TY - JOUR

T1 - Codimension one embeddings of product of three spheres

AU - Lucas, Laércio Aparecido

AU - Saeki, Osamu

N1 - Funding Information:
* Corresponding author. E-mail addresses: llucas@siteplanet.com.br (L.A. Lucas), saeki@math.kyushu-u.ac.jp (O. Saeki). 1 The author has been supported in part by Grant-in-Aid for Scientific Research (No. 13640076), Ministry of Education, Science and Culture, Japan.

PY - 2005/1/1

Y1 - 2005/1/1

N2 - Let f:Sp×Sq×Sr→ Sp+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 Sp+q+r+1-f(Sp×Sq× Sr) 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 S4-f(S1× S1×S1) are homeomorphic to the exterior of an embedded solid torus or Montesinos' twin in S4.

AB - Let f:Sp×Sq×Sr→ Sp+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 Sp+q+r+1-f(Sp×Sq× Sr) 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 S4-f(S1× S1×S1) are homeomorphic to the exterior of an embedded solid torus or Montesinos' twin in S4.

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U2 - 10.1016/j.topol.2003.06.005

DO - 10.1016/j.topol.2003.06.005

M3 - Article

AN - SCOPUS:10144228411

VL - 146-147

SP - 409

EP - 419

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -