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Explorations in Topology

Book Description

Explorations in Topology, Second Edition, provides students a rich experience with low-dimensional topology (map coloring, surfaces, and knots), enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that will help them make sense of future, more formal topology courses.

The book's innovative story-line style models the problem-solving process, presents the development of concepts in a natural way, and engages students in meaningful encounters with the material. The updated end-of-chapter investigations provide opportunities to work on many open-ended, non-routine problems and, through a modified "Moore method," to make conjectures from which theorems emerge. The revised end-of-chapter notes provide historical background to the chapter's ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides ideas for continued research.

Explorations in Topology, Second Edition, enhances upper division courses and is a valuable reference for all levels of students and researchers working in topology.



  • Students begin to solve substantial problems from the start
  • Ideas unfold through the context of a storyline, and students become actively involved
  • The text models the problem-solving process, presents the development of concepts in a natural way, and helps the reader engage with the material

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. Preface
    1. To the Reader
    2. Format of the Chapters
    3. Overview of the Chapters
    4. Success in the Course for Which the Book Is Designed
    5. Instructor’s Manual
  7. Acknowledgments
  8. 1. Acme Does Maps and Considers Coloring Them
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  9. 2. Tours
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  10. 3. Maps Data
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Scene 5
    6. Investigations, Questions, Puzzles, and More
    7. Notes
    8. References
  11. 4. Map Data and Map Coloring
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Investigations, Questions, Puzzles, and More
    5. Notes
    6. References
  12. 5. How to Color a Map with Four Colors
    1. Scene 1
    2. Investigations, Questions, Puzzles, and More
    3. Notes
    4. References
  13. 6. Doughnuts
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  14. 7. The Möbius Strip
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  15. 8. New Worlds: Klein Bottles and Other Surfaces
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Investigations, Questions, Puzzles, and More
    5. Notes
    6. References
  16. 9. Surface Sums and Euler Numbers
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  17. 10. Classification of Surfaces
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Investigations, Questions, Puzzles, and More
    5. Notes
    6. References
  18. 11. Classification (Part II), Existence and Four-Space
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  19. 12. Coloring Maps on Surfaces
    1. Investigations, Questions, Puzzles, and More
    2. Notes
    3. References
  20. 13. Knots
    1. Scene 1
    2. Scene 2
    3. Scene 3
    4. Scene 4
    5. Investigations, Questions, Puzzles, and More
    6. Notes
    7. References
  21. 14. Projects
    1. Project Ideas
    2. I Map Coloring
    3. II Surfaces
    4. III Surfaces and Maps
    5. IV Three-Space
    6. V Knots and Links
    7. VI Miscellaneous
    8. Resources
    9. References
    10. Communicating Project Results