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Cover: 9781461269397 | Topology of Surfaces | L. Christine Kinsey | Taschenbuch | Paperback
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Topology of Surfaces
Taschenbuch von L. Christine Kinsey
Sprache: Englisch

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Beschreibung
" . . . that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore. " Michael Spivak This text was written as an antidote to topology courses such as Spivak It is meant to provide the student with an experience in geomet­ describes. ric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average stu­ dent would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an in­ tegral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed.
" . . . that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore. " Michael Spivak This text was written as an antidote to topology courses such as Spivak It is meant to provide the student with an experience in geomet­ describes. ric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average stu­ dent would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an in­ tegral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed.
Inhaltsverzeichnis
1. Introduction to topology.- 1.1. An overview.- 2. Point-set topology in ?n.- 2.1. Open and closed sets in ?n.- 2.2. Relative neighborhoods.- 2.3. Continuity.- 2.4. Compact sets.- 2.5. Connected sets.- 2.6. Applications.- 3. Point-set topology.- 3.1. Open sets and neighborhoods.- 3.2. Continuity, connectedness, and compactness.- 3.3. Separation axioms.- 3.4. Product spaces.- 3.5. Quotient spaces.- 4. Surfaces.- 4.1. Examples of complexes.- 4.2. Cell complexes.- 4.3. Surfaces.- 4.4. Triangulations.- 4.5. Classification of surfaces.- 4.6. Surfaces with boundary.- 5. The euler characteristic.- 5.1. Topological invariants.- 5.2. Graphs and trees.- 5.3. The euler characteristic and the sphere.- 5.4. The euler characteristic and surfaces.- 5.5. Map-coloring problems.- 5.6. Graphs revisited.- 6. Homology.- 6.1. The algebra of chains.- 6.2. Simplicial complexes.- 6.3. Homology.- 6.4. More computations.- 6.5. Betti numbers and the euler characteristic.- 7. Cellular functions.- 7.1. Cellular functions.- 7.2. Homology and cellular functions.- 7.3. Examples.- 7.4. Covering spaces.- 8. Invariance of homology.- 8.1. Invariance of homology for surfaces.- 8.2. The Simplicial Approximation Theorem.- 9. Homotopy.- 9.1. Homotopy and homology.- 9.2. The fundamental group.- 10. Miscellany.- 10.1. Applications.- 10.2. The Jordan Curve Theorem.- 10.3. 3-manifolds.- 11. Topology and calculus.- 11.1. Vector fields and differential equations in ?n.- 11.2. Differentiable manifolds.- 11.3. Vector fields on manifolds.- 11.4. Integration on manifolds.- Appendix: Groups.- References.
Details
Erscheinungsjahr: 2012
Fachbereich: Geometrie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Undergraduate Texts in Mathematics
Inhalt: x
281 S.
ISBN-13: 9781461269397
ISBN-10: 1461269393
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Kinsey, L. Christine
Auflage: Softcover reprint of the original 1st ed. 1993
Hersteller: Springer New York
Springer US, New York, N.Y.
Undergraduate Texts in Mathematics
Maße: 235 x 155 x 17 mm
Von/Mit: L. Christine Kinsey
Erscheinungsdatum: 03.10.2012
Gewicht: 0,452 kg
Artikel-ID: 106118757
Inhaltsverzeichnis
1. Introduction to topology.- 1.1. An overview.- 2. Point-set topology in ?n.- 2.1. Open and closed sets in ?n.- 2.2. Relative neighborhoods.- 2.3. Continuity.- 2.4. Compact sets.- 2.5. Connected sets.- 2.6. Applications.- 3. Point-set topology.- 3.1. Open sets and neighborhoods.- 3.2. Continuity, connectedness, and compactness.- 3.3. Separation axioms.- 3.4. Product spaces.- 3.5. Quotient spaces.- 4. Surfaces.- 4.1. Examples of complexes.- 4.2. Cell complexes.- 4.3. Surfaces.- 4.4. Triangulations.- 4.5. Classification of surfaces.- 4.6. Surfaces with boundary.- 5. The euler characteristic.- 5.1. Topological invariants.- 5.2. Graphs and trees.- 5.3. The euler characteristic and the sphere.- 5.4. The euler characteristic and surfaces.- 5.5. Map-coloring problems.- 5.6. Graphs revisited.- 6. Homology.- 6.1. The algebra of chains.- 6.2. Simplicial complexes.- 6.3. Homology.- 6.4. More computations.- 6.5. Betti numbers and the euler characteristic.- 7. Cellular functions.- 7.1. Cellular functions.- 7.2. Homology and cellular functions.- 7.3. Examples.- 7.4. Covering spaces.- 8. Invariance of homology.- 8.1. Invariance of homology for surfaces.- 8.2. The Simplicial Approximation Theorem.- 9. Homotopy.- 9.1. Homotopy and homology.- 9.2. The fundamental group.- 10. Miscellany.- 10.1. Applications.- 10.2. The Jordan Curve Theorem.- 10.3. 3-manifolds.- 11. Topology and calculus.- 11.1. Vector fields and differential equations in ?n.- 11.2. Differentiable manifolds.- 11.3. Vector fields on manifolds.- 11.4. Integration on manifolds.- Appendix: Groups.- References.
Details
Erscheinungsjahr: 2012
Fachbereich: Geometrie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Undergraduate Texts in Mathematics
Inhalt: x
281 S.
ISBN-13: 9781461269397
ISBN-10: 1461269393
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Kinsey, L. Christine
Auflage: Softcover reprint of the original 1st ed. 1993
Hersteller: Springer New York
Springer US, New York, N.Y.
Undergraduate Texts in Mathematics
Maße: 235 x 155 x 17 mm
Von/Mit: L. Christine Kinsey
Erscheinungsdatum: 03.10.2012
Gewicht: 0,452 kg
Artikel-ID: 106118757
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