ellis
Email: Graham.Ellis@nuigalway.ie

Address:
School of Mathematics
National University of Irelnd, Galway
Galway
(Ireland)

Abstract

HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.24 was released on 05 December 2019 .
The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis.
This document describes the functions available in HAP. Examples illustrating these functions are available in the HAP tutorial.

Copyright

© 2005-2019 by Graham Ellis

Acknowledgements

The HAP project has been supported by the School of Maths at NUI Galway, various PhD students and postdoctoral researchers at NUI Galway, the Irish Research Council, Science Foundation Ireland, and the EU Marie Curie programme.

Contents

1 Basic functionality for cellular complexes, fundamental groups and homology
2 Basic functionality for ZG-resolutions and group cohomology
3 Basic functionality for homological group theory
4 Basic functionality for parallel computation
5 Resolutions of the ground ring
6 Resolutions of modules
7 Induced equivariant chain maps
8 Functors
9 Chain complexes
10 Sparse Chain complexes
11 Homology and cohomology groups
12 Poincare series
13 Cohomology ring structure
14 Cohomology rings of p-groups (mainly p=2)
15 Commutator and nonabelian tensor computations
16 Lie commutators and nonabelian Lie tensors
17 Generators and relators of groups
18 Orbit polytopes and fundamental domains
19 Cocycles
20 Words in free ZG-modules
21 FpG-modules
22 Meataxe modules
23 G-Outer Groups
24 Cat-1-groups
25 Simplicial groups
26 Coxeter diagrams and graphs of groups
27 Torsion Subcomplexes
28 Simplicial Complexes
29 Cubical Complexes
 29.1  

  29.1-1 ArrayToPureCubicalComplex

  29.1-2 PureCubicalComplex

  29.1-3 FramedPureCubicalComplex

  29.1-4 RandomCubeOfPureCubicalComplex

  29.1-5 PureCubicalComplexIntersection

  29.1-6 PureCubicalComplexUnion

  29.1-7 PureCubicalComplexDifference

  29.1-8 ReadImageAsPureCubicalComplex

  29.1-9 ReadLinkImageAsPureCubicalComplex

  29.1-10 ReadImageSequenceAsPureCubicalComplex

  29.1-11 Size

  29.1-12 Dimension

  29.1-13 WritePureCubicalComplexAsImage

  29.1-14 ViewPureCubicalComplex

  29.1-15 Homology

  29.1-16 Bettinumbers

  29.1-17 DirectProductOfPureCubicalComplexes

  29.1-18 SuspensionOfPureCubicalComplex

  29.1-19 EulerCharacteristic

  29.1-20 PathComponentOfPureCubicalComplex

  29.1-21 ChainComplex

  29.1-22 ChainComplexOfPair

  29.1-23 ExcisedPureCubicalPair

  29.1-24 ChainInclusionOfPureCubicalPair

  29.1-25 ChainMapOfPureCubicalPairs

  29.1-26 ContractPureCubicalComplex

  29.1-27 ContractedComplex

  29.1-28 ZigZagContractedPureCubicalComplex

  29.1-29 ContractCubicalComplex

  29.1-30 DVFReducedCubicalComplex

  29.1-31 SkeletonOfCubicalComplex

  29.1-32 ContractibleSubomplexOfPureCubicalComplex

  29.1-33 AcyclicSubomplexOfPureCubicalComplex

  29.1-34 HomotopyEquivalentMaximalPureCubicalSubcomplex

  29.1-35 HomotopyEquivalentMinimalPureCubicalSubcomplex

  29.1-36 BoundaryOfPureCubicalComplex

  29.1-37 SingularitiesOfPureCubicalComplex

  29.1-38 ThickenedPureCubicalComplex

  29.1-39 CropPureCubicalComplex

  29.1-40 BoundingPureCubicalComplex

  29.1-41 MorseFiltration

  29.1-42 ComplementOfPureCubicalComplex

  29.1-43 PureCubicalComplexToTextFile

  29.1-44 ThickeningFiltration

  29.1-45 Dendrogram

  29.1-46 DendrogramDisplay

  29.1-47 DendrogramToPersistenceMat

  29.1-48 ReadImageAsFilteredPureCubicalComplex

  29.1-49 ComplementOfFilteredPureCubicalComplex

  29.1-50 PersistentHomologyOfFilteredPureCubicalComplex
30 Regular CW-Complexes
31 Knots and Links
32 Knots and Quandles
33 Finite metric spaces and their filtered complexes
34 Commutative diagrams and abstract categories
35 Arrays and Pseudo lists
36 Parallel Computation - Core Functions
37 Parallel Computation - Extra Functions
38 Some functions for accessing basic data
39 Miscellaneous
40 HAP variables that are not yet documented
Index

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