Homotopy type of the complement of an immersion and classification of embeddings of tori

TL;DR

This paper investigates the classification of high-dimensional knotted tori embeddings, introducing an approach that extends results to lower dimensions beyond classical methods.

Contribution

It presents a new approach to classify embeddings of tori in lower dimensions, expanding the understanding beyond the metastable range.

Findings

Explicit description of knotted tori in high dimensions

Extension of classification results to lower dimensions

New methods for embedding theory analysis

Abstract

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.