Topology Illustrated

Topology Illustrated by Peter Saveliev

Note: Below is the draft as of 11/15/15. Because of the heavy editing done since then, some of the sections below had to be removed.

CONTENTS

Preface 1

Chapter 1. Cycles Topology around us 13 Homology classes 15 Topology of graphs 12 Homology groups of graphs 19 Maps of graphs 19 Binary calculus on graphs 11

Chapter 2. Topologies A new look at continuity 10 Neighborhoods and topologies 13 Topological spaces 15 Rules of Topology in Games 16 Topology and Betting games 19 Continuous functions 24 Subspaces 20

Chapter 3. Complexes The algebra of cells 19 Cubical complexes 19 The algebra of oriented cells 14 Simplicial complexes 19 Simplicial homology 19 Simplicial maps 22 Parametric complexes 14

Chapter 4. Spaces Compacta 12 Quotients 20 Cell complexes 24 Triangulations 16 Manifolds 29 Products 29

Chapter 5. Maps Homotopy 27 Cell maps 26 Maps of polyhedra 29 The Euler and Lefschetz numbers 18 Set-valued maps 15

Chapter 6. Forms Discrete forms and cochains 21 Calculus on cubical complexes 23 Cohomology 25 Metric tensor 27

Chapter 7. Flows Metric complexes 12 ODEs 29 PDEs 24 Social choice 26