Definition:Freely Homotopic Loops

Definition
Let $M$ be a topological manifold.

Let $\sigma_0$, $\sigma_1$ be loops in $M$.

Suppose there exists a homotopy $H : \closedint 0 1 \times \closedint 0 1 \to M$ such that:


 * $\forall s \in \closedint 0 1 : \map H {s, 0} = \map {\sigma_0} s$


 * $\forall s \in \closedint 0 1 : \map H {s, 1} = \map {\sigma_1} s$


 * $\forall t \in \closedint 0 1 : \map H {0, t} = \map H {1, t}$

Then $\sigma_0$ and $\sigma_1$ are called freely homotopic.