Topology Illustrated by Peter Saveliev
ISBN-13: 9781495188756
ISBN-10: 1495188752
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
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