Topology Illustrated

Topology Illustrated

<div><span>Please click "Look inside" and read Section 1.1 <span><i>Topology around us</i>.<div></div><br /><br /><div></div></span></span></div><div><span><span></span></span></div><div><span><span></span></span></div><div><span><span></span></span></div><div><span></span></div><div><span>Algebraic topology is the <span>main subject of t</span></span>his 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 <i>Calculus Illustrated. Volume 1: Precalculus</i>). 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.</div><br /><b>CONTENTS</b><br /><ul><li>Chapter 1. Cycles<ul><li> 1. Topology around us</li><li> 2. Homology classes</li><li> 3. Topology of graphs</li><li> 4. Homology groups of graphs</li><li> 5. Maps of graphs</li><li> 6. Binary calculus on graphs</li></ul></li></ul><ul><li> Chapter 2. Topologies<ul><li> 1. A new look at continuity</li><li> 2. Neighborhoods and topologies</li><li> 3. Topological spaces</li><li> 4. Continuous functions</li><li> 5. Subspaces</li></ul></li></ul><ul><li> Chapter 3. Complexes<ul><li> 1. The algebra of cells</li><li> 2. Cubical complexes</li><li> 3. The algebra of oriented cells</li><li> 4. Simplicial complexes</li><li> 5. Simplicial homology</li><li> 6. Simplicial maps</li><li> 7. Parametric complexes</li></ul></li></ul><ul><li> Chapter 4. Spaces<ul><li> 1. Compacta</li><li> 2. Quotients</li><li> 3. Cell complexes</li><li> 4. Triangulations</li><li> 5. Manifolds</li><li> 6. Products</li></ul></li></ul><ul><li> Chapter 5. Maps<ul><li> 1. Homotopy</li><li> 2. Cell maps</li><li> 3. Maps of polyhedra</li><li> 4. The Euler and Lefschetz numbers</li><li> 5. Set-valued maps</li></ul></li></ul><ul><li> Chapter 6. Forms<ul><li> 1. Discrete forms and cochains</li><li> 2. Calculus on cubical complexes</li><li> 3. Cohomology</li><li> 4. Metric tensor</li></ul></li></ul><ul><li> Chapter 7. Flows<ul><li> 1. Metric complexes</li><li> 2. ODEs</li><li> 3. PDEs</li><li> 4. Social choice</li></ul></li></ul>