(Ebook PDF) An Invitation to Computational Homotopy 1st Edition by Graham Ellis ISBN 9780192569417 0192569414 full chapters

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ISBN 10:0192569414

ISBN 13:9780192569417

Author: Graham Ellis

Table of Contents:

• 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

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Graham Ellis,Invitation,Computational Homotopy