Stella & Rose's Books

Contents

• 1 Introduction
• What are Manifolds?
• Why Study Manifolds?
• 2 Topological Spaces
• Topologies
• Bases
• Manifolds
• Problems
• 3 New Spaces from Old
• Subspaces
• Product Spaces
• Quotient Spaces
• Group Actions
• Problems
• 4 Connectedness and Compactness
• Connectedness
• Compactness
• Locally Compact Hausdorff Spaces
• Problems
• 5 Simplicial Complexes
• Euclidean Simplicial Complexes
• Abstract Simplicial Complexes
• Triangulation Theorems
• Orientations
• Combinatorial Invariants
• Problems
• 6 Curves and Surfaces
• Classification of Curves
• Surfaces
• Connected Sums
• Polygonal Presentations of Surfaces
• Classification of Surface Presentations
• Combinatorial Invariants
• Problems
• 7 Homotopy and the Fundamental Group
• Homotopy
• The Fundamental Group
• Homomorphisms Induced by Continuous Maps
• Homotopy Equivalence
• Higher Homotopy Groups
• Categories and Functors
• Problems
• 8 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 Some Group Theory
• Free Products
• Free Groups
• Presentations of Groups
• Free Abelian Groups
• Problems
• Distinguishing Manifolds
• Problems
• 11 Covering Spaces
• Definitions and Basic Properties
• Covering Maps and the Fundamental Group
• The Covering Group
• Problems
• 12 Classification of Coverings
• Covering Homomorphisms
• The Universal Covering Space
• Proper Group Actions
• The Classification Theorem
• Problems
• 13 Homology
• Singular Homology Groups
• Homotopy Invariance
• Homology and the Fundamental Group[
• The Mayer-Vietoris Theorem
• Applications
• The Homology of A Simplicial Complext
• Cohomology
• Problems
• References
• Index