On embedding lifts over a Morse function on a circle : Dedicated to Professor Shyuich Izumiya on the occasion of his 60th birthday (Singularity theory, geometry and topology)
Minoru Yamamoto · Institutional Repositories DataBase (IRDB) · 2013
Let f : S^{ 1} \ri ght arrow \mat hbb{ R} be a Morse function and $\Pi$ : \mat hbb{ R} ^{ 2} \r i ght ar r ow \mat hbb{ R} an orthogonal projection.In [4], Saeki and Takase posed the following problem: Determine those Morse functions f : S^{ 1} \ri ght arrow \mat hbb{ R} which have an embedding \t i l de{ f } : S^{ 1} \r i ght ar r ow \mat hbb{ R} ^{ 2} such that $\Pi$ 0\tilde{f}=f .In this paper, we give a complete answer to this problem and give an application to the existence problem of embedding lifts for fold maps.y around f(q)\in \mathbb{R} such that f has the form yo f=\pm x^{2} .If the singular values of a Morse function f are all distinct, then we call f a stable Morse function.Let f : S^{1}\ri ghtarrow \mathbb{R} be a Morse function and $\Pi $ : \mat hbb{ R} ^{ 2} \r i ght ar r ow \mat hbb{ R} the orthogonal projection defined by $\Pi$(y_{1}, y_{2})=y_{1} .Saeki and Takase [4] pose the following problem: Determine those Morse functions f : S^{1}\ri ghtarrow \mathbb{R} which have an embedding \ t i l d e { f } : S^{1}\ri ght arrow \mat hbb{R}^{2} such that $\Pi$\circ\tilde{f}=f .We call such a map \t i l de{ f } : S^{ 1} \r i ght ar r ow \mat hbb{ R} ^{ 2} an embedding lift over f .In this paper, we give a complete answer to this problem.This paper is organized as follows.In Section 2, we give a necessary and sufficient condition for the existence of an embedding lift \t i l de{ f } : S^{ 1} \r i ght ar r ow \mat hbb{ R} ^{ 2} over a given Morse function