A Combinatorial Introduction to Topology

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Courier Corporation, 1979 - Mathematics - 310 pages
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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

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About the author (1979)

Michael Henle is Professor of Mathematics and Computer Science at Oberlin College and has had two visiting appointments, at Howard University and the Massachusetts Institute of Technology, as well as two semesters teaching in London in Oberlin's own program. He is the author of two books: A Combinatorial Introduction to Topology (W. H. Freeman and Co., 1978, reissued by Dover Publications, 1994) and Modern Geometries: The Analytic Approach (Prentice-Hall, 1996). He is currently editor of The College Mathematics Journal.

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