The classification of certain linked 3-manifolds in 6-space
type: Item
instance of: scholarly article
zbMATH Open document ID: 1347.57026
publication date: 2016-05-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.3918
full work available at URL: 2016-016-001-001.html
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 55Q10
zbMATH DE Number: 6576751
zbMATH Keywords: classification of embeddings
zbMATH Keywords: linked manifolds
zbMATH Keywords: framed cobordism
MaRDI profile type: Publication
arXiv ID: 1408.3918
arXiv classification: math.GT
author: S. N. Avvakumov
title: The classification of certain linked 3-manifolds in 6-space
published in: Moscow Mathematical Journal
review text: \(S^1 \sqcup S^1\) embedded in \(\mathbb R^3\) is usually called a 1-link and \(S^p \sqcup S^p \to \mathbb R^q\) is a \(p\)-link in \(\mathbb R^q\). A very hard general question is the isotopy classification of links, so one puts additional conditions on the embeddings, for example that the embeddings are unknotted in both components; these are called Brunnian embeddings. A. Haefliger obtained a classification of \(S^3 \sqcup S^3 \to S^6\). The present paper treats smooth Brunnian embeddings of \((S^2 \times S^1) \sqcup S^3 \to S^6\) and simultaneously \(S^3 \sqcup S^3 \to S^6\). In this situation a Brunnian embedding is one that is isotopic to a standard one. Therefore, the paper brings explicit constructions of standard embeddings, denoted \(f_{k,m,n}\) in the \((S^2 \times S^1) \sqcup S^3 \to S^6\) case and and \(g_{k,m,n}\) in the case \(S^3 \sqcup S^3 \to S^6\), where \(k,m,n\) are integers such that \(m \equiv n\) (mod 2). Two standard embeddings \(f_{k,m,n}\) and \(f_{k',m',n'}\) are isotopic iff \(k = k', m \equiv m'\) mod (2\(k\)) and \(n \equiv n'\) mod (2\(k\)).NEWLINENEWLINEResults: 1. Any Brunnian embedding \(f: (S^2 \times S^1) \sqcup S^3 \to S^6\) is isotopic to some \(f_{k,m,n}\) such that \(m \equiv n\) mod (2) and two such embeddings are isotopic iff \(k=k'\), \(m \equiv m'\) mod (2\(k\)) and \(n \equiv n'\) mod (2\(k\)). 2. Any Brunnian embedding \(g: S^3\sqcup S^3 \to S^6\) is isotopic to some \(g_{m,n}\) such that \(m \equiv n\) mod (2) and two such embeddings are isotopic iff \(m = m'\) and \(n = n'\).
reviewed by: Ivan Ivanšić
description: scientific article; zbMATH DE number 6576751