Corollary 3.2.6.8. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n > 0$ be an integer. Then the homotopy group $\pi _{n}(X_{s}, x)$ vanishes if and only if $f$ satisfies both of the following conditions:
The group homomorphism $\pi _{n}(f): \pi _{n}(X,x) \rightarrow \pi _{n}(S,s)$ is injective.
The group homomorphism $\pi _{n+1}(f): \pi _{n+1}(X,x) \rightarrow \pi _{n+1}(S,s)$ is surjective.