When is the set of embeddings finite up to isotopy?

TL;DR

This paper investigates when the set of isotopy classes of embeddings of a product of spheres into a higher-dimensional sphere is finite or infinite, providing new criteria based on parity and algebraic conditions.

Contribution

It introduces a novel approach using a group structure and an exact sequence to classify embeddings of S^p x S^q into S^m, extending previous results to new cases.

Findings

The set of embeddings is infinite under specific divisibility conditions.

A new algebraic criterion involving the set FCS determines infiniteness.

The classification reduces to rational homotopy problems.

Abstract

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddings N->S^m finite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS(i,j) is a concrete subset of the integer lattice depending only on the parity of i and j which is defined in the paper. Theorem. Assume that m>2p+q+2 and m<p+3q/2+2. Then the set of isotopy classes of smooth embeddings S^p x S^q -> S^m is infinite if and only if either q+1 or p+q+1 is divisible by 4, or there exists a point (x,y) in the set FCS(m-p-q,m-q) such that (m-p-q-2)x+(m-q-2)y=m-3. Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings S^p x S^q -> S^m to the classification of embeddings S^{p+q} |_| S^q -> S^m and D^p x S^q -> S^m. The latter classification problems are reduced to homotopy ones, which are solved rationally.