Classification of knotted tori

Abstract

For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to {\mathbb{R}}^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge 2(p+q)+\max \{p,q\}+4$. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For $m\ge 2p+q+3$ we introduce an abelian group structure on $E^m(S^p\times S^q)$ and describe this group `up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe $E^m(S^p\times S^q)$. The proof is based on relations between sets $E^m(N)$ for different $N$ and $m$, in particular, on a recent exact sequence of M. Skopenkov.

29 pages, 3 figures, exposition improved, references updated, previous \S2.3 replaced by reference to arXiv:2406.15367