Let $T$ be the complex torus acting on a complex connected algebraic variety $X$ and let $p \colon X\rightarrow Y$ be a good quotient for this action. For any $y\in Y$ we have a sequence $p^{-1}(y) \rightarrow X \rightarrow Y$ which leads to a sequence $\pi (p^{-1}(y))\rightarrow \pi (X)\rightarrow \pi (Y)$ of the corresponding fundamental groups.

Is this sequence an exact sequence?

What i have tried is the following: its easy to see that $im (\pi (p^{-1}(y)) \rightarrow \pi (X))\subset \ker ( \pi (X)\rightarrow \pi (Y)) $ and being fibers of $p$ connected then $\pi(X)\rightarrow \pi(Y)$ is surjective. I've read on a paper that the sequence is in fact exact on $\pi (X)$ but i can't see why $\pi( p^{-1}(y)) \rightarrow \pi(X)$ is injective. sorry if I'm not seeing something trivial.

Thanks in advance.