Geometry and Topology of Manifolds

Surfaces and Beyond

Jointly with Vicente Muñoz and Juan Ángel Rojo, we have published the monograph Geometry and Topology of Manifolds: Surfaces and Beyond, as part of the collection Graduate Studies in Mathematics of the American Mathematical Society

The textbook is oriented to Master’s students and PhD students, with a view towards the interplay between different types of geometric structures on manifolds (topological, differentiable, Riemannian, complex, algebraic, etc.), with special attention to the case of surfaces.

The book can be used as basic bibliography for advanced graduate courses, as it has been done in the Master’s degree in Mathematics at Universidad Complutense de Madrid. Nevertheless, the textbook also contains cornerstone results that may be independently used for courses on differential topology or algebraic topology, like the classification of compact surface, the degree-genus formula, homology and cohomology in various versions, the Gauss-Bonnet theorem or a thorough study of manifolds of constant sectional curvature, among others. Moreover, it also includes a novel proof of the uniformization theorem through the curvature flow and orbifolds.

Geometry and Topology of Manifolds: Surfaces and Beyond.
Graduate Studies in Mathematics (American Mathematical Society). ISBN: 978-1-4704-6132-4

Table of contents

  1. Topological surfaces
    • Topological and smooth manifolds
    • PL structures
    • Planar representations of surfaces
    • Orientability
    • Classification of compact surfaces
  2. Algebraic topology
    • Homotopy theory
    • Covers
    • Singular homology
    • Simplicial homology
    • De Rham cohomology
    • Poincaré duality
  3. Riemannian geometry
    • Riemannian metrics. Curvature. Geodesics
    • Riemannian surfaces and the Gauss-Bonnet theorem
    • Isotropic, symmetric, and homogeneous manifolds
  4. Constant curvature
    • Positive constant curvature
    • Vanishing constant curvature
    • Negative constant curvature
    • Classical geometries
  5.  Complex geometry
    • Complex manifolds
    • Kähler manifolds
    • Complex curves
    • Classification of complex curves
    • Elliptic curves
  6.  Global analysis
    • Conformal structures
    • Hodge theory
    • Metrics of constant curvature
    • The curvature flow