Stratified Morse Theory

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.

1. Stratified Morse Theory.- 2. The Topology of Complex Analytic Varieties and the Lefschetz Hyperplane Theorem.- I. Morse Theory of Whitney Stratified Spaces.- 1. Whitney Stratifications and Subanalytic Sets.- 2. Morse Functions and Nondepraved Critical Points.- 3. Dramatis Personae and the Main Theorem.- 4. Moving the Wall.- 5. Fringed Sets.- 6. Absence of Characteristic Covectors: Lemmas for Moving the Wall.- 7. Local, Normal, and Tangential Morse Data are Well Defined.- 8. Proof of the Main Theorem.- 9. Relative Morse Theory.- 10. Nonproper Morse Functions.- 11. Relative Morse Theory of Nonproper Functions.- 12. Normal Morse Data of Two Morse Functions.- II. Morse Theory of Complex Analytic Varieties.- 0. Introduction.- 1. Statement of Results.- 2. Normal Morse Data for Complex Analytic Varieties.- 3. Homotopy Type of the Morse Data.- 4. Morse Theory of the Complex Link.- 5. Proof of the Main Theorems.- 6. Morse Theory and Intersection Homology.- 7. Connectivity Theorems for q-Defective Pairs.- 8. Counterexamples.- III. Complements of Affine Subspaces.- 0. Introduction.- 1. Statement of Results.- 2. Geometry of the Order Complex.- 3. Morse Theory of ?n.- 4. Proofs of Theorems B, C, and D.- 5. Examples.