Dynamical Systems VIII: Singularity Theory II. Applications by V.I. Arnol'd, J.S. Joel, V.V. Goryunov, O.V. Lyashko, V.A.

By V.I. Arnol'd, J.S. Joel, V.V. Goryunov, O.V. Lyashko, V.A. Vasil'ev

In the 1st quantity of this survey (Arnol'd et al. (1988), hereafter brought up as "EMS 6") we familiar the reader with the fundamental options and strategies of the idea of singularities of delicate mappings and capabilities. This idea has a number of purposes in arithmetic and physics; the following we start describing those applica­ tions. however the current quantity is largely self sustaining of the 1st one: the entire techniques of singularity thought that we use are brought throughout the presentation, and references to EMS 6 are constrained to the quotation of technical effects. even supposing our major target is the presentation of analready formulated conception, the readerwill additionally encounter a few relatively fresh effects, it appears unknown even to experts. We pointout a few of these effects. 2 three within the attention of mappings from C into C in§ three. 6 of bankruptcy 1, we outline the bifurcation diagram of this sort of mapping, formulate a K(n, 1)-theorem for the enhances to the bifurcation diagrams of straightforward singularities, provide the definition of the Mond invariant N within the spirit of "hunting for invariants", and we draw the reader's cognizance to a mode of making similar to a mapping from the corresponding functionality on a manifold with boundary. In§ four. 6 of a similar bankruptcy we introduce the idea that of a versal deformation of a functionality with a nonisolated singularity within the dass of capabilities whose severe units are arbitrary whole intersections of mounted dimension.

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There exists an involution on H which is induced by permuting the sheets of the covering. We denote by H- the anti-invariant part of H. Its rank is Jl. The intersection form on H gives abilinear form on H-. This form is called the intersection form of the projection. Now, after choosing a basis of H-, we can construct a Dynkin diagram describing the intersection form. 4). But if the dimension of V0 is odd (for example, if it is a curve), then we need to be" particularly concerned about the distinction between short and long cycles.

For complete intersectians the analogous process Ieads to a different result. Consider a slightly more general situation. Let f = (g, h): (

The number of these points is equal to the number of spheres in the bouquet to which the quotient § 3. Projections and Left-Right Equivalence ck,eo 51 o-····~·····~ k-z ~ e-z Fig. 21. , the Milnor number J1. of the projection introduced in 2°. In the approach of u from a noncritical value (u = 0) to the J1. critical values along J1. distinguished paths, the Ievel set of the function u on W degenerates on the complex u-axis. At the degenerations J1. semicycles vanish, and hence, J1. homology classes of WfW 0 .

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