Topology and its applications
| Main Author: | Basener, William F., 1973- |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, N.J. :
Wiley-Interscience,
[2006]
|
| Physical Description: |
xxxvii, 339 pages : illustrations ; 25 cm. |
| Series: |
Pure and applied mathematics (John Wiley & Sons : Unnumbered)
|
| Subjects: |
| Item Description: |
Includes bibliographical references (pages 333-336) and index. Preface -- Introduction -- 1.1. Preliminaries -- 1.2. Cardinality -- 1. Continuity -- 1.1. Continuity and open sets in R n -- 1.2. Continuity and open sets in topological spaces -- 1.3. Metric, product, and quotient topologies -- 1.4. Subsets of topological spaces -- 1.5. Continuous functions and topological equivalence -- 1.6. Surfaces -- 1.7. Application : chaos in dynamical systems -- 1.7.1. History of chaos -- 1.7.2. A simple example -- 1.7.3. Notions of chaos -- 2. Compactness -- 2.1. Closed bounded subsets of R -- 2.2. Compact spaces -- 2.3. Identification spaces and compactness -- 2.4. Connectedness and path-connectedness -- 2.5. Cantor sets -- 2.6. Application : compact sets in population dynamics and fractals -- 3. Manifolds and complexes -- 3.1. Manifolds -- 3.2. Triangulations -- 3.3. Classification of surfaces -- 3.3.1. Gluing disks -- 3.3.2. Planar models -- 3.3.3. Classification of surfaces -- 3.4. Euler characteristic -- 3.5. Topological groups -- 3.6. Group actions and orbit spaces -- 3.6.1. Flows on tori -- 3.7.1. Robotic coordination and configuration spaces -- 3.7.2. Geometry of manifolds -- 3.7.3. The topology of the universe -- 4. Homotopy and the winding number -- 4.1. Homotopy and paths -- 4.2. The winding number -- 4.3. Degrees of maps -- 4.4. The Brouwer fixed point theorem -- 4.5. The Borsuk-Ulam theorem -- 4.6. Vector fields and the Poincaré index theorem -- 4.7. Applications 1 -- 4.7.1. The fundamental theorem of algebra -- 4.7.2. Sandwiches -- 4.7.3. Game theory and Nash equilibria -- 4.8. Applications 2 : calculus -- 4.8.1. Vector fields, path integrals, and the winding number -- 4.8.2. Vector fields on surfaces -- 4.8.3. Index theory for n-symmetry fields -- 4.9. Index theory in computer graphics -- 5. Fundamental group -- 5.1. Definition and basic properties -- 5.2. Homotopy equivalence and retracts -- 5.3. The fundamental group of spheres and tori -- 5.4. The Seifert-van Kampen theorem -- 5.4.1. Flowers and surfaces -- 5.4.2. The Seifert-van Kampen theorem -- 5.5. Covering spaces -- 5.6. Group actions and deck transformations -- 5.7. Applications -- 5.7.1. Order and emergent patterns in condensed matter physics -- 6. Homology -- 6.1. [triangle]-complexes -- 6.2. Chains and boundaries -- 6.3. Examples and computations -- 6.4. Singular homology -- 6.5. Homotopy invariance -- 6.6. Brouwer fixed point theorem for D n -- 6.7. Homology and the fundamental group -- 6.8. Betti numbers and the Euler characteristic -- 6.9. Computational homology-- 6.9.1. Computing Betti numbers -- 6.9.2. Building a filtration -- 6.9.3. Persistent homology -- Appendix A : Knot theory -- Appendix B : Groups -- Appendix C : Perspectives in topology -- C.1. Point set topology -- C.2. Geometric topology -- C.3. Algebraic topology -- C.4. Combinatorial topology -- C.5. Differential topology -- References -- Bibliography-- Index. |
|---|---|
| Physical Description: |
xxxvii, 339 pages : illustrations ; 25 cm. |
| Bibliography: |
Includes bibliographical references (pages 333-336) and index. |
| ISBN: |
9780471687559 0471687553 |