Bordism and regular homotopy of low-dimensional immersions
type: Item
instance of: scholarly article
title: Bordism and regular homotopy of low-dimensional immersions
zbMATH Open document ID: 0768.57014
author: John Hughes
published in: Pacific Journal of Mathematics
publication date: 1993-05-16T00:00:00Z
review text: To obtain explicit generators for the groups of regular homotopy classes of immersions of the \(k\)-sphere in \(\mathbb{R}^ n\), the author defines an immersion invariant as an element of the \(k\)-th homotopy group of a Stiefel manifold. Using this invariant, he obtains generators for some groups of low-dimensional immersions. Among other things, he shows: Theorem. The standard (Froissart-Morin) eversion of \(S^ 2\) in \(\mathbb{R}^ 3\) has, as a track, an immersion of \(S^ 2\times [0,1]\) in \(\mathbb{R}^ 4\) whose ends are embedded \(S^ 2\text{s}\). Each of these bounds a 3-ball in \(\mathbb{R}^ 4\). Capping off the track with these 3-balls yields an immersion \(K: S^ 3\to \mathbb{R}^ 4\). Performing the eversion twice and capping off gives an immersion \(E: S^ 3\to \mathbb{R}^ 4\). The immersions \(E\) and \(K\) generate the group of regular homotopy classes of immersions of \(S^ 3\) in \(\mathbb{R}^ 4\).
Mathematics Subject Classification ID: 57R42
zbMATH DE Number: 168269
zbMATH Keywords: generators for the groups of regular homotopy classes of immersions of the \(k\)-sphere in \(\mathbb{R}^ n\)
zbMATH Keywords: homotopy group of a Stiefel manifold
zbMATH Keywords: immersions of \(S^ 3\) in \(\mathbb{R}^ 4\)
reviewed by: Katsuhiro Komiya
MaRDI profile type: Publication
full work available at URL: pjm.1992.156.155
OpenAlex ID: W2035274562
DOI: 10.2140/PJM.1992.156.155
description: scientific article; zbMATH DE number 168269