# oriented cobordism

Two oriented $n$-manifolds^{} $M$ and ${M}^{\prime}$ are called cobordant if there is an oriented $n+1$ manifold with boundary $N$ such that $\partial N=M\coprod {M}^{\prime opp}$ where ${M}^{\prime opp}$ is ${M}^{\prime}$ with orientation reversed. The triple $(N,M,{M}^{\prime})$ is called a oriented cobordism. Cobordism is an equivalence relation^{}, and a very coarse invariant of manifolds. For example, all surfaces are cobordant to the empty set^{} (and hence to each other).

There is a cobordism category, where the objects are manifolds, and the morphisms are cobordisms between them. This category is important in topological .

Title | oriented cobordism |
---|---|

Canonical name | OrientedCobordism |

Date of creation | 2013-03-22 13:56:05 |

Last modified on | 2013-03-22 13:56:05 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 57N70 |

Classification | msc 57Q20 |

Synonym | cobordant |

Synonym | bordism |