Cobordism of Morse maps and its application to map germs

摘要

Let f : M S~l be a Morse map of a closed manifold M into the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifold M has boundary and the map f restricted to the boundary has no critical points. Let g: (R~m, 0) - (R~2, 0), m > 2, be a generic smooth map germ, where the target R~2 is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in a C~°° stable perturbation of g, counted with signs, gives a topological invariant of g.