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.

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