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Stratified Morse Theory

Stratified Morse Theory

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'1. Stratified Morse Theory.- 1.1. Morse-Smale Theory.- 1.2. Morse Theory on Singular Spaces.- 1.3. Two Generalizations of Stratified Morse Theory.- 1.4. What is a Morse Function?.- 1.5. Complex Stratified Morse Theory.- 1.6. Morse Theory and Intersection Homology.- 1.7. Historical Remarks.- 1.8. Remarks on Geometry and Rigor.- 2. The Topology of Complex Analytic Varieties and the Lefschetz Hyperplane Theorem.- 2.1. The Original Lefschetz Hyperplane Theorem.- 2.2. Generalizations Involving Varieties which May be Singular or May Fail to be Closed.- 2.3. Generalizations Involving Large Fibres.- 2.4. Further Generalizations.- 2.5. Lefschetz Theorems for Intersection Homology.- 2.6. Other Connectivity Theorems.- 2.7. The Duality.- 2.8. Historical Remarks.- I. Morse Theory of Whitney Stratified Spaces.- 1. Whitney Stratifications and Subanalytic Sets.- 1.0. Introduction and Historical Remarks.- 1.1. Decomposed Spaces and Maps.- 1.2. Stratifications.- 1.3. Transversality.- 1.4. Local Structure of Whitney Stratifications.- 1.5. Stratified Submersions and Thorn's First Isotopy Lemma.- 1.6. Stratified Maps.- 1.7. Stratification of Subanalytic Sets and Maps.- 1.8. Tangents to a Subanalytic Set.- 1.9. Characteristic Points and Characteristic Covectors of a Map.- 1.10. Characteristic Covectors of a Hypersurface.- 1.11. Normally Nonsingular Maps.- 2. Morse Functions and Nondepraved Critical Points.- 2.0. Introduction and Historical Remarks.- 2.1. Definitions.- 2.2. Existence of Morse Functions.- 2.3. Nondepraved Critical Points.- 2.4. Isolated Critical Points of Analytic Functions.- 2.5. Local Properties of Nondepraved Critical Points.- 2.6. Nondepraved is Independent of the Coordinate System.- 3. Dramatis Personae and the Main Theorem.- 3.0. Introduction.- 3.1. The Setup.- 3.2. Regular Values.- 3.3. Morse Data.- 3.4. Coarse Morse Data.- 3.5. Local Morse Data.- 3.6. Tangential and Normal Morse Data.- 3.7. The Main Theorem.- 3.8. Normal Morse Data and the Normal Slice.- 3.9. Halflinks.- 3.10. The Link and the Halflink.- 3.11. Normal Morse Data and the Halflink.- 3.12. Summary of Homotopy Consequences.- 3.13. Counterexample.- 4. Moving the Wall.- 4.1. Introduction.- 4.2. Example.- 4.3. Moving the Wall: Version 1.- 4.4. Moving the Wall: Version 2.- 4.5. Tangential Morse Data is a Product of Cells.- 5. Fringed Sets.- 5.1. Definition.- 5.2. Connectivity of Fringed Sets.- 5.3. Characteristic Functions.- 5.4. One Parameter Families of Fringed Sets.- 5.5. Fringed Sets Parametrized by a Manifold.- 6. Absence of Characteristic Covectors: Lemmas for Moving the Wall.- 7. Local, Normal, and Tangential Morse Data are Well Defined.- 7.1. Definitions.- 7.2. Regular Values.- 7.3. Local Morse Data, Tangential Morse Data, and Fringed Sets.- 7.4. Local and Tangential Morse Data are Independent of Choices.- 7.5. Normal Morse Data and Halflinks are Independent of Choices.- 7.6. Local Morse Data is Morse Data.- 7.7. The Link and the Halflink.- 7.8. Normal Morse Data is Homeomorphic to the Normal Slice.- 7.9. Normal Morse Data and the Halflink.- 8. Proof of the Main Theorem.- 8.1. Definitions.- 8.2. Embedding the Morse Data.- 8.3. Diagrams.- 8.4. Outline of Proof.- 8.5. Verifications.- 9. Relative Morse Theory.- 9.0. Introduction.- 9.1. Definitions.- 9.2. Regular Values.- 9.3. Relative Morse Data.- 9.4. Local Relative Morse Data is Morse Data.- 9.5. The Main Theorem in the Relative Case.- 9.6. Halflinks.- 9.7. Normal Morse Data and the Halflink.- 9.8. Summary of Homotopy Consequences.- 10. Nonproper Morse Functions.- 10.1. Definitions.- 10.2. Regular Values.- 10.3. Morse Data in the Nonproper Case.- 10.4. Local Morse Data is Morse Data.- 10.5. The Main Theorem in the Nonproper Case.- 10.6. Halflinks.- 10.7. Normal Morse Data and the Halflink.- 10.8. Summary of Homotopy Consequences.- 11. Relative Morse Theory of Nonproper Functions.- 11.1. Definitions.- 11.2. Regula
Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, including its historical development. Some further important topics in the book are: Morse theory, singularities, transversality theory, complex analytic varieties, Lefschetz theorems, connectivity theorems, intersection homology, complements of affine subspaces and combinatorics. The book is designed for all interested students or professionals in this area.

