Book:John M. Lee/Introduction to Topological Manifolds

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John M. Lee: Introduction to Topological Manifolds

Published $2000$, Springer: Graduate Texts in Mathematics

ISBN 1-441-97939-5.


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