Embeddings of S^p × S^q × S^r in S^p+q+r+1

Laércio Aparecido Lucas and Osamu Saeki

Vol. 207 (2002), No. 2, 447–462

DOI: 10.2140/pjm.2002.207.447

Abstract

Let f : S^p × S^q × S^r → S^p+q+r+1,2 ≤ p ≤ q ≤ r, be a smooth embedding. In this paper we show that the closure of one of the two components of S^p+q+r+1 −f(S^p ×S^q ×S^r), denoted by C₁, is diffeomorphic to S^p ×S^q ×D^r+1 or S^p ×D^q+1 ×S^r or D^p+1 ×S^q ×S^r, provided that p + q≠r or p + q = r with r even. We also show that when p + q = r with r odd, there exist infinitely many embeddings which do not satisfy the above property. We also define standard embeddings of S^p × S^q × S^r into S^p+q+r+1 and, using the above result, we prove that if C₁ has the homology of S^p × S^q, then f is standard, provided that q < r.

Milestones

Published: 1 December 2002