# Hopf bundle

Consider ${S}^{3}\subset {\mathbb{R}}^{4}={\u2102}^{2}$. The structure^{} of ${\u2102}^{2}$ gives a map
${\u2102}^{2}-\{0\}\to \u2102{P}^{1}$, the complex projective line by the natural projection^{}.
Since $\u2102{P}^{1}$ is homeomorphic to ${S}^{2}$, by restriction^{} to ${S}^{3}$, we get a
map $\pi :{S}^{3}\to {S}^{2}$. We call this the Hopf bundle.

This is a principal ${S}^{1}$-bundle (http://planetmath.org/PrincipalBundle), and a generator^{} of ${\pi}_{3}({S}^{2})$. From the long exact sequence of the bundle (http://planetmath.org/LongExactSequenceLocallyTrivialBundle):

$$\mathrm{\cdots}{\pi}_{n}({S}^{1})\to {\pi}_{n}({S}^{3})\to {\pi}_{n}({S}^{2})\to \mathrm{\cdots}$$ |

we get that ${\pi}_{n}({S}^{3})\cong {\pi}_{n}({S}^{2})$ for all $n\ge 3$. In particular, ${\pi}_{3}({S}^{2})\cong {\pi}_{3}({S}^{3})\cong \mathbb{Z}$.

Title | Hopf bundle |
---|---|

Canonical name | HopfBundle |

Date of creation | 2013-03-22 13:20:04 |

Last modified on | 2013-03-22 13:20:04 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 55R25 |

Synonym | Hopf fibration |