### 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 C_{1}, 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 C_{1} has the homology of S^{p} × S^{q}, then f is standard, provided that q < r.

Original language | English |
---|---|

Pages (from-to) | 447-462 |

Number of pages | 16 |

Journal | Pacific Journal of Mathematics |

Volume | 207 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

^{p}× S

^{q}× S

^{r}in S

^{p+q+r+1}

*Pacific Journal of Mathematics*,

*207*(2), 447-462. https://doi.org/10.2140/pjm.2002.207.447

**Embeddings of S ^{p} × S^{q} × S^{r} in S^{p+q+r+1} .** / Lucas, Laércio Aparecido; Saeki, Osamu.

Research output: Contribution to journal › Article

^{p}× S

^{q}× S

^{r}in S

^{p+q+r+1}',

*Pacific Journal of Mathematics*, vol. 207, no. 2, pp. 447-462. https://doi.org/10.2140/pjm.2002.207.447

^{p}× S

^{q}× S

^{r}in S

^{p+q+r+1}Pacific Journal of Mathematics. 2002 Jan 1;207(2):447-462. https://doi.org/10.2140/pjm.2002.207.447

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TY - JOUR

T1 - Embeddings of Sp × Sq × Sr in Sp+q+r+1

AU - Lucas, Laércio Aparecido

AU - Saeki, Osamu

PY - 2002/1/1

Y1 - 2002/1/1

N2 - Let f: Sp × Sq × Sr → Sp+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 Sp+q+r+1 - f(Sp × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, 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 Sp × Sq × Sr into Sp+q+r+1 and, using the above result, we prove that if C1 has the homology of Sp × Sq, then f is standard, provided that q < r.

AB - Let f: Sp × Sq × Sr → Sp+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 Sp+q+r+1 - f(Sp × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, 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 Sp × Sq × Sr into Sp+q+r+1 and, using the above result, we prove that if C1 has the homology of Sp × Sq, then f is standard, provided that q < r.

UR - http://www.scopus.com/inward/record.url?scp=0036975403&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036975403&partnerID=8YFLogxK

U2 - 10.2140/pjm.2002.207.447

DO - 10.2140/pjm.2002.207.447

M3 - Article

AN - SCOPUS:0036975403

VL - 207

SP - 447

EP - 462

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -