So this is from Benson Farb's text and I need clarification on his explanation.
QUESTION: So my first question is why is $\gamma \in \Gamma'$, where $\Gamma=\pi_1(S_g^1)$. My second question is why does $[\gamma,x]\in[\Gamma',\Gamma]$ implies that $[\gamma,x]=0\in \wedge^2H$. And why does this imply that $\tau(T_c(x))=0$ where $\tau$ is the Johnson homomorphism.
EDIT: My apologies for the typo. It should have said $\Gamma=\pi_1$ not $\Gamma'$. Also $H=H_1(S_g^1)$.