O. Saeki, STUDYING THE TOPOLOGY OF MORIN SINGULARITIES FROM A GLOBAL VIEWPOINT, Mathematical proceedings of the Cambridge Philosophical Society, 117, 1995, pp. 223-235
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.