Cancellation property of the composition of paths
by Hidenori
Problem Statement
Show that composition of paths satisfies the following cancellation property: If $f_0 \cdot g_0 \simeq f_1 \cdot g_1$ and $g_0 \simeq g_1$ then $f_0 \simeq f_1$.
Solution
By the inverse lemma, $\overline{g_0} \simeq \overline{g_1}$.
We know that $f_0 \cdot g_0 \simeq f_1 \cdot g_1$. As mentioned on P.26, the product operation respects homotopy classes.
\[\begin{align*} (f_0 \cdot g_0) \cdot \overline{g_0} \simeq (f_1 \cdot g_1) \cdot \overline{g_1} &\implies f_0 \cdot (g_0 \cdot \overline{g_0}) \simeq f_1 \cdot (g_1 \cdot \overline{g_1}) \\ &\implies f_0 \cdot c \simeq f_1 \cdot c \\ &\implies f_0 \simeq f_1 & \text{(By reparametrization)} \\ \end{align*}\]Subscribe via RSS