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.
