Preface

The present book arises from notes of a master’s course that the first author delivered from the academic course 2011/12 until 2017/18 for the Master’s Degree in Mathematics at Universidad Complutense de Madrid. The prerequisites are basic courses in linear algebra, elementary topology and algebraic topology, differential geometry, complex analysis, and partial differential equations. This is a somewhat non-standard course on differential topology, that is, its main focus is the geometry and topology of manifolds, trying to touch on many of its ramifications. Being of an introductory nature, it cannot aspire to include all material on the interplay of geometry and topology. On the other hand, we consider it important in a text of this type to give complete proofs of some of the important landmarks in the development of the area. For this reason, we have decided to introduce the different aspects of the theory of manifolds in arbitrary dimension, but then at every chapter we move to give important results and full proofs for the case of dimension 2, that is, surfaces. The title of the book arises from this consideration. The content of the book is distributed into six chapters, following the philosophy of going from the soft to the hard. In Chapter 1, we start with the topological side of the course, introducing manifolds as natural objects to study (natural from the geometrical and from the physical points of view) and setting up the core problem of the classification of manifolds. Smoothmanifolds are introduced, where a differentiable structure on amanifold is the way tomake sense of the concept of differentiation, which is needed to pose (and solve) physical problems on a manifold. The chapter contains complete proofs of two important results in the case of surfaces: the existence of triangulations and the classification of compact surfaces. Chapter 2 introduces the theory of algebraic topology. This is the area of mathematics that constructs algebraic invariants to study the topology of spaces. For manifolds, topology is the way to go from the local to the global. By definition, a manifold is a space that locally looks like a Euclidean space (which is the model of the space in which we live). So locally there is no information on a manifold, and the global