# spheres that are Lie groups

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spheres \PMlinkescapephrasesatisfy

Theorem - The only spheres (http://planetmath.org/Sphere) that are Lie groups^{} are ${S}^{0}$, ${S}^{1}$ and ${S}^{3}$.

$$

*Proof:* $$ It is known that ${S}^{0}$, ${S}^{1}$ and ${S}^{3}$ have a Lie group .

On the other , we have seen in the parent entry (http://planetmath.org/CohomologyOfCompactConnectedLieGroups) that the cohomology groups^{} (http://planetmath.org/DeRhamCohomology) of a compact connected Lie group $G$ satisfy

$${H}^{1}(G;\mathbb{R})=0\u27f9{H}^{3}(G;\mathbb{R})\ne 0$$ |

The result then follows from the fact that the of spheres satisfy ${H}^{1}({S}^{n};\mathbb{R})=0$ and ${H}^{3}({S}^{n};\mathbb{R})=0$ for $n\ne 1,3$. $\mathrm{\square}$

Title | spheres that are Lie groups |
---|---|

Canonical name | SpheresThatAreLieGroups |

Date of creation | 2013-03-22 17:50:02 |

Last modified on | 2013-03-22 17:50:02 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 7 |

Author | asteroid (17536) |

Entry type | Corollary |

Classification | msc 22E99 |

Classification | msc 57T10 |