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The rational classification of links of codimension \(> 2\). (English) Zbl 1300.57032

For positive integers \(m\) and \(p_1, \ldots, p_r<m-2\), the set of links in codimension \(>2\) is the set \(E^m \left( \sqcup_{k=1}^r S^{p_k} \right)\) of smooth isotopy classes of smooth embeddings \(\sqcup_{k=1}^r S^{p_k} \hookrightarrow S^m\). This set is a finitely generated abelian group with respect to componentwise embedded connected summation [A. Haefliger, Topology 1, 241–244 (1962; Zbl 0108.18201)]. For the case \(r=1\), the rank was determined by A. Haefliger [Ann. Math. (2) 83, 402–436 (1966; Zbl 0151.32502)], and it equals \(0\) or \(1\).
The main results are as follows. The authors give an explicit formula for the rank of the abelian group \(E^m \left( \sqcup_{k=1}^r S^{p_k} \right)\), based on which there is a computer application available for the computation of the rank. They give a criterion which determines when the group \(E^m \left( \sqcup_{k=1}^r S^{p_k} \right)\) is finite. They give the similar results for the groups of framed links. They also give applications to the classification of handlebodies and to the computation of certain mapping class groups. The paper is closed by several open questions. The proofs are shown as follows. The group of links \(E^m \left( \sqcup_{k=1}^r S^{p_k} \right)\) splits as the sum of the groups of Brunnian links, for which there is the Haefliger exact sequence [A. Haefliger, Comment. Math. Helv. 41, 51–72 (1966; Zbl 0149.20801)]. They express the homotopy groups in the sequence in terms of the homotopy groups of spheres by using the Hilton-Milnor Theorem, and they convert the problem in terms of Lie algebras.

MSC:

57R52 Isotopy in differential topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
55P62 Rational homotopy theory
17B01 Identities, free Lie (super)algebras
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