# A Combinatorial Introduction to Topology

Courier Corporation, 1979 - Mathematics - 310 pages

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.

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### Contents

 2 Continuous Transformations in the Plane 11 Abstract Point Set Topology 28 6 Sperners Lemma and the Brouwer Fixed Point Theorem 36 7 Phase Portraits and the lndex Lemma 43 8 Winding Numbers 50 9 lsolated Critical Points 54 10 The Poincare lndex Theorem 60 11 Closed lntegral Paths 67
 Homology of Complexes 23 Complexes 132 24 Homology Groups of a Complex 143 25 lnvariance 153 26 Betti Numbers and the Euler Characteristic 159 27 Map Coloring and Regular Complexes 169 28 Gradient Vector Fields 177 29 lntegral Homology 185 30 Torsion and Orientability 192

 12 Further Results and Applications 73 Chapter Three Plane Homology and the Jordan Curve Theorem 13 Polygonal Chains 79 14 The Algebra of Chains on a Grating 84 15 The Boundary Operator 88 16 The Fundamental Lemma 91 17 Alexanders Lemma 97 18 Proof of the Jordan Curve Theorem 100 Chapter Five 103 Chapter Four Surfaces 19 Examples of Surfaces 104 20 The Combinatorial Definition of a Surface 116 21 The Classification Theorem 122 22 Surfaces with Boundary 129
 31 The Poincare lndex Theorem Again 200 Chapter Six Continuous Transformations 32 Covering Spaces 209 33 Simplicial Transformations 221 34 lnvariance Again 228 35 Matrixes 234 36 The Lefschetz Fixed Point Theorem 242 37 Homotopy 251 38 Other Homologies 259 Topics in Point Set Topology 265 41 Compactness Again 279 Suggestions for Further Reading 302 Copyright