Synopses & Reviews
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.
Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics. Lucid coverage of vector fields, surfaces, homology of complexes, much more. Some knowledge of differential equations and multivariate calculus required. Many problems and exercises (some solutions) integrated into the text. 1979 edition. Bibliography.
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.
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.
Includes bibliographical references (p. -304) and index.
Table of Contents
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
24 Homology Groups of a Complex
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
36 The Lefschetz Fixed Point Theorem
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