Simply connectedness of minimal resolution of Kleinian singularities

Question

Is the minimal resolution of Kleinian singularities of type $D_k$ (i.e. the minimal resolution of singularities of the action of the binary dihedral group of order $4(k-2)$ on $ C^2$ simply connected? Is there a reference where their topology is described?

Answer 1

Yes it is simply connected. In general the retraction of $\mathbb C^2$ to $0$ will retract the resolution to the $0$ fiber, which is a tree of $\mathbb{CP}^1$s, hence homotopic to a wedge of $2$-spheres, in this case $k$ of them.