# fundamental group and torus action

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.

• I am not sure what you call a good quotient; I assume this implies that $p$ is smooth and proper, hence topologically a locally trivial fibration; then the result is completely standard (homotopy exact sequence, read any book on algebraic topology).
– abx
Feb 27, 2014 at 15:55
• My definition of good quotient is the following: $p\colon X\rightarrow Y$ is affine and $T$-invariant and the pull-back $p^* \colon \mathcal{O}_Y \rightarrow (p_*\mathcal{O}_X)^T$ is an isomorphism. Feb 27, 2014 at 15:59

## 1 Answer

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 $\pi_2(Y)=0$?

• thanks! one case in which i was interested has this condition! Feb 27, 2014 at 16:18