Topology Illustrated 1st Edition Peter Saveliev by Peter Saveliev 9781495188756, 9781087803463, 1495188752, 1087803462 instant download after payment.

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