# Donaldson’s theorem

Theorem Let $M$ be a smooth simply connected, closed 4- manifold^{}. If the bilinear form^{} induced on ${H}_{2}(M;\bm{Z})$ by the cup product^{} is positive definite^{}, then it may be represented by the identity matrix^{}.

One application is the existence of exotic ${\bm{R}}^{4}$’s. See Donaldson Freedman exotic R4.

Title | Donaldson’s theorem |
---|---|

Canonical name | DonaldsonsTheorem |

Date of creation | 2013-03-22 15:37:30 |

Last modified on | 2013-03-22 15:37:30 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 9 |

Author | whm22 (2009) |

Entry type | Theorem |

Classification | msc 57R12 |

Classification | msc 14J80 |

Related topic | exoticR4s |

Related topic | DonaldsonFreedmanexoticR4 |

Related topic | DonaldsonFreedmanExoticR4 |

Related topic | ExoticR4s |