# 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 $\Omega_{x_{0}}$, 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 $x_{0}$.

Title loop Loop1 2013-03-22 12:16:21 2013-03-22 12:16:21 nerdy2 (62) nerdy2 (62) 5 nerdy2 (62) Definition msc 54-00