# homology sphere

A compact^{} $n$-manifold $M$ is called a homology sphere if its homology is that of the $n$-sphere ${S}^{n}$, i.e. ${H}_{0}(M;\mathbb{Z})\cong {H}_{n}(M;\mathbb{Z})\cong \mathbb{Z}$ and is zero otherwise.

An application of the Hurewicz theorem and homological Whitehead theorem^{} shows that any simply connected homology sphere is in fact homotopy equivalent to ${S}^{n}$, and hence homeomorphic to ${S}^{n}$ for $n\ne 3$, by the higher dimensional equivalent^{} of the Poincaré conjecture.

The original version of the Poincaré conjecture stated that every 3 dimensional homology sphere was homeomorphic to ${S}^{3}$, but Poincaré himself found a counter-example. There are, in fact, a number of interesting 3-dimensional homology spheres.

Title | homology sphere |
---|---|

Canonical name | HomologySphere |

Date of creation | 2013-03-22 13:56:10 |

Last modified on | 2013-03-22 13:56:10 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 4 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 57R60 |