REAL MAP-GERMS WITH GOOD PERTURBATIONS

THIS paper is concerned with the relation between the topology of cert ain real algebraic sets and that of their complexification. Our aim is to determine for which singularities of mappings from surfaces to 3-s pace can the changes in the homology of the complex image resulting fr om a deformation of the mapping be observed in the real image. More pr ecisely, we determine all right-left equivalence classes of map-germs C-2, 0 --> C-3, 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 per turbation''). In fact, the only such classes are the singularities S-1 and H-k (k greater than or equal to 2) (see below for their definitio n). We exhibit real stable perturbations of these with the required pr operty, and give drawings of their images in R(3) (Section 3). This re lative scarcity of singularities with good real perturbations is in sh arp contrast to the case of map-germs R,0 --> R(2),0; here it was show n by A'Campo and Gusein-Sade (independently) in [1] and [6] that such stable perturbations always exist.