Munkres says on pg. 346 that the set of path homotopy classes does not aways form a group under the operation $*$ because the product of two path homotopy classes is not always defined.
What does this mean? Does this mean $[f]*[g]\neq [f*g]$? Or does this mean something else? Could someone please give an example?
On
The path homotopy equivalence relation applies to any two paths $f,g$ with the same start and end points, that is, $f(0)=g(0)$ and $f(1)=g(1)$. Also, the path concatenation function $f*g$ (and the equivalence class analogue $[f]*[g]$) is only defined when the start of the second path matches the end of the first, i.e. $f(1)=g(0)$. Thus if $\pi_{A,B}$ is the set of paths from $A$ to $B$ quotient with the path homotopy relation, then if $A\ne B$, $\pi_{A,B}$ is not a group under $*$ because for any $f\in\pi_{A,B}$, $f*f$ is not defined.
The product of paths, and thus of homotopy classes of paths, is only defined when the terminal point of the first path is the initial point of the seconds.