Complementary Regions for Maps of Surfaces

Abstract

Let F be a closed connected surface, M a closed connected 3-manifold with H1(M,ℤ/2) = 0, and i: F → M a generic map. Then M - i(F) is a union of connected regions, which may be colored black and white by a checkerboard coloring. This coloring induces a color black or white to each cross-cap of i, namely, the color of the majority of the three local regions in its neighborhood. For k ≥ 0, let ak and bk respectively, be the number of black and white components U, with χ(U) = 1 - k. Let Ca, Cb respectively be the number of black and white cross-caps of i. Two more integers attached to i are the number N of triple points of i, and χ = χ(F). In this work, we determine what sets of data ({ak}, {bk}, χ, N, Ca, Cb) may appear in this way.

Dissecting the 2-sphere by immersions

“…His answer was that the number of triple points may be any number which is equal mod 2 to χ(F). Extensions of this result in various directions appear in [7][8][9][10]. Li in [11] asked what are the possible graphs in R 3 that may appear as the intersection set of such immersion.…”