How Surfaces Intersect in Space: An Introduction to Topology (Series on Knots and Everything)

This is a book of pictures that illustrates standard examples in low dimensional topology. The text starts at the most basic level (the intersection of coordinate planes) and gives hands on constructions of beautiful examples in topology: the projective plane, Poincare's example of a homology sphere, lens spaces, knotted surfaces, 2-sphere eversions, and higher dimensional manifolds. The text carefully explains the importance of the examples and the techniques without being bogged down in a morass of technicalities. The book seeks to be an entertaining introduction to topology for the lay-person, and can be used as a tool book for the expert. The book opens with the classification theorem of surfaces and then discusses the building blocks of surfaces mapped into space. These are branch points, double points, and triple points. The second chapter gives various constructions of the projective plane as it intersects itself in ordinary 3-space. The third chapter includes further examples and methods for constructing surfaces in 3-space. In chapter 4, other 3-dimensional spaces that are constructed by gluing standard pieces of ordinary 3-space together are discussed. In chapter 5, knots and surfaces bounded by these 3-spaces are discussed. Chapter 6 contains come current research on embedded surfaces in 4-dimensions. Chapter 7 gives a movie move decomposition of the famous 2-sphere eversion. Chapter 8 contains explorations into higher dimensions.