Let $V\subseteq \mathbb{C}^{n+1}$ be an affine algebraic hypersurface, i.e. a zero set of a polynomial in $n+1$ complex variables. Let $S$ be a small sphere around the origin $0\in \mathbb{C}^{n+1}$ and let $K= S\cap V$ be the link of $0\in V$. Then $K$ is a compact manifold of (real) dimension $2n-1$.
A well-known result of Milnor asserts that $\pi_{i}(K)$ is trivial for $i\leq n-2$. In particular, $H_{i}(K;\mathbb{Z})=0$ for $i\not\in \{0,n-1,n,2n-1\}$. I would like to ask what is known about $H_{n-1}(K)$? Under what conditions is it true that $K$ is a $\mathbb{Q}$-homology sphere? Does the topology of $K$ somehow reflect the fact that the singularity is canonical, terminal etc? I am mostly interested in the following question:
Is it true that the link of a rational singularity is a $\mathbb{Q}$-homology sphere?
I am aware of the fact that algebraic geometers prefer to study the topology of the dual complex of the resolution instead of the topology of the link. Is it because the latter does not capture the algebraic properties of the singularity? I would be very grateful for any examples illustrating this issue.