, Volume 57, Issue 3, pp 235-247

Separation by a codimension-1 map with a normal crossing point

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Letf:M n−1N n be an immersion with normal crossings of a closed orientable (n−1)-manifold into an orientablen-manifold. We show, under a certain homological condition, that iff has a multiple point of multiplicitym, then the number of connected components ofN−f(M) is greater than or equal tom+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 multiplicitym. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.