| Description |
1 online resource (xvii, 433 pages) : illustrations |
| Series |
Graduate texts in mathematics ; 202 |
|
Graduate texts in mathematics ; 202.
|
| Contents |
Machine generated contents note: 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 Manifolds -- Simplicial Complexes -- Problems -- 6. Compact Surfaces -- Surfaces -- Connected Sums of Surfaces -- Polygonal Presentations of Surfaces -- Classification Theorem -- Euler Characteristic -- Orientability -- Problems -- 7. Homotopy and the Fundamental Group -- Homotopy -- Fundamental Group -- Homomorphisms Induced by Continuous Maps -- Homotopy Equivalence -- Higher Homotopy Groups -- Categories and Functors -- Problems -- 8. Circle -- Lifting Properties of the Circle -- 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. 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 -- General Lifting Problem -- Monodromy Action -- Covering Homomorphisms -- Universal Covering Space -- Problems -- 12. Group Actions and Covering Maps -- Automorphism Group of a Covering -- Quotients by Group Actions -- Classification Theorem -- Proper Group Actions -- Problems -- 13. Homology -- Singular Homology Groups -- Homotopy Invariance -- Homology and the Fundamental Group -- 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 |
| Summary |
This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness. This text is designed to be used for an introductory graduate course on the geometry and topology of manifolds. It should be accessible to any student who has completed a solid undergraduate degree in mathematics. The author's book Introduction to Smooth Manifolds is meant to act as a sequel to this book |
| Bibliography |
Includes bibliographical references (pages 407-408) and indexes |
| Notes |
Print version record |
| Subject |
Topological manifolds.
|
|
Variedades topológicas
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Topological manifolds
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Manifolds.
|
| Form |
Electronic book
|
| ISBN |
9781441979407 |
|
1441979409 |
|
