×

zbMATH — the first resource for mathematics

Classification of knotted tori in 2-metastable dimension. (English. Russian original) Zbl 1263.57019
Sb. Math. 203, No. 11, 1654-1681 (2012); translation from Mat. Sb. 203, No. 11, 129-154 (2012).
The classical Knotting Problem asks us how we can describe the set of isotopy classes of embeddings of a given manifold \(N\) into an \(m\)-dimensional sphere \(S^m\). For the specific case of knotted tori, i.e. when \(N=S^p \times S^q\), the classification of isotopy classes is explicitly described for the metastable dimension range \(m\geq p + 3q/2 + 2\), \(p\leq q\); see A. Skopenkov [Comment. Math. Helv. 77, No. 1, 78–124 (2002; Zbl 1012.57035)]. In this paper, the authors consider the classification of isotopy classes of knotted tori in the 2-metastable dimension, i.e. the dimension satisfying \(p+4q/3+2<m<p+3q/2+2\) and \(m>2p+q+2\), and show the result announced by the authors [Russ. Math. Surv. 62, No. 5, 985–987 (2007); translation from Usp. Mat. Nauk 62, No. 5, 165–166 (2007; Zbl 1141.57009)].
The main result is as follows: On the assumption that \(p+4q/3+2<m<p+3q/2+2\) and \(m>2p+q+2\), the set of isotopy classes of smooth embeddings \(S^p \times S^q \to S^m\) is infinite if and only if either \(q+1\) or \(p+q+1\) is divisible by \(4\). The proof is based on an analogue of the exact sequence by U. Koschorke [Math. Ann. 286, No. 4, 753–782 (1990; Zbl 0662.57013)], involving a new invariant called the \(\beta\)-invariant of almost embeddings \(S^p \times S^q \to S^m\). This \(\beta\)-invariant is a generalization of the normal bordism \(\beta\)-invariant of link maps \(S^q \bigsqcup S^{p+q} \to S^m\). The exactness of the sequence is shown by studying the complement of an almost embedding and showing the completeness of the \(\beta\)-invariant, using analogous techniques from link map theory by N. Habegger and U. Kaiser [Topology 37, No. 1, 75–94 (1998; Zbl 0890.57036)].

MSC:
57Q35 Embeddings and immersions in PL-topology
55S37 Classification of mappings in algebraic topology
57Q60 Cobordism and concordance in PL-topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI