STUDYING THE TOPOLOGY OF MORIN SINGULARITIES FROM A GLOBAL VIEWPOINT

Let f: M --> N be a smooth map of a closed n-manifold into a p-manifol d (n greater than or equal to p) having only Morin singularities [17]. We study the topology of such a map and obtain a modulo 2 congruence formula involving the Euler characteristics of M, N, the singular sets and the regular fibres of f. We also consider applications of this fo rmula to the existence problem of maps having only fold singular point s. Stable maps into 3-manifolds are also studied and we obtain a modul e 2 congruence formula involving the swallow tails and the number of t riple points of the discriminant set.