# Poincaré duality

If $M$ is a compact^{}, oriented, $n$-dimensional manifold, then there is a canonical (though non-natural (http://planetmath.org/NaturalTransformation)) isomorphism^{}

$$D:{H}^{q}(M,\mathbb{Z})\to {H}_{n-q}(M,\mathbb{Z})$$ |

(where ${H}^{k}(M,\mathbb{Z})$ is the $k$th homology group^{} of $M$ with integer coefficients and ${H}_{k}(M,\mathbb{Z})$ the $k$th cohomology^{} (http://planetmath.org/DeRhamCohomology) group) for all $q$, which is given by cap product with a generator^{} of ${H}_{n}(M,\mathbb{Z})$
(a choice of a generator here corresponds to an orientation). This isomorphism exists with
coefficients in $\mathbb{Z}/2\mathbb{Z}$ regardless of orientation.

This isomorphism gives a nice interpretation^{} to cup product^{}. If $X,Y$ are transverse submanifolds of $M$, then $X\cap Y$ is also a submanifold. All of these submanifolds represent homology classes of $M$ in the appropriate dimensions^{}, and

$${D}^{-1}([X])\cup {D}^{-1}([Y])={D}^{-1}([X\cap Y]),$$ |

where $\cup $ is cup product, and $\cap $ in intersection^{}, not cap product.

Title | Poincaré duality |
---|---|

Canonical name | PoincareDuality |

Date of creation | 2013-03-22 13:11:36 |

Last modified on | 2013-03-22 13:11:36 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 10 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 55M05 |

Synonym | Poincaré isomorphism |

Related topic | DualityInMathematics |