Loop 2002-02-03 01:00:17 2002-07-24 05:07:09 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A {\em 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 $\Omega_{x_0}$, is known as the loop space of $X$. And one has the homotopy groups $\pi_n(X,x_0) = \pi_{n-1}(\Omega_{x_0},\iota)$, where $\pi_n$ represents the higher homotopy groups, and $\iota$ is the basepoint in $\Omega_{x_0}$ consisting of the constant loop at $x_0$.