Topology Illustrated. Volume 2

Algebraic topology is the main subject of this book that initially follows a two-semester first course in topology. In volume 2, 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. CONTENTS VOLUME 1 Chapter I. 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 II. TOPOLOGIES 1. A new look at continuity 2. Neighborhoods and topologies 3. Topological spaces 4. Continuous functions 5. Subspaces Chapter III. 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 CONTENTS VOLUME 2 Chapter IV. SPACES 1. Compacta 2. Quotients 3. Cell complexes 4. Triangulations 5. Manifolds 6. Products Chapter V. MAPS 1. Homotopy 2. Cell maps 3. Maps of polyhedra 4. The Euler and Lefschetz numbers 5. Set-valued maps Chapter VI. FORMS 1. Discrete forms and cochains 2. Calculus on cubical complexes 3. Cohomology 4. Metric tensor Chapter VII. FLOWS 1. Metric complexes 2. ODEs 3. PDEs 4. Social choice