(Ebook) Topology Illustrated by Peter Saveliev ISBN 9781495188756, 9781087803463, 1495188752, 1087803462

(Ebook) Topology Illustrated by Peter Saveliev ISBN 9781495188756, 9781087803463, 1495188752, 1087803462

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 (extended presentation in Calculus Illustrated. Volume 1: Precalculus). The book contains over 1000 color illustrations and over 1000 exercises. The spreadsheets for the simulations and other supplementary material are found at the author's website.

CONTENTS

Chapter 1. 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 2. Topologies • 1. A new look at continuity • 2. Neighborhoods and topologies • 3. Topological spaces • 4. Continuous functions • 5. Subspaces

Chapter 3. 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

Chapter 4. Spaces • 1. Compacta • 2. Quotients • 3. Cell complexes • 4. Triangulations • 5. Manifolds • 6. Products

Chapter 5. Maps • 1. Homotopy • 2. Cell maps • 3. Maps of polyhedra • 4. The Euler and Lefschetz numbers • 5. Set-valued maps

Chapter 6. Forms • 1. Discrete forms and cochains • 2. Calculus on cubical complexes • 3. Cohomology • 4. Metric tensor

Chapter 7. Flows • 1. Metric complexes • 2. ODEs • 3. PDEs • 4. Social choice