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:

\(\ds \forall s \in \closedint 0 1: \, \)

\(\ds \map H {s, 0}\)

\(=\)

\(\ds \map {\sigma_0} s\)

\(\ds \forall s \in \closedint 0 1: \, \)

\(\ds \map H {s, 1}\)

\(=\)

\(\ds \map {\sigma_1} s\)

\(\ds \forall t \in \closedint 0 1: \, \)

\(\ds \map H {0, t}\)

\(=\)

\(\ds \map H {1, t}\)

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

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Closed Geodesics