Classification of knotted tori in 2-metastable dimension

D. Repovš ; M. B. Skopenkov ; M. Cencelj

Sbornik. Mathematics, Tome 203 (2012) no. 11, pp. 1654-1681

Résumé

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 $N\to S^m$. We study the specific case of knotted tori, that is, the embeddings $S^p\times S^q\to S^m$. The classification of knotted tori up to isotopy in the metastable dimension range $m\geqslant p+\frac32q+2$, $p\leqslant q$, 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: \medskip Theorem Assume that $p+\frac43q+2$ and $m>2p+q+2$. Then 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 $4$. Bibliography: 35 titles.

Détail

Keywords: knotted torus, link, link map, embedding, surgery.