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:

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