### 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}.

Original language | English |
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Pages (from-to) | 409-419 |

Number of pages | 11 |

Journal | Topology and its Applications |

Volume | 146-147 |

DOIs | |

Publication status | Published - Jan 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*146-147*, 409-419. https://doi.org/10.1016/j.topol.2003.06.005

**Codimension one embeddings of product of three spheres.** / Lucas, Laércio Aparecido; Saeki, Osamu.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 146-147, pp. 409-419. https://doi.org/10.1016/j.topol.2003.06.005

}

TY - JOUR

T1 - Codimension one embeddings of product of three spheres

AU - Lucas, Laércio Aparecido

AU - Saeki, Osamu

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.

UR - http://www.scopus.com/inward/record.url?scp=10144228411&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10144228411&partnerID=8YFLogxK

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 -