Book:John M. Lee/Introduction to Topological Manifolds/Second Edition
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John M. Lee: Introduction to Topological Manifolds (2nd Edition)
Published $\text {2011}$
Subject Matter
Contents
- Preface
- $1 \quad$ Introduction
- What Are Manifolds?
- Why Study Manifolds?
- $2 \quad$ Topological Spaces
- Topologies
- Convergence and Continuity
- Hausdorff Spaces
- Bases and Countability
- Manifolds
- Problems
- $3 \quad$ New Spaces from Old
- Subspaces
- Product Spaces
- Disjoint Union Spaces
- Quotient Spaces
- Adjunction Spaces
- Topological Groups and Group Actions
- Problems
- $4 \quad$ Connectedness and Compactness
- Connectedness
- Compactness
- Local Compactness
- Paracompactness
- Proper Maps
- Problems
- $5 \quad$ Cell Complexes
- Cell Complexes and CW Complexes
- Topological Properties of CW Complexes
- Classification of 1-Dimensional Manifolds
- Simplicial Complexes
- Problems
- $6 \quad$ Compact Surfaces
- Surfaces
- Connected Sums of Surfaces
- Polygonal Presentations of Surfaces
- The Classification Theorem
- The Euler Characteristic
- Orientability
- Problems
- $7 \quad$ Homotopy and the Fundamental Group
- Homotopy
- The Fundamental Group
- Homomorphisms Induced by Continuous Maps
- Homotopy Equivalence
- Higher Homotopy Groups
- Categories and Functors
- Problems
- $8 \quad$ The Circle
- Lifting Properties of the Circle
- The Fundamental Group of the Circle
- Degree Theory for the Circle
- Problems
- $9 \quad$ Some Group Theory
- Free Products
- Free Groups
- Presentations of Groups
- Free Abelian Groups
- Problems
- $10 \quad$ The Seifert–Van Kampen Theorem
- Statement of the Theorem
- Applications
- Fundamental Groups of Compact Surfaces
- Proof of the Seifert–Van Kampen Theorem
- Problems
- $11 \quad$ Covering Maps
- Definitions and Basic Properties
- The General Lifting Problem
- The Monodromy Action
- Covering Homomorphisms
- The Universal Covering Space
- Problems
- $12 \quad$ Group Actions and Covering Maps
- The Automorphism Group of a Covering
- Quotients by Group Actions
- The Classification Theorem
- Proper Group Actions
- Problems
- $13 \quad$ Homology
- Singular Homology Groups
- Homotopy Invariance
- Homology and the Fundamental Group
- The Mayer–Vietoris Theorem
- Homology of Spheres
- Homology of CW Complexes
- Cohomology
- Problems
- Appendix $\text{A}$: Review of Set Theory
- Basic Concepts
- Cartesian Products, Relations, and Functions
- Number Systems and Cardinality
- Indexed Families
- Appendix $\text{B}$: Review of Metric Spaces
- Euclidean Spaces
- Metrics
- Continuity and Convergence
- Appendix $\text{C}$: Review of Group Theory
- Basic Definitions
- Cosets and Quotient Groups
- Cyclic Groups
- Notation Index
- Subject Index