Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 10.05.2002, 16:00--17:45, room 311

Knotted Tori
A. Skopenkov

The theory of knotted tori (i. e. isotopy classes of
embeddings $S^p\times S^q\to R^m$) is interesting because it is a
generalization of an important classical theory of links (with two
components of the same dimension). Just as the link theory, the theory of
knotted tori provides interesting examples (of Alexander, Hudson,
Haefliger, Milgram-Rees, Tindell and the author) and new relations with
algebraic topology. Classification of knotted tori can also be considered
as a step towards classification of isotopy classes of arbitrary manifolds   
in $R^m$ (by the Handle Decomposition Theorem). 

Classification theorems for knotted tori (Hudson, Haefliger-Hirsch and the  
author) will be stated. A group structure on the set of knotted tori,  
generators of this set and invariants distinguishing knotted tori will
explicitly be constructed. Many specific corollaries will be presented
(e. g. that $S^1\times S^q$ is unknotted in $\R^{2q-1}$ for $q>5$). Some
open problems in this area will be posed. 

Knowledge of basic notions of topology will be sufficient to understand
main ideas of the talk. Although mostly higher-dimensional knotted
tori will be studied, low-dimensional illustrations and analogues will
always be presented.