Preface 1 Introduction What Are Manifolds Why Study Manifolds 2 Topological Spaces Topologies Convergence and Continuity Hausdorff Spaces Bases and Countability Manifolds Problems 3 New Spaces from Old Subspaces Product Spaces Disjoint Union Spaces Quotient Spaces Adjunction Spaces Topological Groups and Group Actions Problems 4 Connectedness and Compactness Connectedness Compactness Local Compactness Paracompactness Proper Maps Problems 5 Cell Complexes Cell Complexes and CW Complexes Topological Properties of Cw Complexes Classification of 1-Dimensional Manifold Simplicial Complexes Problems 6 Compact Surfaces Surfaces Connected Sums of Surfaces Polygonal Presentations of Surfaces The Classification Theorem The Euler Characteristic Orientability 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 The Circle Lifting Properties of the Circle
The Fundamental Group of the Circle Degree Theory for the Circle Problems 9 Some Group Theory Free Products Free Groups Presentations of Groups Free Abelian Groups Problems 10 The Seifert-Van Kampen Theorem Statement of the Theorem Applications Fundamental Groups of Compact Surfaces Proof of the Seifert-Van Kampen Theorem Problems 11 Covering Maps Definitions and Basic Properties The General Lifting Problem The Monodromy Action Covering Homomorphisms The Universal Covering Space Problems 12 Group Actions and Covering Maps The Automorphism Group of a Covering Ouotients by Group Actions The Classification Theorem Proper Group Actions Problems 13 Homology Singular Homology Groups Homotopy Invariance Homology and the Fundamental Grour The Mayer-Vietoris Theorem Homology of Spheres Homology of CW Complexes Cohomology Problems Appendix A: Review of Set Theory Basic Concepts Cartesian Products,Relations,and Functions Number Systems and Cardinality Indexed Families Appendix B: Review of Metric Spaces Euclidean Spaces Metrics Continuity and Convergence Appendix C: Review of Group Theory Basic Definitions Cosets and Quotient Groups Cyclic Groups
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