An invitation to computational homotopy / Graham Ellis.

Format:

Book

Author/Creator:

Ellis, Graham, author.

Language:

English

Subjects (All):

Homotopy theory.

Physical Description:

1 online resource (550 pages) : illustrations

Edition:

1st ed.

Other Title:

Computational homotopy

Place of Publication:

New York : Oxford University Press, 2019.

Summary:

An Invitation to Computational Homotopy is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques.

Contents:

Cover

An Invitation to Computational Homotopy

Copright

Preface

Contents

List of Figures

List of Tables

List of Algorithms

1 Path Components and the Fundamental Group

1.1 Regular CW-spaces

1.2 Simplicial, Cubical and Permutahedral Complexes

1.3 Path Components and Persistence

1.3.1 Mapper Clustering

1.4 Simple Homotopy

1.5 Non-regular CW-spaces

1.6 The Fundamental Group

1.6.1 Seifert-van Kampen Theorem for Groupoids

1.6.2 The Wirtinger Presentation

1.7 Computing with fp Groups

1.8 Computing with fp Quandles

1.9 Covering Spaces

1.9.1 A Remark on Flat Manifolds

1.10 Cayley Graphs and Presentations

1.11 Exercises

2 Cellular Homology

2.1 Chain Complexes and Euler Integrals

2.2 Euler Characteristics and Group Presentations

2.3 Chain Maps and Homotopies

2.4 Homology over Fields

2.5 Homotopical Data Fitting

2.6 Homology over Principal Ideal Domains

2.7 Excision

2.8 Cohomology Rings

2.8.1 van Kampen Diagrams and Cup Products

2.9 Exercises

3 Cohomology of Groups

3.1 Basic Definitions and Examples

3.2 Small Finite Groups

3.3 Operations on Resolutions

3.3.1 Perturbed Actions

3.4 The Transfer Map

3.5 Finite p-groups

3.5.1 Modular Isomorphism Problem

3.6 Lie Algebras

3.7 Group Cohomology Rings

3.8 Spectral Sequences

3.9 A Test for Cohomology Ring Completion

3.9.1 Computing Kernels of Derivations

3.10 Cohomology Operations

3.10.1 Stiefel-Whitney Classes

3.11 Bredon Homology

3.12 Coxeter Groups

3.13 Exercises

4 Cohomological Group Theory

4.1 Standard Cocycles

4.2 Classification of Group Extensions

4.3 Coefficient Modules

4.4 Crossed Modules

4.5 A Five Term Exact Sequence

4.6 The Nonabelian Tensor Product

4.7 Crossed and relative group extensions

4.8 More on Relative Homology.

4.9 Exercises

5 Cohomology of Homotopy 2-types

5.1 Outline

5.2 The Fundamental Crossed Module

5.2.1 Maps from a Surface to the Projective Plane

5.3 Finite Crossed Modules

Construction of Quasi-isomorphic Representatives

Enumeration of Isomorphism Classes

Establishing Distinct Weak Equivalence Classes

5.4 Simplicial Objects

5.5 The Homological Perturbation Lemma

5.6 Homology of Simplicial Groups

5.7 Exercises

6 Explicit Classifying Spaces

6.1 Review of Constructions

Finite Groups

Groups with Subnormal Series

Crystallographic Groups

Finite Index Subgroups

6.2 Aspherical Groups

6.3 Graphs of Groups

6.3.1 One-relator Groups

6.3.2 The Group SL2(Z[1/m])

6.4 Triangle groups

6.4.1 Cyclic Central Extensions of Triangle Groups

6.4.2 Poincaré's Theorem

6.4.3 Generalized Triangle Groups

6.5 Non-positive Curvature

6.6 Coxeter Groups Revisited

6.7 Artin Groups

6.7.1 Some Cohomology Rings

6.8 Arithmetic Groups

6.9 Exercises

Appendix

A.1 Primer on Topology

A.2 Primer on Category Theory

A.3 Primer on Finitely Presented Groups and Groupoids

A.4 Homology Software

A.5 Software for Group Cohomology

A.6 Parallel Computation

A.7 Installing HAP and Related Software

Bibliography

Index.

Notes:

Includes bibliographical references (pages [507]-520) and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

0-19-256941-4

0-19-187137-0

OCLC:

1128800592