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Topology Illustrated. Volume 1
Paperback
Series: Topology Illustrated, Book 1
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ISBN13: 9798339106746
Publisher: Independently Published
Published: Sep 30 2024
Pages: 296
Weight: 1.52
Height: 0.62 Width: 8.50 Depth: 11.00
Language: English
Publisher: Independently Published
Published: Sep 30 2024
Pages: 296
Weight: 1.52
Height: 0.62 Width: 8.50 Depth: 11.00
Language: English
In the very first few pages, I'll show what makes topology the most interesting part of mathematics. Algebraic topology is the main subject of this book that initially follows a two-semester first course in topology. It furthermore takes the reader to more advanced parts of algebraic topology as well as some applications: the shape of the universe, configuration spaces, digital image analysis, data analysis, social choice, exchange economy. An overview of discrete calculus is also included. The book contains over 1000 color illustrations and over 1000 exercises. CONTENTS VOLUME 1
Chapter I. CYCLES 1. Topology around us 2. Homology classes 3. Topology of graphs 4. Homology groups of graphs 5. Maps of graphs 6. Binary calculus on graphs
Chapter II. TOPOLOGIES 1. A new look at continuity 2. Neighborhoods and topologies 3. Topological spaces 4. Continuous functions 5. Subspaces
Chapter III. Complexes 1. The algebra of cells 2. Cubical complexes 3. The algebra of oriented cells 4. Simplicial complexes 5. Simplicial homology 6. Simplicial maps 7. Parametric complexes CONTENTS VOLUME 2
Chapter IV. SPACES 1. Compacta 2. Quotients 3. Cell complexes 4. Triangulations 5. Manifolds 6. Products
Chapter V. MAPS 1. Homotopy 2. Cell maps 3. Maps of polyhedra 4. The Euler and Lefschetz numbers 5. Set-valued maps
Chapter VI. FORMS 1. Discrete forms and cochains 2. Calculus on cubical complexes 3. Cohomology 4. Metric tensor
Chapter VII. FLOWS 1. Metric complexes 2. ODEs 3. PDEs 4. Social choice
Chapter II. TOPOLOGIES 1. A new look at continuity 2. Neighborhoods and topologies 3. Topological spaces 4. Continuous functions 5. Subspaces
Chapter III. Complexes 1. The algebra of cells 2. Cubical complexes 3. The algebra of oriented cells 4. Simplicial complexes 5. Simplicial homology 6. Simplicial maps 7. Parametric complexes CONTENTS VOLUME 2
Chapter IV. SPACES 1. Compacta 2. Quotients 3. Cell complexes 4. Triangulations 5. Manifolds 6. Products
Chapter V. MAPS 1. Homotopy 2. Cell maps 3. Maps of polyhedra 4. The Euler and Lefschetz numbers 5. Set-valued maps
Chapter VI. FORMS 1. Discrete forms and cochains 2. Calculus on cubical complexes 3. Cohomology 4. Metric tensor
Chapter VII. FLOWS 1. Metric complexes 2. ODEs 3. PDEs 4. Social choice
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