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Produktdetails

Autor: Mark Goresky
ISBN-13: 9783642717161
ISBN: 3642717160
Einband: Book
Seiten: 292
Gewicht: 502 g
Format: 242x170x15 mm
Sprache: Englisch
Autor: Mark Goresky, Robert Macpherson
Inhaltsangabe1. Stratified Morse Theory.- 1.1. Morse-Smale Theory.- 1.2. Morse Theory on Singular Spaces.- 1.3. Two Generalizations of Stratified Morse Theory.- 1.4. What is a Morse Function?.- 1.5. Complex Stratified Morse Theory.- 1.6. Morse Theory and Intersection Homology.- 1.7. Historical Remarks.- 1.8. Remarks on Geometry and Rigor.- 2. The Topology of Complex Analytic Varieties and the Lefschetz Hyperplane Theorem.- 2.1. The Original Lefschetz Hyperplane Theorem.- 2.2. Generalizations Involving Varieties which May be Singular or May Fail to be Closed.- 2.3. Generalizations Involving Large Fibres.- 2.4. Further Generalizations.- 2.5. Lefschetz Theorems for Intersection Homology.- 2.6. Other Connectivity Theorems.- 2.7. The Duality.- 2.8. Historical Remarks.- I. Morse Theory of Whitney Stratified Spaces.- 1. Whitney Stratifications and Subanalytic Sets.- 1.0. Introduction and Historical Remarks.- 1.1. Decomposed Spaces and Maps.- 1.2. Stratifications.- 1.3. Transversality.- 1.4. Local Structure of Whitney Stratifications.- 1.5. Stratified Submersions and Thorn's First Isotopy Lemma.- 1.6. Stratified Maps.- 1.7. Stratification of Subanalytic Sets and Maps.- 1.8. Tangents to a Subanalytic Set.- 1.9. Characteristic Points and Characteristic Covectors of a Map.- 1.10. Characteristic Covectors of a Hypersurface.- 1.11. Normally Nonsingular Maps.- 2. Morse Functions and Nondepraved Critical Points.- 2.0. Introduction and Historical Remarks.- 2.1. Definitions.- 2.2. Existence of Morse Functions.- 2.3. Nondepraved Critical Points.- 2.4. Isolated Critical Points of Analytic Functions.- 2.5. Local Properties of Nondepraved Critical Points.- 2.6. Nondepraved is Independent of the Coordinate System.- 3. Dramatis Personae and the Main Theorem.- 3.0. Introduction.- 3.1. The Setup.- 3.2. Regular Values.- 3.3. Morse Data.- 3.4. Coarse Morse Data.- 3.5. Local Morse Data.- 3.6. Tangential and Normal Morse Data.- 3.7. The Main Theorem.- 3.8. Normal Morse Data and the Normal Slice.- 3.9. Halflinks.- 3.10. The Link and the Halflink.- 3.11. Normal Morse Data and the Halflink.- 3.12. Summary of Homotopy Consequences.- 3.13. Counterexample.- 4. Moving the Wall.- 4.1. Introduction.- 4.2. Example.- 4.3. Moving the Wall: Version 1.- 4.4. Moving the Wall: Version 2.- 4.5. Tangential Morse Data is a Product of Cells.- 5. Fringed Sets.- 5.1. Definition.- 5.2. Connectivity of Fringed Sets.- 5.3. Characteristic Functions.- 5.4. One Parameter Families of Fringed Sets.- 5.5. Fringed Sets Parametrized by a Manifold.- 6. Absence of Characteristic Covectors: Lemmas for Moving the Wall.- 7. Local, Normal, and Tangential Morse Data are Well Defined.- 7.1. Definitions.- 7.2. Regular Values.- 7.3. Local Morse Data, Tangential Morse Data, and Fringed Sets.- 7.4. Local and Tangential Morse Data are Independent of Choices.- 7.5. Normal Morse Data and Halflinks are Independent of Choices.- 7.6. Local Morse Data is Morse Data.- 7.7. The Link and the Halflink.- 7.8. Normal Morse Data is Homeomorphic to the Normal Slice.- 7.9. Normal Morse Data and the Halflink.- 8. Proof of the Main Theorem.- 8.1. Definitions.- 8.2. Embedding the Morse Data.- 8.3. Diagrams.- 8.4. Outline of Proof.- 8.5. Verifications.- 9. Relative Morse Theory.- 9.0. Introduction.- 9.1. Definitions.- 9.2. Regular Values.- 9.3. Relative Morse Data.- 9.4. Local Relative Morse Data is Morse Data.- 9.5. The Main Theorem in the Relative Case.- 9.6. Halflinks.- 9.7. Normal Morse Data and the Halflink.- 9.8. Summary of Homotopy Consequences.- 10. Nonproper Morse Functions.- 10.1. Definitions.- 10.2. Regular Values.- 10.3. Morse Data in the Nonproper Case.- 10.4. Local Morse Data is Morse Data.- 10.5. The Main Theorem in the Nonproper Case.- 10.6. Halflinks.- 10.7. Normal Morse Data and the Halflink.- 10.8. Summary of Homotopy Consequences.- 11. Relative Morse Theory of Nonproper Functions.- 11.1. Definitions.- 11.2. Regular Values.- 11.3. Morse Data in the Relative Nonproper Case.- 11.4. Local Morse Data is Morse Data.-

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Autor: Mark Goresky
ISBN-13:: 9783642717161
ISBN: 3642717160
Erscheinungsjahr: 17.11.2011
Verlag: Springer Berlin Heidelberg
Gewicht: 502g
Seiten: 292
Sprache: Englisch
Auflage Softcover reprint of the original 1st ed. 1988
Sonstiges: Taschenbuch, 242x170x15 mm