Real map-germs with good perturbations

This paper is concerned with the relation between the topology of certain real algebraic sets and that of their complexification. m is to determine for which singularities of mappings from surfaces to 3-space can the changes in the homology of the complex image resulting from a deformation of the mapping be observed in the real image. More precisely, we determine all right-left equivalence classes of map-germs C2, 0 → C3, 0 for which it is possible to find a real form with a real stable perturbation whose image carries the vanishing cohomology of the image of a complex stable perturbation (thus, a “good real perturbation”). In fact, the only such classes are the singularities S1 and Hk (k ≥ 2) (see below for their definition). We exhibit real stable perturbations of these with the required property, and give drawings of their images in R3 (Section 3). elative scarcity of singularities with good real perturbations is in sharp contrast to the case of map-germs R, 0 → R2, 0; here it was shown by AʹCampo and Gusein-Sade (independently) in [1] and [6] that such stable perturbations always exist.