# H-space

A topological space^{} $X$ is said to be an *H-space ^{}* (or

*Hopf-space*) if there exists a continuous binary operation $\phi :X\times X\to X$ and a point $p\in X$ such that the functions $X\to X$ defined by $x\mapsto \phi (p,x)$ and $x\mapsto \phi (x,p)$ are both homotopic

^{}to the identity map via homotopies

^{}that leave $p$ fixed. The element $p$ is sometimes referred to as an ‘identity

^{}’, although it need not be an identity element

^{}in the usual sense. Note that the definition implies that $\phi (p,p)=p$.

Topological groups^{} are examples of H-spaces.

If $X$ is an H-space with ‘identity’ $p$,
then the fundamental group^{} ${\pi}_{1}(X,p)$ is abelian^{}.
(However, it is possible for the fundamental group to be non-abelian^{}
for other choices of basepoint, if $X$ is not path-connected.)

Title | H-space |
---|---|

Canonical name | Hspace |

Date of creation | 2013-03-22 16:18:18 |

Last modified on | 2013-03-22 16:18:18 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 4 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 55P45 |

Synonym | Hopf-space |

Synonym | H space |

Synonym | Hopf space |