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

View the torus as the topological group $S^1\times S^1$ under coordinatewise multiplication, and let $x_0$ be the common basepoint. From the given free homotopy constructThis is continuous, has $K(s,1)=x_0$, and at $s=0,1$ agrees with $\alpha ,\beta$ because $H(0,1)=H(1,1)=x_0$. Thus it exhibits the requested based homotopy explicitly.

Algebraically ${\pi }_1(S^1\times S^1)\cong {\mathbb{Z}}^2$ is Abelian, so conjugate elements are equal. Free and based homotopy classes of loops with a specified common basepoint therefore coincide here.