Codimension one embeddings of product of three spheres

Let f: S-P X S-q X S-r --> S-p+q+1 be a smooth embedding with 1 less than or equal to p less than or equal toq less than or equal to r. For p greater than or equal to 2, the authors have shown that if p + q not equivalent tor, or p + q = r and r is even, then the closure of one of the two components of Sp+q+r+l (_) integral(S-p X S-q x S-r) is diffeomorphic to the product of two spheres and a disk, and that otherwise, there are infinitely many embeddings, called exotic embeddings, which do not satisfy such a property. In this paper, we study the case p = I and construct infinitely many exotic embeddings. We also give a positive result under certain (co)homological hypotheses on the complement. Furthermore, we study the case (p, q, r) = (1, 1, 1) more in detail and show that the closures of the two components of S-4 - integral (S-1 x S-1 x S-1) are homeomorphic to the exterior of an embedded solid torus or Montesinos' twin in S-4. (C) 2004 Elsevier B.V. All rights reserved.