Embeddings of Sp× Sq× Srin Sp+q+r+1

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 x S r ), denoted by C 1 , is diffeomorphic to S p × S q x 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 C 1 has the homology of S p x S q , then f is standard, provided that q < r.