Book:John M. Lee/Introduction to Topological Manifolds
Jump to navigation
Jump to search
John M. Lee: Introduction to Topological Manifolds
Published $\text {2000}$, 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
- Bases
- Manifolds
- Problems
- $3 \quad$ New Spaces from Old
- Subspaces
- Product Spaces
- Quotient Spaces
- Group Actions
- Problems
- $4 \quad$ Connectedness and Compactness
- Connectedness
- Compactness
- Local Compact Hausdorff Spaces
- Problems
- $5 \quad$ Simplical Complexes
- Euclidean Simplical Complexes
- Abstract Simplical Complexes
- Triangulation Theorems
- Orientations
- Combinatorial Invariants
- Problems
- $6 \quad$ Curves and Surfaces
- Classification of Curves
- Surfaces
- Connected Sums
- Polygonal Presentations of Surfaces
- Classification of Surface Presentations
- Combinatorial Invariants
- 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
- $8 \quad$ Circles and Spheres
- The Fundamental Group of the Circle
- Proofs of the Lifting Lemmas
- Fundamental Groups of Spheres
- Fundamental Groups of Product Spaces
- Fundamental Groups of Manifolds
- 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
- Proof of the Theorem
- Distinguishing Manifolds
- Problems
- $11 \quad$ Covering Spaces
- Definitions and Basic Properties
- Covering Maps and the Fundamental Group
- The Covering Group
- Problems
- $12 \quad$ Classification of Coverings
- Covering Homomorphism
- The Universal Covering Space
- The Classification Theorem
- Proper Group Actions
- The Classification Theorem
- Problems
- $13 \quad$ Homology
- Singular Homology Groups
- Homotopy Invariance
- Homology and the Fundamental Group
- The Mayer–Vietoris Theorem
- Applications
- The Homology of a Simplical Complex
- Cohomology
- Problems
- Appendix: Review of Prerequisites
- References
- Index