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