Codimension one embeddings of product of three spheres

Abstract

Let f :S p×S q×S r→S p+q+r+1 be a smooth embedding with 1⩽ p⩽ q⩽ r. For p⩾2, the authors have shown that if p+ q≠ r, or p+ q= r and r is even, then the closure of one of the two components of S p+ q+ r+1 − f( S p × S q × 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=1 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− f( S 1× S 1× S 1) are homeomorphic to the exterior of an embedded solid torus or Montesinos' twin in S 4.