Winding Around: The Winding Number in Topology, Geometry, and Analysis (Student Mathematical Library) by John Roe (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review 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 About the Author John Roe , Pennsylvania State University, State College, PA, USA.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I find this book to be very mixed. For topics I already know a fair amount on, the book sheds a complementary and insightful light. But for everything else I find the book to be all over the place and non-helpful.If you don’t already have a fair amount of knowledge of winding numbers (e.g. from Beardon’s ‘Complex Analysis’ which is much more insightful, or say the first 50 or 100 pages of Fulton’s ‘Algebraic Topology’), look elsewhere.

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