Definition:Multiplication of Homotopy Classes of Paths

Definition
Let $X$ be a topological space.

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

Let $f,g:[0,1]\to X$ be representative paths for $\alpha$ and $\beta$ respectively.

Let $f(1)=g(0)$.

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

Also see

 * Definition:Fundamental Group
 * Homotopic Paths Implies Homotopic Composition