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.



fundamental group and torus action
I assume that in your case, $p^{-1}(y)\rightarrow X\rightarrow Y$ is a fibration. Therefore, there is a long-exact sequence of homotopy groups $$\ldots\rightarrow\pi\_n(p^{-1}(y))\rightarrow\pi\_n(X)\rightarrow\pi\_n(Y)\rightarrow\pi\_{n-1}(p^{-1}(y))\rightarrow\ldots.$$ In particular, we have $$\ldots\rightarrow \pi\_2(Y)\rightarrow\pi\_1(p^{-1}(y))\rightarrow\pi\_1(X)\rightarrow\pi\_1(Y)\rightarrow\pi\_0(p^{-1}(y))\rightarrow\ldots.$$ Since you know $p^{-1}(y)$ to be connected $\pi\_0(p^{-1}(y))=0$. Do you happen to know that $\pi\_2(Y)=0$?


