Introduction to Topological Manifolds by John Lee

• Topology
• Greg Bernhardt
In summary, "Introduction to Topological Manifolds" by John M. Lee is a highly recommended book for graduate students interested in differential geometry. It covers the basics of topological spaces, connectedness and compactness, cell complexes, compact surfaces, homotopy and the fundamental group, group theory, the Seifert-Van Kampen theorem, covering maps, group actions, homology, and includes appendices on set theory, metric spaces, and group theory. The book is well-written and provides clear explanations, but it is assumed that the reader has a background in real analysis and group theory. It is a great supplement for a point-set topology course and a good preparation for Lee's "Smooth Manifolds" book.

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Code:
[LIST]
[*] Preface
[*] Introduction
[LIST]
[*] What Are Manifolds?
[*] Why Study Manifolds?
[/LIST]
[*] Topological Spaces
[LIST]
[*] Topologies
[*] Convergence and Continuity
[*] Hausdorff Spaces
[*] Bases and Countability
[*] Manifolds
[*] Problems
[/LIST]
[*] New Spaces from Old
[LIST]
[*] Subspaces
[*] Product Spaces
[*] Disjoint Union Spaces
[*] Quotient Spaces
[*] Topological Groups and Group Actions
[*] Problems
[/LIST]
[*] Connectedness and Compactness
[LIST]
[*] Connectedness
[*] Compactness
[*] Local Compactness
[*] Paracompactness
[*] Proper Maps
[*] Problems
[/LIST]
[*] Cell Complexes
[LIST]
[*] Cell Complexes and CW Complexes
[*] Topological Properties of CW Complexes
[*] Classification of 1-Dimensional Manifolds
[*] Simplicial Complexes
[*] Problems
[/LIST]
[*] Compact Surfaces
[LIST]
[*] Surfaces
[*] Connected Sums of Surfaces
[*] Polygonal Presentations of Surfaces
[*] The Classification Theorem
[*] The Euler Characteristic
[*] Orientability
[*] Problems
[/LIST]
[*] Homotopy and the Fundamental Group
[LIST]
[*] Homotopy
[*] The Fundamental Group
[*] Homomorphisms Induced by Continuous Maps
[*] Homotopy Equivalence
[*] Higher Homotopy Groups
[*] Categories and Functors
[*] Problems
[/LIST]
[*] The Circle
[LIST]
[*] Lifting Properties of the Circle
[*] The Fundamental Group of the Circle
[*] Degree Theory for the Circle
[*] Problems
[/LIST]
[*] Some Group Theory
[LIST]
[*] Free Products
[*] Free Groups
[*] Presentations of Groups
[*] Free Abelian Groups
[*] Problems
[/LIST]
[*] The Seifert–Van Kampen Theorem
[LIST]
[*] Statement of the Theorem
[*] Applications
[*] Fundamental Groups of Compact Surfaces
[*] Proof of the Seifert–Van Kampen Theorem
[*] Problems
[/LIST]
[*] Covering Maps
[LIST]
[*] Definitions and Basic Properties
[*] The General Lifting Problem
[*] The Monodromy Action
[*] Covering Homomorphisms
[*] The Universal Covering Space
[*] Problems
[/LIST]
[*] Group Actions and Covering Maps
[LIST]
[*] The Automorphism Group of a Covering
[*] Quotients by Group Actions
[*] The Classification Theorem
[*] Proper Group Actions
[*] Problems
[/LIST]
[*] Homology
[LIST]
[*] Singular Homology Groups
[*] Homotopy Invariance
[*] Homology and the Fundamental Group
[*] The Mayer–Vietoris Theorem
[*] Homology of Spheres
[*] Homology of CW Complexes
[*] Cohomology
[*] Problems
[/LIST]
[*] Appendix: Review of Set Theory
[LIST]
[*] Basic Concepts
[*] Cartesian Products, Relations, and Function
[*] Number Systems and Cardinality
[*] Indexed Families
[/LIST]
[*] Appendix: Review of Metric Spaces
[LIST]
[*] Euclidean Spaces
[*] Metrics
[*] Continuity and Convergence
[/LIST]
[*] Appendix: Review of Group Theory
[LIST]
[*] Basic Definitions
[*] Cosets and Quotient Groups
[*] Cyclic Groups
[/LIST]
[*] Notation Index
[*] Subject Index
[*] References
[/LIST]

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Absolutely one of the best topology books out there. Lee is a real master at writing books. He makes everything very clear and his explanations are superb. This book is the perfect book for those who want to go into differential geometry. The results are really focused towards geometry, which means that some material that is important for analysis is left out. For example, Tychonoff's theorem is not covered and neither are nets and filters.
Interested reader absolutely must be comfortable with epsilon-delta proofs and continuity. Some knowledge of metric spaces would be nice as well, although Lee provides an appendix that has everything you need to know. For the later chapters, you will need group theory.

I used this book as a supplement for a point-set topology course that used Munkres. Lee is obviously influenced by Munkres (which he admits in the Intro) so parts are very similar. Although Munkres is a very well-written book, I often liked Lee's explanations more, however he moves through the material at a greater speed (this can be good or bad, depending on your view). He is also focussed on preparing the reader for his Smooth Manifolds book, so he skips a lot of the more fiddly point-set material that Munkres goes into.

1. What is the main focus of "Introduction to Topological Manifolds" by John Lee?

The main focus of this book is to introduce readers to the concept of topological manifolds and their properties, as well as their role in modern mathematics and physics.

2. Who is the target audience for this book?

This book is primarily aimed at advanced undergraduate and graduate students in mathematics and physics, as well as researchers and professionals in these fields who are interested in learning about topological manifolds.

3. What are some of the key topics covered in this book?

Some of the key topics covered in this book include the definition and basic properties of topological manifolds, as well as their various types and constructions, such as smooth manifolds and differential structures. The book also delves into topics such as tangent spaces, vector fields, and differential forms on manifolds.

4. Is prior knowledge of topology required to understand this book?

Some familiarity with basic concepts in topology, such as continuity and compactness, will be helpful in understanding this book. However, the author does provide a review of these concepts in the first chapter for those who may not have a strong background in topology.

5. Are there any practical applications of topological manifolds?

Topological manifolds have a wide range of applications in various fields, such as physics, engineering, and computer science. They are used to model and understand complex systems, such as fluid dynamics and particle interactions, and are also important in the development of algorithms and data structures in computer science.

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