Lifting a generic map of a surface into the plane to an embedding into 4-space
type: Item
instance of: scholarly article
title: Lifting a generic map of a surface into the plane to an embedding into 4-space
zbMATH Open document ID: 1154.57027
author: Minoru Yamamoto
published in: Illinois Journal of Mathematics
publication date: 2008-08-26T00:00:00Z
review text: 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 \(\pi^{-1}([s,t])\).
reviewed by: Tsutomu Yasui
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 5316787
zbMATH Keywords: embedding
zbMATH Keywords: lift
zbMATH Keywords: singularity
zbMATH Keywords: bifurcation point
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5316787