Definition:Multiplication of Homotopy Classes of Paths

Definition

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

Let $\alpha, \beta$ be homotopy classes of paths in $T$.

Let $f, g: \closedint 0 1 \to S$ be representative paths for $\alpha$ and $\beta$ respectively.

Let $\map f 1 = \map g 0$.

The product of the homotopy classes $\alpha$ and $\beta$ is the homotopy class of the concatenated path $f * g$.

Also see

• Definition:Fundamental Group
• Homotopic Paths Implies Homotopic Composition

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $9$: The Fundamental Group: $\S 52$: The Fundamental Group