Let f:Mn-1→Nn be an immersion with normal crossings of a closed orientable (n-1)-manifold into an orientable n-manifold. We show, under a certain homological condition, that if f has a multiple point of multiplicity m, then the number of connected components of N-f(M) is greater than or equal to m+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math.36 (1992), 500-504) and Biasi, Motta and Saeki (Topology Appl.52 (1993), 81-87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicity m. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.
All Science Journal Classification (ASJC) codes
- Geometry and Topology