Классификация вложений торов в 2-метастабильной размерности@@@Classification of knotted tori in the 2-metastable dimension

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings N->S^m. We study the specific case of knotted tori, i. e. the embeddings S^p x S^q -> S^m. The classification of knotted tori up to isotopy in the metastable dimension range m>p+3q/2+3/2, pmetastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that p+4q/3+2 2p+q+2. Then the set of smooth embeddings S^p x S^q -> S^m up to isotopy is infinite if and only if either q+1 or p+q+1 is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new beta-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.