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This book represents a novel approach to differential topology. Its
main focus is to give a comprehensive introduction to the
classification of manifolds, with special attention paid to the
case of surfaces, for which the book provides a complete
classification from many points of view: topological, smooth,
constant curvature, complex, and conformal. Each chapter briefly
revisits basic results usually known to graduate students from an
alternative perspective, focusing on surfaces. We provide full
proofs of some remarkable results that sometimes are missed in
basic courses (e.g., the construction of triangulations on
surfaces, the classification of surfaces, the Gauss-Bonnet theorem,
the degree-genus formula for complex plane curves, the existence of
constant curvature metrics on conformal surfaces), and we give
hints to questions about higher dimensional manifolds. Many
examples and remarks are scattered through the book. Each chapter
ends with an exhaustive collection of problems and a list of topics
for further study. The book is primarily addressed to graduate
students who did take standard introductory courses on algebraic
topology, differential and Riemannian geometry, or algebraic
geometry, but have not seen their deep interconnections, which
permeate a modern approach to geometry and topology of manifolds.
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