Does $[\alpha][\gamma]\tilde{x_0}=[\beta][\gamma]\tilde{x_0}$ imply $[\alpha]\tilde{x_0}=[\beta]\tilde{x_0}$ in a fundamental group $\pi_1(X,x_0)$?

Question

Does $[\alpha][\gamma]\tilde{x_0}=[\beta][\gamma]\tilde{x_0}$ imply $[\alpha]\tilde{x_0}=[\beta]\tilde{x_0}$ in a fundamental group $\pi_1(X,x_0)$?

$\alpha,\beta $ and $\gamma$ are the loops based at $x_0$ and $\tilde{x_0}=p^{-1}(x_0)$ is the lift of $x_0$ through the covering map.

Answer 1

This is not true in general. For example take the covering as in the picture below. Take $\gamma = b$, $\alpha = a\cdot b$, $\beta=b$ and $\tilde{x_0}=y_2$. Then $[\gamma]\tilde{x_0}=y_1$ and then $[\alpha]([\gamma]\tilde{x_0})=[\beta]([\gamma]\tilde{x_0})=y_2$ but $$[\alpha]\tilde{x_0}=y_3\neq y_1=[\beta]\tilde{x_0}.$$