A Guide to the Classification Theorem of Compact Surfaces
Undoubtedly, one of the most beautiful pieces of the mathematical achievements is the classification of compact surfaces. Among other reasons, the result comprises a splendid combinatin of intuition and rigour as well as the extensive use of many geometric and (mainly) topological tools. In addition, this milestone has been the starting point of many essential contemporary mathematical works.
The book under review tackles this central topic. The classification theorem can be found in many excellent classical monographs, mainly as a part of a course on Algebraic Topology, with different approaches and depth. Under this panorama, the motivation of this book is the presentation of the classification theorem as an only topic, giving the required importance to all its view points: intuition, visualization, topological tools, history,… amenable to a wide audience with certain amount of mathematical maturity.
The book is structured in six chapters, guiding the reader from the presentation and intuition of the problema to the complete proof of it. Each chapter provides a good list of references connnecting its topic to the literature. At the end, some appendices complete the work, the history of the classification problem being specially interesting.