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Main Question or Discussion Point
- Author: John M. Lee
- Title: Introduction to Topological Manifolds
- Amazon Link: https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20&tag=pfamazon01-20
- Prerequisities: Real Analysis course involving epsilon-delta and preferebly metric spaces, group theory
- Level: Grad students
Table of Contents:
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
[*] Adjunction 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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