Definition:Homotopy Class/Path

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $f: \closedint 0 1 \to S$ be a path in $T$.

The homotopy class of the path $f$ is the homotopy class of $f$ relative to $\set {0, 1}$.

That is, the equivalence class of $f$ under the equivalence relation defined by path-homotopy.

Also see

• Relative Homotopy is Equivalence Relation

Also known as

The homotopy class of $f$ is also sometimes called as the path class of $f$.

Sources

• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy

Further research is required in order to fill out the details. In particular: usage in "Introduction to Riemannian Manifolds" You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

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