Stratified Morse Theory

ISBN-13: 9783642717161 / Angielski / Miękka / 2011 / 272 str.

Mark Goresky;Robert MacPherson

Opis

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.

Spis treści

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. Regular Values.- 11.3. Morse Data in the Relative Nonproper Case.- 11.4. Local Morse Data is Morse Data.- 11.5. The Main Theorem in the Relative Nonproper Case.- 11.6. Halflinks.- 11.7. Normal Morse Data and the Halflink.- 11.8. Summary of Homotopy Consequences.- 12. Normal Morse Data of Two Morse Functions.- 12.1. Definitions.- 12.2. Characteristic Covectors of the Normal Slice for a Pair of Functions.- 12.3. Characteristic Covectors of a Level.- 12.4. The Quarterlink and Related Spaces.- 12.5. Local Structure of the Normal Slice: The Milnor Fibration.- 12.6. Proof of Proposition 12.5.- 12.7. Monodromy.- 12.8. Monodromy is Independent of Choices.- 12.9. Relative Normal Morse Data for Two Nonproper Functions.- 12.10. Normal Morse Data for Many Morse Functions.- II. Morse Theory of Complex Analytic Varieties.- 0. Introduction.- 1. Statement of Results.- 1.0. Notational Remarks and Basepoints.- 1.1. Relative Lefschetz Theorem with Large Fibres.- 1.1*. Homotopy Dimension with Large Fibres.- 1.2. Lefschetz Theorem with Singularities.- 1.2*. Homotopy Dimension of Nonproper Varieties.- 1.3. Local Lefschetz Theorems.- 1.3*. Local Homotopy Dimension.- 2. Normal Morse Data for Complex Analytic Varieties.- 2.0. Introduction.- 2.1. Nondegenerate Covectors.- 2.2. The Complex Link and Related Spaces.- 2.3. The Complex Link is Independent of Choices.- 2.4. Local Structure of Analytic Varieties.- 2.5. Monodromy, the Structure of the Link, and Normal Morse Data.- 2.6. Relative Normal Morse Data for Nonproper Functions.- 2.7. Normal Morse Data for Two Complex Morse Functions.- 2.A. Appendix: Local Structure of Complex Valued Functions.- 3. Homotopy Type of the Morse Data.- 3.0. Introduction.- 3.1. Definitions.- 3.2. Proper Morse Functions: The Main Technical Result.- 3.3. Nonproper Morse Functions.- 3.4. Relative and Nonproper Morse Functions.- 4. Morse Theory of the Complex Link.- 4.0. Introduction.- 4.1. The Setup.- 4.2. Normal and Tangential Defects.- 4.3. Homotopy Consequences: The Main Theorem.- 4.4. Estimates on Tangential Defects.- 4.5. Estimates on the Normal Defect for Nonsingular X.- 4.5*.Estimates on the Dual Normal Defect for Proper ?.- 4.6. Estimates on the Normal Defect if ? is Finite.- 4.6*.Local Geometry of the Complement of a Subvariety.- 4.A. Appendix: The Levi Form and the Morse Index.- 5. Proof of the Main Theorems.- 5.1. Proof of Theorem 1.1: Relative Lefschetz Theorem with Large Fibres.- 5.1*. Proof of Theorem 1.1*: Homotopy Dimension with Large Fibres.- 5.2. Proof of Theorem 1.2: Lefschetz Theorem with Singularities.- 5.2*. Proof of Theorem 1.2*: Homotopy Dimension of Nonproper Varieties.- 5.3. Proof of Theorem 1.3: Local Lefschetz Theorems.- 5.3*. Proof of Theorem 1.3*: Local Homotopy Dimension.- 5.A. Appendix: Analytic Neighborhoods of an Analytic Set.- 6. Morse Theory and Intersection Homology.- 6.0. Introduction.- 6.1. Intersection Homology.- 6.2. The Set-up and the Bundle of Complex Links.- 6.3. The Variation.- 6.4. The Main Theorem.- 6.5. Vanishing of the Morse Group.- 6.6. Intuition Behind Theorem 6.4.- 6.7. Proof of Theorem 6.4.- 6.8. Intersection Homology of the Link.- 6.9. Intersection Homology of a Stein Space.- 6.10. Lefschetz Hyperplane Theorem.- 6.11. Local Lefschetz Theorem for Intersection Homology.- 6.12. Morse Inequalities.- 6.13. Specialization Over a Curve.- 6.A. Appendix: Remarks on Morse Theory, Perverse Sheaves, and D-Modules.- 7. Connectivity Theorems for q-Defective Pairs.- 7.0. Introduction.- 7.1. q-Defective Pairs.- 7.2. Defective Vectorbundles.- 7.3. Lefschetz Theorems for Defective Pairs.- 7.3*. Homotopy Dimension of Codefective Pairs.- 8. Counterexamples.- III. Complements of Affine Subspaces.- 0. Introduction.- 1. Statement of Results.- 1.1. Notation.- 1.2. The Order Complex.- 1.3. Theorem A: Complements of Affine Spaces.- 1.4. Corollary.- 1.5. Remarks.- 1.6. Theorem B: Moebius Function.- 1.7. Complements of Real Projective Spaces.- 1.8. Complements of Complex Projective Spaces.- 2. Geometry of the Order Complex.- 2.1. I-filtered Stratified Spaces.- 2.2. The Complex C(A).- 2.3. The Homotopy Equivalences.- 2.4. Central Arrangements.- 2.5. Appendix: The Arrangement Maps to the Order Complex.- 3. Morse Theory of ?n.- 3.1. The Morse Function.- 3.2. Intuition Behind the Theorem.- 3.3. Topology Near a Single Critical Point.- 3.4. The Involution.- 3.5. The Morse Function is Perfect.- 3.6. Proof of Theorem A.- 3.7. Appendix: Geometric Cycle Representatives.- 4. Proofs of Theorems B, C, and D.- 4.1. Geometric Lattice.- 4.2. Proof of Theorem B.- 4.3. Complements of Projective Spaces.- 4.4. Proof of Theorem C.- 4.5. Proof of Theorem D.- 5. Examples.- 5.1. The Local Contribution May Occur in Several Dimensions.- 5.2. On the Difference Between Real and Complex Arrangements.