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Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make the material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces and three-dimensional manifolds, fixed points and elementary homotopy theory. The fundamental concepts of point-set topology appear at the end of the book when students can see how this level of abstraction provides a sound logical basis for the geometrical ideas that have come before. This organization exposes students to the exciting world of topology now(!) rather than later. Students using this textbook should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs.

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• Preface
• Acknowledgments
• 1 Deformations
• 2 Knots and links
• 3 Surfaces
• 4 Three-dimensional manifolds
• 5 Fixed points
• 6 The fundamental group
• 7 Metric and topological spaces
• Index

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CHOICE Review

Topology is sexy mathematics, and students want to learn it as soon as they can. If one picks up a standard book on the subject, however, it appears dry, axiomatic, and full of gnarly counterexamples. Messer (Albion College) and Straffin (Beloit College) offer an alternative, expecting only multivariable calculus, linear algebra, and one proof-oriented course as prerequisite. They hope to equip students with principles through repetition, careful examples, and well-chosen proofs that will provide foundations for later explorations of topology. They focus on phenomena that are best studied through topology: knots and links, surfaces and manifolds, fixed points of mappings. This is possible because arguments using combinatorial ideas, such as skein relations, cutting and pasting, and Sperner's lemma, are all accessible if one accepts certain foundational issues as true. The final chapter lays out the basic material for completeness. Issued in the publisher's "Classroom Resource Materials" series, this book provides more than supplementary topics, it fosters the view that topology may be taught without all the heavy foundations. A valuable addition to any undergraduate library. ^BSumming Up: Highly recommended. Lower-division undergraduates through faculty. J. McCleary Vassar College