Classical Topology and Combinatorial Group Theory

Front Cover
Springer Science & Business Media, Mar 25, 1993 - Mathematics - 334 pages
0 Reviews
In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec tions to other parts of mathematics which make topology an important as well as a beautiful subject.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

01 The Fundamental Concepts and Problems of Topology
2
02 Simplicial Complexes
19
03 The Jordan Curve Theorem
26
04 Algorithms
36
05 Combinatorial Group Theory
40
Complex Analysis and Surface Topology
53
11 Riemann Surfaces
54
12 Nonorientable Surfaces
62
52 The Structure Theorem for Finitely Generated Abelian Groups
175
53 Abelianization
181
Curves on Surfaces
185
61 Dehns Algorithm
186
62 Simple Curves on Surfaces
190
63 Simplification of Simple Curves by Homeomorphisms
196
64 The Mapping Class Group of the Torus
206
Knots and Braids
217

13 The Classification Theorem for Surfaces
69
14 Covering Surfaces
80
Graphs and Free Groups
89
21 Realization of Free Groups by Graphs
90
22 Realization of Subgroups
99
Foundations for the Fundamental Group
109
31 The Fundamental Group
110
32 The Fundamental Group of the Circle
116
33 Deformation Retracts
121
34 The SeifertVan Kampen Theorem
124
35 Direct Products
132
Fundamental Groups of Complexes
135
41 Poincares Method for Computing Presentations
136
42 Examples
141
43 Surface Complexes and Subgroup Theorems
156
Homology Theory and Abelianization
169
51 Homology Theory
170
71 Dehn and Schreiers Analysis of the Torus Knot Groups
218
72 Cyclic Coverings
225
73 Braids
233
ThreeDimensional Manifolds
241
81 Open Problems in ThreeDimensional Topology
242
82 Polyhedral Schemata
248
83 Heegaard Splittings
252
84 Surgery
263
85 Branched Coverings
270
Unsolvable Problems
275
91 Computation
276
92 HNN Extensions
285
93 Unsolvable Problems in Group Theory
290
94 The Homeomorphism Problem
298
Bibliography and Chronology
307
Index
319
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page 310 - A practical method for enumerating cosets of a finite abstract group, Proc.
Page iii - I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the visualization of problems from other parts of mathematics -complex analysis (Riemann). mechanics (Poincare), and group theory (Dehn).
Page 314 - Fomenko. AT : The problem of the algorithmic discrimination of the standard three-dimensional sphere. Russian Math.

References to this book

All Book Search results »

About the author (1993)

John Stillwell is professor of mathematics at the University of San Francisco. His many books include "Mathematics and Its History" and "Roads to Infinity".

Bibliographic information