This paper is devoted to the classical Knotting Problem: for a given manifold N and number m describe the set of isotopy classes of embeddings N → Sm. We study the specific case of knotted tori, that is, the embeddings Sp × Sq → Sm. The classification of knotted tori up to isotopy in the metastable dimension range m > p + 3 2 q + 2, p 6 q, was given by Haefliger, Zeeman and A. Skopenkov. We consider the dimensions below the metastable range and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that p+ 4 3 q +2 < mp+ 3 2 q +2 and m > 2p+q +2. Then the set of isotopy classes of smooth embeddings Sp × Sq → Sm is infinite if and only if either q + 1 or p + q + 1 is divisible by 4. © 2012 RAS(DoM) and LMS.
ASJC Scopus subject areas
- Algebra and Number Theory