The rational classification of links of codimension >2

TL;DR

This paper determines the rank and finiteness of the group of links of codimension greater than 2 with specified multi-indices, extending previous results and using Lie algebra theory.

Contribution

It provides a general formula for the rank of the link groups E(p, m) and characterizes when these groups are finite, generalizing prior work by Haefliger.

Findings

Calculated the rank of E(p, m) for general p and m

Established conditions for the finiteness of E(p, m)

Extended results to framed links

Abstract

Fix an integer m and a multi-index p = (p_1, ..., p_r) of integers p_i < m-2. The set of links of codimension > 2, with multi-index p, E(p, m), is the set of smooth isotopy classes of smooth embeddings of the disjoint union of the p_i-spheres into the m-sphere. Haefliger showed that E(p, m) 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 p the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group E(p, m) in general. In particular we determine precisely when E(p,m) 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.