Definition:Multiplication of Homotopy Classes of Paths

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

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

Let $f, g: \left[{0 \,.\,.\, 1}\right] \to S$ be representative paths for $\alpha$ and $\beta$ respectively.

Let $f \left({1}\right) = g \left({0}\right)$.

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