Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 20I / e / Solution

Let $a,b$ be the two oriented circle generators of the fundamental group of a bouquet of circles, ${\pi }_1(S^1\vee S^1)\cong F(a,b)$. TakeTheir loops are freely homotopic: slide the basepoint of the $a$ loop along the path $b$, thereby introducing the route $b^{-1}$ to the original loop and $b$ back. Equivalently the change-of-basepoint construction along a loop realizes conjugation, and the retraced connecting portions can be contracted with a moving marked point.

They are not based homotopic, since their reduced words in the free group are distinct: $a$ has length one and $b^{-1}ab$ has length three with no cancellation. This illustrates that free homotopy remembers conjugacy classes, whereas based homotopy remembers actual elements of the fundamental group.