# exotic R4’s

If $n\ne 4$ then the smooth manifolds^{} homeomorphic to a given topological $n$- manifold, $M$, are parameterized by some discrete algebraic invariant of $M$. In particular there is a unique smooth manifold homeomorphic to ${\mathbb{R}}^{n}$.

By contrast one may choose uncountably many open sets in ${\mathbb{R}}^{4}$, which are all homeomorphic to ${\mathbb{R}}^{4}$, but which are pairwise non-diffeomorphic.

A smooth manifold homeomorphic to ${\mathbb{R}}^{4}$, but not diffeomorphic to it is called an *exotic* ${\mathbb{R}}^{4}$.

Given an exotic ${\mathbb{R}}^{4}$, $E$, we have a diffeomorphism $E\times \mathbb{R}\to {\mathbb{R}}^{5}$. (As there is only one smooth manifold homeomorphic to ${\mathbb{R}}^{5}$). Hence exotic ${\mathbb{R}}^{4}$’s may be identified with closed submanifolds^{} of ${\mathbb{R}}^{5}$. In particular this means the cardinality of the set of exotic ${\mathbb{R}}^{4}$’s is precisely continuum^{}.

Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic ${\mathbb{R}}^{4}$.

Title | exotic R4’s |
---|---|

Canonical name | ExoticR4s |

Date of creation | 2013-03-22 15:37:33 |

Last modified on | 2013-03-22 15:37:33 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 21 |

Author | whm22 (2009) |

Entry type | Definition |

Classification | msc 57R12 |

Classification | msc 14J80 |

Related topic | DonaldsonsTheorem |

Related topic | DonaldsonFreedmanExoticR4 |