Topology Illustrated by Peter Saveliev

Topology Illustrated by Peter Saveliev ISBN-13: 9781495188756 ISBN-10: 1495188752Algebraic 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.CONTENTSChapter 1. Cycles1. Topology around us2. Homology classes3. Topology of graphs4. Homology groups of graphs5. Maps of graphs6. Binary calculus on graphsChapter 2. Topologies1. A new look at continuity2. Neighborhoods and topologies3. Topological spaces4. Continuous functions5. SubspacesChapter 3. Complexes1. The algebra of cells2. Cubical complexes3. The algebra of oriented cells4. Simplicial complexes5. Simplicial homology6. Simplicial maps7. Parametric complexesChapter 4. Spaces1. Compacta2. Quotients3. Cell complexes4. Triangulations5. Manifolds6. ProductsChapter 5. Maps1. Homotopy2. Cell maps3. Maps of polyhedra4. The Euler and Lefschetz numbers5. Set-valued mapsChapter 6. Forms1. Discrete forms and cochains2. Calculus on cubical complexes3. Cohomology4. Metric tensorChapter 7. Flows1. Metric complexes2. ODEs3. PDEs4. Social choice