A guide to the classification theorem for compact surfaces

Abstract

This book aims to give students neither too formal, nor too informal, access to a true classic in algebraic topology: the classification of compact surfaces (up to homeomorphism). Chapter 1 (The Classification Theorem: Informal Presentation) gives an overview of the classification theorem and outlines in which parts of the book missing notions and arguments are presented. Chapter 2 (Surfaces) gives a formal definition of surfaces. In Chapter 3 (Simplices, Complexes, and Triangulations) triangulations are defined via simplicial complexes. Chapter 4 (The Fundamental Group, Orientability) gives a quick introduction to fundamental groups, including a description of the fundamental group of the punctured plane (via winding numbers), and a definition of orientability via the fundamental group. Moreover, surfaces with boundary are introduced in this chapter. In Chapter 5 (Homology Groups), another classical invariant from algebraic topology is introduced: simplicial homology and (very briefly) singular homology. Chapter 6 (The Classification Theorem for Compact Surfaces) contains the proof of the classification theorem, based on manipulations and normal forms of certain cell complexes (similar to the outline in Chapter 1). Furthermore, the book contains several appendices. They cover visualisations of the real projective plane, a standard structural result for finitely generated Abelian groups, basic point-set topology, more historical background on the classification theorem, Thomassen's proof that every surface can be triangulated, and proofs that the authors did not want to include in the main text. The book includes many historical comments throughout and contains lots of pictures—both of the mathematicians involved in the classification theorem and of examples illustrating the mathematical content. Many examples are given that help one understand the basic tools (e.g., there are several nice examples that illustrate simplicial homology). However, mathematically, the book could be more streamlined: the goal of the classification theorem of compact surfaces could be achieved with either the fundamental group or homology. Homology is used as a homeomorphism invariant and computed through triangulations; however, the central fact that simplicial homology of a simplicial complex coincides with singular homology of the geometric realisation is not proved in the book. Similarly, in Chapter 6, where the classification is used to compute fundamental groups of surfaces, the fact that the combinatorial description of the fundamental group and the ordinary fundamental group coincide is not proved. Also, other essential tools from algebraic topology (invariance of domain, the Jordan-Schoenflies theorem) are used, but not much indication on the proofs is given. Categorial language is avoided altogether and some of the definitions seem to be ad hoc (e.g., the cell complexes in Chapter 6; it would be nicer to have a single combinatorial concept attached with surfaces, not both triangulations and cell complexes), and the central notion of triangulation is multiply defined (in Chapter 3 and in Appendix E, which might be confusing). It would have been nice also if the classification up to homotopy equivalence would have been included, and if the classification up to diffeomorphism and the possible Riemannian structures would have been briefly discussed.