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Über dieses Buch

From the reviews: "... My general impression is of a particularly nice book, with a well-balanced bibliography, recommended!"
Mededelingen van Het Wiskundig Genootschap, 1995
"... The authors offer here an up to date guide to the topic and its main applications, including a number of new results. It is very convenient for the reader, a carefully prepared and extensive bibliography ... makes it easy to find the necessary details when needed. The books (EMS 6 and EMS 39) describe a lot of interesting topics. ... Both volumes are a very valuable addition to the library of any mathematician or physicist interested in modern mathematical analysis."
European Mathematical Society Newsletter, 1994



Chapter 1. Critical Points of Functions

One of the most thoroughly studied branches of the theory of singularities is the investigation and classification of degeneracies of critical points of functions. Generic functions have only nondegenerate critical points. More complex singularities vanish under small perturbations, decomposing into nondegenerate ones.
V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev

Chapter 2. Monodromy Groups of Critical Points

Morse theory studies the restructurings, perestroikas, or metamorphoses that the level set f-1(x) of a real function f: M → ℝ, defined on a manifold M, undergoes as x passes through the critical values of f. The Picard-Lefschetz theory is the complex analogue of Morse theory. In the complex case the set of critical values does not divide the range ℂ of a complex-valued function into connected components, and no restructurings occur: all level manifolds close to a critical one are topologically identical. For this reason, in the complex case, rather than passing through a critical value, one has to go around it in the plane ℂ where the function takes its values.
V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev

Chapter 3. Basic Properties of Maps

One of the tasks that must be tackled repetedly in singularity theory is the investigation of how certain properties of map-germs behave with respect to various equivalence relations. Most often the equivalence of two germs means that they belong to the same orbit of the action of some group of diffeomorphisms on a function space.
V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev

Chapter 4. The Global Theory of Singularities

In this chapter we describe topological and numerical characteristics of singular sets of smooth maps, such as cohomology classes dual to sets of critical points and critical values; the invariants of maps defined by these classes; their connections with standard topological characteristics of the source and target manifolds; the structure of spaces of smooth maps without singularities of some given type; restrictions on the number and coexistence of singular points.
V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev


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