Classical Topology and Combinatorial Group Theory

1. Front Matter

   Pages i-xii

   PDF

2. Introduction and Foundations

   • John Stillwell

   Pages 1-51

3. Complex Analysis and Surface Topology

   • John Stillwell

   Pages 53-88

4. Graphs and Free Groups

   • John Stillwell

   Pages 89-107

5. Foundations for the Fundamental Group

   • John Stillwell

   Pages 109-134

6. Fundamental Groups of Complexes

   • John Stillwell

   Pages 135-167

7. Homology Theory and Abelianization

   • John Stillwell

   Pages 169-184

8. Curves on Surfaces

   • John Stillwell

   Pages 185-215

9. Knots and Braids

   • John Stillwell

   Pages 217-240

10. Three-Dimensional Manifolds

    • John Stillwell

    Pages 241-274

11. Unsolvable Problems

    • John Stillwell

    Pages 275-306

12. Back Matter

    Pages 307-335

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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.

• Abelian group
• Group
• Group theory
• Gruppe (Math.)
• Kombinatorik
• Topologie
• Topology

• Department of Mathematics, Monash University, Clayton, Australia

  John Stillwell

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