Explorations in Topology, 2nd Edition

Map Coloring, Surfaces and Knots

Explorations in Topology, 2nd Edition,David Gay,ISBN9780124166486






229 X 152

An innovative exploration into topology, empowering the reader with new approaches to problem solving and mathematical skills

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Key Features

  • 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


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.


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

David Gay

Affiliations and Expertise

Department of Mathematics, University of Arizona, Tucson, AZ, USA

Explorations in Topology, 2nd Edition

CHAPTER 1: ACME makes maps and considers coloring them
CHAPTER 2: ACME adds tours to its services
CHAPTER 3: ACME collects data from maps
CHAPTER 4: ACME gathers more data, proves a theorem, and returns to coloring maps
CHAPTER 5: ACME’s lawyer proves the four color conjecture
CHAPTER 6: ACME adds doughnuts to its repertoire
CHAPTER 7: ACME considers the Möbius strip
CHAPTER 8: ACME creates new worlds --- Klein bottle and other surfaces
CHAPTER 9: ACME makes order out of chaos --- surface sum and Euler numbers
CHAPTER 10: ACME classifies surfaces
CHAPTER 11: ACME encounters the fourth dimension
CHAPTER 12: ACME colors maps on surfaces --- Heawood’s estimate
CHAPTER 13: ACME gets all tied up with knots
CHAPTER 14: Where to go from here --- Projects

Quotes and reviews

"...the tasks that are asked of the reader are challenging and require clear thinking. This text could be an exiting tool for self study or a non-traditional course that is not just based on lectures."--Zentralblatt MATH,  Sep-14

"Each chapter ends with a section marked "Notes", typically about two pages long, which gives a somewhat broader perspective of the material covered in that chapter, typically placing each topic in historical context, and sometimes giving precise definitions and statements of theorems."--MAA.org, May 4, 2014

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