Download e-book for iPad: Algebraic and Geometric Topology, Part 1 by Milgram R. (ed.)

By Milgram R. (ed.)

ISBN-10: 082181432X

ISBN-13: 9780821814321

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Example text

Hence, this set consists of two hypersurfaces. Consequently, its trace on the parameter space of a typical family (called the bifurcation diagram of zeroes of the family) consists of two hypersurfaces. I. 48 ARNOLD Fig. 35 - The bifurcation diagram of zeroes of the boundary singularity B2. EXAMPLE. Let us consider the family of functions on the plane {(x, y)} with a "boundary" x = 0, depending on the parameters ()q, >'2). The bifurcation diagram of zeroes of the family (fig. 35) consists on those values of the parameters for which the circle f = 0 either degenerates (the AI component, >'t = 4>'2) or is tangent to the axis of y (the B I component, >'2 = 0).

I. 8 - The folded and the open umbrellas The folded umbrella is the surface y2 = z3 x 2 (fig. 25). " Fig. 25 - The folded umbrella. It has a smooth semicubical cusped edge and a self-intersection line. Topologically this surface (in real 3-space) has the same structure as the ordinary Cayley umbrella (fig. 7). EXAMPLE. Let us consider a smooth curve in Euclidean (or projective) 3-space. The curvature and the torsion of a generic curve at a generic point are both different from zero (the curve has a quadratic tangency with the tangent line and a cubical tangency with the osculating plane).

SINGULARITIES OF BIFURCATION DIAGRAMS At first glance, the variety of different singularities encountered in different applications produces an impression of a chaos. lg very different phenomena. The coincidence of the answers in apparently different problems is not an accident: there· exist many hidden relations between· apparently unrelated objects in singularity theory. In most of these problems, the object of interest is the bifurcation diagram formed by the values of the parameters for which a "qualitative change" in the objects of the family occurs.

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Algebraic and Geometric Topology, Part 1 by Milgram R. (ed.)

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