# loop

A loop based at ${x}_{0}$ in a topological space^{} $X$ is simply a continuous map $f:[0,1]\to X$ with $f(0)=f(1)={x}_{0}$.

The collection of all such loops, modulo homotopy equivalence^{}, forms a group known as the fundamental group^{}.

More generally, the space of loops in $X$ based at ${x}_{0}$ with the compact-open topology^{}, represented by ${\mathrm{\Omega}}_{{x}_{0}}$, is known as the loop space^{} of $X$. And one has the homotopy groups^{} ${\pi}_{n}(X,{x}_{0})={\pi}_{n-1}({\mathrm{\Omega}}_{{x}_{0}},\iota )$, where ${\pi}_{n}$ represents the higher homotopy groups, and $\iota $ is the basepoint in ${\mathrm{\Omega}}_{{x}_{0}}$ consisting of the constant loop at ${x}_{0}$.

Title | loop |
---|---|

Canonical name | Loop1 |

Date of creation | 2013-03-22 12:16:21 |

Last modified on | 2013-03-22 12:16:21 |

Owner | nerdy2 (62) |

Last modified by | nerdy2 (62) |

Numerical id | 5 |

Author | nerdy2 (62) |

Entry type | Definition |

Classification | msc 54-00 |