Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0003
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples ...
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This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: integral homology of finite groups such as the Mathieu groups, homology of crystallographic groups, homology of nilpotent groups, homology of Coxeter groups, transfer homomorphism, homological perturbation theory, mod-p comology rings of small finite p-groups, Lyndon-Hocshild-Serre spectral sequence, Bokstein operation, Steenrod squares, Stiefel-Whitney classes, Lie algebras, the modular isomorphism problem, and Bredon homology.Less
This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: integral homology of finite groups such as the Mathieu groups, homology of crystallographic groups, homology of nilpotent groups, homology of Coxeter groups, transfer homomorphism, homological perturbation theory, mod-p comology rings of small finite p-groups, Lyndon-Hocshild-Serre spectral sequence, Bokstein operation, Steenrod squares, Stiefel-Whitney classes, Lie algebras, the modular isomorphism problem, and Bredon homology.
Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.001.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This book is an introduction to elementary algebraic topology for students with an interest in computers and computer programming. Its aim is to illustrate how the basics of the subject can be ...
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This book is an introduction to elementary algebraic topology for students with an interest in computers and computer programming. Its aim is to illustrate how the basics of the subject can be implemented on a computer. The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues and it is hoped that the treatment of these will also appeal to readers already familiar with basic theory who are interested in developing computational aspects. The book covers a subset of standard introductory material on fundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules, homotopy 2- types and explicit resolutions for an eclectic selection of discrete groups. It attempts to cover these topics in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of pure algebraic topology itself. The initial chapters include examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approach to the subject. The applied examples in the initial chapters use only low-dimensional and mainly abelian topological tools. Our applications of higher dimensional and less abelian computational methods are currently confined to pure mathematical calculations. The approach taken to computational homotopy is very much based on J.H.C. Whitehead’s theory of combinatorial homotopy in which he introduced the fundamental notions of CW-space, simple homotopy equivalence and crossed module. The book should serve as a self-contained informal introduction to these topics and their computer implementation. It is written in a style that tries to lead as quickly as possible to a range of potentially useful machine computations.Less
This book is an introduction to elementary algebraic topology for students with an interest in computers and computer programming. Its aim is to illustrate how the basics of the subject can be implemented on a computer. The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues and it is hoped that the treatment of these will also appeal to readers already familiar with basic theory who are interested in developing computational aspects. The book covers a subset of standard introductory material on fundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules, homotopy 2- types and explicit resolutions for an eclectic selection of discrete groups. It attempts to cover these topics in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of pure algebraic topology itself. The initial chapters include examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approach to the subject. The applied examples in the initial chapters use only low-dimensional and mainly abelian topological tools. Our applications of higher dimensional and less abelian computational methods are currently confined to pure mathematical calculations. The approach taken to computational homotopy is very much based on J.H.C. Whitehead’s theory of combinatorial homotopy in which he introduced the fundamental notions of CW-space, simple homotopy equivalence and crossed module. The book should serve as a self-contained informal introduction to these topics and their computer implementation. It is written in a style that tries to lead as quickly as possible to a range of potentially useful machine computations.
Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0002
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This chapter introduces more basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: chain complex, chain mapping, ...
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This chapter introduces more basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: chain complex, chain mapping, chain homotopy, homology of a (simplicial or cubical or permutahedral or CW-) space, persistent homology of a filtered space, cohomology ring of a space, van Kampen diagrams, excision. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.Less
This chapter introduces more basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: chain complex, chain mapping, chain homotopy, homology of a (simplicial or cubical or permutahedral or CW-) space, persistent homology of a filtered space, cohomology ring of a space, van Kampen diagrams, excision. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.
Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0006
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical ...
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This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical groups, graphs of groups, special linear groups, triangle groups, generalized triangle groups, Coxeter groups, Artin groups, and arithmetic groups.Less
This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical groups, graphs of groups, special linear groups, triangle groups, generalized triangle groups, Coxeter groups, Artin groups, and arithmetic groups.
Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, ...
More
This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, non-regular CW-complex, simplicial complex, cubical complex, permutahedral complex, simple homotopy, set of path-components, fundamental group, van Kampen’s theorem, knot quandle, Alexander polynomial of a knot, covering space. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.Less
This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, non-regular CW-complex, simplicial complex, cubical complex, permutahedral complex, simple homotopy, set of path-components, fundamental group, van Kampen’s theorem, knot quandle, Alexander polynomial of a knot, covering space. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.
Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0004
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer ...
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This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: explicit cocycles, classification of abelian and nonabelian group extensions, crossed modules, crossed extensions, five-term exact sequences, Hopf’s formula, Bogomolov multipliers, relative central extensions, nonabelian tensor products of groups, and cocyclic Hadamard matrices.Less
This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: explicit cocycles, classification of abelian and nonabelian group extensions, crossed modules, crossed extensions, five-term exact sequences, Hopf’s formula, Bogomolov multipliers, relative central extensions, nonabelian tensor products of groups, and cocyclic Hadamard matrices.
Graham Ellis
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0005
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology
This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using ...
More
This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.Less
This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.
