The rational classification of links of codimension > 2

Diarmuid J. Crowley; Steven C. Ferry; Mikhail Skopenkov

doi:10.1515/FORM.2011.158

The rational classification of links of codimension > 2

Diarmuid J. Crowley, Steven C. Ferry, Mikhail Skopenkov

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let m and p1,.,pr < m - 2 be positive integers. The set of links of codimension > 2, Em(∐k=1 rSPk), is the set of smooth isotopy classes of smooth embeddings ∐k=1 rSPk → Sm. Haefliger showed that Em(∐k=1 rSPk) is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r = 1. For r > 1 and for restrictions on p1,.,pr the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group Em(∐k=1 rSPk) in general. In particular we determine precisely when Em(∐k=1 rSPk) is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras. © de Gruyter 2014.

Original language	English (US)
Pages (from-to)	239-269
Number of pages	31
Journal	Forum Mathematicum
Volume	26
Issue number	1
DOIs	https://doi.org/10.1515/FORM.2011.158
State	Published - Jan 1 2014

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The third author was supported in part by INTAS grant 06-1000014-6277, Moebius Contest Foundation for Young Scientists and Euler Foundation.

ASJC Scopus subject areas

Applied Mathematics
General Mathematics

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10.1515/FORM.2011.158

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title = "The rational classification of links of codimension > 2",

abstract = "Let m and p1,.,pr < m - 2 be positive integers. The set of links of codimension > 2, Em(∐k=1 rSPk), is the set of smooth isotopy classes of smooth embeddings ∐k=1 rSPk → Sm. Haefliger showed that Em(∐k=1 rSPk) is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r = 1. For r > 1 and for restrictions on p1,.,pr the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group Em(∐k=1 rSPk) in general. In particular we determine precisely when Em(∐k=1 rSPk) is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras. {\textcopyright} de Gruyter 2014.",

author = "Crowley, {Diarmuid J.} and Ferry, {Steven C.} and Mikhail Skopenkov",

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AU - Skopenkov, Mikhail

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N2 - Let m and p1,.,pr < m - 2 be positive integers. The set of links of codimension > 2, Em(∐k=1 rSPk), is the set of smooth isotopy classes of smooth embeddings ∐k=1 rSPk → Sm. Haefliger showed that Em(∐k=1 rSPk) is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r = 1. For r > 1 and for restrictions on p1,.,pr the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group Em(∐k=1 rSPk) in general. In particular we determine precisely when Em(∐k=1 rSPk) is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras. © de Gruyter 2014.

AB - Let m and p1,.,pr < m - 2 be positive integers. The set of links of codimension > 2, Em(∐k=1 rSPk), is the set of smooth isotopy classes of smooth embeddings ∐k=1 rSPk → Sm. Haefliger showed that Em(∐k=1 rSPk) is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r = 1. For r > 1 and for restrictions on p1,.,pr the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group Em(∐k=1 rSPk) in general. In particular we determine precisely when Em(∐k=1 rSPk) is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras. © de Gruyter 2014.

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The rational classification of links of codimension > 2

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