A Combinatorial Introduction to Topology

Klappentext The creation of algebraic topology is a major accomplishment of the 20th century. The goal of this book is to show how geometric and algebraic ideas met and grew together into an important branch of mathematics in the recent past. The book also conveys the fun and adventure that can be part of a mathematical investigation. Inhaltsverzeichnis Chapter One Basic Concepts1 The Combinatorial Method2 Continuous Transformations in the Plane3 Compactness and Connectedness4 Abstract Point Set TopologyChapter Two Vector Fields5 A Link Between Analysis and Topology6 Sperner's Lemma and the Brouwer Fixed Point Theorem7 Phase Portraits and the Index Lemma8 Winding Numbers9 Isolated Critical Points10 The Poincaré Index Theorem11 Closed Integral Paths12 Further Results and ApplicationsChapter Three Plane Homology and Jordan Curve Theorem13 Polygonal Chains14 The Algebra of Chains on a Grating15 The Boundary Operator16 The Fundamental Lemma17 Alexander's Lemma18 Proof of the Jordan Curve TheoremChapter Four Surfaces19 Examples of Surfaces20 The Combinatorial Definition of a Surface21 The Classification Theorem22 Surfaces with BoundaryChapter Five Homology of Complexes23 Complexes24 Homology Groups of a Complex25 Invariance26 Betti Numbers and the Euler Characteristic27 Map Coloring and Regular Complexes28 Gradient Vector Fields29 Integral Homology30 Torsion and Orientability31 The Poincaré Index Theorem AgainChapter Six Continuous Transformations32 Covering Spaces33 Simplicial Transformations34 Invariance Again35 Matrixes36 The Lefschetz Fixed Point Theorem37 Homotopy38 Other HomologiesSupplement Topics in Point Set Topology39 Cryptomorphic Versions of Topology40 A Bouquet of Topological Properties41 Compactness Again42 Compact Metric SpacesHints and Answers for Selected ProblemsSuggestions for Further ReadingBibliographyIndex

Inhalt
Chapter One Basic Concepts 1 The Combinatorial Method 2 Continuous Transformations in the Plane 3 Compactness and Connectedness 4 Abstract Point Set Topology Chapter Two Vector Fields 5 A Link Between Analysis and Topology 6 Sperner's Lemma and the Brouwer Fixed Point Theorem 7 Phase Portraits and the Index Lemma 8 Winding Numbers 9 Isolated Critical Points 10 The Poincaré Index Theorem 11 Closed Integral Paths 12 Further Results and Applications Chapter Three Plane Homology and Jordan Curve Theorem 13 Polygonal Chains 14 The Algebra of Chains on a Grating 15 The Boundary Operator 16 The Fundamental Lemma 17 Alexander's Lemma 18 Proof of the Jordan Curve Theorem Chapter Four Surfaces 19 Examples of Surfaces 20 The Combinatorial Definition of a Surface 21 The Classification Theorem 22 Surfaces with Boundary Chapter Five Homology of Complexes 23 Complexes 24 Homology Groups of a Complex 25 Invariance 26 Betti Numbers and the Euler Characteristic 27 Map Coloring and Regular Complexes 28 Gradient Vector Fields 29 Integral Homology 30 Torsion and Orientability 31 The Poincaré Index Theorem Again Chapter Six Continuous Transformations 32 Covering Spaces 33 Simplicial Transformations 34 Invariance Again 35 Matrixes 36 The Lefschetz Fixed Point Theorem 37 Homotopy 38 Other Homologies Supplement Topics in Point Set Topology 39 Cryptomorphic Versions of Topology 40 A Bouquet of Topological Properties 41 Compactness Again 42 Compact Metric Spaces Hints and Answers for Selected Problems Suggestions for Further Reading Bibliography Index

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