HOW SURFACES INTERSECT IN SPACE: AN INTRODUCTION TO TOPOLOGY (2ND EDITION) (Knots and Everything) by J Scott Carter (PDF)

Description

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced.In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space.Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface.Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view.This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.

User’s Reviews

Editorial Reviews: From the Publisher This marvelous book of pictures illustrates thefundamental concepts of geometric topology in a way that is veryfriendly to the reader. The first chapter discusses the meaning ofsurface and space and gives the classification of orientablesurfaces. In the second chapter we are introduced to the Mbiusband and surfaces that can be constructed from this non-orientablepiece of fabric. In chapter 3, we see how curves can fit in surfacesand how surfaces can fit into spaces with these curves on theirboundary. Basic applications to knot theory are discussed andfour-dimensional space is introduced. In Chapter 4 we learn aboutsome 3-dimensional spaces and surfaces that sit inside them. Thesesurfaces help us imagine the structures of the larger space. Chapter5 is completely new! It contains recent results of Cromwell, Izumiyaand Marar. One of these results is a formula relating the rank of asurface to the number of triple points. The other major result is acollection of examples of surfaces in 3-space that have one triplepoint and 6 branch points. These are beautiful generalizations of theSteiner Roman surface. Chapter 6 reviews the movie technique forexamining surfaces in 4-dimensional space. Various movies of the Kleinbottle are presented, and the Carter-Saito movie move theorem isexplained. The author shows us how to turn the 2-sphere inside out bymeans of these movie moves and this illustration alone is well worththe price of the book! In the last chapter higher dimensional spacesare examined from an elementary point of view. This is a guide bookto a wide variety of topics. It will be of value to anyone who wantsto understand the subject by way of examples. Undergraduates,beginning graduate students, and non-professionals will profit fromreading the book and from just looking at the pictures.

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

⭐This book was very impressive for me!This is the kind of the book that beginners in a subject (Topology) should read, before continuing with the “formalistic” university books.But the strongest part of the book is the “fantastic” figures (almost the 50% of the book) of Dr. Carter which puzzle out, with the best way, the concepts presented.

⭐There are hundreds of diagrams of surfaces in the book. I (with assistance from my wife and two teenage children) built virtually every one with scissors, tape, and glue. We were amazed at the number of interesting questions about the shape of space that were raised in the process. In answering them, we have come to look at the world differently. Even an ordinary chair is seen as a movie with elementary changes from frame to frame. My wife made connections with sewing. We got ideas for dozens of science fair projects. We developed the beginnings of an intuition about objects in four dimensions. I will always look back on the period when I read this book as one of the exciting times of my life. The book earns my very highest recommendation.

⭐I have been trying to teach myself topology and just finished the Weeks book on the shape of space. This book in comparison was a bit tougher going and a step towards teaching myself with a standard textbook. I think the previous review must have been a friend of the author because I can’t say I was as enthusiastic about the book (and anyway it is a feat in itself to create paper models from the primative rapidograph drawings the author provides). I found it hard to follow the author’s style and would have preferred more precision in his language. The drawings may be clear to the author but they were very confusing to me. I am not a bad artist and would find it impossible to create real life models from those drawings, many of which are a mess. On the positive side, Professor Carter relates the flavor of topology like the appetizers before a sumptuous banquet. I learned a little bit more about the subject and he shares the results of his personal research which is always good. In sum, it was a good read and probably deserves a reread. It was not a great read.

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