Please click "Look inside" and read Section 1.1 Topology around us.
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
