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Homotopy type of the complement of an immersion and classification of embeddings of tori. (English. Russian original) Zbl 1141.57009
Russ. Math. Surv. 62, No. 5, 985-987 (2007); translation from Usp. Mat. Nauk 62, No. 5, 165-166 (2007).
From the text: We study embeddings \(S_p\times S_q\to S_m\). Such embeddings will be called knotted tori. Links are a classical special case of knotted tori. The investigation of knotted tori is a natural next step after knots and links because of the handle decomposition of an arbitrary manifold.
The set of knotted tori in spaces of sufficiently high dimension, namely, in the metastable range \(m\geq p + 3q/2 + 2\), \(p\leq q\), was explicitly described in [A. Skopenkov, Comment. Math. Helv. 77, No. 1, 78–124 (2002; Zbl 1012.57035)]. The metastable range is a natural bound for the applicability of classical methods in embedding theory. The aim of this note is to present an approach making it possible to obtain results in lower dimensions.

MSC:
57N35 Embeddings and immersions in topological manifolds
57R40 Embeddings in differential topology
57R52 Isotopy in differential topology
55Q52 Homotopy groups of special spaces
55P10 Homotopy equivalences in algebraic topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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