# Singularity links of quotients

Hi there, I have a question concerning the quotient $\mathbb{C}^3/\mu_p$ where $\mu_p=\{\zeta\in \mathbb{C}^3|\zeta^p=1\}$ acts on $\mathbb{C}^3$ via $\zeta(x,y,z)=(\zeta^{-1}x,\zeta^{-p}y,\zeta z)$. What do the links of these singularities look like? Are they real 5-dimensional $\mathbb{Q}$-homology spheres?

Regards,

Nathan

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If $\mathbb{C}^n/G$ has an isolated fixed point at $0$, then the link would be $S^{2n-1}/G$. So it would be a rational homology sphere. Your action has fixed points on the sphere, because of the $\zeta^{-p}y$ factor. You probably meant to something else. – Donu Arapura Nov 28 '10 at 13:36
I think the elements of $\mu_p$ should lie in $\mathbb{C}$ instead of $\mathbb{C}^3$. As Donu suggested, you should note that $\zeta^{-p} = 1$ for all $\zeta \in \mu_p$, so the action on the $y$ coordinate is trivial. – S. Carnahan Nov 28 '10 at 13:51
Sure, $\mu_p$ is in $\mathbb{C}$, sorry for that. Thanks for the help! – Nathan Nov 28 '10 at 17:04