Surfaces in 4-Space
The new volume of the long term project Encyclopaedia of Mathematical Sciences belongs to the sub-series ‘Low-dimensional topology’. The topic described in the book is topology of surfaces in four dimensions. The first chapter describes various diagrammatic methods (motion pictures, normal forms, marked vertex diagrams, surface braids, etc.) and their relations. Constructions of knotted surfaces are discussed in the second chapter. The third chapter introduces a lot of invariants for knotted surfaces. A quandle is a set with a self-distributive binary operation. The quandle cocycle invariants are discussed in the last chapter of the book. There is also a geometric interpretation of quandle homology using coloured cobordisms. Twenty-five pages of quandles and their homology groups are available in the appendix. Ten pages of references are also included. Even though drawings of surfaces in four-dimensional space are impossible, the book is richly illustrated by many drawings and diagrams.