Explorations in Topology

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

Upper division, junior/senior mathematics majors and for high school mathematics teachers; mathematicians/mathematics educators interested/specializing in curriculum development.

Dedication

Preface

To the Reader

Format of the Chapters

Overview of the Chapters

Success in the Course for Which the Book Is Designed

Instructor’s Manual

Acknowledgments

1. Acme Does Maps and Considers Coloring Them

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Investigations, Questions, Puzzles, and More

Notes

References

2. Tours

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Investigations, Questions, Puzzles, and More

Notes

References

3. Maps Data

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Investigations, Questions, Puzzles, and More

Notes

References

4. Map Data and Map Coloring

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Investigations, Questions, Puzzles, and More

Notes

References

5. How to Color a Map with Four Colors

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Investigations, Questions, Puzzles, and More

Notes

References

6. Doughnuts

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Investigations, Questions, Puzzles, and More

Notes

References

7. The Möbius Strip

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Investigations, Questions, Puzzles, and More

Notes

References

8. New Worlds: Klein Bottles and Other Surfaces

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Investigations, Questions, Puzzles, and More

Notes

References

9. Surface Sums and Euler Numbers

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Investigations, Questions, Puzzles, and More

Notes

References

10. Classification of Surfaces

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Investigations, Questions, Puzzles, and More

Notes

References

11. Classification (Part II), Existence and Four-Space

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Investigations, Questions, Puzzles, and More

Notes

References

12. Coloring Maps on Surfaces

Investigations, Questions, Puzzles, and More

Notes

References

13. Knots

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Investigations, Questions, Puzzles, and More

Notes

References

14. Projects

Project Ideas

I Map Coloring

II Surfaces

III Surfaces and Maps

IV Three-Space

V Knots and Links

VI Miscellaneous

Resources

References

Communicating Project Results