I am reading Voisin's book Hodge theorem and complex algebraic geometry II and get stuck at this point. It the proof of Thm 3.22, on page 86. The setting is as follows. (I try to use the same notation)
Let $U'\to W$ be a fiber bundle, $F$ be a fiber (corresponding to $\Delta-\Delta\cap \mathcal Y$ in the book). Then we have a long exact sequence on the homotopy groups:
$$\ldots \pi_2(W) \to \pi_1(F) \to \pi_1(U') \to \pi_1(W) \to \pi_0(F) \to\ldots$$
And in our case $F$ is connected, hence we get
$$\ldots \pi_2(W) \to \pi_1(F) \to \pi_1(U') \to \pi_1(W) \to 1$$
There is another map $p$ (corresponding to $pr_1$ in the book) which is defind on $U'$. By restriction $p$ can be defined on $F$. Since $W$ can be identified with some section $W'$, $p$ also defined on $W$. Therefore we can apply $p_*$ to the sequence, get
$$\ldots p_* \pi_2(W) \to p_*\pi_1(F) \to p_*\pi_1(U') \to p_*\pi_1(W) \to 1$$
My question is
Is this sequence still exact?
Form the contents it seems this is true. (It claims $p_*\pi_1(F) \to p_*\pi_1(U')$ is an isomorphism - in our case $p_* \pi_2(W)$ and $p_*\pi_1(W)$ are trivial) But I don't know why?