Topology Illustrated

This book initially follows a two-semester first course in topology with emphasis on algebraic 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. 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