Softcover ISBN:  9781470461324 
Product Code:  GSM/208 
408 pp 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Sale Price:  $57.85 
Electronic ISBN:  9781470461621 
Product Code:  GSM/208.E 
488 pp 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Sale Price:  $57.85 

Book DetailsGraduate Studies in MathematicsVolume: 208; 2020MSC: Primary 57; 53; 55; 30;
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 GaussBonnet theorem, the degreegenus 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.This book is published in cooperation with Real Sociedád Matematica Española.ReadershipUndergraduate and graduate students interested in teaching and learning the basics of algebraic and differential topology.

Table of Contents

Chapters

Topological surfaces

Algebraic topology

Riemannian geometry

Constant curvature

Complex geometry

Global analysis


Additional Material

Reviews

The book should be wellsuited for several types of graduatelevel geometry courses: The material encompasses manifolds and categories; the basics of homotopy, singular homology, and de Rham cohomology; Riemannian metrics, curvature and uniformization; holomorphic structures; and enough global analysis for Hodge theory, the existence of metrics of constant Gaussian curvature, and basic curvature flow. Each chapter provides useful supplementary reading, from textbooks to original papers, and contains numerous exercises of varying levels of difficulty. There is an extensive index and a comprehensive table of notation indexed by the page where notation first appears.
...The authors provide proofs of results, such as triangulation of surfaces, that in other textbooks are often outsourced, and they provide detailed references for results they do not prove. They have also made an effort to draw 'cultural links' between disparate parts of the material, the way a good instructor might.
Andrew D. Hwang, College of the Holy Cross


Request Exam/Desk Copy

Request Review Copy

Get Permissions
 Book Details
 Table of Contents
 Additional Material
 Reviews

 Request Review Copy
 Request Exam/Desk Copy
 Get Permissions
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 GaussBonnet theorem, the degreegenus 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.
Undergraduate and graduate students interested in teaching and learning the basics of algebraic and differential topology.

Chapters

Topological surfaces

Algebraic topology

Riemannian geometry

Constant curvature

Complex geometry

Global analysis

The book should be wellsuited for several types of graduatelevel geometry courses: The material encompasses manifolds and categories; the basics of homotopy, singular homology, and de Rham cohomology; Riemannian metrics, curvature and uniformization; holomorphic structures; and enough global analysis for Hodge theory, the existence of metrics of constant Gaussian curvature, and basic curvature flow. Each chapter provides useful supplementary reading, from textbooks to original papers, and contains numerous exercises of varying levels of difficulty. There is an extensive index and a comprehensive table of notation indexed by the page where notation first appears.
...The authors provide proofs of results, such as triangulation of surfaces, that in other textbooks are often outsourced, and they provide detailed references for results they do not prove. They have also made an effort to draw 'cultural links' between disparate parts of the material, the way a good instructor might.
Andrew D. Hwang, College of the Holy Cross