Book:John M. Lee/Introduction to Topological Manifolds

Subject Matter

 * Topology
 * Topological Manifolds

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