Stratified Morse Theory

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.