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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)