Lifting a generic map of a surface into the plane to an embedding into 4-space
The main theorem(Theorem 1.1) states that if \(M\) is a closed surface and \(\pi^2_2:{\mathbb R}^4(={\mathbb R}^2\times {\mathbb R}^2)\to {\mathbb R}^2\) is the orthogonal projection, then any stable map \(f:M\to {\mathbb R}^2\) has an embedding lift into \({\mathbb R}^4\). This is a partial answer to the problem proposed at Remark 5.5 in [\textit{V. L. Carrara, M. A. S. Ruas} and \textit{O. Saeki}, Topology Appl. 110, No. 3, 265--287 (2001; Zbl 0971.57036)]. The introduction of this paper contains some related topics to embedding lifts and the last section, \S 4, presents two examples of the constructions along the lines of the proof. The idea of the proof is as follows: For a map \(f\) mentioned above, there exists a generic projection \(\pi :{\mathbb R}^2\to {\mathbb R}^1\) such that \(\pi f:M\to {\mathbb R}^1\) is a stable Morse function [\textit{J. N. Mather}, Math. 98, 226--245 (1973; Zbl 0242.58001)]. Given a closed interval \([s,t]\) containing one bifurcation value in its interior and an embedding lift \(F_s:(\pi f)^{-1}(s)\to \pi^{-1}(s)\times {\mathbb R}^2\), the author construct an embedding lift \((\pi f)^{-1}([s,t])\to \pi^{-1}([s,t])\times {\mathbb R}^2\) extending \(F_s\) by attaching a handle depending on the local form of a bifurcation point in \(\pi^{-1}([s,t])\).
scientific article; zbMATH DE number 5316787