A Combinatorial Introduction to Topology

Michael Henle

ISBN 10: 0486679667 / ISBN 13: 9780486679662

Editore: Dover Publications Inc., 2003

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Excellent text covers vector fields, plane homology and the Jordan Curve Theorem, surfaces, homology of complexes, more. Problems and exercises. Some knowledge of differential equations and multivariate calculus required. Bibliography. 1979 edition.

Codice articolo CIN0486679667G

Dati bibliografici

• Titolo: A Combinatorial Introduction to Topology
• Editore: Dover Publications Inc.
• Data di pubblicazione: 2003
• Lingua: Inglese
• ISBN 10: 0486679667
• ISBN 13: 9780486679662
• Rilegatura: Paperback
• Condizione: Good

Descrizione articolo

Riassunto

The creation of algebraic topology is a major accomplishment of 20th-century mathematics. 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.
Combinatorial topology has a wealth of applications, many of which result from connections with the theory of differential equations. As the author points out, "Combinatorial topology is uniquely the subject where students of mathematics below graduate level can see the three major divisions of mathematics &; analysis, geometry, and algebra &; working together amicably on important problems."
To facilitate understanding, Professor Henle has deliberately restricted the subject matter of this volume, focusing especially on surfaces because the theorems can be easily visualized there, encouraging geometric intuition. In addition, this area presents many interesting applications arising from systems of differential equations. To illuminate the interaction of geometry and algebra, a single important algebraic tool &; homology &; is developed in detail.

Written for upper-level undergraduate and graduate students, this book requires no previous acquaintance with topology or algebra. Point set topology and group theory are developed as they are needed. In addition, a supplement surveying point set topology is included for the interested student and for the instructor who wishes to teach a mixture of point set and algebraic topology. A rich selection of problems, some with solutions, are integrated into the text.

Contenuti

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