How Surfaces Intersect in Space

Written mostly through diagrams, this book introduces the core ideas of geometric topology to readers who want to see the subject rather than only read proofs. It opens by asking what a surface and a space are, then classifies orientable surfaces. The second chapter turns to the Möbius band and the non-orientable surfaces that can be assembled from it.

The middle chapters move into how curves sit on surfaces and how surfaces can meet a space along boundary curves. They cover the basics of knot theory and bring in four-dimensional space. Later, three-dimensional spaces and the surfaces embedded in them are used to help the reader picture the larger structures involved.

The newest chapter presents recent results by Cromwell, Izumiya and Marar. One result is a formula linking a surface's rank to its number of triple points. Another gives a family of surfaces in three-dimensional space, each with one triple point and six branch points, generalizing the Steiner Roman surface. A separate chapter reviews the movie technique for studying surfaces in four dimensions, with several movies of the Klein bottle and the Carter-Saito movie move theorem. It also shows how the 2-sphere can be turned inside out using these movie moves.

The final chapter surveys higher-dimensional spaces at an elementary level and serves as a guide to a wide range of topics. The book is aimed at undergraduates, beginning graduate students and non-specialists who want to learn the subject through examples and pictures.