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Geometry and Topology of Manifolds: Surfaces and Beyond

Vicente Muñoz, Ángel González-Prieto, and Juan Ángel Rojo
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics
[Reviewed by
Andrew D. Hwang
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Geometry and Topology of Manifolds: Surfaces and Beyond is a self-descriptively titled textbook and reference. Though definitions are usually given for manifolds of arbitrary finite dimension, proofs and examples concentrate mostly on real two-manifolds.
The book should be well-suited for several types of graduate-level geometry course: 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. One consequence of this strategy, however, is that the narrative threads tend to be lengthy. A reader wanting to learn a new subject can locate
the relevant section easily but may need to skim backward to pick up context. In other words, the book seems likely to be an admirable reference, but its utility to each reader is likely to grow with time as more parts of the book become familiar.
Andrew D. Hwang is an associate professor of mathematics at the College of the Holy Cross.