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By Milgram R. (ed.)

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Extra resources for Algebraic and Geometric Topology, Part 1

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The Maxwell set is the projection of the set 2A 1 of selfintersections of the discriminant. The caustics of the boundary singularities have generically 3 components: they are the projections of the cusped edges of the two components (AI and B t ) of the discriminant and the projection of the intersection of these components. , EXAMPLE. The caustic of the singularity F4 is shown in fig. 38. Fig. 38 - The caustic of the reflection group F4 . Let us consider a generic surface with a generic boundary in Euclidean 3-space.

Givental [11] has proved that this set of isotropic mappings is open in the set of all isotropic mappings, but not the density. 9 - The singularities of the projections and of the apparent contours Let us consider a generic smooth surface in the usual 3-space; The shape we see depends on the point of view. What we see is the apparent contour of the projection to the plarle (to the retina surface). 2). Consequently, an apparent contour has generically no singularities besides the semicubical cusp points (and the transversal self-intersection points).

The union of these three surfaces is locally diffeomorphic to the caustic of F4 (at a neighbourhood of the curvature centre on the normal to the surface at the point of tangency of the boundary with the curvature line), the surface and the boundary being generic (LG. Shcherbak [15]). 2 - Stability boundary A system of linear differential equation x = Ax is stable, if the real parts of the eigenvalues of the Operator A are all negative. The stable systems form the stability domain in the space of linear operators A.

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