**Student Mathematical Library**

Volume: 76;
2015;
269 pp;
Softcover

MSC: Primary 55;
Secondary 57; 47; 58

**Print ISBN: 978-1-4704-2198-4
Product Code: STML/76**

List Price: $52.00

Individual Price: $41.60

**Electronic ISBN: 978-1-4704-2625-5
Product Code: STML/76.E**

List Price: $49.00

Individual Price: $39.20

#### Supplemental Materials

# Winding Around: The Winding Number in Topology, Geometry, and Analysis

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*John Roe*

The winding number is one of the most basic invariants in topology. It measures the number of times a moving point \(P\) goes around a fixed point \(Q\), provided that \(P\) travels on a path that never goes through \(Q\) and that the final position of \(P\) is the same as its starting position. This simple idea has far-reaching applications. The reader of this book will learn how the winding number can

- help us show that every polynomial equation has a root (the fundamental theorem of algebra),
- guarantee a fair division of three objects in space by a single planar cut (the ham sandwich theorem),
- explain why every simple closed curve has an inside and an outside (the Jordan curve theorem),
- relate calculus to curvature and the singularities of vector fields (the Hopf index theorem),
- allow one to subtract infinity from infinity and get a finite answer (Toeplitz operators),
- generalize to give a fundamental and beautiful insight into the topology of matrix groups (the Bott periodicity theorem).

All these subjects and more are developed starting only from mathematics that is common in final-year undergraduate courses.

This book is published in cooperation with Mathematics Advanced Study Semesters

#### Readership

Undergraduates and beginning graduate students interested in (and trying to learn) ideas concentrated around the notion of the winding number and its appearance in such areas of mathematics as analysis, differential geometry, and topology.

#### Reviews & Endorsements

This book covers a lot of ground. But it does so in a clear and careful manner that would make a terrific read for the prepared undergraduate. It is a study in how an intuitive idea can transport one into some deep waters of mathematics, and that is an important story to tell.

-- John McCleary, Mathematical Reviews

People who teach university-level mathematics for a living often find themselves reading lots of books on the subject. But even for the book-lovers among us, after you've just read about ten linear algebra texts, all of which look like they were stamped from the same cookie cutter, the process can occasionally wear thin. It's very pleasant, then, to stumble across a book that is genuinely unique, that addresses a topic in a way not found elsewhere, and that teaches you something that you didn't know before. It's even nicer when the book in question does a really good job of it, as is the case with the book under review. ...Roe's writing style is succinct, but clear and quite elegant; I could practically hear a British accent as I read the book. This clarity of writing and the numerous appendices help make the book accessible.

-- MAA Online

#### Table of Contents

# Table of Contents

## Winding Around: The Winding Number in Topology, Geometry, and Analysis

- Cover Cover11
- Title page iii4
- Contents v6
- Foreword: MASS and REU at Penn State University ix10
- Preface xi12
- Chapter 1. Prelude: Love, hate, and exponentials 116
- Chapter 2. Paths and homotopies 1530
- Chapter 3. The winding number 2742
- Chapter 4. Topology of the plane 4964
- Chapter 5. Integrals and the winding number 7388
- Chapter 6. Vector fields and the rotation number 101116
- Chapter 7. The winding number in functional analysis 121136
- Chapter 8. Coverings and the fundamental group 139154
- Chapter 9. Coda: The Bott periodicity theorem 169184
- Appendix A. Linear algebra 181196
- Appendix B. Metric spaces 203218
- Appendix C. Extension and approximation theorems 217232
- Appendix D. Measure zero 223238
- Appendix E. Calculus on normed spaces 229244
- Appendix F. Hilbert space 239254
- Appendix G. Groups and graphs 249264
- Bibliography 261276
- Index 265280
- Other titles in this series 271286
- Back Cover Back Cover1287