Classification of Knotted tori

Many interesting examples of embeddings are embeddings S^p x S^q -> R^m, i.e. knotted tori . A classification of knotted tori is a natural next step (after the link theory and the classification of embeddings of highly-connected manifolds ) towards classification of embeddings of arbitrary manifolds. Since the general Knotting Problem is very hard, it is very interesting to solve it for the important particular case of knotted tori. Recent classification results for knotted tori give some insight or even precise information concerning arbitrary manifolds and reveal new interesting relations to algebraic topology.
A description of the set of smooth isotopy classes of smooth embeddings S^p x S^q -> R^m was known only for p=0, m>q+2 or for 2m>3p+3q+3. For m>2p+q+2 we introduce a group structure on this set and describe this group up to an extension problem (in terms of homotopy groups of spheres and Stiefel manifolds). In the proof we use a recent exact sequence of M. Skopenkov. www.mccme.ru/~skopenko