Concerning the problem circled in red:
Would this work as the homotopy rel $\dot I$?
$F(t_1, t_2) = \begin{cases} \sigma(e_1), & \text{if $t_1=0,1$ or $t_2=1$} \\ (\sigma_0 * \sigma_1^{-1}) * \sigma_2[(1-t_2)t_1 + t_2], & \text{otherwise} \end{cases}$
And also, how would you use theorem $1.6$ here to show this? I can see by part three I can show that it's nullhomotopic rel $\{1\}$ if we define $f ': S \rightarrow X$ by $f'(e^{2 \pi i t_1}) = f(t_1)$, where $f(t)= (\sigma_0 * \sigma_1^{-1}) * \sigma_2(t)$ and the homotopy as $G(t_1, t_2)$ as $(e^{2 \pi i t_1}, t_2) \rightarrow f'((e^{2 \pi i t_1})^{(1-t_2)})$, but that doesn't show it's rel $\{0,1\}$.
