Memoirs of the American Mathematical Society 2003; 268 pp; softcover Volume: 163 ISBN10: 0821832808 ISBN13: 9780821832806 List Price: US$84 Individual Members: US$50.40 Institutional Members: US$67.20 Order Code: MEMO/163/778
 The Milnor number is a powerful invariant of an isolated, complex, affine hypersurface singularity. It provides data about the local, ambient, topologicaltype of the hypersurface, and the constancy of the Milnor number throughout a family implies that Thom's \(a_f\) condition holds and that the local, ambient, topologicaltype is constant in the family. Much of the usefulness of the Milnor number is due to the fact that it can be effectively calculated in an algebraic manner. The Lê cycles and numbers are a generalization of the Milnor number to the setting of complex, affine hypersurface singularities, where the singular set is allowed to be of arbitrary dimension. As with the Milnor number, the Lê numbers provide data about the local, ambient, topologicaltype of the hypersurface, and the constancy of the Lê numbers throughout a family implies that Thom's \(a_f\) condition holds and that the Milnor fibrations are constant throughout the family. Again, much of the usefulness of the Lê numbers is due to the fact that they can be effectively calculated in an algebraic manner. In this work, we generalize the Lê cycles and numbers to the case of hypersurfaces inside arbitrary analytic spaces. We define the LêVogel cycles and numbers, and prove that the LêVogel numbers control Thom's \(a_f\) condition. We also prove a relationship between the Euler characteristic of the Milnor fibre and the LêVogel numbers. Moreover, we give examples which show that the LêVogel numbers are effectively calculable. In order to define the LêVogel cycles and numbers, we require, and include, a great deal of background material on Vogel cycles, analytic intersection theory, and the derived category. Also, to serve as a model case for the LêVogel cycles, we recall our earlier work on the Lê cycles of an affine hypersurface singularity. Readership Graduate students and research mathematicians interested in several complex variables and analytic spaces. Table of Contents Part I. Algebraic Preliminaries: Gap Sheaves and Vogel Cycles  Introduction
 Gap sheaves
 Gap cycles and Vogel cycles
 The LêIomdineVogel formulas
 Summary of Part I
Part II. Lê Cycles and Hypersurface Singularities  Introduction
 Definitions and basic properties
 Elementary examples
 A handle decomposition of the Milnor fibre
 Generalized LêIomdine formulas
 Lê numbers and hyperplane arrangements
 Thom's \(a_f\) condition
 Aligned singularities
 Suspending singularities
 Constancy of the Milnor fibrations
 Another characterization of the Lê cycles
Part III. Isolated Critical Points of Functions on Singular Spaces  Introduction
 Critical avatars
 The relative polar curve
 The link between the algebraic and topological points of view
 The special case of perverse sheaves
 Thom's \(a_f\) condition
 Continuous families of constructible complexes
Part IV. NonIsolated Critical Points of Functions on Singular Spaces  Introduction
 LêVogel cycles
 LêIomdine formulas and Thom's condition
 LêVogel cycles and the Euler characteristic
 Appendix A. Analytic cycles and intersections
 Appendix B. The derived category
 Appendix C. Privileged neighborhoods and lifting Milnor fibrations
 References
 Index
