### Abstract

We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into 5-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of 3-manifolds M
^{3}
, the following phenomenon occurs: there exists a codimension two immersion of the 3-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of M
^{3}
. This involves introducing and studying an equivalence relation on the set of spin structures on M
^{3}
. Their associated μ-invariants also play an important role.

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

Pages (from-to) | 5049-5061 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 354 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 1 2002 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*354*(12), 5049-5061. https://doi.org/10.1090/S0002-9947-02-03070-2

**Spin structures and codimension two embeddings of 3-manifolds up to regular homotopy.** / Saeki, Osamu; Takase, Masamichi.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 354, no. 12, pp. 5049-5061. https://doi.org/10.1090/S0002-9947-02-03070-2

}

TY - JOUR

T1 - Spin structures and codimension two embeddings of 3-manifolds up to regular homotopy

AU - Saeki, Osamu

AU - Takase, Masamichi

PY - 2002/12/1

Y1 - 2002/12/1

N2 - We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into 5-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of 3-manifolds M 3 , the following phenomenon occurs: there exists a codimension two immersion of the 3-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of M 3 . This involves introducing and studying an equivalence relation on the set of spin structures on M 3 . Their associated μ-invariants also play an important role.

AB - We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into 5-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of 3-manifolds M 3 , the following phenomenon occurs: there exists a codimension two immersion of the 3-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of M 3 . This involves introducing and studying an equivalence relation on the set of spin structures on M 3 . Their associated μ-invariants also play an important role.

UR - http://www.scopus.com/inward/record.url?scp=0036898325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036898325&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-02-03070-2

DO - 10.1090/S0002-9947-02-03070-2

M3 - Article

VL - 354

SP - 5049

EP - 5061

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -