A Combinatorial Introduction to Topology

Portada
Courier Corporation, 1 de gen. 1994 - 310 pàgines

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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Continguts

Continuous Transformations in the Plane
11
4 Abstract Point Set Topology
28
6 Sperners Lemma and the Brouwer Fixed Point Theorem
36
7 Phase Portraits and the Index Lemma
43
8 Winding Numbers
48
9 Isolated Critical Points
54
10 The Poincaré Index Theorem
60
11 Closed Integral Paths
67
Homology of Complexes 23 Complexes
132
24 Homology Groups of a Complex
143
25 Invariance
153
26 Betti Numbers and the Euler Characteristic
159
27 Map Coloring and Regular Complexes
169
28 Gradient Vector Fields
176
29 Integral 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
Proof of the Jordan Curve Theorem
100
Chapter Four Surfaces
103
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 Poincaré Index Theorem Again
200
Chapter Six Continuous Transformations 32 Covering Spaces
209
33 Simplicial Transformations
221
34 Invariance Again
228
35 Matrixes
234
36 The Lefschetz Fixed Point Theorem
242
Homotopy
251
Other Homologies
259
Topics in Point Set Topology
265
Compactness Again
279
Suggestions for Further Reading
302
Copyright

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