Definition 4

Let X be a topological space and let $c$ and $c'$ be two loops in X. A free homotopy joining $c$ to $c'$ means a homotopy $\sigma$ joining $c$ to $c'$ such that $\sigma (0, s) =\sigma (1, s)$ for every $s\in \mathbf{I}$. One says that $c$ is freely homotopic to $c'$ if there exists a free homotopy joining $c$ to $c'$.

The free homotopies joining $c$ to $c'$ correspond to the paths joining $c$ to $c'$ in the space Ω(X) of loops in X. Consequently, the relation “$c$ is freely homotopic to $c'$” is an equivalence relation in Ω(X) whose equivalence classes are the arcwise connected components of Ω(X).