# homeotopy

Let $X$ be a topological Hausdorff space. Let $\mathrm{Homeo}(X)$ be the group of homeomorphisms $X\to X$, which can be also turn into a topological space^{} by means of the compact-open topology^{}. And let ${\pi}_{k}$ be the k-th homotopy group^{} functor.

Then the k-th homeotopy is defined as:

$${\mathscr{H}}_{k}(X)={\pi}_{k}(\mathrm{Homeo}(X))$$ |

that is, the group of homotopy classes of maps ${S}^{k}\to \mathrm{Homeo}(X)$. Which is different from ${\pi}_{k}(X)$, the group of homotopy classes of maps ${S}^{k}\to X$.

One important result for any low dimensional topologist is that for a surface $F$

$${\mathscr{H}}_{0}(F)=\mathrm{Out}({\pi}_{1}(F))$$ |

which is the $F$’s extended mapping class group.

Reference

G.S. McCarty, Homeotopy groups, Trans. A.M.S. 106(1963)293-304.

Title | homeotopy |

Canonical name | Homeotopy |

Date of creation | 2013-03-22 15:41:54 |

Last modified on | 2013-03-22 15:41:54 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 17 |

Author | juanman (12619) |

Entry type | Definition |

Classification | msc 20F38 |

Synonym | mapping class group |

Related topic | isotopy^{} |

Related topic | group |

Related topic | homeomorphism |

Related topic | Group |

Related topic | Isotopy |

Related topic | Homeomorphism |