Stratified Morse Theory by Mark Goresky (PDF)

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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. Regular Values.- 11.3. Morse Data in the Relative Nonproper Case.- 11.4. Local Morse Data is Morse Data.-

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