Geometry, Topology and Physics, Second EditionDifferential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. 
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This book is good for supplemental material, but not to learn from. It attempts to be self contained, but some topics (for example singular homology and many physics topics) are dealt with in such a terse and hand waving fashion that there is no hope of understanding what is going on unless you are already comfortable with the topic. Other areas (like basic differential geometry) are given better treatment.
What this book is great for is a ton of worked out examples, and I would highly recommend it as a supplement to other more detailed and rigorous texts, or as a decent overview (if you don't need to know the material well enough to work with it).
Review: Geometry, Topology and Physics
User Review  Brian Powell  GoodreadsA survey of highermathematics necessary for General Relativity, Quantum Field Theory, and beyond. Nakahara will not provide a deep understanding of the many topics covered, but he explains just ... Read full review
Contents
Quantum Physics  1 
3  81 
Homology Groups  93 
Homotopy Groups  121 
Manifolds  169 
de Rham Cohomology Groups  226 
Riemannian Geometry  244 
torsion tensor  256 
Fibre Bundles  348 
Connections on Fibre Bundles  374 
Characteristic Classes  419 
Index Theorems  453 
Anomalies in Gauge Field Theories  501 
528  
560  
Complex Manifolds  308 